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Physical principles of membrane mechanics, membrane domain formation, and cellular signal transduction
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Physical principles of membrane mechanics, membrane domain formation, and cellular signal transduction
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Content
PHYSICAL PRINCIPLES OF MEMBRANE MECHANICS,
MEMBRANE DOMAIN FORMATION, AND CELLULAR
SIGNAL TRANSDUCTION
by
Carlos D. Alas
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
December 2023
Copyright 2024 Carlos D. Alas
Acknowledgements
I give thanks to the various sources of funding which supported my thesis research:
Nation Science Foundation Grants: DMR-1554716, MCB-2202087, DMR-2051681,
MCB-2202087.
USC Center of Advanced Research Computing (CARC).
USC Graduate School DIA Fellowship.
With heartfelt gratitude to those who profoundly influenced my academic journey I give thanks:
I extend a deep appreciative thank you to my father, Vinicio Alas, who, from an early
age, taught me about computers, instilled in me a curiosity in math and science, discipline, and encouraged higher education, supporting me throughout childhood and early
higher education. My deepest appreciation also goes to my mother, Evelyn Alas, whose
resilience during her battle with cancer initiated my existential exploration. This challenging period sparked my interest in philosophy, leading me to trace human knowledge, read
and discuss philosophy, and research historical narratives shaping modern perspectives.
ii
I give thanks to my older brother, Vinicio Jr., whose interest in history, science, and computers shaped my own. I also thank my little sister, Sarah, a positive light in our family
and, in many ways, the glue that kept us together.
I have profound gratitude for the work of ancient Greek thinkers—Socrates, Plato,
Aristotle, and Euclid—, philosophers Descartes and Kant, and physicists Galileo Galilei,
Isaac Newton, and Albert Einstein. Their inspiration guided me to pose metaphysical
questions about cognition, reality, and the formation of individual perception which gives
shape to one’s experiences. This exploration extended to understanding how our bodies
gather sensory data and respond to physical stimuli. These luminaries provided me with
examples in the devotion of the pursuit of absolute truths through critical inquiry, evidence
gathering, and rigorous hypothesis testing. This foundation ultimately steered my path
into the realm of cellular-level perception.
Special thanks to my professors at Antelope Valley Community College (AVC):
Dr. Alexandra Schroer, my chemistry professor, who gave me my first real introduction
to the physical sciences and supported me with guidance and encouragement.
Dr. Jason Bowen, my first physics professor, who, after just a few lectures, convinced
me there was no journey more rewarding than that in mastering physics. To this day, his
lecturing style of suspense and awe, his words of encouragement, and guidance hold a
dear place in my heart. I’ll always remember our discussions about differentiating between
the dream state and reality, emphasizing the convincing continuity in our shared reality
over any dream.
iii
Dr. Zia Nisani, for inviting me into his research group to perform experiments for
measuring the tail strike kinematics of the largest scorpions native to the United States,
Hadrurus Arizonensis.
Christos Valiotis, who actively encouraged me to be a role model for younger generations through participation as a judge and event designer in Cal Tech’s Science Olympiad
competition for high school students and other outreach through AVC’s STEM club over
many years.
I made many friends through the STEM club and my job as a tutor at AVC’s Learning
Center who helped shape my path, and I give thanks to them as well: Victoria Rose, Henry
Zamora, Joseph John Barker, Mireille Sandjong (Mimi), Leidy Villarreal, Jamie Jones, and
Thomas Musgrove.
At California Polytechnic State University of San Luis Obispo (Cal Poly SLO), I extend
thanks to:
Dr. Robert Echols (Dr. Bob), for always making time for me, despite his large responsibility as the physics department chair, to discuss philosophical and abstract ideas about
reality, consciousness, and perception, based on current research in modern physics.
Also, I thank Dr. Bob for taking me into his research group to do some of the most interesting experiments I’ll ever do in my life, involving microwaves in a resonant cavity to
generate propulsion.
Dr. Jon Fernsler, for fond memories of pure joy and excitement in discussing theory
and experiments in active matter, his help in earning a summer Research Experience
for Undergraduates at the University of Colorado, Boulder, and his advice on choosing
graduate schools.
iv
Dr. Karl Saunders, for being the greatest lecturer in electromagnetism and for sharing
with me his wisdom in regards to choosing a mentor in graduate school.
Carmen Castaneda, for giving me room and board in her home throughout my first year
at Cal Poly SLO and connecting me with her expansive network all over San Luis Obispo,
including connecting me with Jenny Cruz in the Physics department, upon hearing of my
past tutoring experience at AVC. I was truly lucky to have met Carmen as her help was
pivotal to much of my success at Cal Poly SLO.
Jenny Cruz, for her belief in me, interviewing and hiring me as one of her tutors in
the Physics department even though I was a stranger to the department since I was a
transfer, her steadfast support in administrative tasks, and encouragement which helped
build my confidence professionally.
Sebastian Pardo, for being the best housemate and friend a physics undergraduate
could have, with whom I spent countless hours discussing physics and philosophy.
Most of all, Dr. Tatiana Kuriabova, for instilling in me a passion for computational
and theoretical biophysics. I surveyed many research groups and was very indecisive
about which field in Physics I would specialize in until I worked with Tatiana. Tatiana gave
me the most fun and interesting problems that were in the field of bacteria swimming
dynamics in thin films to solve and the space to grow as an independent researcher. My
research with her became my senior thesis work for my Bachelor’s of Science degree in
Physics, which eventually led to my first peer-reviewed publication [C. Alas, T. R. Powers,
and T. Kuriabova. Swimming of microorganisms in quasi-two-dimensional membranes.
Journal of Fluid Mechanics, Volume 911, A35, 2021.]. Working with Tatiana gave me the
confidence to pursue theory in biophysics and made me realize I had many strengths in
v
applying computational methods to solve problems. She also provided me with a first
example on how theorists approach and model phenomena. After working with Tatiana, I
knew I belonged in the field of computational and theoretical biophysics. Tatiana also gave
me the best advice on the value of a PhD education and educated me on my options after
earning a PhD, including gifting me a very valuable book called "A PhD is not enough" by
Peter J. Feibelman, which gave me great insight into the journey of a PhD in physics and
reasonable expectations for what I could do with my PhD after. Tatiana is also the best
lecturer in classical mechanics, and I owe her much gratitude for her very effective style
of teaching the fundamentals, which helped me succeed in graduate school.
To a few others:
A heartfelt thanks to Dr. Cheol Park at the Soft Materials Research Center, University
of Colorado, Boulder for his support and levity throughout my study of hydrodynamic flow
in liquid crystal films. His guidance and encouragement were invaluable in shaping my
academic journey.
I give thanks to the Margeat and Doucet labs from the French National Centre for
Scientific Research (CNRS) who met with our group to discuss the experimental phenomenology of emerin nanodomain assembly at the nuclear envelope, every other Tuesday morning.
I give thanks to my orange tabby feline friend Jeremy, who has given my wife and I
endless joy and inspired us with his deep commitment to studying door mechanics and
high-pitched sound waves at wee hours every morning. I often stare at Jeremy’s patterned
coat of fur and wonder if the pattern comes from a Turing mechanism. Interestingly, a
recent study [C. B. Kaelin, K. A. McGowan, and G. S. Barsh. Nature Communications,
vi
Volume 12, 5127, 2021.] has identified candidate activator and inhibitor genes in cats
that may have reaction and diffusion properties for yielding cat patterns from a Turing
mechanism. It would be interesting to model cat patterns someday.
I give thanks to Dr. Yuriy Bakach, who managed me during my internship at Travelers,
for letting me run with my ideas and supporting me as I developed them. From Yuriy, I
learned a great deal about data science and being a professional in industry. I couldn’t
have had the incredible success I had that summer without Yuriy. With his guidance, I
was able to obtain an amazing offer to return for full-time employment after completing
my graduate studies. The success he helped me accomplish gave me outstanding confidence in myself as a data scientist that I will always be grateful for.
From the University of Southern California (USC), I give many thanks to:
Betty Byers, for her steadfast, precise, and vast administrative support. Betty is a
multifaceted individual and was generally the first person I would reach out to when I
needed help with things around the physics department.
Christina Tasulis Williams, was also another person I could count on to figure out how
to do things around the physics department. I came to find Christina keeps the department
running efficiently.
Avishuman Ray, my friend and research group mate, who has spent countless hours
with me discussing kinetic Monte Carlo simulations of emerin and other phenomena,
which he will use in his work to study the dynamics of emerin.
Hoa Trinh, a dear friend of mine who I met through the PhD physics program, with
whom I teamed up with often to tackle difficult courses together, and with whom I strategize about navigating the job market.
vii
Armen Tokadjian, a dear friend of mine who I would team up with to tackle difficult
courses together, and from time to time meet up with to play billiards and talk about tech.
Marco Olguin, for his technical support in using CARC.
Dr. Krzysztof Pilch, for his role as Faculty Graduate Advisor during my first few years
in graduate school, and help in learning about the dual degree program which allowed me
to supplement my PhD physics degree electives with courses in computer science for my
Master’s degree in Computer Science. I also thank him for letting me win in billiards after
I was down a hundred dollars :).
Dr. Rosa Di Felice, for her guidance in research during my first year in the PhD program. She taught me a lot about molecular dynamics simulations and Gromacs software
for implementing such simulations.
Dr. James Boedicker, for convincing me to join the PhD program at USC, helping me
obtain a USC graduate school DIA fellowship which provided me with additional financial
support, for all the background tasks he helped me with in his role as my Faculty Graduate Advisor during my latter years in graduate school, and for participating in both my
Qualification Exam and PhD Thesis Defense committees.
Dr. Aiichiro Nakano, for his lectures on methods in computational physics, highperformance computation, and other topics in computer science. I really appreciated the
structure of his courses, especially in regards to course projects which gave me space
to run with my ideas. From these projects, I learned a lot and applied what I learned
throughout my thesis work. Aiichiro also gave me great advice on coursework towards
viii
my Masters in Computer Science with a specialization in High-Performance Computation. I also thank Aiichiro for participating in both my Qualification Exam and PhD Thesis
Defense committees.
Dr. Peter Chung, for his participation in my Qualification Exam committee.
Dr. Peter Foster, for his participation in my PhD Thesis Defense committee.
Dr. Osman Kahraman, for his contributions to our model for thermosensing through
membrane mechanics and for showing me how to run code for the finite element method
calculations which we used to benchmark our boundary value method [see C. D. Alas and
C. A. Haselwandter. Dependence of protein-induced lipid bilayer deformations on protein
shape. Phys. Rev. E, 107:024403, 2023.].
Dr. Fabien Pinaud, for his various contributions to our work on modeling the selfassembly of emerin nanodomains at the inner nuclear membrane, both in modeling and
experiments. I thank Fabien for his suggestion on modeling the stability of oligomers [See
our published work: C. D. Alas and C. A. Haselwandter. Dependence of protein-induced
lipid bilayer deformations on protein shape. Phys. Rev. E, 107:024403, 2023.]. I also
thank Fabien for his participation in both my Qualification Exam and PhD Thesis Defense
committees.
Most of all, my dear friend and PhD advisor Dr. Christoph A. Haselwandter, for his
belief in me, his deep commitment to mentoring me in research, my development as a
professional, and to seeing me succeed. Upon first meeting with Christoph, he spoke very
eloquently about membrane proteins and their various roles in cellular biology, including
giving cells and the organisms they compose the ability to perceive and adapt to the external world. His detailed descriptions captured my imagination. I can say Christoph has and
ix
always will continue to capture my imagination. Over the past five years, Christoph has
strategically fostered my development into a professional, including guiding me in forming ideas and hypotheses in biophysics, constructing quantitative physical models to describe biological phenomena, writing manuscripts, rebutting peer-reviewers, reporting the
progress to our collaborators, reporting our research through presentations at research
conferences, mentoring newer students in our lab, recommending me as a peer reviewer
to journals, writing grant proposals, and managing my time to meet multiple deadlines.
Christoph has also given me crucial advice in regards to choosing a career path after
graduate school. It was Christoph who suggested I look into pursuing data science and
always encouraged me to take advantage of the physics department’s dual degree program to study and earn a Master’s degree in Computer Science for technical skills and
knowledge that would transfer directly to industry. Even though I know I still have a lot
to learn, I have never been more confident in myself as a physicist and professional, and
this is largely due to Christoph. I am deeply honored to have had the opportunity to learn
from Christoph.
Last of all, but not least, I have to thank the love of my life and best friend, my wife
Zhibek Mukhamedina-Alas, who has supported me throughout my PhD journey. Zhibek
also worked part-time to help support us all while pursuing her own academic studies.
Since meeting Zhibek, she has brought much structure and stability to my life and, in
many ways, helped me mature as a person. She is my north star, guiding me and helping
me work towards a future for both of us. I’m very grateful to have had her support during
this very exciting, challenging, and rewarding journey. This accomplishment is as much
hers as it is mine.
x
Carlos D. Alas
December 2023
xi
Table of Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Proteins and phospholipids: the building blocks of cell membranes . . . . . 1
1.2 Coordinators of cellular responses: the multifaceted functions of transmembrane proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 The influence of membrane proteins on membrane shape . . . . . . 6
1.2.2 Temperature sensing in cells: the convergence of membrane
mechanics and function . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Collective protein behavior in specialized membrane domains . . . . 11
1.3 Shortcomings of former approaches for calculating protein-induced lipid
bilayer deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter 2: A boundary value method for lipid bilayer deformations . . . . . . . 18
2.1 Modeling protein-induced lipid bilayer thickness deformations . . . . . . . . 20
2.1.1 Continuum elasticity theory of lipid bilayer deformations . . . . . . . 21
2.1.2 Modeling protein shape . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Boundary value method for bilayer thickness deformations . . . . . . . . . . 33
2.2.1 Formulation and validation of the boundary value method . . . . . . 34
2.2.2 Nonuniform boundary point distributions . . . . . . . . . . . . . . . . 40
Chapter 3: Dependence of protein-induced lipid bilayer deformations on
protein shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Analytic approximation of the bilayer thickness deformation energy . . . . . 48
3.2 Dependence of bilayer thickness deformation energy on protein shape . . . 54
3.2.1 Constant bilayer-protein boundary conditions . . . . . . . . . . . . . 55
3.2.2 Variations in protein hydrophobic thickness . . . . . . . . . . . . . . 60
3.2.3 Variations in bilayer-protein contact slope . . . . . . . . . . . . . . . 62
3.3 Transitions in protein organization and shape . . . . . . . . . . . . . . . . . 64
3.3.1 Self-assembly of protein oligomers . . . . . . . . . . . . . . . . . . . 65
xii
3.3.2 Transitions in protein conformational state . . . . . . . . . . . . . . . 68
Chapter 4: Thermosensing through membrane mechanics . . . . . . . . . . . . 72
4.1 Modeling the effect of temperature changes on
protein-induced bilayer deformations . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Modeling transitions in protein shape . . . . . . . . . . . . . . . . . . . . . . 76
4.2.1 Chemoreceptor trimers . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.2 MscL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.3 Piezo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Temperature-sensing through chemoreceptors and ion channels . . . . . . 86
4.3.1 Chemoreceptor activation . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.2 MscL gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3.3 Piezo gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 Connection to experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4.1 Chemoreceptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4.2 MscL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4.3 Piezo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Chapter 5: Physical mechanism for the self-assembly of emerin nanodomains at the inner nuclear membrane . . . . . . . . . . . . . . . 110
5.1 Reaction-diffusion equations and linear stability analysis . . . . . . . . . . . 111
5.2 Physical model of emerin nanodomains . . . . . . . . . . . . . . . . . . . . 116
5.2.1 Reaction kinetics of emerin nanodomains . . . . . . . . . . . . . . . 122
5.2.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3.1 Wild-type system under no mechanical stress . . . . . . . . . . . . . 127
5.3.1.1 Crowding effects due to emerin monomers . . . . . . . . . 130
5.3.2 Wild-type system under mechanical stress . . . . . . . . . . . . . . . 131
5.3.3 Q133H mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.4 P183H mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3.4.1 Spontaneous dimerization . . . . . . . . . . . . . . . . . . . 137
5.3.5 ∆95-99 mutant system under no mechanical stress . . . . . . . . . . 138
5.3.6 ∆95-99 mutant system under mechanical stress . . . . . . . . . . . . 139
Chapter 6: Overview and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 142
6.1 Overview and conclusions of Chapter 2 . . . . . . . . . . . . . . . . . . . . 142
6.2 Overview and conclusions of Chapter 3 . . . . . . . . . . . . . . . . . . . . 144
6.3 Overview and conclusions of Chapter 4 . . . . . . . . . . . . . . . . . . . . 146
6.4 Overview and conclusions of Chapter 5 . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A Supplemental material for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 178
A.1 Computational implementation of the boundary value method . . . . 178
A.2 Numerical precision . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
xiii
B Supplemental material for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 183
B.1 Axisymmetric bilayer midplane deformations . . . . . . . . . . . . . . 183
C Supplemental material for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . 185
C.1 Molecule distributions in emerin nanodomains from spatially
heterogeneous diffusion coefficients . . . . . . . . . . . . . . . . . . 185
C.1.1 Free diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 186
C.1.2 Diffusion in crowded membranes . . . . . . . . . . . . . . . 189
xiv
List of Figures
1.1 Illustration (from Ref. [3]) depicting various membrane proteins within lipid
bilayer environments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Illustration (adapted from Ref. [10]) depicting a transduction process
initiated by a transmembrane protein. In particular, in this illustration, a
signaling molecule binds to a receptor protein in the plasma membrane
seperating the cell’s cytoplasm from the extracellular environment, which
sets off a signal transduction pathway leading to a response that involves
the activation of cellular process. . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Illustration (from Ref. [41]) depicting the change in lipid bilayer hydrophobic
thickness as MscL transitions from its closed state to its opened state. . . . 8
1.4 Schematic (from Ref. [12]) depicting the reorganization of emerin at the
NE in response to mechanical stress. Increased lateral mobility at the
INM is initiated by emerin monomer unbinding from nuclear actin and BAF,
facilitating LEM domain interactions with binding sites along the intrinsically
disordered region of other emerin molecules. This controlled process
leads to the formation of emerin oligomers at SUN1 LINC complexes,
subsequently stabilized by lamin A/C. . . . . . . . . . . . . . . . . . . . . . 13
1.5 Diagrams (from Ref. [145]) of the general system domain discetization
schemes utilized in (a) BVM, (b) FDM, and (c) FEM [145]. . . . . . . . . . . 15
xv
2.1 Protein-induced lipid bilayer thickness deformations for selected families
of protein shapes: (a) Clover-leaf protein cross section with five-fold
symmetry, constant protein hydrophobic thickness, and zero bilayerprotein contact slope, (b) polygonal protein cross section with six-fold
symmetry, constant protein hydrophobic thickness, and constant bilayerprotein contact slope U
′ = 0.3, (c) clover-leaf protein cross section with
three-fold symmetry, a five-fold symmetric (sinusoidal) variation in protein
hydrophobic thickness, and zero bilayer-protein contact slope, and (d)
polygonal protein cross section with seven-fold symmetry, constant protein
hydrophobic thickness, and a three-fold symmetric (sinusoidal) variation in
the bilayer-protein contact slope. The color map and purple surfaces show
the positions of the upper and lower lipid bilayer leaflets, respectively. The
bilayer-protein boundaries are color-coded according to their symmetries
(see also Fig. 2.3 in Sec. 2.1). For panels (a) and (c) we used ϵ = 0.2 and
ϵ = 0.3 in Eq. (2.17), respectively, and for panels (b) and (d) we used P = 5
in Eqs. (2.18) and (2.19). All bilayer surfaces were calculated using the
reference parameter values in Sec. 2.1 and the BVM for protein-induced
lipid bilayer thickness deformations described in Sec. 2.2. . . . . . . . . . . 19
2.2 Notation used for the calculation of protein-induced lipid bilayer thickness
deformations in (a) angled and (b) side views. As an example, we consider
here a membrane protein with a non-circular (clover-leaf) bilayer-protein
boundary curve, C(θ), constant hydrophobic thickness, W(θ) = W0, and
zero bilayer-protein contact slope, U
′
(θ) = 0. The positions of the upper
and lower lipid bilayer leaflets are denoted by h+ and h−, from which the
bilayer midplane and bilayer thickness deformation fields h and u can be
obtained via Eqs. (2.1) and (2.2), respectively. We denote one-half the
unperturbed bilayer thickness by a, resulting in a hydrophobic mismatch
U = W/2 − a at the bilayer-protein interface. The unit vectors ˆt and ˆn
denote the directions tangential and perpendicular (pointing towards the
protein) to the bilayer-protein boundary, respectively. . . . . . . . . . . . . . 22
2.3 Cross sections of cylindrical protein shapes (left-most panels) and (a)
clover-leaf and (b) polygonal protein shapes (right panels). The clover-leaf
protein cross sections in panel (a) are obtained from Eq. (2.17) with
ϵ = 0.07, 0.14, 0.21, 0.28, and 0.35 (left to right) and s = 1, 2, 3, 4, and 5 (top
to bottom), with ϵ = 0 yielding a circular protein cross section. Note that
the clover-leaf protein cross sections with s = 1 only show small deviations
from the corresponding circular protein cross section obtained with ϵ = 0
in Eq. (2.17) (dashed curves) for the values of ϵ considered here. The
polygonal protein cross sections in panel (b) are obtained from Eq. (2.18)
with P = 1, 2, 3, , 4, and 5 (left to right) and s = 4, 5, 6, 7, and 8 (top to
bottom). As a guide to the eye, these polygonal protein cross sections are
inscribed in circles obtained with P = 0 in Eq. (2.18) (dashed curves). . . . 32
xvi
2.4 Percentage difference between the exact bilayer thickness deformation field
along the bilayer-protein boundary and the bilayer thickness deformation
field obtained from the BVM solution, ηb
′ in Eq. (2.25), as a function of the
number of terms in the Fourier-Bessel series in Eq. (2.20) with Eq. (2.21)
for (a) uniformly distributed points along the bilayer-protein boundary and
(b) the boundary point distributions implied by the APD method (see
Sec. 2.2.2). For both panels we considered three-fold clover-leaf protein
shapes (s = 3) in Eq. (2.17) with the indicated values of ϵ, Rλ¯ ≈ 2.3 nm,
and the constant Uλ¯ = 0.3 nm and U¯′ = 0 and set τ = 0. In panel (b) we
used, for ease of comparison, the same gap factor Ω = 0.25 in Eq. (2.28)
for all curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Percentage difference between exact analytic and FEM (red curves) or
BVM (blue curves) solutions for the bilayer thickness deformation energy,
ηG in Eq. (2.27), as a function of the average edge size ⟨L⟩ used in the
FEM solution (upper axes) or the number of terms in Eq. (2.20) with
Eq. (2.21) used in the BVM solution (lower axes) for (a) a cylindrical protein
with Rλ¯ = 2.3 nm and Uλ¯ = 0.3 nm and (b) a crown-shaped protein with
Rλ¯ = 2.3 nm, U¯
0λ = −0.1 nm, βλ¯ = 0.5 nm, and w = 5 in Eq. (2.15). We
set U¯′ = 0 and τ = 0 for both panels. . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Illustration of the APD method used to increase the numerical efficiency of
BVM solutions for (a) a three-fold clover-leaf protein shape (s = 3) and (b)
a four-fold clover-leaf protein shape (s = 4) in Eq. (2.17). The boundary
points used for the BVM solutions are indicated by blue dots. In panel (a)
we set ϵ = 0.38, N = 31, and Ω = 0.62 for the gap length ¯ℓ in Eq. (2.28). In
panel (b) we set ϵ = 0.30, N = 42, and Ω = 0.72. For both panels we set
Rλ¯ ≈ 2.3 nm and τ = 0. To achieve an approximately periodic distribution
of boundary points for even s, we duplicated in panel (b) the boundary
point at the right-most apex, and slightly offset the resulting two boundary
points along the bilayer-protein interface (see main text). The values of N
in panels (a) and (b) were chosen for illustrative purposes. We generally
employ values of N greater than those considered here so as to meet the
numerical precision criteria imposed here (see main text). . . . . . . . . . . 42
2.7 Comparing BVM and FEM solutions for the elastic energy of clover-leaf
protein-induced bilayer thickness deformations. (a) Bilayer thickness
deformation energy, G¯ in Eq. (2.6), with τ = 0, obtained using BVM
and FEM solutions for u¯ in Eq. (2.2) and (b) corresponding percentage
difference between the BVM and FEM solutions for G¯, µ
′
G in Eq. (2.29),
for the clover-leaf protein shapes in Eq. (2.17) as a function of ϵ with the
indicated values of s, Rλ¯ ≈ 2.3 nm, Uλ¯ = 0.3 nm, and U¯′ = 0. For the FEM
solutions we employed an average edge size ⟨L⟩ ≈ 0.1 nm. . . . . . . . . . 44
xvii
2.8 Convergence of BVM solutions for the elastic energy of polygon proteininduced bilayer thickness deformations. (a) Percentage difference
between the exact bilayer thickness deformation field along the bilayerprotein boundary and the bilayer thickness deformation field obtained
from the BVM solution, ηb
′ in Eq. (2.25), as a function of the number of
terms, N, in the Fourier-Bessel series in Eq. (2.20) with Eq. (2.21) for the
boundary point distributions implied by the APD method (see Sec. 2.2.2),
using the indicated values of P in Eq. (2.18) and (2.19) with s = 5. (b)
Percentage difference between the FEM and BVM solutions for G¯, µ
′
G
in Eq. (2.29), where we calculated the BVM solutions for the polygon
protein shapes in Eqs. (2.18) and (2.19) as a function of P and with the
indicated values of s, Rλ¯ ≈ 2.3 nm, Uλ¯ = 0.3 nm, U¯′ = 0, N = 750, and
Ω ≈ 0.32, 0.24, 0.18, 0.16, and 0.16, for symmetries s = 4, 5, 6, 7, and 8,
respectively, in Eq. (2.28), while for the FEM solutions we employed an
average edge size ⟨L⟩ ≈ 0.1 nm and used true polygon shapes, in contrast
to those implied by Eqs. (2.18) and (2.19) with finite P. For all BVM and
FEM solutions depicted here, we set τ = 0. . . . . . . . . . . . . . . . . . . 46
3.1 Color maps of the bilayer thickness deformation footprints due to cloverleaf protein shapes with (a) R¯ = 1 and (b) R¯ = 10 in Eq. (2.17) for
s = 5, ϵ = 0.2, Uλ¯ = 0.3 nm in Eq. (2.15), and U¯′ = 0. Panels (c)
and (d) show the mean curvature in units of 1/λ, H¯ = λH, associated
with the thickness deformation fields in panels (a) and (b), respectively,
while panels (e) and (f) show the corresponding mean curvature maps
obtained for Uλ¯ = −0.3 nm in Eq. (2.15) rather than Uλ¯ = 0.3 nm. We set
2¯aλ = 3.2 nm and τ = 0 for all panels. All results were obtained through
the BVM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Line tension along the bilayer-protein boundary, Λ¯ in Eq. (3.4), as a function
of θ for (a) the protein shape in Fig. 3.1(a) and (b) the protein shape in
Fig. 3.1(b), calculated using the same parameter values as in Fig. 3.1. The
red dashed lines show the average of Λ( ¯ θ) in Eq. (3.4) over the interval
0 ≤ θ ≤ 2π/5, ⟨Λ¯⟩. The yellow dashed lines show Λ¯
analy = G¯
analy/Γ¯, where
G¯
analy is given by Eq. (3.1) and Γ¯ is the protein circumference in Eq. (3.3). . 53
3.3 Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the
BVM (see Sec. 2.2) as a function of protein size R¯ for clover-leaf protein
shapes with (a) s = 3, (b) s = 4, and (c) s = 5 in Eq. (2.17) with the
indicated values of ϵ, and (d) polygonal protein shapes with the indicated
values of s and P = 5 in Eq. (2.18). For all panels we set Uλ¯ = 0.3 nm and
U¯′ = 0 and τ = 0. The insets show the signed percent error ξG in Eq. (3.5)
for the corresponding analytic approximations G¯
analy in Eq. (3.1). . . . . . . 56
xviii
3.4 Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the
BVM (see Sec. 2.2) as a function of lipid chain length m in Eq. (2.4) for (a)
clover-leaf protein shapes with ϵ = 0.3 and the indicated values of s, (b)
clover-leaf protein shapes with s = 5 and the indicated values of ϵ, and
(c) polygonal protein shapes with the indicated values of s and P = 5 in
Eq. (2.18). For all panels we set τ = 0, U¯′ = 0, W λ ¯ = 3.8 nm in Eq. (2.13),
and Rλ¯ ≈ 2.3 nm. The insets show the signed percent error ξG in Eq. (3.5)
for the corresponding analytic approximations G¯
analy in Eq. (3.1). We
always have
U¯
> 0 for the m-discretization used here. . . . . . . . . . . . 57
3.5 Bilayer thickness deformation profile u¯ due to a protein with a circular cross
section as a function of the radial distance from the protein center, r¯ = r/λ,
obtained from the exact analytic solution in Eq. (2.8) with Eq. (2.9) for the
indicated values of U¯′
. We set Uλ¯ = 0.3 nm, τ = 0, and Rλ¯ = 2.3 nm. . . . . 58
3.6 Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the
BVM (see Sec. 2.2) as a function of the bilayer-protein contact slope U¯′
for (a) clover-leaf protein shapes with ϵ = 0.3 and the indicated values of
s, (b) clover-leaf protein shapes with s = 5 and the indicated values of ϵ,
and (c) polygonal protein shapes with the indicated values of s and P = 5,
and cylindrical protein shapes with a circular cross section of radius R¯.
For all panels we set Rλ¯ = 2.3 nm, Uλ¯ = 0.3 nm, and τ = 0. The insets
show the signed percent error ξG in Eq. (3.5) for the corresponding analytic
approximations G¯
analy in Eq. (3.1). . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the
BVM (see Sec. 2.2) as a function of the periodicity in protein hydrophobic
thickness, w in Eq. (2.15), for (a) the clover-leaf protein shapes in Eq. (2.17)
with s = 2 and the indicated values of ϵ, (b) the clover-leaf protein shapes
in Eq. (2.17) with s = 3 and the indicated values of ϵ, and (c) the polygonal
protein shapes in Eq. (2.18) with the indicated values of s and P = 5. For
all panels we set Rλ¯ = 2.3 nm, U¯
0λ = −0.1 nm, βλ¯ = 0.5 nm, U¯′ = 0, and
τ = 0. The red dashed lines indicate the asymptotic scaling ∼ w
3
. The
insets show the signed percent error ξG in Eq. (3.5) for the corresponding
analytic approximations G¯
analy in Eq. (3.1). In panel (d) we show color
maps of the protein-induced bilayer thickness deformations associated
with ϵ = 0.4 in panel (a) at (i) w = 2 and (ii) w = 3, with ϵ = 0.4 in panel (b)
at (iii) w = 3 and (iv) w = 4, and with s = 4 in panel (c) at (v) w = 2 and (vi)
w = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
xix
3.8 Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the
BVM (see Sec. 2.2) as a function of the periodicity in the bilayer-protein
contact slope, v in Eq. (2.16), for (a) the clover-leaf protein shapes in
Eq. (2.17) with s = 2 and the indicated values of ϵ, (b) the clover-leaf
protein shapes in Eq. (2.17) with s = 3 and the indicated values of ϵ, and
(c) the polygonal protein shapes in Eq. (2.18) with the indicated values of s
and P = 5. For all panels we set τ = 0, Rλ¯ = 2.3 nm, Uλ¯ = 0.3 nm, U¯′
0 = 0,
and γ¯ = 0.3. The red dashed lines indicate the asymptotic scaling ∼ v. The
insets show the signed percent error ξG in Eq. (3.5) for the corresponding
analytic approximations G¯
analy in Eq. (3.1). In panel (d) we show color
maps of the protein-induced bilayer thickness deformations associated
with ϵ = 0.4 in panel (a) at (i) v = 2 and (ii) v = 4, with ϵ = 0.4 in panel (b)
at (iii) v = 3 and (iv) v = 4, and with s = 4 in panel (c) at (v) v = 2 and (vi)
v = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9 Difference between the lipid bilayer thickness deformation energies
associated with protein oligomers of symmetry s and their corresponding
s monomers, ∆G¯, calculated using the BVM (see Sec. 2.2) as a function
of (a) the lipid chain length m in Eq. (2.4) and (b) the (constant) bilayeroligomer contact slope U
′
in Eq. (2.14) for a variety of clover-leaf (solid
curves) and polygonal (dashed curves) shapes of the protein oligomers.
We took the protein monomers to have circular cross sections with
U
′ = 0 and used the indicated values of s, with ϵ = 0.3 for the clover-leaf
oligomer shapes in Eq. (2.17) and P = 5 for the polygonal oligomer
shapes in Eq. (2.18). For both panels, we set τ = 0. We set Rλ¯ = 1 nm
for the monomer radii, and used identical cross-sectional areas of the
oligomers and their corresponding monomers. We set U
′ = 0 in panel
(a), 2¯aλ = 3.2 nm in panel (b), and used W λ ¯ = 3.8 nm for the protein
monomers and oligomers in all panels. The schematics in the insets
illustrate transitions between monomers and oligomers for selected
oligomeric shapes. The plots in the insets show the difference in the
oligomerization energies obtained from the analytic approximation G¯
analy
in Eq. (3.1) and the BVM, ∆G¯
ξ = ∆G¯
analy − ∆G¯, for each curve in the
main panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
xx
3.10 Difference between the lipid bilayer thickness deformation energies
associated with the final and initial protein shapes indicated in the insets,
∆G¯, calculated using the BVM (see Sec. 2.2) as a function of (a,b) the
lipid chain length m in Eq. (2.4) and (c,d) the (constant) bilayer-protein
contact slope U
′
in Eq. (2.14). The values of ϵ associated with each
clover-leaf shape in Eq. (2.17) are indicated in the insets, while for the
polygonal protein shapes we set P = 5. We set U
′ = 0 in panels (a,b)
and 2¯aλ = 3.2 nm in panels (c,d), and used W λ ¯ = 3.8 nm and τ = 0 for
all panels. The cross sections of all protein shapes considered here have
area πR¯2 with Rλ¯ = 2.3 nm. The plots in the insets show the differences
in the protein transition energies obtained from the analytic approximation
G¯
analy in Eq. (3.1) and the BVM, ∆G¯
ξ = ∆G¯
analy − ∆G¯, for each curve in
the main panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Schematic views of our hydrophobic shape model for chemoreceptor
trimers. The molecular model of the chemoreceptor trimer in panel (a) is
taken from Ref. [45] and the adjacent clover boundary curve was derived
from Eq. (2.17) with Ron/off = 3.1 nm, son/off = 3, and ϵon/off = 0.2 for both
the on and off states. In panel (b), the decrease in chemoreceptor trimer
hydrophobic thickness when activated is illustrated (not to scale) with
Won = 4.05 nm and Woff = 4.21 nm in Eq. (2.13) for the on and off states. . . 79
4.2 Schematic views of our hydrophobic shape model for MscL. The molecular
models of MscL’s closed and open states in panel (a) are taken from
Ref. [59] and the superimposed clover boundary curves were derived from
Eq. (2.17) with Roff ≈ 2.27 nm and Ron ≈ 3.49 nm, son/off = 5, and ϵoff = 0.22
and ϵon = 0.11 for both the off (closed) and on (open) states. In panel (b),
we show our model for MscL which ignores any change in hydrophobic
thickness (not to scale) with Won = Woff = 3.8 nm in Eq. 2.13 for the on
and off states. In panel (c), we show our model for MscL with a decrease
in hydrophobic thickness (not to scale) when activated with Won = 2.5 nm
and Woff = 3.8 nm in Eq. 2.13 for the on and off states. . . . . . . . . . . . . 81
4.3 Cross-sectional view of Piezo-induced membrane deformations (adapted
from Ref. [107]). The Piezo dome resembles a spherical cap with a fixed
area Scap = 450 nm2
. Key parameters include R (radius of curvature),
h = h0 and r = r0 (central pore axis and radial coordinates at s = 0), s
(arclength along Piezo’s membrane footprint profile, s = 0 at the dome
interface, s > 0 away from the dome), and α (cap angle). Deformations are
assumed to diminish towards a flat membrane shape for large s. . . . . . . 84
xxi
4.4 Estimates of the lipid bilayer deformation contribution to the chemoreceptor
activation energy obtained using Eq. (2.3) with τ = 0, are depicted for
(a) DOPC and (b) EcoC membranes, as a function of temperature. Solid
lines represent calculations employing the clover-leaf chemoreceptor
trimer cross-section shape model described in Sec. 4.2.1, while dashed
lines incorporate a cylinder cross-section shape with chemoreceptor
trimer on and off states possessing equivalent cross-section areas to
the corresponding clover cross-section shapes. The color legend below
the panels indicates which parameters (a, Kℓ
b
, or Kt) were assigned the
temperature relations in Eqs. (4.1)–(4.3) for DOPC lipid bilayers [116]
and, by omission, which of these parameters were held constant at
their respective values at room temperature Trm = 25◦C. In (b) EcoC
membranes, a was modified using a0 = 2.45 nm in Eq. (4.1). In panel
(b), the lightly shaded region depicts the clover model solutions, with a
50% variation in m in Eq. (4.1), and the overlapping darker shaded region
represents solutions with variations in ε by 50%. . . . . . . . . . . . . . . . . 88
4.5 Estimates of the lipid bilayer deformation contribution to MscL’s activation
energy obtained using Eq. (2.3) with τ = 0, are depicted for (a,c) DOPC
and (b,d) EcoC membranes, as a function of temperature. In panels (a,b)
we set Woff = Won = 3.8 nm in Eq. (2.13), and in panels (c,d) we set
Woff = 3.8 nm and Won = 2.5 nm. In all panels, solid lines represent
calculations employing the clover-leaf MscL cross-section shape models
for MscL’s open (on) and closed (off) states described in Sec. 4.2.2, while
dashed lines incorporate a cylinder cross-section shape with MscL opened
(on) and closed (off) states possessing equivalent cross-section areas to
the corresponding clover cross-section shapes. The color legend below
the panels indicates which parameters (a, Kℓ
b
, or Kt) were assigned the
temperature relations in Eqs. (4.1)–(4.3) for DOPC lipid bilayers [116]
and, by omission, which of these parameters were held constant at
their respective values at room temperature Trm = 25◦C. In (b,d) EcoC
membranes, a was modified using a0 = 2.45 nm in Eq. (4.1). In panels
(b,d), the lightly shaded region depicts the clover model solutions, with a
50% variation in m in Eq. (4.1), and the overlapping darker shaded region
represents solutions with variations in ε by 50%. . . . . . . . . . . . . . . . . 92
xxii
4.6 In panels (a,b), we depict estimates of MscL’s activation energy in Eq. (4.5)
in an EcoC membrane as a function of temperature, where we set the
membrane tension τ to the values indicated by the color legend beneath
panel (c). In panels (b,d) we show the opening channel probability in
Eq. (4.4) for MscL in an EcoC membrane as a function of τ , where we set
the temperature T to the values indicated in the color legend underneath
panel (d), and the shaded regions. The shaded regions in panels (b,d)
denote the range of solutions for 50% variations in m in Eq. (4.1) and ε
in Eq. (4.2) as indicated in the greyscale legend underneath panel (d).
In panels (a,b) we set Woff = Won = 3.8 nm, and in panels (c,d) we set
Woff = 3.8 nm and Won = 2.5 nm in Eq. (2.13). In panels (a,b) we set
∆Gp = 55 kBTrm, and in panels (c,d) we set ∆Gp = 0 in Eq. (4.5). . . . . . . 95
4.7 Estimates of (a) the change in deformation energy associated with the
lipid bilayer surrounding the Piezo dome, ∆GM
ℓ
, (b) the change in the
energy associated with the change in the Piezo dome’s in-plane bilayer
area under membrane tension, ∆Gτ
ℓ,cap, (c) the change in the bending
energies associated with the lipid bilayer component of the Piezo dome,
∆Gb
ℓ,cap (green curves), and the protein component of the Piezo dome,
∆Gb
p,cap (purple curves), and (d) the activation energy of Piezo at the
membrane tension values indicated by the color legends and as functions
of temperature. For our estimates of ∆GM
ℓ
(T), in panel (a), we used
Eq. (B.1) in the arc-length representation (see Sec. B.1 for details) with
the boundary conditions in Eqs. (4.6)–(4.8), which we evaluated with
Scap = 450 nm2 and Roff(T) in Eq. (4.9). For our estimates of ∆Gτ
ℓ,cap(T), in
panel (b), we used Eq. (4.10) which we evaluated with Scap = 450 nm2 and
Roff(T) in Eq. (4.9). For our estimates of ∆Gb
ℓ,cap(T), in panel (c), we used
Eq. (4.11), which we evaluated with Kℓ
b
(T) in Eq. (4.2), Scap = 450 nm2
,
and Roff(T) in Eq. (4.9). For our estimates of ∆Gb
p,cap(T), in panel (c), we
used Eq. (4.12), which we evaluated with the K
p
b
(T) that is indicated by
the legend underneath all of the panels, R
p
0 = 10.2 nm, Scap = 450 nm2
,
and Roff(T) in Eq. (4.9). To evaluate Roff(T) in Eq. (4.9), we used
R
p
0 = 10.2 nm, Kℓ
b
(T) in Eq. (4.2), and the K
p
b
(T) that is indicated by the
legend underneath all of the panels. For our estimates of ∆G(T), in panel
(d), we used ∆G(T) = ∆GM
ℓ
(T) + ∆Gτ
ℓ,cap(T) + ∆Gb
ℓ,cap(T) + ∆Gb
p,cap(T). . . 96
4.8 Estimates of channel opening probability in Eq. (4.4) for Piezo as a
function of membrane tension and at the indicated values of temperature
T, assuming (a) K
p
b
(T) = 20 kBTrm, (b) K
p
b
(T) = Kℓ
b
(T), and (c)
K
p
b
(T) = Kℓ
b
(2ε → ε, T), with Kb(T) calculated by Eq. (4.2). To evaluate
∆G = ∆GM
ℓ + ∆Gτ
ℓ,cap + ∆Gb
ℓ,cap + ∆Gb
p,cap in Eq. (4.4) we followed the
caption of Fig. 4.7 to calculate all of its various contributions. The shaded
regions denote the range of solutions that include 50% variations in ε about
ε = 7 × 10−21 J at the temperatures indicated by the color legend in each
panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
xxiii
5.1 Table comparing experimental data on emerin [12] (orange) and predictions
of our reaction-diffusion model (emerin nanodomain diameter, ℓΦ, and
ratio of the fraction of emerin nanodomain area covered by I and A
complexes relative to that of the wild type system under no mechanical
stress, FΦ/F WT
Φ ) (red) for the various emerin systems in Fig. 5.2 and
the ∆95-99 system under no mechanical stress. For ∆95-99 systems
under no mechanical stress, emerin nanodomains were not observed to
self-assemble in experiments [12] and were not predicted to self-assemble
by our model. So, for the ∆95-99 system under no mechanical stress, we
specify “null" for FΦ/F WT
Φ and ℓΦ. . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Colormaps depict numerical solutions for A, I, and (A + I) (overlay)
calculated from our emerin nanodomain self-assembly model (see
Sec. 5.2.2 for numerical implementation), where I and A denote fields
for the fraction of the INM area locally covered by I and A complexes,
respectively. The colorbar scale indicates the values of the fields A, I,
or (A + I). We show model solutions for wild-type systems (a) with and
(b) without mechanical stress, (c) Q133H systems without mechanical
stress, (d) P183H systems without mechanical stress, and (e) ∆95-99
systems under mechanical stress. Diffusion coefficients νI and νA are as
indicated in Fig. 5.1 with νA = νslow and νI = νfast. The reaction rates f1,
f2, g1, g2, and g3 utilized in our calculations are discussed in (a) Sec. 5.2,
(b) Sec. 5.3.2, (c) Sec. 5.3.3, (d) Sec. 5.3.4, and (e) Sec. 5.3.6. The
colormaps depict numerical solutions at the corresponding time t = 100 τ
(see Sec. 5.2.2), with (a) τ ≈ 6 s, (b) τ ≈ 17 s, (c) τ ≈ 3 s, (d) τ ≈ 50 s, (e)
τ ≈ 74 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.1 Boundary error ηb
′ in Eq. (2.25) in BVM calculations (see Sec. 2.2) for
clover-leaf protein shapes with (a) s = 1 and ϵ = 0.54 in Eq. (2.17) and
(b) s = 3 and ϵ = 0.38 in Eq. (2.17) as a function of the gap factor Ω in
Eq. (2.28). We set R ≈ 2.3 nm, U = 0.3 nm, and U
′ = 0. For ease of
comparison we used, for each curve, the indicated, fixed values of N in
Eq. (2.20) with Eq. (2.21). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.2 (a) Lipid bilayer thickness deformation energy G in Eq. (2.26) calculated
using the BVM (see Sec. 2.2) and (b) corresponding boundary error ηb
′ in
Eq. (2.25) for clover-leaf protein shapes as a function of the bit precision
employed in the numerical computations. We used the indicated values of
s and ϵ in Eq. (2.17), and R ≈ 2.3 nm, U = 0.3 nm, and U
′ = 0. For ease
of comparison we used for s = 1 the fixed values N = 20 in Eq. (2.20) with
Eq. (2.21) and Ω = 0.726 in Eq. (2.28), N = 48 and Ω = 0.62 for s = 2,
N = 72 and Ω = 0.52 for s = 3, N = 90 and Ω = 0.552 for s = 4, and
N = 125 and Ω = 0.45 for s = 5. . . . . . . . . . . . . . . . . . . . . . . . . . 182
xxiv
C.1 Table comparing experimental data [12] (orange), estimated parameters
(blue), free diffusion model predictions (fraction of emerin molecules
that are slow diffusers, ρslow, and relative emerin molecule density inside
nanodomains, ⟨Nslow⟩/⟨Nfast⟩) (green) wild-type emerin systems under no
mechanical stress (“WT") and mechanical stress (“WT; force"), Q133H
and P183H mutant emerin systems under no mechanical stress, and
∆95-99 mutant emerin systems under no mechanical stress (“∆95-99") and
mechanical stress (“∆95-99; force"). . . . . . . . . . . . . . . . . . . . . . . 188
C.2 Diffusion-only models applied to wild-type emerin systems under no
mechanical stress. Steady-state (a) fractions of emerin molecules inside
emerin nanodomains, ρslow, and (b) relative densities of emerin molecules
inside emerin nanodomains, ⟨Nslow⟩/⟨Nfast⟩, as a function of the global
fractional INM area covered by emerin molecules, ⟨N⟩, assuming free
diffusion (green curves) and diffusion with steric constraints linear in the
local fractional INM area covered by emerin molecules N (purple curves).
All results were obtained through direct solution of the ME (see Ref. [227]),
with the analytical solutions shown in (a) Eq. (C.1) and (b) Eq. (C.2). . . . . 191
xxv
Abstract
Over recent years, a diverse range of experiments have provided much quantitative
data on the role of membrane proteins in cellular signal transduction and the adaptation
of cells to dynamic environments. Membrane proteins exhibit diverse molecular mechanisms for sensing stimuli, initiating signaling pathways through structural changes, and engaging in collective signaling activities. In particular, protein clustering into domains dedicated to specialized functions provides important mechanisms for cell membrane organization. Furthermore, the perturbation of the lipid bilayer by membrane proteins is thought
to play an important role in membrane protein function. This thesis comprises a set of interconnected studies that employ theoretical physics to investigate fundamental aspects
of the intricate coupling between membrane proteins and cellular responses to stimuli.
Chapter 1 provides a general introduction to cell membranes, membrane proteins, and the
diverse functions they serve in cell membranes, and as well as the theory of membrane
mechanics. Chapter 2 introduces a novel boundary value method (BVM) that bridges
structural biology with membrane elasticity theory, enabling the analytic determination of
protein-induced lipid bilayer deformations, even for non-circular protein cross-sections, in
xxvi
excellent agreement with finite element solutions. Inspired by our BVM, Chapter 3 formulates a simple analytic approximation of the bilayer thickness deformation energy associated with general protein shapes and shows that, for modest deviations from rotational
symmetry, this analytic approximation is in good agreement with BVM solutions. The BVM
and analytical approximation are utilized to explore how variations in protein shape influence elastic bilayer thickness deformations. Our findings reveal that alterations in protein
shape induce changes to the lipid bilayer deformation energy exceeding 10 kBT, which
may have important implications for protein conformational changes and protein oligomerization processes. Chapter 4 examines the interplay between membrane mechanics and
thermosensing, revealing how, mediated by lipid bilayer properties such as hydrophobic
thickness and bending rigidity, temperature changes influence the conformational transitions of membrane proteins. We thus investigate the fundamental principles underlying
the coordination of thermosensing and mechanosensing in living systems. Chapter 5 explores the physical principles underlying the self-assembly of emerin nanodomains at the
inner nuclear membrane, which may shed new light on the role of emerin nanodomains
in mechanotransduction. By employing a comprehensive modeling approach rooted in
the Turing mechanism of nonequilibrium pattern formation, we develop a simple model
quantifying the intricate reaction-diffusion properties of proteins and their nuclear binding partners. On this basis, we provide insight into the wild-type properties of emerin
nanodomains and their response to applied forces, as well as the mechanisms underlying the observed defects in the self-assembly of emerin nanodomains for mutated forms
of emerin associated with Emery-Dreifuss muscular dystrophy. Chapter 6 provides an
overview and conclusions from our studies, and suggests potential future directions of
xxvii
research inspired by our findings. By integrating these diverse research strands, our work
contributes to a deeper understanding of the fundamental principles governing membrane
mechanics and pattern formation, with implications for both physics and biology.
Most of the material described in this thesis is/will be discussed in the following publications:
I. C. D. Alas and C. A. Haselwandter. Dependence of protein-induced lipid bilayer
deformations on protein shape. Phys. Rev. E, 107:024403, 2023.
II. C. D. Alas, O. Kahraman, and C. A. Haselwandter. Thermosensing through membrane mechanics, (expected submission in early 2024).
III. C. D. Alas, F. Pinaud, and C. A. Haselwandter. Physical mechanism for the selfassembly of emerin nanodomains at the inner nuclear membrane, (expected submission
in 2023).
xxviii
Chapter 1
Introduction
This chapter provides an introduction to the research topics addressed in this PhD
thesis. We start by exploring the fundamental components of cell membranes in Section 1.1. In Section 1.2, we dive into various intricate aspects of transmembrane proteins
in cell membranes, including their influence on membrane shape (Section 1.2.1), their
responsiveness to physical stimuli such as temperature (Section 1.2.2), and their collective response to mechanical stress (Section 1.2.3). Finally, Section 1.3 motivates the
boundary value method (BVM) for calculating protein-induced membrane deformations
developed as part of this PhD thesis.
1.1 Proteins and phospholipids: the building blocks of
cell membranes
Cell membranes maintain cellular integrity, and serve as dynamic barriers for nutrient,
signal, and waste exchange with the external environment [1]. Comprising membrane
1
Figure 1.1: Illustration (from Ref. [3]) depicting various membrane proteins within lipid
bilayer environments.
proteins and phospholipids, these structures facilitate vital cellular functions. Phospholipids form bilayers that interlock with membrane proteins, as illustrated in Figure 1.1,
ensuring structural integrity, while membrane proteins govern essential processes such
as ion transport, signal transduction, and membrane shape regulation [2].
Phospholipids possess an amphiphilic nature, with hydrophobic tail chains repelling
water and hydrophilic head groups favorably interacting with it [4]. This amphiphilic property drives the self-assembly of lipid aggregates when introduced into an aqueous environment. The specific type of lipid aggregate formed depends on factors such as hydrocarbon chain characteristics, ionic conditions, and temperature. For instance, doublechained lipids with a large head group area tend to prefer, energetically, bilayer structures.
These self-assembled lipid aggregates typically exhibit characteristic lateral sizes on the
order of magnitude of micrometers while being only a few nanometers in the thickness.
2
Transmembrane proteins exhibit a consistent structural framework composed of three
fundamental constituents [5]. First and foremost, the transmembrane segments are pivotal for anchoring the protein within the membrane. These segments traverse the lipid
bilayer, showing hydrophobic amino acid residues that establish favorable interactions
with the hydrophobic core of the lipid bilayer. Their role extends to providing structural
stability and ensuring the protein’s secure integration within the lipid bilayer. Furthermore, membrane proteins encompass extracellular domains that project into the external
environment. These domains frequently serve as sites for interactions with neighboring
cells, specific ligands, or extracellular molecules. Consequently, membrane proteins actively engage in essential processes such as cell signaling, adhesion, and recognition,
contributing significantly to cellular function [6, 7]. In addition to extracellular domains,
intracellular domains project into the cellular interior. These segments play multifaceted
roles, often involving intracellular signal transduction and interactions with various components within the cell. Intriguingly, they can also be instrumental in anchoring the protein
to the cell’s cytoskeleton, thereby enhancing the protein’s structural stability.
It is noteworthy that the precise composition and arrangement of the above constituent
elements of membrane proteins can exhibit substantial variations among different membrane proteins. This diversity allows membrane proteins to fulfill a wide spectrum of
functions, underscoring their significance in various cellular processes. A profound understanding of the structural organization of membrane proteins is indispensable for unraveling their contributions to cellular functions and the maintenance of cell membrane
integrity.
3
1.2 Coordinators of cellular responses: the multifaceted
functions of transmembrane proteins
The strategic placement of transmembrane proteins empowers them to fulfill a wide
range of functions, including cellular gatekeeping activities such as the regulation of
molecular transport, and pivotal roles as receptors for various signaling pathways [4].
A primary function of transmembrane proteins is the precise regulation of molecular
transport across cell membranes [4, 8]. This function is facilitated by various types of
transmembrane proteins, including ion channels, transporters, and pumps, responsible for
maintaining the delicate balance of ions, small molecules, and nutrients inside and outside
the cell. They act as gatekeepers, controlling the passage of substances to ensure the
proper functioning of cellular processes. For example, ion channels enable the controlled
flow of ions, which is vital for maintaining the electrochemical balance necessary for nerve
impulses and muscle contraction.
Beyond their gatekeeping duties, transmembrane proteins also serve as primary receptors for numerous signaling pathways, as illustrated in Figure 1.2, allowing cells to
detect and respond to various external stimuli [4, 6, 9]. Upon binding specific ligands,
such as hormones or neurotransmitters, transmembrane receptors initiate cascades of
intracellular events, culminating in the activation of essential cellular processes, such as
gene expression, cell growth, and differentiation. Conformational changes triggered by
ligand binding enable the transmission of signals across the cell membrane, ensuring the
coordination of complex cellular activities.
4
Figure 1.2: Illustration (adapted from Ref. [10]) depicting a transduction process initiated
by a transmembrane protein. In particular, in this illustration, a signaling molecule binds
to a receptor protein in the plasma membrane seperating the cell’s cytoplasm from the
extracellular environment, which sets off a signal transduction pathway leading to a response that involves the activation of cellular process.
Moreover, transmembrane proteins contribute to the organization of specialized membrane domains critical for cellular signal transduction [11, 12]. Through their collective
actions, these proteins form defined clusters or domains within the cell membrane, facilitating the initiation of intricate signaling pathways and the recruitment of various signaling
molecules. This protein clustering enables the efficient coordination of cellular responses
to diverse environmental stimuli, ensuring the proper regulation of cellular processes, including adaptation to mechanical stress. Some notable examples are synaptic protein
5
Figure 1.2: Illustration (adapted from Ref. [10]) depicting a transduction process initiated
by a transmembrane protein. In particular, in this illustration, a signaling molecule binds
to a receptor protein in the plasma membrane seperating the cell’s cytoplasm from the
extracellular environment, which sets off a signal transduction pathway leading to a response that involves the activation of cellular process.
Moreover, transmembrane proteins contribute to the organization of specialized membrane domains critical for cellular signal transduction [11, 12]. Through their collective
actions, these proteins form defined clusters or domains within the cell membrane, facilitating the initiation of intricate signaling pathways and the recruitment of various signaling
molecules. This protein clustering enables the efficient coordination of cellular responses
to diverse environmental stimuli, ensuring the proper regulation of cellular processes, including adaptation to mechanical stress. Some notable examples are synaptic protein
domains in neuronal membranes, which regulate synaptic transmission essential for cognitive function and learning, and inner nuclear membrane (INM) emerin nanodomains in
mammalian cells, which facilitate the regulation of signals between the nucleoskeleton
and the cytoskeleton vital for cellular adaptation to mechanical stress.
5
Transmembrane proteins also play a significant role in detecting physical stimuli, responding to mechanical forces, temperature changes, and osmotic pressure variations
[8, 12, 13]. In particular, transmembrane proteins can undergo conformational changes
in response to these stimuli, activating various cellular processes and signaling cascades
[6, 14–16]. By sensing alterations in the cell’s external environment, transmembrane proteins contribute to maintaining cellular homeostasis, enabling cells to adapt and respond
to environmental fluctuations.
The multifaceted functionality of transmembrane proteins underscores their indispensable role in maintaining cellular homeostasis and regulating complex cellular processes.
Their intricate structural arrangement within the cellular membrane enables them to act
as gatekeepers for molecular transport, as receptors for various signaling pathways, and
as sensors for physical stimuli, ensuring cells can adapt and respond to environmental
changes. The collective behavior of transmembrane proteins in forming specialized domains plays a critical role in efficiently regulating cellular function and maintaining cellular
integrity. Understanding the intricate functionality of transmembrane proteins offers valuable insights into the complex mechanisms governing cellular life and may lead to the
development of novel therapeutic strategies for various diseases and disorders.
1.2.1 The influence of membrane proteins on membrane shape
Membrane proteins spanning the lipid bilayer are characterized by large hydrophobic
regions that approximately match up with the thickness of the lipid bilayer hydrophobic
6
core [1, 4, 17–20]. However, distinct membrane proteins often show distinct hydrophobic thicknesses, and transitions in protein conformational state can change the protein’s
hydrophobic thickness. Figure 1.3, for instance, illustrates the gating of the mechanosensitive ion channel of large conductance (MscL), for which the change in protein shape is
thought to impact the lipid bilayer hydrophobic thickness of the surrounding membrane
area. Moreover, the lipid composition in cell membranes tends to be highly heterogeneous, with distinct lipids often showing distinct unperturbed lipid bilayer thicknesses. As
a result, membrane proteins are generally expected to show a (modest) hydrophobic mismatch with the surrounding lipid bilayer, resulting in protein-induced lipid bilayer thickness
deformations [21–31]. Membrane proteins may also deform the lipid bilayer membrane
without perturbing the lipid bilayer thickness [32–38] as, for instance, in the case of bilayer midplane (curvature) deformations (see Appendix. B.1). The energy cost of such
protein-induced lipid bilayer deformations depends on the protein shape and conformational state, the lipid composition, membrane mechanical properties such as membrane
tension, as well as membrane organization, and can thus regulate, or even determine,
membrane protein function. Membrane elasticity theory provides a beautiful framework
for the quantitative description of protein-induced lipid bilayer deformations with, at least
in the most basic models, all physical parameters being determined directly from experiments [21–25, 27–34, 36–40]. As a result, membrane elasticity theory yields definite
predictions for the energy cost of protein-induced lipid bilayer deformations and, hence,
the coupling between lipid bilayer mechanics and membrane protein function, allowing
direct comparisons between theoretical predictions and experimental measurements.
7
Figure 1.3: Illustration (from Ref. [41]) depicting the change in lipid bilayer hydrophobic
thickness as MscL transitions from its closed state to its opened state.
Over the past two decades, breakthroughs in membrane protein crystallography and,
more recently, cryo-electron microscopy have yielded enormous insight into the shape
of membrane proteins [42–53]. Despite the large diversity in protein shape revealed by
these experiments, mathematical difficulties associated with the description of proteininduced lipid bilayer deformations for general protein shapes have meant that the elasticity theory of bilayer-protein interactions has largely been limited to idealized, rotationally
symmetric protein shapes. However, membrane protein shape may have important consequences for membrane morphology, membrane elastic properties, membrane curvature sensing and mechanosensing, the lateral organization and orientation of membrane
proteins, bilayer-mediated protein interactions, and the regulation of protein function [30,
8
54–61]. This PhD thesis aims to address these issues by providing a versatile mathematical methodology allowing for the description of bilayer-protein interactions for the protein
shapes observed in structural studies (see Chapters 2 and 3).
1.2.2 Temperature sensing in cells: the convergence of membrane
mechanics and function
Living organisms are inherently attuned to their ever-changing surroundings, relying
on a plethora of environmental cues to orchestrate vital biological processes. Among
these cues, temperature is a crucial factor, exerting a significant influence on cellular
physiology. The capacity to detect and respond to temperature variations is essential
for survival, enabling organisms to thrive in diverse thermal environments. For instance,
many cells employ temperature as a critical determinant in decision-making processes.
Many microorganisms utilize temperature as a cue for optimal growth conditions, adjusting
not only their metabolic activities but also their motion to flourish within specific thermal
niches [13, 62–69]. Moreover, extreme cold temperatures can inhibit vital cellular processes, leading organisms to employ thermosensory mechanisms to evade or adapt to
adverse conditions [70–79]. Conversely, excessive high temperatures can pose a severe threat to cell viability, necessitating rapid responses to mitigate the damaging effects
of extreme warmth [16, 80–84]. In recent decades, a diverse range of experiments have
significantly advanced our understanding of how organisms perceive and respond to environmental cues, particularly temperature fluctuations. This progress has been punctuated
9
by the identification of key molecular players, including temperature and touch transmembrane protein sensors, that play pivotal roles in these sensory processes. While these
insights have provided valuable pieces of the puzzle, the precise physical mechanisms
governing temperature sensing at the molecular level continue to be elusive.
In recent decades, extensive research has uncovered the mechanical properties of
cellular membranes, revealing a coupling between protein function and membrane mechanics [8, 14, 15, 22, 23, 27, 38, 46, 47, 49, 59, 85–112]. These specialized proteins,
often called mechanosensors, couple to membrane mechanical properties such as membrane thickness and rigidity. Additionally, various lines of experimental research suggest
that biological membranes, typically considered soft materials, exhibit substantial changes
as temperatures rise, making them easier to deform and thinner [15, 113–117]. While our
quantitative understanding of the effect of temperature changes on membrane deformability continues to evolve, available evidence, albeit limited, permits a basic quantification of
the relationship between temperature and membrane mechanical properties within physiologically relevant temperature ranges. As we shall discuss in this thesis (see Chapter 4),
this interplay between protein functionality and membrane mechanics, influenced by temperature, suggests that cells might possess the inherent capacity to detect temperature
variations through membrane mechanics. On this basis, we develop a basic framework
allowing quantification of the potential consequences of temperature fluctuations on membrane mechanics and protein conformational states. In the scope of our investigation,
membrane elasticity theory [21, 23, 27, 32–34, 59, 102, 118, 119], which we discuss in
Chapter 2, serves to establish a direct link between the mechanics of the lipid bilayer and
the functional behavior of membrane proteins. Notably, different conformational states
10
of membrane proteins often yield distinct protein-induced membrane deformations, with
resultant changes in the energy of bilayer-protein interactions. On this basis, we employ membrane elasticity theory to connect measured temperature-dependent changes
in bilayer mechanical properties to transitions in protein conformational state.
1.2.3 Collective protein behavior in specialized membrane domains
Proteins, both individually and collectively, are fundamental for a cell’s ability to sense
and respond to physical stimuli. An example of this is the organized clustering of proteins into specialized domains, a prevalent occurrence in cells. These protein-rich domains, composed of an assortment of proteins, often act as hubs for specific cellular
functions. Take neurons, for instance. Neuron synaptic protein domains trigger complex signal pathways, transmitting information through neurotransmitter release, receptor
binding, synaptic function, vesicle recycling, structural integrity, and material transport,
ultimately supporting learning, memory, and neural communication [9, 11, 120–122]. The
precise arrangement and organization of these domains are critical for dynamic and regulated synaptic communication. Any disruption or dysregulation in this intricate protein
network can lead to various neurological disorders and cognitive impairments. In our exploration of collective protein behavior, we place a specific focus on the self-assembly of
emerin nanodomains located at the INM and their role in initiating signaling pathways in
response to mechanical stress.
Emerin, a nuclear membrane protein within mammalian cells, is a fundamental component of the INM [12]. It plays a pivotal role in mechanotransduction, as illustrated in
11
Figure 1.4, a process through which mechanical forces are detected and converted into
biochemical signals. This function is intricately connected to emerin’s role as a key link
between the plasma membrane, the cytoskeleton, and the nucleoskeleton.
Emerin is a key component of the Linker of Nucleoskeleton and Cytoskeleton (LINC)
complex. This complex acts as a bridge, connecting the nuclear lamina inside the nucleoplasm to the cytoskeleton on the cytoplasmic side. The LINC complex has the ability
to sense and relay mechanical signals from the plasma membrane to the cell nucleus,
which, consequently, can be perceived by emerin.
Emerin can affect gene expression involved in the processes of mechanotransduction. For example, by influencing genes such as β-catenin and Lmo7, emerin can affect
cytoskeletal dynamics and cell shape [123–126]. Numerous studies have highlighted
the significance of emerin’s intrinsically disordered region in performing many of its vital
functions. This structural flexibility allows emerin to adopt various conformations, form
oligomers, and engage with multiple partners at the INM [12, 127–139]. Mutations in
emerin or its absence have been correlated with abnormal responses of the nuclear envelope to mechanical stress, ultimately resulting in Emery-Dreifuss muscular dystrophy
(EDMD).
A recent study utilizing single-molecule tracking and super-resolution fluorescence
microscopy has provided intriguing insights into the steady-state distributions and mobilities of wild-type and mutated emerin at the INM under various conditions, including
mechanical stress [12]. In particular, these experiments revealed two distinct distributions
of emerin species at the INM, slow and fast diffusers, and that emerin generally forms
12
Figure 1.4: Schematic (from Ref. [12]) depicting the reorganization of emerin at the NE in
response to mechanical stress. Increased lateral mobility at the INM is initiated by emerin
monomer unbinding from nuclear actin and BAF, facilitating LEM domain interactions with
binding sites along the intrinsically disordered region of other emerin molecules. This
controlled process leads to the formation of emerin oligomers at SUN1 LINC complexes,
subsequently stabilized by lamin A/C.
13
stable nanodomains of elevated emerin concentrations which are maintained through interactions with emerin and other nuclear binding partners (NBPs) [e.g., SUN1, lamin A/C,
barrier-to-autointegration factor (BAF), and nuclear actin]. Mutations of emerin and mechanical stress were found to perturb the distributions of emerin and its oligomerization
potential.
The INM spatial pattern of emerin nanodomains of increased concentrations and the
distinction between slower and faster diffusing emerin species resembles the properties
of molecular domains self-assembled through a Turing mechanism in an activator-inhibitor
reaction-diffusion model [9, 11, 120–122, 140–144]. In this model, inhibitors, which diffuse rapidly, act to restrain increased molecular concentrations via steric constraints. On
the other hand, activators diffuse at a slower pace compared to inhibitors but activate elevated molecular concentrations of both inhibitors and other activators. In Chapter 5, we
show that the self-assembly of stable emerin nanodomains may be attributed to the selfstabilization of slow-diffusing, activating emerin-complexes and fast-diffusing, inhibiting
emerin-complexes at the INM, coupled with the steric repulsion of the inhibitors.
1.3 Shortcomings of former approaches for calculating
protein-induced lipid bilayer deformations
While it is relatively straightforward to derive analytical solutions for bilayer deformations induced by idealized proteins with approximately circular cross-sections [4], structural biology has revealed that membrane proteins often deviate significantly from this
14
Figure 1.5: Diagrams (from Ref. [145]) of the general system domain discetization
schemes utilized in (a) BVM, (b) FDM, and (c) FEM [145].
intricately shaped system domains, necessitating a substantial number of elements or grid
points for high accuracy. Implementing parallelized computing structures to handle such
situations can help to alleviate these issues but adds complexity to the process. These
challenges become particularly evident when dealing with bilayer deformations with large
decay lengths, demanding extensive computational resources and often rendering FEM
and FDM impractical.
BVM solutions, characterized by their focus on boundary conditions and minimal discretization [see Fig. 1.5(a)], offer several distinct advantages over FEM and FDM. Notably,
16
Figure 1.5: Diagrams (from Ref. [145]) of the general system domain discetization
schemes utilized in (a) BVM, (b) FDM, and (c) FEM [145].
idealized shape. As a result, a more precise assessment of protein-induced bilayer deformations, incorporating the complex protein shapes observed in structural biology, typically relies on numerical methods. In previous studies [30], both finite element methods
(FEM) and finite difference methods (FDM) have been employed to address bilayer deformations caused by proteins of arbitrary shapes. However, these methods show various
limitations and, in some cases, are unable to provide accurate solutions. The BVM developed in this thesis (see Chapter 2) provides an alternative approach for calculating
protein-induced bilayer deformations, and allows accurate calculation of protein-induced
bilayer deformations for proteins with arbitrary shape. Our investigation reveals that the
BVM offers several advantages over FEM and FDM when it comes to addressing lipid
bilayer deformations.
FEM is a versatile numerical technique employed to solve partial differential equations
by dividing the computational domain into smaller elements, often triangular in shape
[see Fig. 1.5(c)]. It excels in addressing complex problems characterized by intricate geometries, diverse material properties, and complex boundary conditions. However, FEM
15
can be computationally intensive, particularly when dealing with numerous elements due
to large sized systems and complex boundary geometries. Its implementation often demands expertise in tasks such as mesh generation, element selection, and boundary
condition specification to achieve exceptional accuracy. It is noteworthy that, in the case
of protein-induced lipid bilayer deformations, standard FEM software is often not suitable,
as it requires a non-standard approach [30]. Conversely, FDM adopts a grid-based approach, discretizing the domain and approximating derivatives using finite differences [30]
[see Fig. 1.5(b)]. FDM, like FEM, produces limited success when dealing with large and
intricately shaped system domains, necessitating a substantial number of elements or grid
points for high accuracy. Implementing parallelized computing structures to handle such
situations can help to alleviate these issues but adds complexity to the process. These
challenges become particularly evident when dealing with bilayer deformations with large
decay lengths, demanding extensive computational resources and often rendering FEM
and FDM impractical.
BVM solutions, characterized by their focus on boundary conditions and minimal discretization [see Fig. 1.5(a)], offer several distinct advantages over FEM and FDM. Notably,
their simplicity in problem setup is a key strength. BVM involves the definition of boundary conditions, which are often known in advance, reducing the necessity for an extensive
grid or mesh. This streamlined process enhances computational efficiency and simplifies
implementation. In the context of bilayer deformations, BVM excels in scenarios where
significant bilayer deformation attenuation lengths are expected. This advantage arises
from BVM’s discretization scheme, which relies solely on the boundary geometry rather
than the size of the system enclosed by the boundary, a key contrast with FEM and FDM.
16
Consequently, BVM imposes a considerably lighter computational load. Another distinctive feature of BVM is its capacity to yield analytic solutions, in contrast to the numerical
solutions provided by FEM and FDM.
Nonetheless, the very simplicity that makes BVM advantageous can, in certain scenarios, become a constraint. Its strong computational reliance on boundary shape can
result in significant requirements on computational resources, such as situations involving
many proteins. In such instances FEM, which tends to show a computational efficiency
that is only mildly dependent on boundary geometry, is the optimal choice. In Chapter 2
we demonstrate, through rigourous benchmarks, that for many of the types of protein
shapes observed in structural biology, a BVM, often effortlessly, generates accurate analytic (if complicated) solutions for the protein-induced lipid bilayer deformations and their
associated energies.
17
Chapter 2
A boundary value method for lipid bilayer deformations
The objective of this chapter is to develop, describe, and test a straightfoward and
easy to implement BVM suited for the construction of analytic solutions of protein-induced
lipid bilayer deformations for protein shapes with arbitrarily large deviations from a circular cross section. This BVM allows for constant as well as variable boundary conditions
along the bilayer-protein interface. In particular, we consider here, as test cases for the
BVM, four generic classes of protein shapes breaking the rotational symmetry of proteininduced lipid bilayer thickness deformations, which are illustrated in Fig. 2.1. Inspired by
observed molecular structures of membrane proteins [5, 146], we consider two classes
of non-circular membrane protein cross sections: Clover-leaf [see Fig. 2.1(a)] and polygonal [see Fig. 2.1(b)] protein shapes. Furthermore, we allow for variations in the bilayerprotein hydrophobic mismatch [see Fig. 2.1(c)] as well as in the bilayer-protein contact
slope [see Fig. 2.1(d)] along the bilayer-protein interface. Such variations in the bilayerprotein boundary conditions can arise, on the one hand, as inherent features of the protein
structure or, on the other hand, as a result of, for instance, the binding of small peptides,
such as spider toxins, or other molecules along the bilayer-protein interface [5, 22, 146,
18
3
FIG. 1: Surface views of thickness deformation induced by a bilayer membrane embedded protein of cylindrical shape. (a) Isometric
view. (b) Lateral view.
it is therefore convenient to represent the positions of
the two lipid bilayer leaflets in the Monge parametrization of surfaces, h± = h±(x, y), with Cartesian coordinates (x, y) (see Fig. 2). Furthermore, it is instructive to
express h+(x, y) and h(x, y) in terms of the midplane
deformation field h = h(x, y),
h = h+ + h
2 , (1)
and in terms of the thickness deformation field u =
u(x, y),
u = h+ h 2a
2 , (2)
where a is one-half the unperturbed lipid bilayer thickness (Fig. 2). The value of a depends on, for instance,
the tail length of the lipid species under consideration,
and can be measured directly in experiments [3, 24, 27].
The membrane elasticity theory describing the shape
Figure 2.1: Protein-induced lipid bilayer thickness deformations for selected families of
protein shapes: (a) Clover-leaf protein cross section with five-fold symmetry, constant
protein hydrophobic thickness, and zero bilayer-protein contact slope, (b) polygonal protein cross section with six-fold symmetry, constant protein hydrophobic thickness, and
constant bilayer-protein contact slope U
′ = 0.3, (c) clover-leaf protein cross section with
three-fold symmetry, a five-fold symmetric (sinusoidal) variation in protein hydrophobic
thickness, and zero bilayer-protein contact slope, and (d) polygonal protein cross section with seven-fold symmetry, constant protein hydrophobic thickness, and a three-fold
symmetric (sinusoidal) variation in the bilayer-protein contact slope. The color map and
purple surfaces show the positions of the upper and lower lipid bilayer leaflets, respectively. The bilayer-protein boundaries are color-coded according to their symmetries (see
also Fig. 2.3 in Sec. 2.1). For panels (a) and (c) we used ϵ = 0.2 and ϵ = 0.3 in Eq. (2.17),
respectively, and for panels (b) and (d) we used P = 5 in Eqs. (2.18) and (2.19). All bilayer
surfaces were calculated using the reference parameter values in Sec. 2.1 and the BVM
for protein-induced lipid bilayer thickness deformations described in Sec. 2.2.
147]. For each of these four classes of protein shapes we use the BVM to obtain the
energy cost of protein-induced lipid bilayer thickness deformations, and test these results
against corresponding numerical solutions obtained through the FEM for bilayer thickness
deformations [30, 60, 61]. Our BVM reproduces available analytic solutions for proteins
19
with circular cross section and yields, for proteins with non-circular cross section, excellent agreement with the numerical, finite element solutions. Crucially, our BVM does not
suffer from the membrane areal domain size limitations exhibited by FEM and FDM, with
a computational demand that only scales with the complexity of the bilayer boundary geometry, making it the preferred choice for addressing bilayer deformations with potentially
long decay lengths.
This chapter is organized as follows. Section 2.1 summarizes the elasticity theory of
protein-induced lipid bilayer thickness deformations. In Sec. 2.2 we describe in detail the
BVM for bilayer thickness deformations, test this BVM against FEM solutions, and discuss
how the BVM can be used to calculate protein-induced lipid bilayer thickness deformations, and their associated elastic energy, for general protein shapes. We summarize the
conclusions of our work and discuss limitations and further potential applications of our
BVM in Sec. 6.1.
2.1 Modeling protein-induced lipid bilayer thickness deformations
The preferred hydrophobic thickness of lipid bilayers depends strongly on the lipid
chain length [1, 4, 17–20] while different membrane proteins, and even different conformational states of the same membrane protein, often have distinct hydrophobic thicknesses.
For membrane proteins that offer a rigid interface to the lipid bilayer and show a modest hydrophobic mismatch with the unperturbed lipid bilayer, the lipid bilayer thickness is
20
expected to deform in the vicinity of the membrane protein so as to achieve hydrophobic matching at the bilayer-protein interface [21–31]. The resulting protein-induced lipid
bilayer thickness deformations can result in a pronounced dependence of the protein conformational state, and protein function, on lipid chain length [14, 23, 102, 118, 148, 149].
The purpose of this section is to summarize the elasticity theory of protein-induced lipid
bilayer thickness deformations [21–25, 27–30, 39, 58, 59]. We first outline the standard
elasticity theory of lipid bilayer thickness deformations (see Sec. 2.1.1). We then describe
how protein shape couples to lipid bilayer thickness, and discuss the models of protein
shape considered in Chapters 2–4 (see Sec. 2.1.2).
2.1.1 Continuum elasticity theory of lipid bilayer deformations
Lipid bilayer thickness deformations tend to decay rapidly, with a characteristic decay
length ≈ 1 nm [22, 28]. When modeling protein-induced lipid bilayer thickness deformations it is therefore convenient to represent the positions of the two lipid bilayer leaflets in
the Monge parameterization of surfaces, h± = h±(x, y), with Cartesian coordinates (x, y)
(see Fig. 2.2). It is instructive to express h+(x, y) and h−(x, y) in terms of the midplane
deformation field h = h(x, y),
h =
h+ + h−
2
, (2.1)
and in terms of the thickness deformation field u = u(x, y),
u =
h+ − h− − 2a
2
, (2.2)
21
3
troduced by the Bessel functions in our deformation field
expressions and software to curve these issues and optimize our code’s speed in C++. We also give approximate ranges of protein shapes for which double precision
floating point numbers are sucient for the BVM to accumulate negligible floating point error. In appendix B,
we discuss optimizing the BVM through adaptive point
distributions and how the optimization is dependent on
the protein shape and the number of terms used to approximate the deformation field.
FEM dicult to implement for general protein shapes
with varying boundary conditions.
II. MODELING PROTEIN-INDUCED BILAYER
THICKNESS DEFORMATIONS
The preferred hydrophobic thickness of lipid bilayers
depends strongly on the lipid tail length [21–24] while
di↵erent membrane proteins, and even di↵erent conformational states of the same membrane protein, often have distinct hydrophobic thicknesses. Since membrane proteins do, in general, o↵er a rigid interface to
the lipid bilayer, the lipid bilayer thickness tends to
deform in the vicinity of membrane proteins so as to
achieve hydrophobic matching at the bilayer-protein interface [3, 10, 19, 25–27]. The resulting protein-induced
lipid bilayer thickness can result in a pronounced dependence of protein conformational state, and protein function, on lipid tail length [5, 10, 19, 28–30]. The purpose of this section is to summarize the elasticity theory
of protein-induced lipid bilayer thickness deformations
[3, 6, 10, 16, 25, 26, 31]. We first summarize the standard
elasticity theory of lipid bilayer thickness deformations
(see Sec. II A). In Sec. II B we then describe how protein
shape couples to lipid bilayer thickness, and discuss the
models of protein shape considered in this article.
A. Elasticity theory of bilayer thickness
deformations
Bilayer thickness deformations tend to decay rapidly,
with a typical decay length ⇡ 1 nm [27, 31]. When modeling protein-induced lipid bilayer thickness deformations,
it is therefore convenient to represent the positions of
the two lipid bilayer leaflets in the Monge parametrization of surfaces, h± = h±(x, y), with Cartesian coordinates (x, y) (see Fig. ??). Furthermore, it is instructive
to express h+(x, y) and h(x, y) in terms of the midplane
deformation field h = h(x, y),
h = h+ + h
2 , (1)
and in terms of the thickness deformation field u =
u(x, y),
u = h+ h 2a
2 , (2)
FIG. 2: Surface views of thickness deformation induced by a
bilayer membrane embedded protein of cylindrical shape. (a)
Isometric view. (b) Lateral view.
where a is one-half the unperturbed lipid bilayer thickness (Fig. ??). The value of a depends on, for instance,
the tail length of the lipid species under consideration,
and can be measured directly in experiments [3, 24, 27].
The membrane elasticity theory describing the shape
of lipid bilayers [2, 32–34] dates back to the classic work
of W. Helfrich [7], P. B. Canham [35], E. A. Evans [36],
and H. W. Huang [10]. Interestingly, one finds that the
elastic energies governing h and u in Eqs. (1) and (2)
decouple from each other to leading order [6, 37]. In
the most straightforward model of bilayer-protein interactions [3, 10, 16, 17, 25–27, 32–34, 38], the energy
cost of protein-induced bilayer midplane deformations
is then captured by the Helfrich-Canham-Evans energy
[7, 35, 36], and the energy cost of protein-induced bilayer
thickness deformations is given by [3, 6, 10, 16, 25, 26, 31]
G = 1
2
Z
dxdy
Kb
r2u
2
+ Kt
⇣u
a
⌘2
, (3)
where the integral runs over the (in-plane) lipid bilayer
surface, Kb is the lipid bilayer bending rigidity, and Kt is
Figure 2.2: Notation used for the calculation of protein-induced lipid bilayer thickness deformations in (a) angled and (b) side views. As an example, we consider here a membrane
protein with a non-circular (clover-leaf) bilayer-protein boundary curve, C(θ), constant hydrophobic thickness, W(θ) = W0, and zero bilayer-protein contact slope, U
′
(θ) = 0. The
positions of the upper and lower lipid bilayer leaflets are denoted by h+ and h−, from which
the bilayer midplane and bilayer thickness deformation fields h and u can be obtained via
Eqs. (2.1) and (2.2), respectively. We denote one-half the unperturbed bilayer thickness
by a, resulting in a hydrophobic mismatch U = W/2 − a at the bilayer-protein interface.
The unit vectors ˆt and ˆn denote the directions tangential and perpendicular (pointing towards the protein) to the bilayer-protein boundary, respectively.
22
where a is one-half the unperturbed lipid bilayer thickness (Fig. 2.2). The value of a
depends on, for instance, the chain length of the lipid species under consideration, and
can be directly measured in experiments [21, 22, 40].
The membrane elasticity theory describing the shape of lipid bilayers [4, 150–152]
dates back to the classic work of W. Helfrich [32], P. B. Canham [33], E. A. Evans [34],
and H. W. Huang [23]. Interestingly, one finds that the elastic energies governing h and
u in Eqs. (2.1) and (2.2) decouple from each other to leading order [27, 39]. In the most
straightforward model of bilayer-protein interactions [21–25, 27–30, 150–154], the energy
cost of protein-induced lipid bilayer midplane deformations is then captured by the classic Helfrich-Canham-Evans energy [32–34], and the energy cost of protein-induced lipid
bilayer thickness deformations is given by [21, 23–25, 27–30]
G =
1
2
Z
dxdy
Kb (2H)
2 + Kt
u
a
2
+ τ (∇u)
2 + 2τ
u
a
, (2.3)
where the integral runs over the (in-plane) lipid bilayer surface, Kb is the lipid bilayer
bending rigidity, the mean curvature H =
1
2∇2u, Kt
is the bilayer thickness deformation
modulus, and τ is the lateral membrane tension.
The terms Kb (∇2u)
2
, Kt (u/a)
2
, τ (∇u)
2
, and 2τ
u
a
in Eq. (2.3) provide lowest-order
descriptions of the energy cost of bilayer bending, the compression/expansion of the bilayer hydrophobic core, changes in the projection of the bilayer area onto the reference
plane used in the Monge representation, and stretching deformations tangential to the
leaflet surfaces, respectively. Equation (2.3) has been successfully employed to describe
bilayer-protein interactions in a variety of experimental systems [4, 14, 21–23, 27, 28, 30,
23
59, 102, 118, 155–157]. In general, the protein-induced lipid bilayer thickness deformations captured by Eq. (2.3) compete with protein-induced bilayer midplane deformations
[22, 32–38, 54–56, 107, 153, 154, 158]. Depending on the specific bilayer-protein system
under consideration, both contributions to the elastic energy of bilayer-protein interactions
may need to be considered [22, 27]. We also assume in Eq. (2.3) that the lipids forming
the bilayer have zero intrinsic curvature. A nonzero lipid intrinsic curvature could also be
included in the formalism employed here [24, 25, 119]. Furthermore, the elastic energy
of protein-induced lipid bilayer deformations involves, in general, contributions due to lipid
tilt deformations [32, 56, 159–161], in addition to contributions due to bilayer thickness
and bilayer midplane deformations. Finally, we note that one could extend Eq. (2.3) to include a bending term associated with the Gaussian membrane curvature. Previous work
on bilayer-protein interactions indicates that Gaussian curvature contributions to Eq. (2.3)
tend to be negligible in experiments [27, 119]. We employ here Eq. (2.3) as a simple
model system for investigating the effect of protein shape on the elastic energy of proteininduced lipid bilayer deformations while noting that, as illustrated above, Eq. (2.3) can be
extended and modified in a variety of ways.
Similarly as the unperturbed lipid bilayer thickness 2a, the effective parameters Kb and
Kt
in Eq. (2.3) characterizing the elastic properties of the bilayer membrane depend on
the lipid composition, and can be directly measured in experiments [21, 22, 40]. Typical
values of Kb, Kt
, and a for cell membranes are Kb = 20 kBT, Kt = 60 kBT/nm2
, and
a = 1.6 nm [28, 40, 114, 116]. In Chapters 2 and 3 we use these values of Kb, Kt
, and
a; in Chapter 4 we will assign to Kb, Kt
, and a temperature dependent relations. When
studying the dependence of protein-induced bilayer thickness deformations on lipid chain
24
length we follow Refs. [27, 40, 162] and assume, for simplicity, a linear dependence of a
on lipid chain length:
a =
1
2
(0.13m + 1.7) nm . (2.4)
The integer m in Eq. (2.4) denotes the lipid chain length (number of carbon atoms comprising each lipid chain), with the approximate range 13 ≤ m ≤ 22 for phospholipids in
cell membranes [1, 17, 19, 40]. For simplicity, we take Kb and Kt
to be independent of m
while noting that, in general, Kb and Kt may have a (weak) dependence on m [40].
The effective parameters Kb, Kt
, and a in Eq. (2.3) yield the characteristic length scale
λ =
a
2Kb
Kt
1/4
, (2.5)
which corresponds to the characteristic decay length of bilayer thickness deformations
[27]. As alluded to above, we have λ ≈ 1 nm [22, 28]. Similarly, the bilayer bending
rigidity Kb defines a characteristic energy scale in Eq. (2.3). It is therefore convenient to
recast the bilayer thickness deformation energy in Eq. (2.3) in terms of the characteristic
spatial and energy scales, λ and Kb. In Chapters 2 and 3, we use a dimensionless form
of Eq. (2.3) such that GK¯
b → G, xλ¯ → x, yλ¯ → y, uλ¯ → u, aλ¯ → a, K¯
tKb/λ2 → Kt
, and
τK¯ b/λ2 → τ , resulting in
G¯ =
1
2
Z
dxd¯ y¯
h
∇¯ 2u¯
2
+ ¯u
2 + ¯τ
∇¯ u¯
2
+ 2¯τ
u¯
a¯
i
, (2.6)
where ∇ ≡ ¯ λ∇.
We assume that, for a given protein conformational state, the dominant bilayer thickness deformation field u¯(¯x, y¯) minimizes Eq. (2.6) subject to suitable boundary conditions
[21, 23–25, 27, 28, 30]. The Euler-Lagrange equation associated with Eq. (2.6) is given
by
∇¯ 2 − ν¯+
∇¯ 2 − ν¯−
u¯ = −
τ¯
a¯
(2.7)
with ν¯± =
1
2
τ¯ ± i
√
4 − τ¯
2
, where i is the imaginary unit, and u¯ is composed of a general
solution u¯g (¯x, y¯) and a particular solution u¯p = −τ/¯ a¯. To construct the general solution of
Eq. (2.7) for protein-induced bilayer thickness deformations it is useful to transform (¯x, y¯)
to the dimensionless polar coordinates (¯r, θ) with the protein center as the origin of the
polar coordinate system. Assuming that protein-induced bilayer thickness deformations
form a localized membrane footprint u¯g → 0 as r¯ → ∞ [23, 118, 119], in which case
Eq. (2.7) yields [30, 58, 59]
u¯ (¯r, θ) + τ¯
a¯
= ¯f
+ (¯r, θ) + ¯f
− (¯r, θ) , (2.8)
where the Fourier-Bessel series
¯f
±(¯r, θ) = A
±
0 K0
√
ν¯±r¯
+
X∞
n=1
A
±
n Kn
√
ν¯±r¯
cos (nθ) + B
±
n Kn
√
ν¯±r¯
sin (nθ)
, (2.9)
in which the Kn are the modified Bessel functions of the second kind [163] and the values of the coefficients A
±
0
, A±
n
, and B±
n are determined by the bilayer-protein boundary
condition
The bilayer thickness deformation energy in Eq. (2.6) is conveniently evaluated for
the stationary bilayer thickness deformation field in Eq. (2.8) by noting that, via Eq. (2.7),
Eq. (2.6) can be transformed to a line integral along the bilayer-protein boundary C¯ [27,
30, 59] (Fig. 2.2). For simplicity, we thereby take the bilayer-protein boundary to be specified by the polar curve ¯r = C¯(θ)ˆr, where ˆr is the radial unit vector pointing away from the
protein center. We thus have
G¯ =
1
2
Z 2π
0
dθ ¯l ˆn ·
h
∇¯ u¯∇¯ 2u¯ −
u¯ +
τ¯
a¯
∇¯ 3u¯ + ¯τ
u¯ +
τ¯
a¯
∇¯ u¯
i
r¯=C¯(θ)
+ G¯∞ , (2.10)
where the line element ¯l =
q
C¯(θ)
2
+
C¯′
(θ)
2
, the unit vector ˆn is normal to the tangent
of ¯r = C¯(θ)ˆr and points towards the protein (Fig. 2.2), and G¯∞ = − (¯τ/a¯)
2
R
dxd¯ y¯. The
constant term G¯∞ emerges from the relaxation of the “loading" device producing tension
(τ > 0) and diverges under the assumption of an asymptotically flat, infinite membrane.
Since G¯∞ does not contribute to the energy cost of protein-induced bilayer thickness
deformations we shift G¯ so as to subtract G¯∞ from G¯, G¯ − G¯∞ → G¯. Note that the term
in brackets in Eq. (2.10) may be interpreted as a bilayer-protein line tension along the
bilayer-protein boundary [27, 30, 102]. The normal vector ˆn in Eq. (2.10) is obtained by
differentiating the bilayer-protein boundary curve ¯r = C¯(θ)ˆr with respect to θ and rotating
the resulting tangent vector by π/2 so as to point towards the protein,
ˆn =
−C¯(θ)ˆr + C¯′
(θ)
ˆθ
¯l
, (2.11)
27
where we have noted that the (counterclockwise) angular unit vector ˆθ = dˆr/dθ in polar coordinates (Fig. 2.2). Equation (2.10) with Eq. (2.11) allows calculation of G¯ in
Eq. (2.6) and, hence, G in Eq. (2.3) along a one-dimensional curve rather than over a
two-dimensional surface, which provides a computationally efficient method for evaluating G¯.
2.1.2 Modeling protein shape
The coefficients A
±
0
, A±
n
, and B±
n
in Eq. (2.9) are fixed by the boundary conditions at the
bilayer-protein interface. The general mathematical form of these boundary conditions,
which encode the key protein properties governing protein-induced lipid bilayer thickness
deformations, follows from the calculus of variations [164, 165]. Based on previous work
on protein-induced bilayer thickness deformations [21–26, 118], we assume that the lipid
bilayer thickness deforms in the vicinity of membrane proteins so as to achieve hydrophobic matching at the bilayer-protein interface. We thus have the boundary condition
u¯(¯r, θ)
r¯=C¯(θ)
+
τ¯
a¯
= U¯(θ) + τ¯
a¯
, (2.12)
where the bilayer-protein hydrophobic mismatch
U¯(θ) = 1
2
W¯ (θ) − 2¯a
, (2.13)
in which W(θ) = λW¯ (θ) is the protein hydrophobic thickness along the bilayer-protein
boundary (Fig. 2.2). For large enough magnitudes of U, membrane proteins or lipids may
28
expose parts of their hydrophobic regions to water, which would amount to an offset of W¯
in Eq. (2.13). For a given membrane protein, W(θ) can be estimated from the molecular
structure of the membrane protein [21, 22, 30, 59, 60] and/or computer simulations [31,
166]. We explore here protein-induced bilayer thickness deformations for generic models
of W(θ) inspired by the molecular structure of the mechanosensitive channel of large
conductance (MscL) [42, 91, 167, 168].
In addition to Eq. (2.12), it is also necessary to specify boundary conditions on the
(normal) derivative of u at the bilayer-protein interface [164, 165]. The appropriate choice
for these boundary conditions has been a matter of debate, and is likely to depend on the
specific system under consideration [21–25, 27–31, 39, 56, 102, 118, 160, 161, 169]. We
generally focus on the fixed-value boundary condition
ˆn · ∇¯ u¯(¯r, θ)
r¯=C¯(θ)
= U¯′
(θ), (2.14)
but also explore choices for U¯′
(θ) minimizing the bilayer thickness deformation energy.
One may physically interpret fixed-value boundary conditions on the derivative of u as
corresponding to scenarios in which the lipid leaflet surfaces are normal to the protein hydrophobic surface at the bilayer-protein boundary [27], while natural boundary conditions
minimizing the bilayer thickness deformation energy permit arbitrary slopes of u [118]. A
more detailed molecular model of the gradients of the lipid bilayer leaflets at bilayer-protein
interfaces can be developed by explicitly taking into account lipid tilt [26, 161, 170]. We
allow for constant as well as varying U¯′
(θ) in Eq. (2.14).
29
For a (hypothetical) membrane protein with a perfectly circular cross section C¯(θ) =
R¯, where R¯ is the (dimensionless) protein radius, and constant U¯ and U¯′
, the bilayerprotein boundary conditions in Eqs. (2.12) and (2.14) are azimuthally symmetric about
the protein center, and the resulting protein-induced bilayer thickness deformations also
show azimuthal symmetry about the protein center [23, 27, 102, 118]. Equations (2.12)
and (2.14) suggest three, not mutually exclusive, modes for protein structures to break
rotational symmetry, and to hence endow protein-induced bilayer thickness deformations
with a non-trivial structure [58, 59]. First, the value of U¯ in Eq. (2.12) or, second, the
value of U¯′
in Eq. (2.14) may vary along the bilayer-protein interface. To explore generic
effects of varying U¯ or U¯′ on protein-induced bilayer thickness deformations we consider
the bilayer-protein hydrophobic mismatch
U¯(θ) = U¯
0 + β¯ cos(wθ) (2.15)
and the bilayer-protein contact slope
U¯′
(θ) = U¯′
0 + ¯γ cos(vθ), (2.16)
where U¯
0 and U¯′
0 denote the average bilayer-protein hydrophobic mismatch and bilayerprotein contact slope, β¯ and γ¯ denote the amplitudes of the perturbations about these
average values, and w and v denote the protein symmetries associated with variations in
U¯ and U¯′
. In Chapters 2 and 3, we set U¯
0λ = −0.1 nm and βλ¯ = 0.5 nm in Eq. (2.15) for
all calculations involving a modulation in the bilayer-protein hydrophobic mismatch, and
30
U¯′
0 = 0 and γ¯ = 0.3 in Eq. (2.16) for all calculations involving a modulation in the bilayerprotein contact slope. For all scenarios considered in Chapters 2 and 3 for which we
keep U¯ or U¯′ constant along the bilayer-protein interface we set, unless stated otherwise,
Uλ¯ = 0.3 nm or U¯′ = 0. The values of U and U
′ employed here are in line with previous
work on MscL and gramicidin channels [28, 31, 42, 91, 118].
Angular variations in C¯(θ) along the bilayer-protein boundary ¯r = C¯(θ)ˆr provide, in
addition to Eqs. (2.15) and (2.16), a third mode for a protein structure to break azimuthal
symmetry of protein-induced bilayer thickness deformations about the protein center. Inspired by molecular structures of tetrameric and pentameric MscL [42, 59, 60, 171] and
other membrane proteins [5, 146], we consider here two generic classes of protein shapes
breaking rotational symmetry. On the one hand, we consider clover-leaf protein cross sections specified by
C¯(θ) = R¯
1 + ϵ cos(sθ)
, (2.17)
where ϵ parameterizes the magnitude of deviations from a circular protein cross section,
ϵ = 0 for circular protein cross sections, and s denotes the symmetry of the boundary
curve [see Figs. 2.2(a) and 2.3(a)]. On the other hand, we consider (rounded) polygonal
protein cross sections specified by the series
C¯(θ) = A¯
R¯
vuut
" X
P
p=−P
cos(sp + 1)θ
(sp + 1)2
#2
+
" X
P
p=−P
sin(sp + 1)θ
(sp + 1)2
#2
, (2.18)
where larger P yield sharper polygonal corners with P = 0 for circular protein cross
sections, A¯
R¯ is a rescaling factor chosen so as to ensure that the polygons are inscribed
31
6
✏ = 0 ✏ = 0.07 ✏ = 0.14 ✏ = 0.21 ✏ = 0.28 ✏ = 0.35
(a)
My q An hmm Mom ow
Boo B. Good 3%
yo
Bk Mn BBB mid MMM
B. •% % BE kg MARSH µ now was DM
* % % % %
AM _rM pm now →
✓ B. B. Dog and
(b)
P = 0 P = 1 P = 2 P = 3 P = 4 P = 5
FIG. 3: Cross-sectional shapes of (a) clover-leaf proteins–from
top to bottom we have monomers, dimers, trimers, tetramers, and
pentamers, respectively and from left to right we have ✏ = 0, 0.1,
0.3, 0.5, 0.7, 0.9, respectively– and (b) polygonal proteins–from
top to bottom we have tetramers, pentamers, hexamers,
heptamers, and octamers, respectively and from left to right we
have P = 1, 2, 3, 4, 5, respectively.
where larger P yield sharper polygonal corners with
P = 0 for circular protein cross sections, A¯
R¯ is a rescaling factor chosen so as to ensure that the polygons are
inscribed in circles of radius R¯, and s denotes the polygonal symmetry [15, 49] [see Figs. 2(b) and ??(b)]. We
have
A¯
R¯ = R¯
PP
p=P
1
(sp+1)2
. (19)
We set P = 5 in Eqs. (18) and (19). For the scenarios considered here we found that an increase in P
beyond P = 5 only resulted in minor shifts in the bilayer thickness deformation energy. For all the calculations described here we use R¯ ⇡ 2.3/, which approximately corresponds to the observed size of a closed state
of MscL [6, 12].
III. BOUNDARY VALUE METHOD FOR
BILAYER THICKNESS DEFORMATIONS
In this section we introduce a BVM for bilayer thickness deformations, which allows the calculation of the
protein-induced bilayer thickness deformations and associated energy of bilayer thickness deformations for arbitrary protein shapes. We will use the BVM described
here to calculate the bilayer thickness deformation field
u¯(¯r, ✓) in Eq. (8) and the resulting bilayer thickness deformation energy G in Eq. (3) via Eq. (10) for arbitrary
bilayer-protein boundary shapes ¯r = C¯(✓)ˆr and boundary conditions U¯(✓) and U¯0
(✓) in Eqs. (12) and (14). We
first provide a general formulation of the BVM for bilayer
thickness deformations, and validate this BVM against
exact analytic solutions and numerical solutions obtained
using FEM (see Sec. III A). We then discuss how the numerical performance of the BVM for bilayer thickness
deformations can be improved by employing nonuniform
boundary point distributions (see Sec. III C).
A. Formulation and validation of the boundary
value method
The BVM for bilayer thickness deformations takes the
analytic solution for ¯u(¯r, ✓) in Eq. (8) as its starting
point, and assumes that the infinite series in this general solution can be truncated at some finite order N:
u¯ (¯r, ✓) ⇡ ¯f +
N (¯r, ✓) + ¯f
N (¯r, ✓) , (20)
where
¯f ±
N (¯r, ✓) =A±
0 K0 (
p⌫¯±r¯) +
X
N
n=1
A±
n Kn (
p⌫¯±r¯) cos (n✓) + B±
n Kn (
p⌫¯±r¯) sin (n✓)
.
(21)
The solution in Eq. (20) with Eq. (21) contains the 4N +2
unknown constants A±
0 , A±
n , and B±
n . In the BVM for bilayer thickness deformations, we fix these coecients by
imposing the boundary conditions in Eqs. (12) and (14)
at 2N + 1 points along the bilayer-protein interface. For
now, we take these boundary points to be uniformly
distributed along the bilayer-protein interface, with an
identical arc length separation between boundary points
along the bilayer-protein interface. We return to the distribution of boundary points in Sec. III C.
From the boundary conditions in Eqs. (12) and (14)
we have
u¯(¯r, ✓j )
r¯=C¯(✓j ) = U¯ (✓j ) , (22)
ˆn · r¯ u¯(¯r, ✓j )
r¯=C¯(✓j ) = U¯0 (✓j ) , (23)
in which j = 1, 2,..., 2N + 1 denote the boundary
points along the bilayer-protein interface employed for
Figure 2.3: Cross sections of cylindrical protein shapes (left-most panels) and (a) cloverleaf and (b) polygonal protein shapes (right panels). The clover-leaf protein cross sections
in panel (a) are obtained from Eq. (2.17) with ϵ = 0.07, 0.14, 0.21, 0.28, and 0.35 (left to
right) and s = 1, 2, 3, 4, and 5 (top to bottom), with ϵ = 0 yielding a circular protein
cross section. Note that the clover-leaf protein cross sections with s = 1 only show small
deviations from the corresponding circular protein cross section obtained with ϵ = 0 in
Eq. (2.17) (dashed curves) for the values of ϵ considered here. The polygonal protein
cross sections in panel (b) are obtained from Eq. (2.18) with P = 1, 2, 3, , 4, and 5 (left
to right) and s = 4, 5, 6, 7, and 8 (top to bottom). As a guide to the eye, these polygonal
protein cross sections are inscribed in circles obtained with P = 0 in Eq. (2.18) (dashed
curves).
32
in circles of radius R¯, and s denotes the polygonal symmetry [59, 172] [see Fig. 2.3(b)].
We have
A¯
R¯ =
R¯
PP
p=−P
1
(sp+1)2
. (2.19)
Unless stated otherwise, we set P = 5 in Eqs. (2.18) and (2.19). For the scenarios
considered here we found that an increase in P beyond P = 5 only resulted in minor
shifts in the bilayer thickness deformation energy (see Figure 2.8). For all the calculations
described in Chapters 2 and 3 we use Rλ¯ ≈ 2.3 nm in Eqs. (2.17) and (2.19), which
approximately corresponds to the observed size of a closed state of MscL [27, 42].
2.2 Boundary value method for bilayer thickness deformations
In this section we introduce a BVM for bilayer thickness deformations, which allows
calculation of protein-induced bilayer thickness deformations, and their associated elastic
energy, for general protein shapes. In the following sections we use this BVM to calculate the bilayer thickness deformation field u¯(¯r, θ) in Eq. (2.8), and the resulting bilayer
thickness deformation energy G in Eq. (2.3), for the clover-leaf and polygonal protein
shapes ¯r = C¯(θ)ˆr in Eqs. (2.17) and (2.18) and the boundary conditions U¯(θ) and U¯′
(θ)
in Eqs. (2.15) and (2.16). We first provide a general formulation of the BVM for bilayer
thickness deformations, and validate this BVM against exact analytic and FEM solutions
(see Sec. 2.2.1). We then discuss how the numerical performance of the BVM for bilayer thickness deformations can be improved by employing an adaptive point distribution
33
(APD) that results in a nonuniform distribution of boundary points for non-circular protein
cross sections (see Sec. 2.2.2). For simplicity, we set τ = 0 in all of our calculations in
Chapters 2 and 3, but we continue discussing our mathematical equations assuming a
finite τ for completeness.
2.2.1 Formulation and validation of the boundary value method
The BVM for bilayer thickness deformations takes the analytic solution for
u¯(¯r, θ) + τ¯
a¯
in Eq. (2.8) as its starting point, and assumes that the infinite series in this general solution
can be truncated at some finite order N:
u¯ (¯r, θ) + τ¯
a¯
≈ ¯f
+
N (¯r, θ) + ¯f
−
N (¯r, θ) , (2.20)
where
¯f
±
N (¯r, θ) = A
±
0 K0
√
ν¯±r¯
+
X
N
n=1
A
±
n Kn
√
ν¯±r¯
cos (nθ) + B
±
n Kn
√
ν¯±r¯
sin (nθ)
. (2.21)
The solution in Eq. (2.20) with Eq. (2.21) contains the 4N + 2 unknown constants A
±
0
,
A±
n
, and B±
n
. In the BVM for bilayer thickness deformations, we fix these coefficients by
imposing the boundary conditions in Eqs. (2.12) and (2.14) at 2N + 1 boundary points
along the bilayer-protein interface. For now, we take these boundary points to be uniformly distributed along the bilayer-protein interface, with a constant arc length separating
adjacent boundary points along the bilayer-protein interface. We return to the distribution
of boundary points in Sec. 2.2.2.
From the boundary conditions in Eqs. (2.12) and (2.14) we have
u¯(¯r, θj )
r¯=C¯(θj )
+
τ¯
a¯
= U¯ (θj ) + τ¯
a¯
, (2.22)
ˆn · ∇¯ u¯(¯r, θj )
r¯=C¯(θj )
= U¯′
(θj ) , (2.23)
in which j = 1, 2, . . . , 2N+1 denote the boundary points along the bilayer-protein interface,
where C¯(θ) = R¯ for proteins with a circular cross section, C¯(θ) is as in Eq. (2.17) for cloverleaf protein shapes, and C¯(θ) is as in Eq. (2.18) for polygonal protein shapes (Fig. 2.3).
Equations (2.22) and (2.23) amount to a linear system of equations
Ax = b , (2.24)
where the vector x has dimension 4N + 2 and contains the unknown constants A
±
0
, A±
n
,
and B±
n
, the 4N + 2 components of the vector b contain the boundary conditions on the
right-hand sides of Eqs. (2.22) and (2.23), and A is a square matrix of order 4N + 2 that
has the coefficients of the constants A
±
0
, A±
n
, and B±
n on the left-hand sides of Eqs. (2.22)
and (2.23) as its entries. Equation (2.24) can be solved efficiently using the extensive
numerical methods available for the solution of matrix equations. We employed here LU
decomposition with partial pivoting to solve Eq. (2.24) for x [173, 174].
To quantify numerical errors in our BVM solutions it is useful to compute, based on the
calculated A
±
0
, A±
n
, and B±
n
in Eq. (2.20) with Eq. (2.21), the values of
u¯ (¯r, θ) + τ¯
a¯
and
nˆ · ∇¯ u¯ (¯r, θ) along the bilayer-protein boundary for a given set of reference points distinct
from the boundary points employed for the BVM solution in Eq. (2.24). We compile these
3
computed boundary values of
u¯(¯r, θ) + τ¯
a¯
and nˆ · ∇¯ u¯ (¯r, θ) in a vector b˜, and compare b˜
to the corresponding exact boundary values b
′ mandated by the boundary conditions in
Eqs. (2.12) and (2.14),
ηb
′ = 100 ×
||b˜ − b
′
||L2
||b′
||L2
, (2.25)
where || · ||L2 is the L
2 norm [173]. For all the results shown in this thesis, we used vectors b
′ and b˜ with 800 components [400 components each for
U¯(θ) + τ¯
a¯
and U¯′
(θ)] in
Eq. (2.25), which we chose for a given protein shape so as to yield reference points with a
uniform spacing in θ over the interval 0 ≤ θ ≤ 2π. Figure 2.4(a) shows ηb
′ in Eq. (2.25) for
the clover-leaf shapes in Eq. (2.17) with s = 3 and various values of ϵ. As expected, we
find that ηb
′ tends to decrease with increasing N in Eq. (2.20) with Eq. (2.21), indicating
that a greater accuracy of BVM solutions is obtained at larger N. The local minima of ηb
′
in Fig. 2.4(a) correspond to values of N that are multiples of s, which suggests that the accuracy of the BVM is improved if N matches the protein symmetry. Figure 2.4(b) indicates
that the convergence of BVM solutions with increasing N can be improved substantially
through an APD that allows for a nonuniform distribution of boundary points, which we
discuss in Sec. 2.2.2.
We performed our BVM calculations in C++ using the arbitrary precision numerical library Arb [175]. Unless stated otherwise, we allowed for sufficient numerical precision so
that the boundary error ηb
′ ≤ 0.1% in Eq. (2.25) and we obtained changes in G and ηb
′ of
no more than 10−5% as the numerical precision was increased. We generated all figures in
MATLAB [176]. To speed up our calculations, we multi-threaded some of the source code
7
convenient to perform calculations in numerical precision
greater than double precision (64 bits). However, many
protein cross-section shapes aren’t expected to deviate
largely from a circular shape. In Appendix B, we provide
examples of clover-leaf protein shapes for which the deformation energy calculations by the computational implementation of the BVM presented here can be safely
carried out in double precision.
To quantify numerical errors in our BVM solutions it
is useful to compute, based on the calculated A±
0 , A±
n ,
and B±
n in Eq. (20) with Eq. (21), the values of ¯u(¯r, ✓)
along the bilayer-protein boundary for a given set of reference points distinct from the boundary points employed
for the BVM solution in Eq. (24). We compile these
computed boundary values of ¯u(¯r, ✓) in a vector b˜, and
compare b˜ to the corresponding exact boundary values
b0 mandated by the boundary conditions in Eqs. (12)
and (14),
⌘b0 = 100 ⇥ ||b˜ b0
||L2
||b0
||L2
, (25)
where ||·||L2 is the L2 norm [53]. For all the results shown
in this article, we used vectors b0 and b˜ with 800 components (400 components for U¯(✓) and U¯0
(✓)) in Eq. (25),
which we chose for a given protein shape so as to yield
reference points with a uniform spacing in ✓ over the interval 0 ✓ 2⇡.
Figure 4(a) shows ⌘b0 in Eq. (25) for the clover-leaf
shapes in Eq. (17) with s = 3 and various values of ✏.
As expected, we find that ⌘b0 tends to decrease with increasing N in Eq. (20) with Eq. (21), indicating that a
greater accuracy of BVM solutions is obtained at larger
N. The local minima of ⌘b0 in Fig. 4(a) correspond to values of N that are multiples of s, which suggests that the
accuracy of the BVM is improved if N matches the protein symmetry. Figure 4(b) shows that the convergence
of BVM solutions with increasing N can be improved
substantially through an APD that allows for a nonuniform distribution of boundary points, which we discuss
in Sec. III B.
In our BVM calculations we evaluate the bilayer thickness deformation energy G in Eq. (3) by numerically computing Eq. (10) using the same 400 reference points employed to calculate ⌘b0 in Eq. (25). To this end, we approximate G¯ in Eq. (10) through Eq. (20) with Eq. (21),
G¯ ⇡ i
2
Z 2⇡
0
d✓¯l
U¯0 U¯ ˆn · r¯ ¯f +
N ¯f
N
r¯=C¯(✓) , (26)
where we have used Eq. (7) and the boundary conditions
in Eqs. (12) and (14). We have confirmed that, within the
numerical accuracy used here, identical results for ⌘b0 and
G¯ are obtained with more than 400 reference points. Note
that Eq. (26) must evaluate to a real number and l is real,
which means that the remaining terms in the integrand
in Eq. (26) evaluate to a purely imaginary number.
We validated the BVM for bilayer thickness deformations against exact analytic solutions obtained for proFIG. 4. Percentage di↵erence between the exact bilayer thickness deformation field along the bilayer-protein boundary
and the bilayer thickness deformation field obtained from the
BVM solution, ⌘b0 in Eq. (25), as a function of the number
of terms in the Fourier-Bessel series in Eq. (20) with Eq. (21)
for (a) uniformly distributed points along the bilayer-protein
boundary and (b) the boundary point distributions implied by
the APD method (see Sec. III B). For both panels we considered three-fold clover-leaf protein shapes (s = 3) in Eq. (17)
with the indicated values of ✏, R¯ ⇡ 2.3 nm, and the constant
U¯ = 0.3 nm and U¯0 = 0. In panel (b) we used, for ease of
comparison, the same gap factor ⌦ = 0.25 in Eq. (28) for all
curves.
teins with circular cross sections [9, 12, 15, 28, 32, 33] and
against FEM solutions [15, 25, 26] (see Fig. 5). In particular, we consider in Fig. 5 cylindrical membrane proteins
with constant U and U0 [see Fig. 5(a)], for which the
analytic solution of bilayer thickness deformations simply amounts to the zeroth-order terms in Eq. (20) with
Eq. (21) [9, 12, 28]. Furthermore, we consider in Fig. 5
crown-shaped membrane proteins with circular cross section, constant U0
, and the periodically varying U(✓) in
Eq. (15) [see Fig. 5(b)], for which the analytic solution
is obtained at order N = w in Eq. (20) with Eq. (21)
[15, 32, 33]. We quantified the level of agreement between the BVM and FEM solutions and the corresponding analytic solutions through the percentage di↵erence
in the calculated bilayer thickness deformation energy G
Figure 2.4: Percentage difference between the exact bilayer thickness deformation field
along the bilayer-protein boundary and the bilayer thickness deformation field obtained
from the BVM solution, ηb
′ in Eq. (2.25), as a function of the number of terms in the
Fourier-Bessel series in Eq. (2.20) with Eq. (2.21) for (a) uniformly distributed points along
the bilayer-protein boundary and (b) the boundary point distributions implied by the APD
method (see Sec. 2.2.2). For both panels we considered three-fold clover-leaf protein
shapes (s = 3) in Eq. (2.17) with the indicated values of ϵ, Rλ¯ ≈ 2.3 nm, and the constant
Uλ¯ = 0.3 nm and U¯′ = 0 and set τ = 0. In panel (b) we used, for ease of comparison, the
same gap factor Ω = 0.25 in Eq. (2.28) for all curves.
of the Arb library [177]. Appendix A.1 provides a more in-depth description of our computational implementation of the BVM, and discusses possible issues with the numerical
solution of Eq. (2.24) arising from floating point errors and numerical instabilities. For the
polygonal protein shapes considered here, with P = 5 in Eq. (2.18) with Eq. (2.19), and
37
for the clover-leaf protein shapes considered here with large s and/or large ϵ in Eq. (2.17)
we found it convenient to perform the BVM calculations with numerical precision greater
than double precision (64 bits). In Appendix A.2 we illustrate the extent to which double
precision calculations could be used to approximate the BVM results described here.
In our BVM calculations we evaluate the bilayer thickness deformation energy G in
Eq. (2.3) by numerically computing Eq. (2.10) using the same 400 reference points employed to calculate ηb
′ in Eq. (2.25). To this end, we approximate G¯ in Eq. (2.10) through
Eq. (2.20) with Eq. (2.21),
G¯ ≈
1
4
Z 2π
0
dθ¯l
h
U¯′ −
U¯ +
τ¯
a¯
ˆn · ∇¯
i
ν¯+
¯f
+
N + ¯ν−
¯f
−
N
r¯=C¯(θ)
, (2.26)
where we have used Eq. (2.7) and the boundary conditions in Eqs. (2.12) and (2.14). We
have confirmed that, within the numerical accuracy used here, identical results for ηb
′ and
G¯ are obtained with more than 400 reference points. Note that ¯l in Eq. (2.26) is real and
that Eq. (2.26) must evaluate to a real number, which means that the remaining terms in
the integrand in Eq. (2.26) evaluate to a real number; in the scenarios considered here,
we find ¯f
+
N and ¯f
−
N are complex conjugates so
¯f
+
N − ¯f
−
N
evaluates to a purely imaginary
number and
¯f
+
N + ¯f
−
N
evaluates to a purely real number.
We validated the BVM for bilayer thickness deformations against exact analytic solutions obtained for proteins with circular cross sections [23, 27, 30, 58, 59, 118] and
against FEM solutions [30, 60, 61] (see Fig. 2.5). In particular, we consider in Fig. 2.5
cylindrical membrane proteins with constant U and U
′
[see Fig. 2.5(a)], for which the exact analytic solution of bilayer thickness deformations simply amounts to the zeroth-order
8
FIG. 5. Percentage di↵erence between exact analytic and
FEM (red curves) or BVM (blue curves) solutions for the
bilayer thickness deformation energy, ⌘G in Eq. (27), as a
function of the average edge size hLi used in the FEM solution
(upper axes) or the number of terms in Eq. (20) with Eq. (21)
used in the BVM solution (lower axes) for (a) a cylindrical
protein with R¯ = 2.3 nm and U¯ = 0.3 nm and (b) a crownshaped protein with R¯ = 2.3 nm, U¯0 = 0.1 nm, ¯ =
0.5 nm, and w = 5 in Eq. (15). We set U¯0 = 0 for both
panels.
in Eq. (3),
⌘G = 100⇥
G Ganaly
Ganaly
, (27)
where Ganaly denotes the analytic solution [9, 12, 15, 28,
32, 33] and G denotes the corresponding BVM or FEM
solutions. We found, as expected, excellent numerical
agreement between the BVM and analytic solutions for
N = 0 [Fig. 5(a)] or N w [Fig. 5(b)] within floating point error. The FEM solutions in Fig. 5 are, up to
their expected numerical precision [15], in good agreement with the analytic and BVM solutions, with the
agreement improving with decreasing average edge size
hLi in the FEM grid. For both cylindrical and crownshaped proteins, we have ⌘G ⇡ 0.01% for hLi ⇡ 0.1 nm
in the FEM solutions in Fig. 5.
B. Nonuniform boundary point distributions
As illustrated in Figs. 4 and 5, the BVM can provide
a highly accurate method for calculating protein-induced
bilayer thickness deformations. However, Fig. 4(a) also
shows that, for large enough deviations from a circular protein cross section, accurate BVM solutions require a large number of terms in the Fourier-Bessel series in Eq. (20) with Eq. (21). For non-circular protein cross sections, the numerical performance of the
BVM can be improved substantially by choosing suitable,
nonuniform boundary point distributions. In particular,
we found that boundary point distributions that assign
more points to, as viewed from the lipid bilayer, concave
boundary regions yield a more rapid convergence of G
with increasing N. This can be understood intuitively
by noting that, in the concave regions of a boundary
curve, di↵erent sections of the boundary curve can produce overlapping bilayer thickness deformation fields, inducing protein self-interactions. One expects that higherorder terms in the Fourier-Bessel series in Eq. (20) with
Eq. (21) are required to capture such interactions [15, 32].
To assign more boundary points to the concave boundary regions of clover-leaf and (finite-P) polygonal protein
shapes, we employ an APD of the BVM boundary points.
In the APD method, we distribute the 2N + 1 boundary
points such that boundary points are always assigned
to the apex points along the bilayer-protein boundary
curves furthest away from the protein center (see Fig. 6).
We distribute the remaining boundary points along the
sections of the bilayer-protein boundary curves that are
an arclength ¯l ¯` away from the apex points, such that
these points are uniformly spaced in arclength, with the
gap length
¯` = ⌦ ¯
2s , (28)
where the gap factor ⌦ satisfies 0 < ⌦ < 1, ¯ is the
(dimensionless) protein circumference, and s is the symmetry of the clover-leaf or polygonal protein shape [see
also Eqs. (17) and (18)] (Fig. 6). For even values of s,
we consider in our APD method the general solution
in Eq. (20) with Eq. (21) for N = sM/2 with integers
M 3. For odd values of s, we allow in Eq. (20) with
Eq. (21) for N = sM/2 for even integers M 3, and for
N = (sM 1) /2 for odd integers M 3. To achieve an
approximately periodic distribution of boundary points
for even s and for odd s with even M, we found it convenient to duplicate one of the apex boundary points, with
a slight o↵set in the duplicated boundary points by an
equal arclength distance from the apex [Fig. 6(b)]. For
greater numerical accuracy, this distance from the apex
could be optimized so as to reduce the boundary error
⌘b0 in Eq. (25), but we found it sucient here to set it
equal to one-half the mean arclength spacing between
the other boundary points. Unless stated otherwise, we
use the APD method for all BVM calculations described
in this article, fixing ⌦ in Eq. (28) and N in Eq. (20)
Figure 2.5: Percentage difference between exact analytic and FEM (red curves) or BVM
(blue curves) solutions for the bilayer thickness deformation energy, ηG in Eq. (2.27), as
a function of the average edge size ⟨L⟩ used in the FEM solution (upper axes) or the
number of terms in Eq. (2.20) with Eq. (2.21) used in the BVM solution (lower axes) for (a)
a cylindrical protein with Rλ¯ = 2.3 nm and Uλ¯ = 0.3 nm and (b) a crown-shaped protein
with Rλ¯ = 2.3 nm, U¯
0λ = −0.1 nm, βλ¯ = 0.5 nm, and w = 5 in Eq. (2.15). We set U¯′ = 0
and τ = 0 for both panels.
terms in Eq. (2.20) with Eq. (2.21) [23, 27, 118]. Furthermore, we consider in Fig. 2.5
crown-shaped membrane proteins with circular cross section, constant U
′
, and the periodically varying U(θ) in Eq. (2.15) [see Fig. 2.5(b)], for which the exact analytic solution
is obtained at order N = w in Eq. (2.20) with Eq. (2.21) [30, 58, 59]. We quantified the
level of agreement between the BVM and FEM solutions and the corresponding exact
39
analytic solutions through the percentage difference in the calculated bilayer thickness
deformation energy G in Eq. (2.3),
ηG = 100×
G − Ganaly
Ganaly
, (2.27)
where Ganaly denotes the analytic solution [23, 27, 30, 58, 59, 118] and G denotes
the corresponding BVM or FEM solutions. We found, as expected, excellent numerical
agreement between the BVM and the aforementioned exact analytic solutions for N = 0
[Fig. 2.5(a)] or N ≥ w [Fig. 2.5(b)] within floating point error. The FEM solutions in Fig. 2.5
are, up to their expected numerical precision [30], in good agreement with the exact analytic and BVM solutions, with the agreement improving with decreasing average edge size
⟨L⟩ in the FEM grid. For both cylindrical and crown-shaped membrane proteins, we have
ηG ≈ 0.01% for ⟨L⟩ ≈ 0.1 nm in the FEM solutions in Fig. 2.5.
2.2.2 Nonuniform boundary point distributions
As illustrated in Figs. 2.4 and 2.5, the BVM can provide a highly accurate method
for calculating protein-induced bilayer thickness deformations. However, Fig. 2.4(a) also
shows that, for large enough deviations from a circular protein cross section, accurate
BVM solutions require a large number of terms in the Fourier-Bessel series in Eq. (2.20)
with Eq. (2.21). For non-circular protein cross sections, the numerical performance of the
BVM can be improved substantially by choosing suitable, nonuniform boundary point distributions. In particular, we found that boundary point distributions that assign more points
40
to, as viewed from the lipid bilayer, concave boundary regions yield a more rapid convergence of G with increasing N. This can be understood intuitively by noting that, in the concave regions of a boundary curve, different sections of the boundary curve can produce
overlapping bilayer thickness deformation fields, inducing protein self-interactions. One
expects that higher-order terms in the Fourier-Bessel series in Eq. (2.20) with Eq. (2.21)
are required to capture such interactions [30, 58].
To assign more boundary points to the concave boundary regions of clover-leaf and
(finite-P) polygonal protein shapes, we employ an APD of the BVM boundary points. In
the APD method, we distribute the 2N + 1 boundary points such that boundary points
are always assigned to the apex points along the bilayer-protein boundary curves furthest
away from the protein center (see Fig. 2.6). We distribute the remaining boundary points
along the sections of the bilayer-protein boundary curves that are an arc length ¯l ≥ ¯ℓ away
from the apex points such that these points are uniformly spaced in arc length, with the
gap length
¯ℓ = Ω Γ¯
2s
, (2.28)
where the gap factor Ω satisfies 0 < Ω < 1, Γ¯ is the (dimensionless) protein circumference,
and s is the symmetry of the clover-leaf or polygonal protein shape [see also Eqs. (2.17)
and (2.18)] (Fig. 2.6). For even values of s, we consider in our APD method the general
solution in Eq. (2.20) with Eq. (2.21) for N = sM/2 with integers M ≥ 3. For odd values
of s, we allow in Eq. (2.20) with Eq. (2.21) for N = sM/2 for even integers M ≥ 3, and
for N = (sM − 1) /2 for odd integers M ≥ 3. To achieve an approximately periodic distribution of boundary points for even s and for odd s with even M, we found it convenient
41
Figure 2.6: Illustration of the APD method used to increase the numerical efficiency of
BVM solutions for (a) a three-fold clover-leaf protein shape (s = 3) and (b) a four-fold
clover-leaf protein shape (s = 4) in Eq. (2.17). The boundary points used for the BVM
solutions are indicated by blue dots. In panel (a) we set ✏ = 0.38, N = 31, and ⌦ = 0.62 for
the gap length ¯` in Eq. (2.28). In panel (b) we set ✏ = 0.30, N = 42, and ⌦ = 0.72. For both
panels we set R¯ ⇡ 2.3 nm and ⌧ = 0. To achieve an approximately periodic distribution
of boundary points for even s, we duplicated in panel (b) the boundary point at the rightmost apex, and slightly offset the resulting two boundary points along the bilayer-protein
interface (see main text). The values of N in panels (a) and (b) were chosen for illustrative
purposes. We generally employ values of N greater than those considered here so as to
meet the numerical precision criteria imposed here (see main text).
to duplicate one of the apex boundary points, with a slight offset in the duplicated boundary points by an equal arc length distance from the apex [see Fig. 2.6(b)]. For greater
numerical accuracy, this distance from the apex could be optimized so as to reduce the
boundary error ⌘b0 in Eq. (2.25), but we found it sufficient here to set it equal to one-half
42
Figure 2.6: Illustration of the APD method used to increase the numerical efficiency of
BVM solutions for (a) a three-fold clover-leaf protein shape (s = 3) and (b) a four-fold
clover-leaf protein shape (s = 4) in Eq. (2.17). The boundary points used for the BVM
solutions are indicated by blue dots. In panel (a) we set ϵ = 0.38, N = 31, and Ω = 0.62 for
the gap length ¯ℓ in Eq. (2.28). In panel (b) we set ϵ = 0.30, N = 42, and Ω = 0.72. For both
panels we set Rλ¯ ≈ 2.3 nm and τ = 0. To achieve an approximately periodic distribution
of boundary points for even s, we duplicated in panel (b) the boundary point at the rightmost apex, and slightly offset the resulting two boundary points along the bilayer-protein
interface (see main text). The values of N in panels (a) and (b) were chosen for illustrative
purposes. We generally employ values of N greater than those considered here so as to
meet the numerical precision criteria imposed here (see main text).
to duplicate one of the apex boundary points, with a slight offset in the duplicated boundary points by an equal arc length distance from the apex [see Fig. 2.6(b)]. For greater
numerical accuracy, this distance from the apex could be optimized so as to reduce the
boundary error ηb
′ in Eq. (2.25), but we found it sufficient here to set it equal to one-half
the arc length spacing between the boundary points in the concave boundary regions.
Unless stated otherwise, we used the APD method for all BVM calculations described in
42
this article, fixing Ω in Eq. (2.28) and N in Eq. (2.20) with Eq. (2.21) such that the boundary error ηb
′ ≤ 0.1% in Eq. (2.25) and we obtained changes in G and ηb
′ of no more than
10−5% as the numerical precision was increased.
As illustrated in Fig. 2.4(b) for clover-leaf protein shapes, the APD method employed
here improves considerably the convergence of the BVM with increasing N, particularly
for proteins that show substantial deviations from a circular cross section. As a result, a
given numerical accuracy of BVM solutions can be achieved with smaller N. We note that,
for proteins with (discrete) rotational symmetry, the Fourier-Bessel series in Eq. (2.20) with
Eq. (2.21) must show the same symmetry. Indeed, in our BVM calculations we find that,
within the numerical precision employed here, the coefficients of terms in Eq. (2.20) with
Eq. (2.21) that break the protein symmetry take values equal to zero. On this basis one
could, for a given protein symmetry, further improve the numerical efficiency of the BVM
by using the protein symmetry to remove some of the terms in the Fourier-Bessel series
in Eq. (2.20) with Eq. (2.21). For the scenarios considered here the BVM was efficient
enough so as not to require such further refinement.
Figure 2.7 illustrates the calculation of the bilayer thickness deformation energy, G in
Eq. (2.3), using the BVM with APD for the clover-leaf protein shapes in Eq. (2.17) with
various protein symmetries, s, and deviations from a circular protein cross section, ϵ. As
expected [59, 61], we find in Fig. 2.7(a) that G increases with increasing s and ϵ. We also
show in Fig. 2.7 the corresponding results obtained from the FEM with an average edge
43
11
FIG. 7. Comparing BVM and FEM solutions for the elastic
energy of protein-induced bilayer thickness deformations. (a)
Bilayer thickness deformation energy, G¯ in Eq. (6), obtained
using BVM and FEM solutions for ¯u in Eq. (2) and (b) corresponding percentage di↵erence between the BVM and FEM
solutions for G¯, µ0
G in Eq. (29), for the clover-leaf protein
shapes in Eq. (17) as a function of ✏ with the indicated values
of s, R¯ ⇡ 2.3 nm, U¯ = 0.3 nm, and U¯0 = 0. For the FEM
solutions we employed an average edge size hLi ⇡ 0.1 nm.
which we denote by ⇤¯analy. As expected, Fig. 9 shows
that the variations in ⇤¯ are more pronounced for smaller
clover-leaf protein shapes. We also find in Fig. 9 that
h⇤¯i < ⇤¯analy, with a larger
h⇤¯i ⇤¯analy
for smaller R¯
in Fig. 9.
Interestingly, we can have ⇤¯ < 0 in Fig. 9(a) for the
smaller clover-leaf protein shape in Fig. 8(a), while ⇤ > 0
in Fig. 9(b) for the larger clover-leaf protein shape in
Fig. 8(b). The regime with ⇤ < 0 in Fig. 9(a) can be
understood by noting that, with a constant U > ¯ 0 and
U¯0 = 0, ⇤ in Eq. (33) corresponds to minus the change
in the mean curvature in the direction perpendicular to
the protein boundary. For the points along the cloverleaf boundary closest and furthest away from the protein center, ˆn is aligned with the radial direction and
points towards the protein center. For the points along
the clover-leaf boundary shape in Figs. 8(a) and 8(b) furthest away from the protein center the mean curvature is
negative with the sign convention used here and decreases
10
FIG. 8. Color maps of the thickness deformation footprints
due to clover-leaf protein shapes with (a) R¯ = 1 and (b)
R¯ = 10 in Eq. (17) with s = 5, ✏ = 0.2, U¯ = 0.3/, and
U¯0 = 0. Both thickness deformation footprints were calculated through the BVM. The unperturbed lipid bilayer thickness is given by 2¯a = 3.2/.
Fig. 7(b) we quantify the agreement between our BVM
and FEM results through the percentage di↵erence in G,
µ0
G = 100⇥
GBVM GFEM
GFEM
. (29)
We find in Fig. 7 that the BVM and FEM solutions yield
excellent agreement for the energy of protein-induced bilayer thickness deformations for non-circular as well as
circular protein cross sections, with the level of agreement
between BVM and FEM solutions being in line with the
accuracy of FEM solutions expected from Fig. 5.
IV. ANALYTIC ESTIMATES OF THE BILAYER
THICKNESS DEFORMATION ENERGY FOR
ARBITRARY PROTEIN SHAPES
For membrane inclusions with circular cross section,
the solution for the thickness deformation field ¯u(¯r, ✓) in
Eq. (8) with Eq. (9) and the bilayer thickness deformation energy in Eq. (10) yield exact analytic expressions
for the energy of protein-induced bilayer thickness deformations for arbitrary (angular) variations in the bilayerprotein boundary conditions [18, 33, 44, 56]. The purpose
of this section is to develop, on this basis, a simple analytic scheme for estimating the energy of protein-induced
bilayer thickness deformations for proteins of arbitrary
shape. In Sec. V we show that these simple analytic
estimates agree remarkably well with the corresponding
BVM solutions for proteins of various shapes and sizes.
For a single membrane inclusion with circular cross
section and arbitrary (angular) variations in U(✓) and
U0
(✓), the exact solution of the Euler-Lagrange equation
in Eq. (7) is given by Eq. (8) with Eq. (9), and the corresponding bilayer thickness deformation energy follows
from Eq. (10) [18, 33, 44, 56]. For the choices for U(✓)
and U0
(✓) in Eqs. (15) and (16), we thus find the bilayer
thickness deformation energy
G¯analy = ⇡R¯analy(¯⌫+ ⌫¯)
U¯ 2
0 E¯0 + U¯02
0 F¯0 + U¯0U¯0
0H¯0
+
1
2
⇣
¯2E¯w + ¯2F¯v + wv¯ ¯H¯w
⌘
r¯=R¯analy
,
w, v 6= 0 ,
(30)
where R¯analy is the radius of the circular protein cross
section, wv is the Kronecker delta, and we’ve defined
D¯ q = Kq(
p⌫¯+r¯)@r¯Kq(
p⌫¯r¯) Kq(
p⌫¯r¯)@r¯Kq(
p⌫¯+r¯),
E¯q = @r¯Kq(
p⌫¯+r¯) · @r¯Kq(
p⌫¯r¯)
D¯ q
,
F¯q = Kq(
p⌫¯+r¯)Kq(
p⌫¯r¯)
D¯ q
,
H¯q = Kq(
p⌫¯+r¯)@r¯Kq(
p⌫¯r¯) + Kq(
p⌫¯r¯)@r¯Kq(
p⌫¯+r¯)
D¯ q
,
(31)
with Kq as the modified Bessel function of the second
kind and of arbitrary q-th order and @r¯Kq is its partial derivative with respect to ¯r; the 0th order terms in
Eq. (30) are the contribution due to the circular crosssection shape and the remaining terms are due to variations in the boundary conditions. We use here Eq. (30)
to analytically estimate the bilayer thickness deformation energy associated with proteins of arbitrary (noncircular) cross section. To this end, we choose R¯analy
in Eq. (30) such that the circumference of the circular
membrane inclusion considered in Eq. (30) is equal to
the circumference of the membrane protein under consideration,
R¯analy = ¯
2⇡ , (32)
where, for the clover-leaf and polygonal boundary curves
in Eqs. (17) and (18) with Eq. (19), the protein circumference follows from
¯ =
Z 2⇡
0
d✓¯l . (33)
FIG. 8. Color maps of the bilayer thickness deformation footprints due to clover-leaf protein shapes with (a) R¯ = 1 and
(b) R¯ = 10 in Eq. (17) with s = 5, ✏ = 0.2, U¯ = 0.3 nm,
and U¯0 = 0. Panels (c) and (d) show the mean curvature in
units of 1/, H¯ = H, associated with the thickness deformation fields in panels (a) and (b), respectively, while panels
(e) and (f) show the corresponding mean curvature maps obtained for U¯0 = 0.3 nm rather than U¯0 = 0.3 nm. We
set 2¯a = 3.2 nm for all panels. All results were obtained
through the BVM.
in magnitude as one radially moves away from the protein boundary, yielding ⇤¯ > 0 [Figs. 8(c,d)]. In contrast,
for the points along the clover-leaf boundary shape closest to the protein center in Fig. 8(a) [but not Fig. 8(b)],
the mean curvature is approximately zero at the protein
boundary and decreases as one radially moves away from
the protein boundary, yielding ⇤¯ < 0. With a di↵erent
sign convention for the mean curvature or a protein with
a constant U < 0 rather than U > 0, analogous considerations apply [Figs. 8(e,f)]. Thus, protein self-interactions
can e↵ectively lower the energy cost of protein-induced
lipid bilayer thickness deformations, in analogy to the energetically favorable bilayer-thickness-mediated protein
interactions found for identical membrane proteins in
close enough proximity [8, 10, 11, 15, 25, 26, 32, 58–65].
Figure 2.7: Comparing BVM and FEM solutions for the elastic energy of clover-leaf
protein-induced bilayer thickness deformations. (a) Bilayer thickness deformation energy,
G¯ in Eq. (2.6), with τ = 0, obtained using BVM and FEM solutions for u¯ in Eq. (2.2) and
(b) corresponding percentage difference between the BVM and FEM solutions for G¯, µ
′
G
in Eq. (2.29), for the clover-leaf protein shapes in Eq. (2.17) as a function of ϵ with the
indicated values of s, Rλ¯ ≈ 2.3 nm, Uλ¯ = 0.3 nm, and U¯′ = 0. For the FEM solutions we
employed an average edge size ⟨L⟩ ≈ 0.1 nm.
size ⟨L⟩ ≈ 0.1 nm. In Fig. 2.7(b) we quantify the agreement between our BVM and FEM
results through the percentage difference in G,
µ
′
G = 100×
GBVM − GFEM
GFEM
, (2.29)
44
where GBVM and GFEM correspond to the values of G in Eq. (2.3) obtained through the
BVM and the FEM [30, 60, 61], respectively. Figure 2.8(a), shows the percentage error of
the bilayer thickness deformation and contact slope fields at the protein-bilayer interface
obtained from the BVM solution, ηb
′ in Eq. (2.25), with APD as a function of the number
of terms in the Fourier-Bessel series in Eq. (2.20) with Eq. (2.21) for the pentamer (s =
5) polygon protein shapes using the P values indicated, showing convergence with an
increasingly smaller error with an increasing number of terms in the Fourier-Bessel series
in Eq. (2.20) with Eq. (2.21). The local minima of ηb
′ in Fig. 2.8(a) correspond to values
of N that are multiples of s, which suggests that the accuracy of the BVM is improved
if N matches the protein symmetry. We also performed calculations using the FEM, but
for true polygon protein shapes, rather than those implied by Eqs. (2.18) and (2.19) with
finite P. As such, the polygon protein shapes used in the FEM solutions differ slightly from
those in the BVM solutions. As expected we find increasing agreement between the FEM
and BVM solutions with increasing P in Eqs. (2.18) and (2.19) as these are expected to
yield true polygonal protein shapes in the limit of P → ∞ [see Figure 2.8(b)]. Our results
indicate that using P = 5 is sufficient for our BVM solutions to agree remarkably well with
the FEM solutions. We find in Figs. 2.7 and 2.8 that the BVM and FEM solutions yield
excellent agreement for the energy of protein-induced bilayer thickness deformations for
non-circular as well as circular protein cross sections, with the level of agreement between
BVM and FEM solutions being in line with the accuracy of the FEM solutions expected
from Fig. 2.5.
45
Figure 2.8: Convergence of BVM solutions for the elastic energy of polygon proteininduced bilayer thickness deformations. (a) Percentage difference between the exact bilayer thickness deformation field along the bilayer-protein boundary and the bilayer thickness deformation field obtained from the BVM solution, ⌘b0 in Eq. (2.25), as a function
of the number of terms, N, in the Fourier-Bessel series in Eq. (2.20) with Eq. (2.21) for
the boundary point distributions implied by the APD method (see Sec. 2.2.2), using the
indicated values of P in Eq. (2.18) and (2.19) with s = 5. (b) Percentage difference
between the FEM and BVM solutions for G¯, µ0
G in Eq. (2.29), where we calculated the
BVM solutions for the polygon protein shapes in Eqs. (2.18) and (2.19) as a function of
P and with the indicated values of s, R¯ ⇡ 2.3 nm, U¯ = 0.3 nm, U¯0 = 0, N = 750, and
⌦ ⇡ 0.32, 0.24, 0.18, 0.16, and 0.16, for symmetries s = 4, 5, 6, 7, and 8, respectively, in
Eq. (2.28), while for the FEM solutions we employed an average edge size hLi ⇡ 0.1 nm
and used true polygon shapes, in contrast to those implied by Eqs. (2.18) and (2.19) with
finite P. For all BVM and FEM solutions depicted here, we set ⌧ = 0.
46
Figure 2.8: Convergence of BVM solutions for the elastic energy of polygon proteininduced bilayer thickness deformations. (a) Percentage difference between the exact bilayer thickness deformation field along the bilayer-protein boundary and the bilayer thickness deformation field obtained from the BVM solution, ηb
′ in Eq. (2.25), as a function
of the number of terms, N, in the Fourier-Bessel series in Eq. (2.20) with Eq. (2.21) for
the boundary point distributions implied by the APD method (see Sec. 2.2.2), using the
indicated values of P in Eq. (2.18) and (2.19) with s = 5. (b) Percentage difference
between the FEM and BVM solutions for G¯, µ
′
G in Eq. (2.29), where we calculated the
BVM solutions for the polygon protein shapes in Eqs. (2.18) and (2.19) as a function of
P and with the indicated values of s, Rλ¯ ≈ 2.3 nm, Uλ¯ = 0.3 nm, U¯′ = 0, N = 750, and
Ω ≈ 0.32, 0.24, 0.18, 0.16, and 0.16, for symmetries s = 4, 5, 6, 7, and 8, respectively, in
Eq. (2.28), while for the FEM solutions we employed an average edge size ⟨L⟩ ≈ 0.1 nm
and used true polygon shapes, in contrast to those implied by Eqs. (2.18) and (2.19) with
finite P. For all BVM and FEM solutions depicted here, we set τ = 0.
46
Chapter 3
Dependence of protein-induced lipid bilayer
deformations on protein shape
In this chapter we apply our BVM to survey the dependence of protein-induced lipid
bilayer thickness deformations on protein shape. This chapter is organized as follows.
In Sec. 3.1, inspired by the BVM (see Chapter 2), we develop a simple analytic scheme
for estimating the energy of protein-induced lipid bilayer thickness deformations for membrane proteins with non-circular cross sections. In Secs. 3.2 and 3.3 we test this analytic
approximation against BVM solutions. In Sec. 3.2, we survey the dependence of the bilayer thickness deformation energy on membrane protein shape, while in Sec. 3.3 we
explore some implications of these results for the self-assembly of protein oligomers and
transitions in protein conformational state. A summary and conclusions are provided in
Sec. 6.2.
47
3.1 Analytic approximation of the bilayer thickness deformation energy
For membrane inclusions with circular cross section, the solution for the thickness
deformation field u¯(¯r, θ) in Eq. (2.8) with Eq. (2.9) and the bilayer thickness deformation
energy in Eq. (2.10) yield exact analytic expressions for the energy of protein-induced bilayer thickness deformations for arbitrary (angular) variations in the bilayer-protein boundary conditions [23–25, 30, 58, 59, 61]. The purpose of this section is to develop, on this
basis, a simple analytic scheme for estimating the energy of protein-induced bilayer thickness deformations for membrane proteins with non-circular cross sections. In Sec. 3.2
we show that, for many protein shapes, these simple analytic estimates agree remarkably
well with the corresponding BVM solutions.
As in Chapter 2, it is convenient to recast the bilayer thickness deformation energy in
Eq. (2.3) in terms of the characteristic spatial and energy scales, the bilayer thickness deformation decay length scale λ and the bilayer bending ridigity Kb. Thus, consistent with
Chapter 2, we reformulate Eq. (2.3) with the following parameter substitutions: GK¯
b → G,
xλ¯ → x, yλ¯ → y, uλ¯ → u, aλ¯ → a, K¯
tKb/λ2 → Kt
, and τK¯ b/λ2 → τ . We maintain
specific values for U¯
0λ = −0.1 nm and βλ¯ = 0.5 nm in Eq. (2.15) in calculations involving
variations in the bilayer-protein hydrophobic mismatch. Furthermore, we set U¯′
0 = 0 and
γ¯ = 0.3 in Eq. (2.16) in calculations involving variations in the bilayer-protein contact slope.
In scenarios where we maintain U¯ or U¯′ constant along the bilayer-protein interface, we
set Uλ¯ = 0.3 nm or U¯′ = 0, unless otherwise specified. These parameter values align
48
with previous studies on MscL and gramicidin channels [28, 31, 42, 91, 118]. Additionally,
unless explicitly stated, we adopt a value of P = 5 in Eqs. (2.18) and (2.19), and we assume Rλ¯ ≈ 2.3 nm in Eqs. (2.17) and (2.19), roughly corresponding to the observed size
of a closed state of MscL [27, 42]. For the sake of simplicity, we exclude considerations
of lateral membrane tension in this chapter by setting τ = 0.
For a single membrane inclusion with circular cross section and arbitrary (angular)
variations in U(θ) and U
′
(θ), the exact solution of the Euler-Lagrange equation in Eq. (2.7)
is given by Eq. (2.8) with Eq. (2.9), and the corresponding bilayer thickness deformation
energy follows from Eq. (2.10) [30, 58, 59, 61]. For the choices for U(θ) and U
′
(θ) in
Eqs. (2.15) and (2.16), and assuming membrane tension τ = 0, one thus finds the bilayer
thickness deformation energy
G¯
analy = πR¯
analy(¯ν+ − ν¯−)
U¯ 2
0 E¯
0 + U¯′2
0 F¯
0 + U¯
0U¯′
0H¯
0
+
1
2
β¯2E¯
w + ¯γ
2F¯
v + δwvβ¯ γ¯H¯
w
r¯=R¯
analy
(3.1)
with v > 0 and w > 0, where R¯
analy is the radius of the circular protein cross section, δwv
is the Kronecker delta, and we have defined
D¯
q = Kq(
√
ν¯+r¯)∂r¯Kq(
√
ν¯−r¯) − Kq(
√
ν¯−r¯)∂r¯Kq(
√
ν¯+r¯),
E¯
q =
[∂r¯Kq(
√
ν¯+r¯)] [∂r¯Kq(
√
ν¯−r¯)]
D¯
q
,
F¯
q =
Kq(
√
ν¯+r¯)Kq(
√
ν¯−r¯)
D¯
q
,
H¯
q =
Kq(
√
ν¯+r¯)∂r¯Kq(
√
ν¯−r¯) + Kq(
√
ν¯−r¯)∂r¯Kq(
√
ν¯+r¯)
D¯
q
,
(3.2)
49
where q = 0, 1, . . . , Kq denotes the q
th order modified Bessel function of the second
kind, and ∂r¯ denotes the partial derivative with respect to r¯. The E0, F0, and H0 terms
in Eq. (3.1) are the contributions to G¯
analy due to the constant U¯
0 and U¯′
0
in Eqs. (2.15)
and (2.16), while the remaining terms encapsulate the effects of the variations in U(θ) and
U
′
(θ) in Eqs. (2.15) and (2.16) on G¯
analy. We use here Eq. (3.1) to analytically estimate
the energy of protein-induced bilayer thickness deformations for membrane proteins with
non-circular cross sections. To this end, we choose R¯
analy in Eq. (3.1) such that the
circumference of the circular membrane inclusion considered in Eq. (3.1) is equal to the
circumference of the membrane protein under consideration,
R¯
analy =
Γ¯
2π
, (3.3)
where, for the clover-leaf and polygonal boundary curves in Eqs. (2.17) and (2.18) with
Eq. (2.19), the protein circumference Γ follows from Γ =¯
R 2π
0
dθ¯l, where, as in Eq. (2.10),
¯l is the (dimensionless) line element.
The analytic estimate of the thickness deformation energy in Eq. (3.1) captures, for
the choice of R¯
analy in Eq. (3.3), effects related to the overall shape of membrane proteins. However, Eq. (3.1) does not capture effects due to strong local variations in the
protein cross section. For instance, the clover-leaf shapes in Eq. (2.17) can give, for large
enough ϵ and s, protein cross sections with pronounced invaginations. If the protein size
R is comparable to the decay length of bilayer thickness deformations, λ in Eq. (2.5),
such protein invaginations can yield overlaps in the protein-induced lipid bilayer thickness deformations due to different portions of the bilayer-protein interface, resulting in
50
10
the level of agreement between BVM and FEM solutions
being in line with the accuracy of the FEM solutions
expected from Fig. 5.
IV. ANALYTIC APPROXIMATION OF THE
BILAYER THICKNESS DEFORMATION
ENERGY
For membrane inclusions with circular cross section,
the solution for the thickness deformation field ¯u(¯r, ✓)
in Eq. (8) with Eq. (9) and the bilayer thickness deformation energy in Eq. (10) yield exact analytic expressions for the energy of protein-induced bilayer thickness
deformations for arbitrary (angular) variations in the
bilayer-protein boundary conditions [9–11, 15, 28, 35, 36].
The purpose of this section is to develop, on this basis,
a simple analytic scheme for estimating the energy of
protein-induced bilayer thickness deformations for membrane proteins with non-circular cross sections. In Sec. V
we show that, for many protein shapes, these simple analytic estimates agree remarkably well with the corresponding BVM solutions.
For a single membrane inclusion with circular cross
section and arbitrary (angular) variations in U(✓) and
U0
(✓), the exact solution of the Euler-Lagrange equation
in Eq. (7) is given by Eq. (8) with Eq. (9), and the corresponding bilayer thickness deformation energy follows
from Eq. (10) [15, 28, 35, 36]. For the choices for U(✓)
and U0
(✓) in Eqs. (15) and (16), one thus finds the bilayer
thickness deformation energy
G¯analy = ⇡R¯analy(¯⌫+ ⌫¯)
U¯ 2
0 E¯0 + U¯02
0 F¯0 + U¯0U¯0
0H¯0
+
1
2
✓
¯2E¯w + ¯2F¯v + wv¯ ¯H¯w
◆
r¯=R¯analy
(30)
with v > 0 and w > 0, where R¯analy is the radius of the
circular protein cross section, wv is the Kronecker delta,
and we have defined
D¯ q = Kq(
p⌫¯+r¯)@r¯Kq(
p⌫¯r¯) Kq(
p⌫¯r¯)@r¯Kq(
p⌫¯+r¯),
E¯q = [@r¯Kq(
p⌫¯+r¯)] [@r¯Kq(
p⌫¯r¯)]
D¯ q
,
F¯q = Kq(
p⌫¯+r¯)Kq(
p⌫¯r¯)
D¯ q
,
H¯q = Kq(
p⌫¯+r¯)@r¯Kq(
p⌫¯r¯) + Kq(
p⌫¯r¯)@r¯Kq(
p⌫¯+r¯)
D¯ q
,
(31)
where q = 0, 1,... , Kq denotes the qth order modified
Bessel function of the second kind, and @r¯ denotes the
partial derivative with respect to ¯r. The zeroth order
terms in Eq. (30) are the contributions to G¯analy due
to the constant U¯0 and U¯0
0 in Eqs. (15) and (16), while
the remaining terms encapsulate the e↵ects of the variations in U(✓) and U0
(✓) in Eqs. (15) and (16) on G¯analy.
FIG. 8: Color maps of the bilayer thickness deformation footprints due to clover-leaf protein shapes with (a) R¯ = 1 and (b)
R¯ = 10 in Eq. (17) for s = 5, ✏ = 0.2, U¯ = U¯0 = 0.3 nm in
Eq. (15), and U¯0 = 0. Panels (c) and (d) show the mean curvature in units of 1/, H¯ = H, associated with the thickness
deformation fields in panels (a) and (b), respectively, while
panels (e) and (f) show the corresponding mean curvature
maps obtained for U¯ = U¯0 = 0.3 nm in Eq. (15) rather
than U¯ = U¯0 = 0.3 nm. We set 2¯a = 3.2 nm for all panels.
All results were obtained through the BVM.
We use here Eq. (30) to analytically estimate the bilayer
thickness deformation energy of membrane proteins with
non-circular cross sections. To this end, we choose R¯analy
in Eq. (30) such that the circumference of the circular
membrane inclusion considered in Eq. (30) is equal to
the circumference of the membrane protein under consideration,
R¯analy = ¯
2⇡ , (32)
where, for the clover-leaf and polygonal boundary curves
in Eqs. (17) and (18) with Eq. (19), the protein circumference follows from ¯ = R 2⇡
0 d✓¯l, where, as in Eq. (10),
¯l is the (dimensionless) line element.
The analytic estimate of the thickness deformation energy in Eq. (30) captures, for the choice of R¯analy in
Figure 3.1: Color maps of the bilayer thickness deformation footprints due to clover-leaf
protein shapes with (a) R¯ = 1 and (b) R¯ = 10 in Eq. (2.17) for s = 5, ϵ = 0.2, Uλ¯ = 0.3 nm
in Eq. (2.15), and U¯′ = 0. Panels (c) and (d) show the mean curvature in units of 1/λ, H¯ =
λH, associated with the thickness deformation fields in panels (a) and (b), respectively,
while panels (e) and (f) show the corresponding mean curvature maps obtained for Uλ¯ =
−0.3 nm in Eq. (2.15) rather than Uλ¯ = 0.3 nm. We set 2¯aλ = 3.2 nm and τ = 0 for all
panels. All results were obtained through the BVM.
protein self-interactions [see Fig. 3.1(a)]. As R¯ is increased, these overlaps in proteininduced bilayer thickness deformations become less pronounced [see Fig. 3.1(b)]. Depending on the value of R¯, one thus obtains distinct distributions of the mean curvature
51
of u¯ about the protein [see Figs. 3.1(c,d)], which also depend on the value and sign of U¯
[see Figs. 3.1(e,f)].
To quantify the protein self-interactions suggested by Fig. 3.1 it is useful to define,
based on Eq. (2.10), the line tension along the bilayer-protein interface,
Λ¯ ≡
U¯′
(θ)∇¯ 2u¯ − U¯(θ)ˆn · ∇¯ 3u¯
r¯=C¯(θ)
, (3.4)
where we used Eqs. (2.12) and (2.14) with lateral membrane tension τ = 0. In Figs. 3.2(a)
and 3.2(b) we compare, for the protein shapes in Figs. 3.1(a) and 3.1(b) with constant
U > ¯ 0 and U¯′ = 0, the line tensions Λ¯ in Eq. (3.4) and their average values ⟨Λ¯⟩ to
the corresponding Λ¯ associated with G¯
analy in Eq. (3.1), which we denote by Λ¯
analy. As
expected, Fig. 3.2 shows that the variations in Λ¯ are more pronounced for smaller cloverleaf protein shapes. We also find in Fig. 3.2 that ⟨Λ¯⟩ < Λ¯
analy, with a larger
⟨Λ¯⟩ − Λ¯
analy
for smaller R¯ in Fig. 3.2.
Interestingly, we can have Λ¯ < 0 in Fig. 3.2(a) for the smaller clover-leaf protein
shape in Fig. 3.1(a), while Λ¯ > 0 in Fig. 3.2(b) for the larger clover-leaf protein shape
in Fig. 3.1(b). The regime with Λ¯ < 0 in Fig. 3.2(a) can be understood by noting that, with
a constant U > ¯ 0 and U¯′ = 0, Λ¯ in Eq. (3.4) is directly proportional to the change in the
mean curvature of u¯ at the protein boundary, in the direction perpendicular to the proteinbilayer boundary and into the bilayer (−ˆn). For the points along the clover-leaf boundary
closest and furthest away from the protein center, ˆn is anti-parallel with the radial direction ˆr. For the points along the clover-leaf boundary shape in Figs. 3.1(a) and 3.1(b)
52
12
FIG. 9. Line tension along the bilayer-protein boundary, ⇤¯
in Eq. (33), as a function of ✓ for (a) the protein shape
in Fig. 8(a) and (b) the protein shape in Fig. 8(b), calculated using the same parameter values as in Fig. 8. The red
dashed lines show the average of ⇤¯(✓) in Eq. (33) over the
interval 0 ✓ 2⇡/5, h⇤¯i. The yellow dashed lines show
⇤¯analy = G¯analy/¯, where G¯analy is given by Eq. (30) and ¯ i s
the protein circumference in Eq. (32).
V. DEPENDENCE OF BILAYER THICKNESS
DEFORMATION ENERGY ON PROTEIN SHAPE
In this section we survey the dependence of the bilayer thickness deformation energy in Eq. (3) on the
shape of membrane proteins. In particular, we allow for
three distinct, not mutually exclusive, modes for breaking rotational symmetry about the protein center (see
also Sec. II B). In Sec. V A we take the bilayer-protein
boundary conditions to be constant along the protein
circumference, but allow for protein cross sections that
break rotational symmetry about the protein center. In
Sec. V B we explore the e↵ect of variations in the protein
hydrophobic thickness on protein-induced bilayer thickness deformations. Finally, in Sec. V C we study proteininduced bilayer thickness deformations for proteins that
show variations in the bilayer-protein contact slope along
the bilayer-protein boundary. To test the analytic approximation of the bilayer thickness deformation energy
described in Sec. IV we compare, for all three scenarios
considered in Secs. V A–V C, our BVM results to the corresponding analytic estimates by computing the signed
percent error
⇠G = 100 ⇥
G¯analy G¯
G¯ , (34)
where G¯ is the thickness deformation energy in Eq. (6)
obtained through the BVM and the corresponding analytic estimate G¯analy is given by Eq. (30) with Eq. (31).
A. Constant bilayer-protein boundary conditions
In Fig. 10 we consider the energy of proteininduced bilayer thickness deformations for clover-leaf [see
Figs. 10(a,b,c)] and polygonal [see Fig. 10(d)] protein
shapes as a function of protein size R¯ with a constant
U¯ 6= 0 and U¯0 = 0. Previous work on the bilayer thickness deformations induced by proteins with circular cross
section [12, 30] suggests that, for R¯ 1, G¯ increases approximately linearly with R¯. We find in Fig. 10 that we
also approximately have G¯ / R¯ for non-circular protein
cross sections, with the (positive) constant of proportionality depending on the protein shape. The analytic estimates G¯analy obtained from Eq. (30) approximate G¯
in Fig. 10 within approximately 10%, with particularly
small magnitudes of the signed percent error ⇠G¯ for the
polygonal protein shapes in Fig. 10(d). Note that for
protein size R comparable to the decay length we generally have ⇠G¯ > 0 in Fig. 10, indicating that protein
self-interactions tend to lower the energy cost of proteininduced bilayer thickness deformations.
The energy cost of protein-induced bilayer thickness
deformations depends crucially on the unperturbed lipid
bilayer thickness, which can be varied by changing the
lipid chain length m in Eq. (4) [7, 8, 24, 29]. In Fig. 11
we plot G¯ for clover-leaf [see Figs. 11(a,b)] and polygonal
[see Fig. 11(c)] protein shapes as a function of lipid chain
length m with U¯0 = 0. We used a protein hydrophobic thickness W¯ = 3.8 nm, which matches the unperturbed lipid bilayer thickness for m ⇡ 16. In Fig. 11(a)
we consider clover-leaf protein shapes with di↵erent symmetries and the same value of ✏, while in Fig. 11(b) we
consider clover-leaf protein shapes with di↵erent values
of ✏. Similarly as in Fig. 10 we find that deviations from
a circular protein cross section increase G¯. Similarly as
in Fig. 10, the dependence of G¯ on m in Fig. 11 is very
well captured by the analytic approximation G¯analy in
Eq. (30), suggesting that the increase in G¯ for clover-leaf
and polygonal protein cross sections compared to circular
protein cross sections results primarily from the increase
in the length of the bilayer-protein boundary ¯.
In Figs. 10 and 11 we set, in line with previous work on
gramicidin channels and MscL [9, 12, 30], U0 = 0. However, the most suitable choice for the boundary conditions on the gradient of u at the bilayer-protein interface
has been a matter of debate [16, 28, 42], and may depend
Figure 3.2: Line tension along the bilayer-protein boundary, Λ¯ in Eq. (3.4), as a function of
θ for (a) the protein shape in Fig. 3.1(a) and (b) the protein shape in Fig. 3.1(b), calculated
using the same parameter values as in Fig. 3.1. The red dashed lines show the average
of Λ( ¯ θ) in Eq. (3.4) over the interval 0 ≤ θ ≤ 2π/5, ⟨Λ¯⟩. The yellow dashed lines show
Λ¯
analy = G¯
analy/Γ¯, where G¯
analy is given by Eq. (3.1) and Γ¯ is the protein circumference in
Eq. (3.3).
furthest away from the protein center the mean curvature is negative with the sign convention used here and decreases in magnitude as one radially moves away from the protein
boundary, yielding Λ¯ > 0 [Figs. 3.1(c,d)]. In contrast, for the points along the clover-leaf
boundary shape closest to the protein center in Fig. 3.1(a) [but not Fig. 3.1(b)], the mean
curvature is approximately zero at the protein boundary and decreases as one radially
53
moves away from the protein boundary, yielding Λ¯ < 0. With a different sign convention
for the mean curvature or a protein with a constant U < ¯ 0 rather than U > ¯ 0, analogous
considerations apply [Figs. 3.1(e,f)]. Thus, protein self-interactions can effectively lower
the energy cost of protein-induced lipid bilayer thickness deformations, in analogy to the
energetically favorable bilayer-thickness-mediated protein interactions found for identical
membrane proteins in close enough proximity [22, 24, 25, 30, 41, 57, 58, 60, 61, 157,
161, 178–183].
3.2 Dependence of bilayer thickness deformation energy
on protein shape
In this section we survey the dependence of the bilayer thickness deformation energy
in Eq. (2.3) on the shape of membrane proteins. In particular, we allow for three distinct,
not mutually exclusive, modes for breaking rotational symmetry about the protein center (see also Sec. 2.1.2). In Sec. 3.2.1 we take the bilayer-protein boundary conditions
to be constant along the protein circumference, but allow for protein cross sections that
break rotational symmetry about the protein center. In Sec. 3.2.2 we explore the effect of
variations in the protein hydrophobic thickness on protein-induced bilayer thickness deformations. Finally, in Sec. 3.2.3 we study protein-induced bilayer thickness deformations for
proteins that show variations in the bilayer-protein contact slope along the bilayer-protein
boundary. To test the analytic approximation of the bilayer thickness deformation energy
described in Sec. 3.1 we compare, for all three scenarios considered in Secs. 3.2.1–3.2.3,
54
our BVM results to the corresponding analytic estimates by computing the signed percent
error
ξG = 100 ×
G¯
analy − G¯
G¯
, (3.5)
where G¯ is the thickness deformation energy in Eq. (2.6) obtained through the BVM and
the corresponding analytic estimate G¯
analy is given by Eq. (3.1) with Eq. (3.2).
3.2.1 Constant bilayer-protein boundary conditions
In Fig. 3.3 we consider the energy of protein-induced bilayer thickness deformations
for clover-leaf [see Figs. 3.3(a,b,c)] and polygonal [see Fig. 3.3(d)] protein shapes as a
function of protein size R¯ with a constant U¯ ̸= 0 and U¯′ = 0. Previous work on the
lipid bilayer thickness deformations induced by proteins with circular cross section [27,
102] suggests that, for R¯ ≫ 1, G¯ increases approximately linearly with R¯. We find in
Fig. 3.3 that we also approximately have G¯ ∝ R¯ for non-circular protein cross sections,
with the (positive) constant of proportionality depending on the protein shape. The analytic
estimates G¯
analy obtained from Eq. (3.1) match G¯ in Fig. 3.3 within approximately 10%,
with particularly small magnitudes of the signed percent error ξG¯ for the polygonal protein
shapes in Fig. 3.3(d). Note that for protein sizes R comparable to the decay length λ we
generally have ξG¯ > 0 in Fig. 3.3, indicating that protein self-interactions tend to lower the
energy cost of protein-induced bilayer thickness deformations in Fig. 3.3.
The energy cost of protein-induced bilayer thickness deformations depends crucially
on the unperturbed lipid bilayer thickness, which can be varied by changing the lipid
chain length m in Eq. (2.4) [14, 21, 22, 40]. In Fig. 3.4 we plot G¯ for clover-leaf [see
55
Figure 3.3: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of protein size R¯ for clover-leaf protein shapes with (a) s = 3,
(b) s = 4, and (c) s = 5 in Eq. (2.17) with the indicated values of ✏, and (d) polygonal
protein shapes with the indicated values of s and P = 5 in Eq. (2.18). For all panels we
set U¯ = 0.3 nm and U¯0 = 0 and ⌧ = 0. The insets show the signed percent error ⇠G in
Eq. (3.5) for the corresponding analytic approximations G¯analy in Eq. (3.1).
Figs. 3.4(a,b)] and polygonal [see Fig. 3.4(c)] protein shapes as a function of the lipid
chain length m with U¯0 = 0. We used a protein hydrophobic thickness W¯ = 3.8 nm,
which matches the unperturbed lipid bilayer thickness for m ⇡ 16. In Fig. 3.4(a) we consider clover-leaf protein shapes with different symmetries s and the same value of ✏, while
in Fig. 3.4(b) we consider clover-leaf protein shapes with different values of ✏ and the
same symmetry s. Similarly as in Fig. 3.3 we find that deviations from a circular protein
56
Figure 3.3: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of protein size R¯ for clover-leaf protein shapes with (a) s = 3,
(b) s = 4, and (c) s = 5 in Eq. (2.17) with the indicated values of ϵ, and (d) polygonal
protein shapes with the indicated values of s and P = 5 in Eq. (2.18). For all panels we
set Uλ¯ = 0.3 nm and U¯′ = 0 and τ = 0. The insets show the signed percent error ξG in
Eq. (3.5) for the corresponding analytic approximations G¯
analy in Eq. (3.1).
Figs. 3.4(a,b)] and polygonal [see Fig. 3.4(c)] protein shapes as a function of the lipid
chain length m with U¯′ = 0. We used a protein hydrophobic thickness W λ ¯ = 3.8 nm,
which matches the unperturbed lipid bilayer thickness for m ≈ 16. In Fig. 3.4(a) we consider clover-leaf protein shapes with different symmetries s and the same value of ϵ, while
in Fig. 3.4(b) we consider clover-leaf protein shapes with different values of ϵ and the
same symmetry s. Similarly as in Fig. 3.3 we find that deviations from a circular protein
cross section increase G¯. Furthermore, similarly as in Fig. 3.3, the dependence of G¯ on m
56
Figure 3.4: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of lipid chain length m in Eq. (2.4) for (a) clover-leaf protein
shapes with ✏ = 0.3 and the indicated values of s, (b) clover-leaf protein shapes with s = 5
and the indicated values of ✏, and (c) polygonal protein shapes with the indicated values
of s and P = 5 in Eq. (2.18). For all panels we set ⌧ = 0, U¯0 = 0, W¯ = 3.8 nm in
Eq. (2.13), and R¯ ⇡ 2.3 nm. The insets show the signed percent error ⇠G in Eq. (3.5) for
the corresponding analytic approximations G¯analy in Eq. (3.1). We always have
U¯
> 0 for
the m-discretization used here.
cross section increase G¯. Furthermore, similarly as in Fig. 3.3, the dependence of G¯ on m
in Fig. 3.4 is very well captured by the analytic approximation G¯analy in Eq. (3.1), suggesting that the increase in G¯ for clover-leaf and polygonal protein cross sections compared
to circular protein cross sections results primarily from the increase in the length of the
bilayer-protein boundary ¯ due to deviations from a circular protein cross section.
57
Figure 3.4: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of lipid chain length m in Eq. (2.4) for (a) clover-leaf protein
shapes with ϵ = 0.3 and the indicated values of s, (b) clover-leaf protein shapes with s = 5
and the indicated values of ϵ, and (c) polygonal protein shapes with the indicated values
of s and P = 5 in Eq. (2.18). For all panels we set τ = 0, U¯′ = 0, W λ ¯ = 3.8 nm in
Eq. (2.13), and Rλ¯ ≈ 2.3 nm. The insets show the signed percent error ξG in Eq. (3.5) for
the corresponding analytic approximations G¯
analy in Eq. (3.1). We always have
U¯
> 0 for
the m-discretization used here.
in Fig. 3.4 is very well captured by the analytic approximation G¯
analy in Eq. (3.1), suggesting that the increase in G¯ for clover-leaf and polygonal protein cross sections compared
to circular protein cross sections results primarily from the increase in the length of the
bilayer-protein boundary Γ¯ due to deviations from a circular protein cross section.
In Figs. 3.3 and 3.4 we set, in line with previous work on gramicidin channels and
MscL [23, 27, 102], U
′ = 0. As noted in Sec. 2.1, however, the most suitable choice for
57
13
20
BIBLIOGRAPHY 30
Figure 21: Optimal gap lengths, , converge in increasing the
number of Fourier-Basset terms used in the truncated series
representations of the thickness deformation fields. Boundary
condition error calculations for (a) monomer clover-leaf proteins,
with = 0.54 and (b) trimer clover-leaf proteins with = 0.38 are
shown here.
FIG. 18: Boundary error ⌘b0 in Eq. (25) in BVM
calculations (see Sec. III) for clover-leaf protein shapes
with (a) s = 1 and ✏ = 0.54 in Eq. (17) and (b) s = 3
and ✏ = 0.38 in Eq. (17) as a function of the gap factor
⌦ in Eq. (28). We set R ⇡ 2.3 nm, U = 0.3 nm, and
U0 = 0. For ease of comparison we used the indicated,
fixed values of N in Eq. (20) with Eq. (21).
2013), 2nd ed., ISBN 9780815344506.
[4] P. Yeagle, The membranes of cells (Academic Press is
an imprint of Elsevier, London, 2016), ISBN 978-0-12-
800047-2.
[5] E. Sezgin, I. Levental, S. Mayor, and C. Eggeling, Nat.
Rev. Mol. Cell Biol. 18, 361 (2017).
[6] T. Sych, Y. M´ely, and W. R¨omer, Philosophical Transactions of the Royal Society B: Biological Sciences 373,
20170117 (2018), URL http://dx.doi.org/10.1098/
rstb.2017.0117.
[7] O. S. Andersen and R. E. Koeppe, Annual Review of Biophysics and Biomolecular Structure 36,
107–130 (2007), URL http://dx.doi.org/10.1146/
annurev.biophys.36.040306.132643.
[8] R. Phillips, T. Ursell, P. Wiggins, and P. Sens, Nature 459, 379–385 (2009), URL http://dx.doi.org/10.
1038/nature08147.
[9] H. Huang, Biophysical Journal 50, 1061–1070 (1986),
URL http://dx.doi.org/10.1016/s0006-3495(86)
83550-0.
[10] N. Dan, P. Pincus, and S. A. Safran, Langmuir 9,
2768–2771 (1993), URL http://dx.doi.org/10.1021/
la00035a005.
[11] N. Dan, A. Berman, P. Pincus, and S. A. Safran, Journal
de Physique II 4, 1713–1725 (1994), URL http://dx.
doi.org/10.1051/jp2:1994227.
[12] P. Wiggins and R. Phillips, Biophysical Journal 88,
880–902 (2005), URL http://dx.doi.org/10.1529/
biophysj.104.047431.
[13] T. S. Ursell, J. Kondev, D. Reeves, P. A. Wiggins, and
R. Phillips, in Mechanosensitivity in Cells and Tissues
1: Mechanosensitive Ion Channels, edited by A. Kamkin
and I. Kiseleva (Springer Verlag, 2008), pp. 37–70.
[14] S. Mondal et al., Biophy. J. 101, 2092 (2011).
[15] O. Kahraman, P. D. Koch, W. S. Klug, and C. A.
FIG. 19: Bilayer thickness deformation energy G¯ in
Eq. (6) calculated using the BVM (see Sec. III) as a
function of lipid chain length m in Eq. (4) for (a)
clover-leaf protein shapes with ✏ = 0.3 and the indicated
values of s, (b) clover-leaf protein shapes with s = 5 and
the indicated values of ✏, and (c) polygonal protein
shapes with the indicated values of s and P = 5 in
Eq. (18). For all panels we set U¯0 = 0, W¯ = 3.8 nm in
Eq. (13), and R¯ ⇡ 2.3 nm. The insets show the signed
percent error ⇠G in Eq. (34) for the corresponding
analytic approximations G¯analy in Eq. (30). [s rather
than s in (c); Extend y-range slightly so as to fully show
curves at 0.]
FIG. 11. Bilayer thickness deformation energy G¯ in Eq. (6)
calculated using the BVM (see Sec. III) as a function of lipid
chain length m in Eq. (4) for (a) clover-leaf protein shapes
with ✏ = 0.3 and the indicated values of s, (b) clover-leaf
protein shapes with s = 5 and the indicated values of ✏, and
(c) polygonal protein shapes with the indicated values of s and
P = 5 in Eq. (18). For all panels we set U¯0 = 0, W¯ = 3.8 nm
in Eq. (13), and R¯ ⇡ 2.3 nm. The insets show the signed
percent error ⇠G in Eq. (34) for the corresponding analytic
approximations G¯analy in Eq. (30). We always have |U¯| > 0
for the discretization of the domain of m used for plotting the
curves in all panels here.
consider clover-leaf protein shapes with di↵erent values
of ✏. Similarly as in Fig. 10 we find that deviations from
a circular protein cross section increase G¯. Similarly as
in Fig. 10, the dependence of G¯ on m in Fig. 11 is very
well captured by the analytic approximation G¯analy in
Eq. (30), suggesting that the increase in G¯ for clover-leaf
and polygonal protein cross sections compared to circular
FIG. 12. Bilayer thickness deformation profile ¯u due to a
protein with circular cross section as a function of the radial
distance from the protein center, ¯r = r/, obtained from the
exact analytic solution in Eq. (8) with Eq. (9) for the indicated
values of U¯0
. We set U¯ = 0.3 nm and R¯ = 2.3 nm.
protein cross sections results primarily from the increase
in the length of the bilayer-protein boundary ¯.
In Figs. 10 and 11 we set, in line with previous work on
gramicidin channels and MscL [9, 12, 30], U0 = 0. However, the most suitable choice for the boundary conditions on the gradient of u at the bilayer-protein interface
has been a matter of debate [16, 28, 42], and may depend
on the particular membrane protein and lipid species under consideration. In particular, U0 may di↵er from zero
or vary along the bilayer-protein interface, or the gradient of u at the bilayer-protein interface may satisfy natural boundary conditions with U0 being adjusted so as to
minimize the bilayer thickness deformation energy. As
illustrated in Fig. 12 for a membrane protein with circular protein cross section and constant U¯ = 0.3 nm, the
value of U0 can have a substantial e↵ect on the shape of
protein-induced lipid bilayer thickness deformations. In
particular, for U0 ⇡ 0.3 protein-induced bilayer thickness
deformations are seen to decay rapidly in Fig. 12. PlotFigure 3.5: Bilayer thickness deformation profile u¯ due to a protein with a circular cross
section as a function of the radial distance from the protein center, r¯ = r/λ, obtained from
the exact analytic solution in Eq. (2.8) with Eq. (2.9) for the indicated values of U¯′
. We set
Uλ¯ = 0.3 nm, τ = 0, and Rλ¯ = 2.3 nm.
the boundary conditions on the gradient of u at the bilayer-protein interface has been a
matter of debate [21–25, 27–31, 39, 102, 118], and is likely to depend on the particular
membrane protein and lipid species under consideration. In particular, U
′ may differ from
zero or vary along the bilayer-protein interface, or U
′ may satisfy natural boundary conditions with U
′ being adjusted so as to minimize the bilayer thickness deformation energy.
As illustrated in Fig. 3.5 for a membrane protein with circular cross section and constant
Uλ¯ = 0.3 nm, the value of U
′ can have a substantial effect on the shape of protein-induced
lipid bilayer thickness deformations. In particular, for U
′ ≈ 0.3 in Fig. 3.5 protein-induced
lipid bilayer thickness deformations are seen to decay rapidly.
Plotting G¯ as a function of U
′
for the scenario in Fig. 3.5 (see Fig. 3.6), we find that
G¯ is minimal for U
′ ≈ 0.28 with, as suggested by G¯
analy in Eq. (3.1), an approximately
quadratic dependence of G¯ on U
′
. Allowing for non-circular protein cross sections we
58
Figure 3.6: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of the bilayer-protein contact slope U¯0 for (a) clover-leaf protein
shapes with ✏ = 0.3 and the indicated values of s, (b) clover-leaf protein shapes with s = 5
and the indicated values of ✏, and (c) polygonal protein shapes with the indicated values
of s and P = 5, and cylindrical protein shapes with a circular cross section of radius
R¯. For all panels we set R¯ = 2.3 nm, U¯ = 0.3 nm, and ⌧ = 0. The insets show the
signed percent error ⇠G in Eq. (3.5) for the corresponding analytic approximations G¯analy
in Eq. (3.1).
quadratic dependence of G¯ on U0
. Allowing for non-circular protein cross sections we
find that the optimal U0 depends strongly, for large enough ✏, on the symmetry of cloverleaf protein shapes [see Figs. 3.6(a,b)], but only weakly on the symmetry of polygonal
protein shapes [see Fig. 3.6(c)]. Note that, for clover-leaf protein shapes, the optimal
U0 tend to shift towards U0 ⇡ 0 compared to proteins with circular cross section. This
59
Figure 3.6: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of the bilayer-protein contact slope U¯′
for (a) clover-leaf protein
shapes with ϵ = 0.3 and the indicated values of s, (b) clover-leaf protein shapes with s = 5
and the indicated values of ϵ, and (c) polygonal protein shapes with the indicated values
of s and P = 5, and cylindrical protein shapes with a circular cross section of radius
R¯. For all panels we set Rλ¯ = 2.3 nm, Uλ¯ = 0.3 nm, and τ = 0. The insets show the
signed percent error ξG in Eq. (3.5) for the corresponding analytic approximations G¯
analy
in Eq. (3.1).
find that the optimal U
′ depends strongly, for large enough ϵ, on the symmetry of cloverleaf protein shapes [see Figs. 3.6(a,b)], but only weakly on the symmetry of polygonal
protein shapes [see Fig. 3.6(c)]. Note that, for clover-leaf protein shapes, the optimal
U
′
tend to shift towards U
′ ≈ 0 compared to proteins with circular cross section. This
can be understood by noting that, for clover-leaf protein shapes, the effective reduction in
the size of the membrane footprint brought about by U
′ ̸= 0 competes with contributions
to the bilayer thickness deformation energy due to protein self-interactions. Conversely,
59
polygonal protein shapes only show weak self-interactions, resulting in minor shifts in
the optimal U
′ compared to proteins with circular cross section. Finally, we note that the
analytic estimates G¯
analy in Eq. (3.1) tend to become less accurate for larger U
′
, with up
to |ξG| ≈ 60% for the clover-leaf and polygonal shapes considered here [Fig. 3.6(insets)].
3.2.2 Variations in protein hydrophobic thickness
Membrane proteins are, in general, expected to show variations in their hydrophobic
thickness along the bilayer-protein interface [184, 185]. For oligomeric membrane proteins, variations in protein hydrophobic thickness are expected to be periodic so as to reflect the protein symmetry. We employ here the sinusoidal variations of U(θ) in Eq. (2.15)
as a generic model of variations in protein hydrophobic thickness, in which we denote the
periodicity of U(θ) by w. We focus, for now, on zero bilayer-protein contact slopes, U
′ = 0
in Eq. (2.16), but return to the effects of angular variations in U
′
in Sec. 3.2.3.
Figure 3.7 shows that variations in U(θ) can have a strong impact on the energy cost of
protein-induced bilayer thickness deformations, for non-circular as well as circular protein
cross sections. The analytic estimate G¯
analy in Eq. (3.1) is seen to approximately capture
G¯ in Fig. 3.7, but tends to become less accurate as the protein cross section exhibits
greater deviations from a circular shape, with up to |ξG| ≈ 50% for the clover-leaf and
polygonal protein shapes considered here [Fig. 3.7(insets)]. Note that, for large enough
w, G¯ in Fig. 3.7 scales approximately as w
3
for all protein cross sections considered. This
can be understood from G¯
analy in Eq. (3.1) by noting that E¯
w ∼ w
3 at large w, and γ¯ = 0 if
U
′ = 0.
60
Figure 3.7: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of the periodicity in protein hydrophobic thickness, w in
Eq. (2.15), for (a) the clover-leaf protein shapes in Eq. (2.17) with s = 2 and the indicated
values of ✏, (b) the clover-leaf protein shapes in Eq. (2.17) with s = 3 and the indicated
values of ✏, and (c) the polygonal protein shapes in Eq. (2.18) with the indicated values of
s and P = 5. For all panels we set R¯ = 2.3 nm, U¯0 = 0.1 nm, ¯ = 0.5 nm, U¯0 = 0, and
⌧ = 0. The red dashed lines indicate the asymptotic scaling ⇠ w3. The insets show the
signed percent error ⇠G in Eq. (3.5) for the corresponding analytic approximations G¯analy
in Eq. (3.1). In panel (d) we show color maps of the protein-induced bilayer thickness
deformations associated with ✏ = 0.4 in panel (a) at (i) w = 2 and (ii) w = 3, with ✏ = 0.4
in panel (b) at (iii) w = 3 and (iv) w = 4, and with s = 4 in panel (c) at (v) w = 2 and (vi)
w = 3.
w, G¯ in Fig. 3.7 scales approximately as w3 for all protein cross sections considered. This
can be understood from G¯analy in Eq. (3.1) by noting that E¯w ⇠ w3 at large w, and ¯ = 0 if
U0 = 0.
61
Figure 3.7: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of the periodicity in protein hydrophobic thickness, w in
Eq. (2.15), for (a) the clover-leaf protein shapes in Eq. (2.17) with s = 2 and the indicated
values of ϵ, (b) the clover-leaf protein shapes in Eq. (2.17) with s = 3 and the indicated
values of ϵ, and (c) the polygonal protein shapes in Eq. (2.18) with the indicated values of
s and P = 5. For all panels we set Rλ¯ = 2.3 nm, U¯
0λ = −0.1 nm, βλ¯ = 0.5 nm, U¯′ = 0, and
τ = 0. The red dashed lines indicate the asymptotic scaling ∼ w
3
. The insets show the
signed percent error ξG in Eq. (3.5) for the corresponding analytic approximations G¯
analy
in Eq. (3.1). In panel (d) we show color maps of the protein-induced bilayer thickness
deformations associated with ϵ = 0.4 in panel (a) at (i) w = 2 and (ii) w = 3, with ϵ = 0.4
in panel (b) at (iii) w = 3 and (iv) w = 4, and with s = 4 in panel (c) at (v) w = 2 and (vi)
w = 3.
While, broadly speaking, variations in protein hydrophobic thickness are seen to increase G¯ in Fig. 3.7 for all protein cross sections considered, the interplay of U(θ) and the
shape of the protein cross section can yield comparatively favorable or unfavorable scenarios. For instance, depending on whether adjacent regions of the bilayer-protein boundaries in clover-leaf protein shapes yield bilayer thickness deformations of the same sign
[see panels (i) and (iii) in Fig. 3.7(d)] or distinct signs [see panels (ii) and (iv) in Fig. 3.7(d)],
61
protein self-interactions can decrease or increase the energy of protein-induced bilayer
thickness deformations. For polygonal protein shapes, we find that scenarios in which the
maxima or minima of U(θ) coincide with the corners of the polygonal shapes [see panel
(v) in Fig. 3.7(d)] tend to be unfavorable from an energetic perspective, as compared
to scenarios in which the extrema of U(θ) tend to occur along the polygonal faces [see
panel (vi) in Fig. 3.7(d)]. However, compared to the clover-leaf protein shapes considered
in Fig. 3.7, the bilayer thickness deformation energy associated with the polygonal protein
shapes in Fig. 3.7 depends only weakly on the interplay between U(θ) and the shape of
the protein cross section.
3.2.3 Variations in bilayer-protein contact slope
Similarly as the variations in U(θ) considered in Sec. 3.2.2, U
′
(θ) in Eq. (2.16) will
generally vary along the bilayer-protein interface. Such variations could come about, for
instance, through the protein structure or the binding of peptides to some sections of the
bilayer-protein interface [22, 147]. Alternatively, if the (normal) gradient of u¯ obeys natural
boundary conditions at the bilayer-protein interface, a non-circular protein cross section
or variations in U(θ) may effectively induce variations in U
′
(θ). We employ here the simple
model of U
′
(θ) in Eq. (2.16) to explore the effect of variations in U
′
(θ) on the energy cost
of protein-induced lipid bilayer thickness deformations. For simplicity we thereby use a
constant U >¯ 0.
Figure 3.8 illustrates the impact of variations in U
′
(θ) on the energy cost of proteininduced lipid bilayer thickness deformations. Similarly as in Fig. 3.7, the analytic estimate
62
Figure 3.8: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of the periodicity in the bilayer-protein contact slope, v in
Eq. (2.16), for (a) the clover-leaf protein shapes in Eq. (2.17) with s = 2 and the indicated
values of ✏, (b) the clover-leaf protein shapes in Eq. (2.17) with s = 3 and the indicated
values of ✏, and (c) the polygonal protein shapes in Eq. (2.18) with the indicated values
of s and P = 5. For all panels we set ⌧ = 0, R¯ = 2.3 nm, U¯ = 0.3 nm, U¯0
0 = 0, and
¯ = 0.3. The red dashed lines indicate the asymptotic scaling ⇠ v. The insets show the
signed percent error ⇠G in Eq. (3.5) for the corresponding analytic approximations G¯analy
in Eq. (3.1). In panel (d) we show color maps of the protein-induced bilayer thickness
deformations associated with ✏ = 0.4 in panel (a) at (i) v = 2 and (ii) v = 4, with ✏ = 0.4 in
panel (b) at (iii) v = 3 and (iv) v = 4, and with s = 4 in panel (c) at (v) v = 2 and (vi) v = 4.
or variations in U(✓) may effectively induce variations in U0
(✓). We employ here the simple
model of U0
(✓) in Eq. (2.16) to explore the effect of variations in U0
(✓) on the energy cost
of protein-induced lipid bilayer thickness deformations. For simplicity we thereby use a
constant U >¯ 0.
63
Figure 3.8: Bilayer thickness deformation energy G¯ in Eq. (2.6) calculated using the BVM
(see Sec. 2.2) as a function of the periodicity in the bilayer-protein contact slope, v in
Eq. (2.16), for (a) the clover-leaf protein shapes in Eq. (2.17) with s = 2 and the indicated
values of ϵ, (b) the clover-leaf protein shapes in Eq. (2.17) with s = 3 and the indicated
values of ϵ, and (c) the polygonal protein shapes in Eq. (2.18) with the indicated values
of s and P = 5. For all panels we set τ = 0, Rλ¯ = 2.3 nm, Uλ¯ = 0.3 nm, U¯′
0 = 0, and
γ¯ = 0.3. The red dashed lines indicate the asymptotic scaling ∼ v. The insets show the
signed percent error ξG in Eq. (3.5) for the corresponding analytic approximations G¯
analy
in Eq. (3.1). In panel (d) we show color maps of the protein-induced bilayer thickness
deformations associated with ϵ = 0.4 in panel (a) at (i) v = 2 and (ii) v = 4, with ϵ = 0.4 in
panel (b) at (iii) v = 3 and (iv) v = 4, and with s = 4 in panel (c) at (v) v = 2 and (vi) v = 4.
G¯
analy in Eq. (3.1) is seen to approximately capture G¯ in Fig. 3.8, but tends to become less
accurate with increasing deviation of the protein cross section from a circular shape, with
up to |ξG| ≈ 40% for the clover-leaf and polygonal shapes considered here [Fig. 3.8(insets)]. Note that, for large enough v, G¯ in Fig. 3.7 scales approximately linearly with v,
independent of the protein cross section considered. This can be understood from G¯
analy
in Eq. (3.1) by noting that F¯
v ∼ v at large v. In analogy to Fig. 3.7 we find in Fig. 3.8 that,
broadly speaking, variations in U
′
(θ) increase G¯ for all protein cross sections considered.
63
However, the interplay of U
′
(θ) and the shape of the protein cross section can yield, similarly as in Fig. 3.7, comparatively favorable or unfavorable scenarios. In particular, for
the clover-leaf protein shapes in Figs. 3.8(a) and 3.8(b), it tends to be energetically favorable for the minima of U¯′
(θ) to coincide with the minima of C¯(θ), so as to make protein
self-interactions more favorable, and the maxima of U¯′
(θ) to coincide with the maxima of
C¯(θ), so as to reduce the protein’s membrane footprint. This configuration is achieved,
for instance, when v = s [see panels (i) and (iii) in Fig. 3.8(d)]. Conversely, it tends to be
energetically unfavorable for the minima of U¯′
(θ) to coincide with the maxima of C¯(θ), and
vice versa, or for U¯′
(θ) and C¯(θ) to be out of phase [see panels (ii) and (iv) in Fig. 3.8(d)].
For the polygonal protein shapes in Fig. 3.8(c), particularly favorable configurations tend
to be achieved when the minima of U¯′
(θ) fall on the polygonal faces, rather than on the
corners of the polygonal shapes [see panels (v) and (vi) in Fig. 3.8(d)].
3.3 Transitions in protein organization and shape
Section 3.2 demonstrates that protein-induced lipid bilayer thickness deformations
show a strong dependence on protein shape, and that changes in protein shape can
bring about changes in the bilayer thickness deformation energy > 10 kBT in magnitude.
In the present section we suggest possible implications of these results for the biophysical
properties of membrane proteins. In particular, Sec. 3.3.1 explores the energetic contribution of lipid bilayer thickness deformations to the self-assembly of protein monomers
into protein oligomers, and how changes in bilayer-protein interactions could destabilize
64
protein oligomers. In Sec. 3.3.2 we investigate the effect of lipid bilayer thickness deformations on transitions in protein conformational state that involve substantial changes in
protein shape.
3.3.1 Self-assembly of protein oligomers
Complex molecular architectures of membrane proteins often arise from self-assembly
of small protein subunits (monomers) into protein oligomers [5, 146]. While entropic effects are generally expected to oppose the self-assembly of membrane protein oligomers,
bilayer-protein interactions can favor or oppose protein oligomerization depending on the
lipid composition, protein shape, and membrane deformation mode considered [22, 24,
25, 30, 41, 57, 58, 60, 61, 157, 158, 161, 170, 178–183, 186–191]. In particular, the
thermodynamic competition between different oligomeric states of membrane proteins
depends crucially on how the energy per protein subunit changes with protein oligomeric
state. If the hydrophobic thickness of the protein oligomers or monomers differs from
the unperturbed hydrophobic thickness of the surrounding lipid bilayer, one set of contributions to the oligomerization energy is expected to arise from protein-induced bilayer
thickness deformations [22, 24, 25, 30, 41, 57, 58, 60, 61, 157, 178–183, 191]. Other
potential contributions to the oligomerization energy can arise, for instance, from lipid tilt
deformations [161, 186, 187].
We illustrate here, in the context of protein-induced lipid bilayer thickness deformations, how contributions to the oligomerization energy due to bilayer-protein interactions
can be calculated through the BVM. For simplicity, we thereby consider a protein oligomer
65
of symmetry s with a clover-leaf or polygonal cross section, and take the s (identical) competing protein monomers to have circular cross sections with the same total area as the
protein oligomer, and no interactions between the monomers. Furthermore, we assume
that the protein oligomers and monomers show constant values of U and U
′ along the
bilayer-protein interface, with identical U for the protein oligomers and monomers and
U
′ = 0 for the protein monomers. These assumptions could easily be lifted to describe
more complex scenarios.
Figure 3.9 shows the difference between the (dimensionless) bilayer thickness deformation energies associated with protein oligomers and their corresponding monomers,
∆G¯, as a function of the lipid chain length m [see Fig. 3.9(a)] and the bilayer-oligomer
contact slope U
′
[see Fig. 3.9(b)] for a variety of shapes of the oligomer cross section.
The insets in Fig. 3.9 show the differences in the oligomerization energies obtained from
the analytic approximation G¯
analy in Eq. (3.1) and the BVM, ∆G¯
ξ = ∆G¯
analy − ∆G¯. Equation (3.1) is seen to provide, for modest magnitudes of U and U
′
, good estimates of the
oligomerization energy. We generally have ∆G < 0 in Fig. 3.9(a), indicating that proteininduced lipid bilayer thickness deformations support oligomerization. This can be understood from G¯
analy by noting that the protein oligomers in Fig. 3.9(a) have a smaller circumference than their corresponding monomers. Interestingly, Fig. 3.9(b) shows that ∆G can
become positive for large enough magnitudes of U
′
for the protein oligomer, which is thus
destabilized. Such a change in U
′ could be achieved, for instance, through a transition
in the conformational state of the oligomer or the binding of peptides to the oligomer [22,
147]. Figure 3.9 therefore suggests that protein-induced bilayer thickness deformations
66
17
FIG. 16. Di↵erence between the bilayer thickness deformation energies associated with protein oligomers of symmetry
s and their corresponding s monomers, G¯, calculated using the BVM (see Sec. III) as a function of (a) the lipid
chain length m in Eq. (4) and (b) the (constant) bilayeroligomer contact slope U0 in Eq. (14) for a variety of cloverleaf (solid curves) and polygonal (dashed curves) shapes of
the protein oligomers. We took the protein monomers to
have circular cross sections with U0 = 0 and used the indicated values of s, with ✏ = 0.3 for the clover-leaf oligomer
shapes in Eq. (17) and P = 5 for the polygonal oligomer
shapes in Eq. (18). We set R¯ = 1 for the monomer radii,
and used identical cross-sectional areas of the oligomers and
their corresponding monomers. We set U0 = 0 in panel (a),
2¯a = 3.2 nm in panel (b), and used W¯ = 3.8 nm for the
protein monomers and oligomers in all panels. We always
have |U¯| > 0 for the discretization of the domain of m used
for plotting the curves in panel (a).The schematics in the insets illustrate transitions between monomers and oligomers
for selected oligomeric shapes. The plots in the insets show
the di↵erence in the oligomerization energies obtained from
the analytic approximation G¯analy in Eq. (30) and the BVM,
G¯⇠ = G¯analy G¯, for each curve in the main panels.
cross section. The insets in Fig. 16 show the di↵erences
in the oligomerization energies obtained from the analytic approximation G¯analy in Eq. (30) and the BVM,
G¯⇠ = G¯analy G¯. Equation (30) is seen to provide, for modest magnitudes of U and U0
, good estimates of the oligomerization energy. We generally have
G < 0 in Fig. 16(a), indicating that protein-induced
bilayer thickness deformations support oligomerization.
This can be understood from G¯analy by noting that the
protein oligomers in Fig. 16(a) have a smaller circumference than their corresponding monomers. Interestingly,
Fig. 16(b) shows that G can become positive for large
enough magnitudes of U0 for the protein oligomer, thus
destabilizing the protein oligomer. Such a change in U0
could be achieved, for instance, through a transition in
the conformational state of the oligomer or the binding
of peptides to the oligomer [8, 68]. Figure 16 thus suggests that protein-induced bilayer thickness deformations
could assist both in the assembly and disassembly of protein oligomers, and contribute > 10 kBT to the energy
budget of oligomer assembly or disassembly.
B. Transitions in protein conformational state
To perform their biological functions, membrane proteins often have to transition between di↵erent conformational states. Such transitions in protein conformational state can be accompanied by changes in the crosssectional shape of proteins producing, in turn, changes in
protein-induced membrane deformations. Proteins that
deform the bilayer hydrophobic thickness can thus be
regulated by bilayer properties, such as the bilayer hydrophobic thickness [7, 8, 29, 72]. We illustrate here how
the BVM can be used to calculate the contribution of
bilayer thickness deformations to the energy di↵erence
between two protein states with distinct cross-sectional
shapes. For simplicity, we thereby take the two states of
the membrane protein to show identical U and U0 that
are constant along the bilayer-protein interface, and to
have cross-sectional shapes with the same area. These
assumptions could easily be lifted to provide detailed
models of specific conformational transitions in membrane proteins, which may also involve more than just
two protein states.
Figure 17 shows the di↵erence between the bilayer
thickness deformation energies associated with the final
and initial protein shapes indicated in the insets, G¯,
as a function of the lipid chain length m [see Figs. 17(a)
and 17(b)] and the bilayer-protein contact slope U0 [see
Figs. 17(c) and 17(d)]. The insets in Fig. 17 show the
corresponding di↵erences in the protein transition energies obtained from the analytic approximation G¯analy in
Eq. (30) and the BVM, G¯⇠ = G¯analy G¯. Similarly as in Fig. 16, Eq. (30) is seen to provide, for modest magnitudes of U and U0
, good estimates of G¯ in
Fig. 17. In Figs. 17(a) and 17(c) we consider idealized
scenarios in which the initial protein shape shows a circular cross section, while the final state corresponds to
a clover-leaf or polygonal shape. We find that bilayer
thickness deformations generally inhibit such transitions
in protein shape, G¯ 0, which is easily understood
from G¯analy in Eq. (30) by noting that these transitions in
protein shape are accompanied by an increase in protein
circumference. In Fig. 17(b) we study G¯ for transitions
Figure 3.9: Difference between the lipid bilayer thickness deformation energies associated with protein oligomers of symmetry s and their corresponding s monomers, ∆G¯,
calculated using the BVM (see Sec. 2.2) as a function of (a) the lipid chain length m in
Eq. (2.4) and (b) the (constant) bilayer-oligomer contact slope U
′
in Eq. (2.14) for a variety of clover-leaf (solid curves) and polygonal (dashed curves) shapes of the protein
oligomers. We took the protein monomers to have circular cross sections with U
′ = 0
and used the indicated values of s, with ϵ = 0.3 for the clover-leaf oligomer shapes in
Eq. (2.17) and P = 5 for the polygonal oligomer shapes in Eq. (2.18). For both panels, we
set τ = 0. We set Rλ¯ = 1 nm for the monomer radii, and used identical cross-sectional
areas of the oligomers and their corresponding monomers. We set U
′ = 0 in panel (a),
2¯aλ = 3.2 nm in panel (b), and used W λ ¯ = 3.8 nm for the protein monomers and oligomers
in all panels. The schematics in the insets illustrate transitions between monomers and
oligomers for selected oligomeric shapes. The plots in the insets show the difference in
the oligomerization energies obtained from the analytic approximation G¯
analy in Eq. (3.1)
and the BVM, ∆G¯
ξ = ∆G¯
analy − ∆G¯, for each curve in the main panels.
67
could assist both in the assembly and disassembly of protein oligomers, and contribute
> 10 kBT to the energy budget of membrane protein oligomer assembly or disassembly.
3.3.2 Transitions in protein conformational state
To perform their biological functions, membrane proteins often have to transition between different conformational states. Such transitions in protein conformational state can
be accompanied by changes in the cross-sectional shape of membrane proteins producing, in turn, changes in protein-induced lipid bilayer deformations. Membrane proteins
can thus be regulated by lipid bilayer properties, such as the bilayer hydrophobic thickness [14, 21, 22, 192]. We illustrate here how the BVM can be used to calculate the
contribution of lipid bilayer thickness deformations to the energy difference between two
protein states with distinct cross-sectional shapes. For simplicity, we thereby take the
two states of the membrane protein to show identical U and U
′ with U and U
′ both being
constant along the bilayer-protein interface, and to have cross-sectional shapes with the
same area. These assumptions could easily be lifted to provide detailed models of specific conformational transitions in membrane proteins, which may also involve more than
just two protein states.
Figure 3.10 shows the difference between the lipid bilayer thickness deformation energies associated with the final and initial protein shapes indicated in the insets, ∆G¯,
as a function of the lipid chain length m [see Figs. 3.10(a) and 3.10(b)] and the bilayerprotein contact slope U
′
[see Figs. 3.10(c) and 3.10(d)]. The insets in Fig. 3.10 show
68
Figure 3.10: Difference between the lipid bilayer thickness deformation energies associated with the final and initial protein shapes indicated in the insets, G¯, calculated using
the BVM (see Sec. 2.2) as a function of (a,b) the lipid chain length m in Eq. (2.4) and (c,d)
the (constant) bilayer-protein contact slope U0 in Eq. (2.14). The values of ✏ associated
with each clover-leaf shape in Eq. (2.17) are indicated in the insets, while for the polygonal
protein shapes we set P = 5. We set U0 = 0 in panels (a,b) and 2¯a = 3.2 nm in panels
(c,d), and used W¯ = 3.8 nm and ⌧ = 0 for all panels. The cross sections of all protein
shapes considered here have area ⇡R¯2 with R¯ = 2.3 nm. The plots in the insets show
the differences in the protein transition energies obtained from the analytic approximation
G¯analy in Eq. (3.1) and the BVM, G¯⇠ = G¯analy G¯, for each curve in the main panels.
same area. These assumptions could easily be lifted to provide detailed models of specific conformational transitions in membrane proteins, which may also involve more than
just two protein states.
69
Figure 3.10: Difference between the lipid bilayer thickness deformation energies associated with the final and initial protein shapes indicated in the insets, ∆G¯, calculated using
the BVM (see Sec. 2.2) as a function of (a,b) the lipid chain length m in Eq. (2.4) and (c,d)
the (constant) bilayer-protein contact slope U
′
in Eq. (2.14). The values of ϵ associated
with each clover-leaf shape in Eq. (2.17) are indicated in the insets, while for the polygonal
protein shapes we set P = 5. We set U
′ = 0 in panels (a,b) and 2¯aλ = 3.2 nm in panels
(c,d), and used W λ ¯ = 3.8 nm and τ = 0 for all panels. The cross sections of all protein
shapes considered here have area πR¯2 with Rλ¯ = 2.3 nm. The plots in the insets show
the differences in the protein transition energies obtained from the analytic approximation
G¯
analy in Eq. (3.1) and the BVM, ∆G¯
ξ = ∆G¯
analy − ∆G¯, for each curve in the main panels.
the corresponding differences in the protein transition energies obtained from the analytic approximation G¯
analy in Eq. (3.1) and the BVM, ∆G¯
ξ = ∆G¯
analy − ∆G¯. Similarly as
in Fig. 3.9, Eq. (3.1) is seen to provide, for modest magnitudes of U and U
′
, good estimates of ∆G¯ in Fig. 3.10. In Figs. 3.10(a) and 3.10(c) we consider idealized scenarios in
which the initial protein shape shows a circular cross section, while the final protein state
corresponds to a clover-leaf or polygonal protein shape. We find that bilayer thickness
deformations generally inhibit such transitions in protein shape, ∆G¯ ≥ 0, which is easily
69
understood from G¯
analy in Eq. (3.1) by noting that these transitions in protein shape are
accompanied by an increase in protein circumference. In Fig. 3.10(b) we study ∆G¯ for
transitions between proteins with non-circular cross sections. We thereby arranged the
initial and final protein states such that ∆G¯ ≤ 0. Similarly as in Figs. 3.10(a) and 3.10(c),
the results in Fig. 3.10(b) can be understood by noting that the transitions in protein shape
in Fig. 3.10(b) are accompanied by a decrease in protein circumference. Note, in particular, that the energetically favorable protein shapes in Fig. 3.10(b) tend to correspond to
polygonal protein shapes or clover-leaf shapes with small ϵ.
Finally, we consider in Fig. 3.10(d) scenarios where the sign of ∆G¯ does not necessarily follow from the relative protein circumferences of the initial and final protein shapes,
and may not be captured by Ganaly in Eq. (3.1) for all the values of U and U
′ considered
here. In particular, for the dotted teal, green, and red curves in Fig. 3.10(d) we chose the
initial and final protein shapes so that their circumferences lie within 1% of each other, and
the remaining protein shapes so that the final protein shape has a circumference that is
substantially smaller than the circumference associated with the initial protein state, by at
least 6%. The former sets of protein shapes yield a change of sign in ∆G¯ with U
′
. Three
of the latter sets of protein shapes, corresponding to the teal, purple, and dotted purple
curves in Fig. 3.10(d), always yield ∆G¯ ≤ 0, which can again be understood from G¯
analy
in Eq. (3.1), while the fourth, corresponding to the pink curve in Fig. 3.10(d), can yield
a change of sign in ∆G¯ with U
′
. In analogy to Fig. 3.9(b) this suggests that, for certain
protein shapes, modification of U
′
in a given (stable) protein conformational state through,
for instance, peptide binding [22, 147] could trigger, mediated by protein-induced bilayer
thickness deformations, a change in the protein conformational state. We note, however,
70
that for the protein shapes considered in Fig. 3.10(d) ∆G exceeds zero by not more than
a few kBT.
71
Chapter 4
Thermosensing through membrane mechanics
This chapter proposes and develops a physical mechanism, based on protein-membrane mechanics, for the ability of cells to sense temperature changes, and illustrates and
tests this mechanism in the context of three distinct bacterial and eukaryotic membrane
proteins. In particular, we systematically explore the impact of temperature changes on
the energetic cost of protein-induced lipid bilayer deformations. We begin by introducing a simple and straightforward temperature-dependent protein-membrane mechanical
model, grounded in empirical data on phospholipid membranes (Sec. 4.1). This model
lays the foundation for understanding how temperature influences protein-induced bilayer
deformations. Next, we delve into how to model transitions in protein shape (Sec. 4.2),
providing a framework for studying how temperature affects protein functionality. We
then present the results of our model (Sec. 4.3), offering insights into the temperaturedependent activation energy of key sensory proteins, including bacterial chemoreceptors,
MscL, and Piezo. In our discussion section (Sec. 4.4), we explore the implications of
our findings in greater detail and consider their significance. Finally, in our concluding
remarks (Sec. 6.3), we synthesize our findings, derive conclusions, and propose future
72
avenues for research within the domain of cellular thermosensation. We use, here, the
term “protein sensor" to refer to a diverse range of temperature-responsive membrane
proteins. In particular, we consider protein sensors that are ion channels, which transition
between closed and open conformations, and chemoreceptors, which transition between
on (ligand bound) and off (no ligand bound) states.
4.1 Modeling the effect of temperature changes on
protein-induced bilayer deformations
We build our description of bilayer-protein interactions, on the established membranemechanical framework [21–23, 27, 28, 32–34, 49, 59, 102, 107, 110, 118, 119, 155,
156] discussed in Chapter 2 and Appendix B.1. For the purposes of this chapter it is
convenient to use dimensional units G/Kb → G¯, x/λ → x¯, y/λ → y¯, u/λ → u¯, a/λ → a¯,
Ktλ
2/Kb → K¯
t
, and τλ2/Kb → τ¯. In particular, a variety of experiments have shown
that bilayer mechanical properties change with temperature [15, 114–117]. A linear fit of
experimental data on DOPC lipid bilayer thickness versus temperature (from Table 1 in
Ref. [116]) yields the lipid bilayer half-thickness
a = −mT + a0, (4.1)
with m = 0.0025 nm/K, and a0 = 2.10 nm for DOPC lipid bilayers*. According to Eq. (4.1)
and the associated experiments on DOPC lipid bilayers, a decreases from a ≈ 1.4 nm
*The variable m in Eq. (4.1) should not be mistaken for the variable m representing the lipid chain length
in Eq. (2.4)—in particular, the lipid chain length Eq. (2.4) is a dimensionless integer.
73
to a ≈ 1.3 nm in the temperature range T = 10–50◦C. Similarly, experimental data on
the temperature dependence of the bending rigidity Kℓ
b of DOPC phospholipid bilayers
suggests the phenomenological relation [116]
Kℓ
b = Kℓ
b,rme
ε
kB
(
1
T − 1
Trm
)
, (4.2)
where kB is Boltzmann’s constant, Kℓ
b,rm = 20 kBTrm is the bilayer bending rigidity at room
temperature, and ε = 7 × 10−21 J, with the energy scale kBTrm = 4.11 × 10−21 J for the
room temperature Trm = 25◦C; here we use the superscript ℓ to denote the lipid bilayer
bending rigidity rather than the bending rigidity of the protein which we discuss, in the
case of Piezo ion channels, in Sec. 4.2.3. In the temperature range T = 10–50◦C, Kℓ
b
decreases from Kℓ
b ≈ 22 kBTrm to Kℓ
b ≈ 18 kBTrm. The above measurements of a and Kℓ
b
allow estimation of the area deformation modulus via
Ka = 6
Kℓ
b
a
2
, (4.3)
in accordance with the polymer brush model of the lipid bilayer [40, 116]. Assuming
incompressibility of the lipid tail volume [193], we set, here, the thickness deformation
modulus Kt = Ka. This assumption breaks down at extremely high or low temperatures
at which lipid tails may undergo phase transitions and under high pressure, conditions
affecting the volume and packing of lipid tails. In the temperature range T = 10–50◦C, Kt
then varies from Kt ≈ 68 kBTrm/nm2
to Kt ≈ 63 kBTrm/nm2
.
74
While the values of Kℓ
b and Kt measured for DOPC bilayers [116] are close to those
measured for other phospholipid bilayers [22], bilayer hydrophobic thicknesses can vary
to a greater extent depending on membrane composition. For instance, while (m, a0) =
(0.0025 nm/K, 2.10 nm) in Eq. (4.1) was found to correspond to DOPC bilayers, (m, a0) =
(0.0025 nm/K, 2.45 nm) in Eq. (4.1) yields a lipid bilayer half-thickness a ≈ 1.7 nm at
T = 25◦C, which is consistent with measurements of the lipid bilayer hydrophobic core
half-thickness of the E. coli cytoplasmic (EcoC) membrane [194]. Furthermore, m is
generally also expected to vary with lipid composition. For instance, in Bacillus subtilis
cytoplasmic membranes the membrane thickness was measured to linearly decrease with
temperature with (m, a0) ≈ (0.0087 nm/K, 5.5 nm) [15].
To estimate the dependence of protein-induced lipid bilayer deformations on temperature changes we insert the empirical relations in Eqs. (4.1)–(4.3) into membrane elasticity
theory discussed in Chapter 2. For simplicity, we thereby focus on the simple special
cases with U = constant and U
′ = 0 [23, 32, 118], while noting that the BVM approach
developed in Chapter 2 could be applied to relax these assumptions. For all our calculations involving protein-induced lipid bilayer thickness we employ the BVM developed in
Chapter 2, yielding errors of about 0.1% or less in G. For our calculations involving midplane deformations, which we assume to be axisymmetric here, we apply the formalism
described in Appendix B.1.
75
4.2 Modeling transitions in protein shape
Changes in protein conformation are often accompanied by changes in protein shape,
which can in turn result in changes in protein-induced lipid bilayer deformations. Seminal
advances in experimental techniques such as x-ray crystallography and cryo-electron microscopy have significantly enhanced our understanding of membrane protein structure
and the interaction of proteins with lipid bilayers [42–45, 48–53]. Though, in general, there
can be intermediary states between deactivated and activated states of protein sensors,
in our discussion we depict protein sensor activation as a straightforward transition from
an off (closed) conformation to an on (open) conformation. While we recognize that this
portrayal is, in general, a gross oversimplification [27, 59, 195], it effectively captures the
core aspects of the temperature-dependent phenomena under consideration.
To describe the competition between open (on) and closed (off) states of protein sensors, a straightforward two-state Boltzmann model has proven to be useful for the systems
considered here. At the core of this model is the channel opening probability:
Po =
1
1 + e∆G/kBT
, (4.4)
where ∆G is the energy difference between the open and closed states of the bilayerprotein system. In general, both the lipid bilayer (∆Gℓ) and protein (∆Gp) components
contribute to the total transition energy ∆G between any two protein conformations:
∆G = ∆Gℓ + ∆Gp , (4.5)
76
where, from here on out, we distinguish the lipid bilayer deformation energy and protein
internal energy with subscripts ℓ and p, respectively, and refer to the system state energy
as G, without any subscripts (or superscripts). Recent studies suggest that ∆Gp may
change significantly with temperature [50–52, 195–206]. Our focus here is to provide estimates of temperature-dependence changes in DeltaGℓ
, to put into place a framework
for determing whether DeltaGp or DeltaGℓ are dominant for a particular model system. In
the following subsections we introduce the hydrophobic shape parameters for chemoreceptors and the ion channels MscL and Piezo, which we use as inputs for our calculations
of ∆Gℓ
.
4.2.1 Chemoreceptor trimers
Traditionally, chemoreceptors have been recognized for their pivotal role in chemotaxis, a process enabling organisms, including bacteria like Escherichia coli, to effectively
navigate their environments by detecting and responding to changes in chemical concentrations [207]. Numerous bacteria navigate a diverse range of environments using
whip-like appendages known as flagella, and their movement is orchestrated by signals
received from transmembrane chemoreceptors. This complex sensory system operates
through a sophisticated signaling pathway [6, 45, 208–212]. When a chemoattractant
binds to a chemoreceptor, it initiates a conformational change in the receptor. This, in
turn, inhibits CheA, a kinase protein. CheA stops autophosphorylating and subsequently
stops transferring the phosphate group to another protein called CheY. Dephosphorylated
CheY does not bind to the bacterial flagellar motor and does not induce a change in its
77
rotation direction. This lack of change in flagellar rotation direction ultimately controls the
bacterium’s swimming behavior, allowing it to move towards favorable substances and
away from harmful ones with remarkable precision in diverse environments.
Chemoreceptors have a conical shape and therefore induce midplane/curvature deformations in the surrounding lipid bilayer. However, a previous study [57] suggests that
contributions due to bilayer midplane deformations typically amount to much less than
1 kBTrm, an order of magnitude or more smaller than the contribution from bilayer thickness deformations. In the context of chemoreceptors, we therefore focus on proteininduced bilayer thickness deformations. Structural protein shape data collected by previous experiments suggest that the hydrophobic thicknesses of chemoreceptor on (ligand
bound) and off (no ligand bound) states are about Woff = 4.21 nm and Won = 4.05 nm
[57, 213]. Thus, both states lead to a hydrophobic thickness mismatch U in Eq. (2.13)
with U < a for bilayer half-thicknesses a = 1.35 nm and a = 1.7 nm, which approximately
correspond to DOPC [116] and EcoC [194] membranes, respectively. This suggests that
chemoreceptor-induced bilayer thickness deformations u satisfy |u| < a and |∇u| < 1,
and we may use Eq. (2.3) to accurately estimate bilayer deformation contributions to the
chemoreceptor activation energy.
To utilize Eq. (2.3), we specify the boundary conditions in Eqs. (2.12)–(2.14), namely U
and the protein-bilayer cross-section boundary curve C(θ) set by the shape of chemoreceptors. In particular, we consider chemoreceptor proteins that are oligomers composed
of three smaller dimer proteins [45]. Based on cryoelectron tomography images, a simple coarse-grained model suggests that these chemoreceptor trimers exhibit a roughly
three-leaf clover cross-sectional shape, with their hydrophobic thickness decreasing upon
78
Figure 4.1: Schematic views of our hydrophobic shape model for chemoreceptor trimers.
The molecular model of the chemoreceptor trimer in panel (a) is taken from Ref. [45] and
the adjacent clover boundary curve was derived from Eq. (2.17) with Ron/o↵ = 3.1 nm,
son/o↵ = 3, and ✏on/o↵ = 0.2 for both the on and off states. In panel (b), the decrease in
chemoreceptor trimer hydrophobic thickness when activated is illustrated (not to scale)
with Won = 4.05 nm and Wo↵ = 4.21 nm in Eq. (2.13) for the on and off states.
During the transition from the closed (“off") state to the open (“on") state, structural
studies suggest MscL changes its cross-sectional area Ap from Ap,o↵ = ⇡R02
o↵ to Ap,on =
⇡R02
on with, approximately, R0
o↵ = 2.3 nm and R0
on = 3.5 nm. The shape of closed, pentameric MscL, C(✓), resembles a 5-leaf clover with s = 5 and ✏ = 0.22 in Eq. (2.17).
Based on previous structural models, we take here the open state of MscL to have a
five-leaf clover cross-sectional shape C(✓) with s = 5 and ✏ = 0.11 [59].
The alteration in MscL’s cross-sectional dimensions and the closed-state hydrophobic thickness, measured at around Woff = 3.8 nm using its resolved structure [42, 27,
59], is well-documented. However, the precise hydrophobic thickness of its open state is
still undetermined. Though several studies have suggested a likely decrease in MscL’s
hydrophobic thickness upon activation [42, 91, 208, 59], the extent of this reduction remains undetermined. To address this uncertainty, we explore two distinct possibilities.
79
Figure 4.1: Schematic views of our hydrophobic shape model for chemoreceptor trimers.
The molecular model of the chemoreceptor trimer in panel (a) is taken from Ref. [45] and
the adjacent clover boundary curve was derived from Eq. (2.17) with Ron/off = 3.1 nm,
son/off = 3, and ϵon/off = 0.2 for both the on and off states. In panel (b), the decrease in
chemoreceptor trimer hydrophobic thickness when activated is illustrated (not to scale)
with Won = 4.05 nm and Woff = 4.21 nm in Eq. (2.13) for the on and off states.
activation [57, 213]. This behavior resembles that of a button or switch. We describe
the cross-section shape of clover-leaf proteins using Eq. (2.17) with the estimated shape
parameters ϵ = 0.2, R = 3.1 nm, and s = 3 for both chemoreceptor off and on states [57,
213]. We show schematics of our shape model for chemoreceptor trimer on and off states
in Figure 4.1.
4.2.2 MscL
Mechanosensitive channels are vital components for the cellular membranes, serving
a crucial role in responding to mechanical stresses and preserving cell integrity. Among
these channels, MscL emerges as a pivotal figure in the cellular physiology of bacteria.
The prominence of MscL becomes evident when bacteria face a sudden drop in external
osmolarity, leading to rapid water influx and subsequent cell swelling [8]. As this water
influx causes the cell’s volume to expand, the lipid bilayer of the cell membrane stretches
and distorts, resulting in increased membrane tension. In response to this mechanical
challenge, MscL assumes a pivotal role in preventing cell rupture and potential cell death.
79
MscL possesses the remarkable ability to sense alterations in membrane tension, forming
a channel within the lipid bilayer akin to a safety valve. When the membrane tension
exceeds a critical threshold due to hypoosmotic shock, MscL undergoes a conformational
change, creating a pore-like structure within the membrane. This newly formed pore
expedites the rapid efflux of solutes, including ions and osmolytes, which in turn reduces
internal osmolarity and facilitates the exit of water, thus restoring the cell to its normal
volume. Notably, MscL’s opening is swift and reversible, ensuring its effectiveness under
various mechanical stress conditions.
For MscL, we focus on its homopentamer homolog found in Myobacterium Tuberculosis. While MscL’s closed state has been solved using x-ray crystallography techniques, its
open state is still unresolved. However, several structural models have been proposed,
analyzed, and show promising agreement with experimental data [14, 27, 59, 89, 91, 102,
168, 214].
During the transition from the closed (“off") state to the open (“on") state, structural
studies suggest MscL changes its cross-sectional area Ap from Ap,off = πR′2
off to Ap,on =
πR′2
on with, approximately, R′
off = 2.3 nm and R′
on = 3.5 nm. The shape of closed, pentameric MscL, C(θ), resembles a 5-leaf clover with s = 5 and ϵ = 0.22 in Eq. (2.17).
Based on previous structural models, we take here the open state of MscL to have a
five-leaf clover cross-sectional shape C(θ) with s = 5 and ϵ = 0.11 [59].
The alteration in MscL’s cross-sectional dimensions and the closed-state hydrophobic thickness, measured at around Woff = 3.8 nm using its resolved structure [27, 42,
59], is well-documented. However, the precise hydrophobic thickness of its open state is
still undetermined. Though several studies have suggested a likely decrease in MscL’s
80
Figure 4.2: Schematic views of our hydrophobic shape model for MscL. The molecular
models of MscL’s closed and open states in panel (a) are taken from Ref. [59] and the
superimposed clover boundary curves were derived from Eq. (2.17) with Ro↵ ⇡ 2.27 nm
and Ron ⇡ 3.49 nm, son/o↵ = 5, and ✏o↵ = 0.22 and ✏on = 0.11 for both the off (closed) and
on (open) states. In panel (b), we show our model for MscL which ignores any change
in hydrophobic thickness (not to scale) with Won = Wo↵ = 3.8 nm in Eq. 2.13 for the on
and off states. In panel (c), we show our model for MscL with a decrease in hydrophobic
thickness (not to scale) when activated with Won = 2.5 nm and Wo↵ = 3.8 nm in Eq. 2.13
for the on and off states.
cap of area Scap = 450 nm2, protruding into the cytoplasm, and with a radius of curvature
R. Piezo is believed to gate through a membrane dome mechanism [49, 209, 107, 110,
111], where membrane tension stabilizes Piezo towards a flatter state. In its closed state,
Piezo thereby exhibits a smaller in-plane area compared to its flatter state when open,
akin to MscL gating which also exhibits an in-plane area expansion upon opening from
81
Figure 4.2: Schematic views of our hydrophobic shape model for MscL. The molecular
models of MscL’s closed and open states in panel (a) are taken from Ref. [59] and the
superimposed clover boundary curves were derived from Eq. (2.17) with Roff ≈ 2.27 nm
and Ron ≈ 3.49 nm, son/off = 5, and ϵoff = 0.22 and ϵon = 0.11 for both the off (closed) and
on (open) states. In panel (b), we show our model for MscL which ignores any change
in hydrophobic thickness (not to scale) with Won = Woff = 3.8 nm in Eq. 2.13 for the on
and off states. In panel (c), we show our model for MscL with a decrease in hydrophobic
thickness (not to scale) when activated with Won = 2.5 nm and Woff = 3.8 nm in Eq. 2.13
for the on and off states.
hydrophobic thickness upon activation [42, 59, 91, 214], the extent of this reduction remains undetermined. To address this uncertainty, we explore two distinct possibilities.
First, we focus on the change in the cross-sectional shape of MscL, ignoring the possible change in its hydrophobic thickness [102]. In this case, we maintain the thickness at
Won = 3.8 nm for the open state. in its open state MscL has a decreased hydrophobic
thickness, Won = 2.5 nm [42, 59, 91, 214]. We show schematics of our shape model for
MscL on and off states in Figure 4.2.
For MscL, the lipid bilayer contribution ∆Gℓ
to the gating energy ∆G accounts for
changes in the deformation of the lipid bilayer surrounding MscL, ∆GM
ℓ
. It also accounts for the work performed on the lipid bilayer under membrane tension as MscL
expands its cross-sectional area to open up a pore in the membrane, ∆Gτ
ℓ = −τ∆Ap
with ∆Ap ≈ 22 nm. Estimates of ∆GM
ℓ
[27] suggest that the impact of MscL-induced bilayer midplane deformations on the gating energy of MscL is less than 1 kBTrm, with the
bilayer thickness deformation contribution being an order of magnitude or more larger.
81
Similarly to chemoreceptors, the hydrophobic thickness of MscL is expected to be near
that of DOPC and EcoC membranes. We therefore focus here on contributions to the gating energy of MscL due to small protein-induced bilayer thickness deformations. Thus, we
utilize Eq. (2.3), with the hydrophobic shape parameters discussed above, incorporated
into the boundary conditions outlined in Eqs. (2.12)–(2.14) with Eq. (2.17), to estimate
∆GM
ℓ
.
4.2.3 Piezo
Since its discovery in 2010 [215], Piezo, a mechanosensor of eukaryotes, has emerged
as a pivotal player in numerous physiological processes. For example, Piezo is located
within the membranes of endothelial cells lining blood vessels [216]. When blood pressure rises from an acute increase in blood flow, causing increased shear stress on vessel
walls, Piezo is activated due to elevated membrane tension. This activation appears to set
off a cascade of events beginning with an increase in intracellular calcium concentration
[217]. Piezo-induced calcium influx triggers intracellular signaling pathways leading to the
production and release of nitric oxide (NO) by endothelial cells. NO, a potent vasodilator,
relaxes smooth muscle cells in blood vessel walls, resulting in vasodilation. This, in turn,
leads to reduced peripheral resistance and the regulation of blood pressure.
Recently, an in-depth investigation into the free membrane shape of lipid bilayer vesicles containing Piezo has yielded excellent agreement between the membrane footprint
model [32, 38, 49, 107, 110, 111] and experimental data [110]—without any free parameters. These findings underscore specific characteristics of Piezo, including its intrinsic
82
curvature, membrane footprint, low stiffness, and expansive surface area. These features
collectively contribute to Piezo’s remarkable ability for low-threshold, high-sensitivity mechanical gating [111], providing compelling evidence of the intricate connection between
Piezo’s functionality and membrane mechanical properties.
Piezo’s shape [218] has been determined through cryoelectron tomography experiments in lipid vesicles of different sizes, and a theoretical analysis [111] provided predictions for Piezo’s shape in cell membranes. In its closed state, Piezo is described as
having an intrinsic protein radius of curvature R
p
0 = 10.2 nm and a room temperature bending rigidity K
p
b
(T = Trm) ≈ 20 kBTrm. Piezo bends the lipid bilayer to form a dome shape,
composed of 25% protein and 75% lipid bilayer, that can be approximated as a spherical
cap of area Scap = 450 nm2
, protruding into the cytoplasm, and with a radius of curvature
R [111]. Piezo is believed to gate through a membrane dome mechanism [49, 107, 110,
111, 218], where membrane tension stabilizes Piezo towards a flatter state. In its closed
state, Piezo thereby exhibits a smaller in-plane area compared to its flatter state when
open, akin to MscL gating which also exhibits an in-plane area expansion upon opening
from tension.
The boundary conditions at the Piezo dome-free membrane interface can be expressed in terms of Piezo’s dome shape parameters R and Scap [see also Eq. (B.1) in
Appendix B.1], the cap angle
α = cos−1
1 −
Scap
2πR2
, (4.6)
83
Figure 4.1: Chemoreceptor’s membrane footprint contribution to chemoreceptor’s activation energy estimated using Eq. (2.6) with ⌧ = 0 in (a) DOPC and (b) EcoC membranes
as a function of membrane temperature. Calculations using the clover-leaf cross-section
shape model described in Sec. 4.2.1 are shown in solid lines while the dashed lines incorporated a cylinder shape model with the protein on and off states having equivalent
cross-section areas to the corresponding clover cross-section shapes. The color legend
beneath the panels indicates which parameters were assigned the temperature dependencies of Eqs. (4.1)–(4.3) for DOPC lipid bilayers [116], where in (b) EcoC membrane,
a was modified using a0 = 2.45 nm in Eq. (4.1), and a, K`
b , and Kt were assigned their
respective values at room temperature Trm = 25C whenever held constant. In panel (b),
the lightly shaded region depicts clover model GM solutions in which we varied m in
Eq. (4.1) by 50% and the overlapping darker shaded region represents GM with variations in " by 50%.
Following previous work [110, 111], we take the closed and open states of the Piezo dome
to have the same area, with a radius of curvature
Ro↵ = Rp
0(1 + 3K`
b/Kp
b ) (4.9)
81
Figure 4.3: Cross-sectional view of Piezo-induced membrane deformations (adapted from
Ref. [107]). The Piezo dome resembles a spherical cap with a fixed area Scap = 450 nm2
.
Key parameters include R (radius of curvature), h = h0 and r = r0 (central pore axis and
radial coordinates at s = 0), s (arclength along Piezo’s membrane footprint profile, s = 0
at the dome interface, s > 0 away from the dome), and α (cap angle). Deformations are
assumed to diminish towards a flat membrane shape for large s.
in-plane cap radius
r0 = R sin α , (4.7)
and central pore axis coordinate of the interface of the Piezo dome and surrounding membrane
h0 = −R cos α . (4.8)
Following previous work [110, 111], we take the closed and open states of the Piezo dome
to have the same area, with a radius of curvature
Roff = R
p
0
(1 + 3Kℓ
b
/Kp
b
) (4.9)
in the closed state and Ron = ∞ in the open state. Figure 4.3 depicts the Piezo dome
shape model and the membrane deformation profile using the arclength parametrization.
84
In our model, the activation energy of Piezo can be decomposed into several contributions. There are energetic contributions associated with flattening the deformations in the
surrounding lipid bilayer ∆GM
ℓ and the flattening of the Piezo dome ∆Gcap. We estimate
∆GM
ℓ using Eq. (B.1).
∆Gcap can be further decomposed into three separate contributions. One contribution comes from the change in the bilayer’s in-plane area, under membrane tension, in
flattening the Piezo dome,
∆G
τ
cap = −τ∆Acap , (4.10)
where ∆Acap = S
2
cap/4πR2
off. Another contribution comes from bending the lipid bilayer
part of the Piezo dome in to a flat configuration,
∆G
b
ℓ,cap = −0.75
Kℓ
b
2
Scap
2
Roff 2
, (4.11)
with lipid bilayer bending rigidity Kℓ
b
. Kℓ
b
varies with temperature according to the relationship in Eq. (4.2).
The final contribution comes from bending Piezo’s arms into a flat configuration,
∆G
b
p,cap = 0.25
K
p
b
2
Scap"
2
R
p
0
2
−
2
R
p
0
−
2
Roff 2
#
, (4.12)
with the piezo protein bending rigidity K
p
b
. K
p
b was previously measured to be similar to Kℓ
b
at room temperature [111], so we maintain K
p
b
(T = Trm) = Kℓ
b,rm = 20 kBTrm. Currently, to
our knowledge, there is no available data on how K
p
b
changes with temperature. However,
in the study of soft materials, it is generally anticipated that material rigidity decreases with
85
increasing temperature, particularly over the range T = 10–50◦C. Given the uncertainty
regarding the impact of changes in temperature on K
p
b
, we examined several different scenarios. The simplest approach is to assume K
p
b
is independent of temperature [maintain
K
p
b
(T) = 20 kBTrm]. Alternatively, since K
p
b = Kℓ
b at room temperature, this may indicate
temperature effects K
p
b
similarly as Kℓ
b
, so we also explore K
p
b
(T) = Kℓ
b
(T). Additionally,
we also consider the possibility of K
p
b
having a stronger dependence on temperature than
Kℓ
b
, in particular, twice as strong. So we set K
p
b
(T) = Kℓ
b
(2ε → ε, T)
†
.
4.3 Temperature-sensing through chemoreceptors and
ion channels
4.3.1 Chemoreceptor activation
Based on the phenomenological relations in Eqs. (4.1)–(4.3), we have employed a simple membrane-mechanical model to estimate how changes in temperature might impact
the activation of chemoreceptor trimers. Specifically, we have calculated the lipid bilayer
deformation energy contribution ∆Gℓ
from the membrane surrounding the chemoreceptor
to the transition from its off to on state. Though contributions due to membrane tension,
namely changes in the bilayer’s in-plane area and stretching deformations tangential to
the bilayer leaflet surfaces in Eq. (2.3), can in general vary with temperature [113], their effect on the chemoreceptor activation energy is expected to be negligible in comparison to
†Kℓ
b
(2ε → ε, T) = Kℓ
b,rme
2ε
kB
(
1
T − 1
Trm
)
.
86
that of the hydrophobic thickness mismatch [57]. For this reason we ignore contributions
due to membrane tension by setting τ = 0.
We find that ∆Gℓ decreases by roughly 8 kBTrm (equivalent to a decrease of approximately 14%) as temperature rises within the range of T = 10–50◦C in a DOPC bilayer [illustrated by the solid red curve in Fig. 4.4(a)]. To measure the impact of protein shape on
these results, we performed analogous calculations using a circular cross-section with an
area equivalent to that of the clover-leaf protein cross-section. The model of the chemoreceptor trimer with a circular cross-section yields a comparable shift in activation energy
due to temperature increases over the same range, T = 10–50◦C [represented by the red
dashed curve in Fig. 4.4(a)], signifying that our results do not depend significantly on the
variations in cross-section shape of chemoreceptor trimers.
Furthermore, the decrease in ∆Gℓ with rising temperature implies that the energy of
the chemoreceptor trimer in the off state increases relative to its on state as temperature
increases, ultimately favoring the chemoreceptor trimer’s on state. The physical origin of
these results lies in the significant energetic penalty associated with the chemoreceptor
trimer hydrophobic thickness mismatch with the bilayer’s unperturbed hydrophobic core
in Eq. (2.3) (see Sec. 3.1 for details) [57, 59]. Note that if, the sole effect of temperature
was a reduction in bending rigidity, Kℓ
b
, [as depicted by the cyan curve in Fig. 4.4(a)]
or a decrease in thickness deformation modulus, Kt
, [illustrated by the purple curve in
Fig. 4.4(a)], then an increase in temperature would produce the opposite effect, favoring
the off state with increasing temperature. In contrast, if rising temperature only caused a
reduction in bilayer thickness, 2a, the chemoreceptor trimer would remain biased towards
87
Figure 4.4: Estimates of the lipid bilayer deformation contribution to the chemoreceptor
activation energy obtained using Eq. (2.3) with τ = 0, are depicted for (a) DOPC and
(b) EcoC membranes, as a function of temperature. Solid lines represent calculations
employing the clover-leaf chemoreceptor trimer cross-section shape model described in
Sec. 4.2.1, while dashed lines incorporate a cylinder cross-section shape with chemoreceptor trimer on and off states possessing equivalent cross-section areas to the corresponding clover cross-section shapes. The color legend below the panels indicates which
parameters (a, Kℓ
b
, or Kt) were assigned the temperature relations in Eqs. (4.1)–(4.3) for
DOPC lipid bilayers [116] and, by omission, which of these parameters were held constant at their respective values at room temperature Trm = 25◦C. In (b) EcoC membranes,
a was modified using a0 = 2.45 nm in Eq. (4.1). In panel (b), the lightly shaded region
depicts the clover model solutions, with a 50% variation in m in Eq. (4.1), and the overlapping darker shaded region represents solutions with variations in ε by 50%.
88
the on state. Thus, the predicted bias of chemoreceptors towards the on state can be
explained from the decrease in bilayer hydrophobic thickness with increasing temperature.
The dependence of the temperature variations on membrane thickness follows directly
from the bilayer deformation energy’s quadratic dependence on hydrophobic thickness
mismatch U in Eq. (2.3) (see Chapter 3). The hydrophobic thicknesses of chemoreceptor trimers are larger than those of DOPC membranes, with the off state having a larger
protein hydrophobic thickness compared to the on state. As the bilayer thins with rising
temperature, U increases for both chemoreceptor off and on states by the same amount.
However, the change in U
2
is greater for the chemoreceptor off state, resulting in a more
substantial alteration in the energetic cost for the bilayer-chemoreceptor off state relative
to the on state. Consequently, membrane thinning leads to a bias towards the chemoreceptor’s on state‡
. From this standpoint, chemoreceptor trimer switching conceptually
resembles the gating of mechanosensitive ion channels, where the gating tension was
observed to depend on lipid tail length [14].
Up to this point, we have assumed that the bilayer is DOPC. However, the measured
hydrophobic thicknesses of DOPC bilayers [116] are less than the estimated hydrophobic
thickness of the EcoC membrane. To account for an EcoC membrane, we have modified the relation in Eq. (4.1) as outlined in Sec. 4.1. Subsequently, we have recalculated the switching energy curves displayed in Fig. 4.4(a) [see Fig. 4.4(b)]. Our results
in EcoC membranes exhibit qualitative similarity to those in DOPC membranes, with a
‡A similar biasing would emerge if 2a > Woff < Won or Woff > 2a > Won. If Woff > 2a > Won, a
decrease in a would reduce the hydrophobic mismatch, in Eq. (2.13), of the chemoreceptor trimer on state
while increasing the hydrophobic mismatch of the off state, which would yield a more substantial biasing
towards the chemoreceptor trimer on state. This implies that Woff > Won ensures that the decrease in
membrane thickness with increasing temperature biases chemoreceptor trimers to the on state.
89
bias towards the chemoreceptor’s on state at higher temperatures stemming from membrane thinning. Our model’s predicted temperature-dependent ∆Gℓ
for chemoreceptors
in EcoC membranes decreases by approximately 6 kBTrm (equivalent to a decrease of
approximately 27%) over the temperature range T = 10–50◦C [demonstrated by the solid
red curve in Fig. 4.4(b)]. The predicted temperature-induced biasing of chemoreceptors
towards their on state persists even under perturbations to the temperature-dependent
relationship strength parameters, m = 0.0025nm/K [indicated by the light shaded region
in Fig.4.4(a)], and ε = 7 × 10−21J [depicted by the dark shaded region in Fig.4.4(a)], in
Eqs. (4.1)–(4.2), by up to 50%. This indicates that comparable outcomes are anticipated
in membranes with somewhat different lipid compositions.
4.3.2 MscL gating
Much like in the case of chemoreceptors, we have utilized Eqs. (2.3), (2.11)–(2.14),
(2.17), and (4.1)–(4.3), but this time incorporating the protein shape models for MscL [58,
59, 102], detailed in Section 4.2.2, to estimate the lipid bilayer deformation contribution
∆Gℓ
towards MscL’s gating energy.
To focus on the temperature dependent bilayer mechanical parameters in Eqs. (4.1)–
(4.3) (a, Kℓ
b
, Kt), we ignored membrane tension by setting τ = 0. Thus, we set ∆Gℓ =
∆GM
ℓ
|τ=0. Given that the cross-section boundary curve of MscL’s open (on) state has a
greater circumference than that of its off state, ∆Gℓ > 0 (see Sec. 3.3.2 for details). If
MscL’s hydrophobic thickness does not change when activated and MscL’s hydrophobic
thickness is greater than that of the lipid bilayer’s unperturbed thickness, such as in DOPC
90
and EcoC membranes, the hydrophobic thickness mismatch U in Eq. (2.13) increases as
the membrane’s unperturbed hydrophobic thickness decreases with increasing temperature. We also have the approximate scaling ∆Gℓ ∼ U
2
(see Sec. 3.1). We find that, due to
an increase in hydrophobic thickness mismatch, ∆Gℓ
increases with increasing temperature. Note that if, the only effect of temperature was a decrease in bending rigidity, Kℓ
b
, [as
depicted by the cyan curves in Figs. 4.5(a) and (b)] or a decrease in thickness deformation modulus, Kt
, [illustrated by the purple curves in Fig. 4.5(a) and (b)], then an increase
in temperature would produce the opposite effect, favoring the open (on) state with increasing temperature. In contrast, if increasing temperature only caused a decrease in
the bilayer’s unperturbed thickness, 2a, MscL would remain biased towards the closed
(off) state. Thus, our model predicts a bias in MscL towards the closed (off) state which
can be explained from the decrease in the bilayer’s unperturbed hydrophobic thickness
with increasing temperature §
. Since DOPC membranes being thinner than EcoC bilayers, MscL has a larger hydrophobic mismatch in a DOPC bilayer. So ∆Gℓ
increases by a
larger amount with increasing temperature in DOPC membranes. In particular, we found
∆Gℓ
increases by about 14 kBTrm in a DOPC bilayer, compared to 4 kBTrm in an EcoC
bilayer, with increasing temperature over the range T = 10–50◦C.
If we, instead, assume Won = 2.5 nm [58, 59], we have Woff > 2a > Won in DOPC and
EcoC membranes. So as the membrane hydrophobic thickness decreases, U increases
with respect to the closed state and decreases with respect to the open state. Given
the approximate scaling Gℓ ∼ U
2
(see Sec. 3.1), as membrane hydrophobic thickness
§Conversely, if the bilayer’s unperturbed thickness were greater than MscL, U would decrease as the
bilayer thins with increasing temperature, reducing ∆Gℓ.
91
Figure 4.5: Estimates of the lipid bilayer deformation contribution to MscL’s activation energy obtained using Eq. (2.3) with ⌧ = 0, are depicted for (a,c) DOPC and (b,d) EcoC
membranes, as a function of temperature. In panels (a,b) we set Wo↵ = Won = 3.8 nm
in Eq. (2.13), and in panels (c,d) we set Wo↵ = 3.8 nm and Won = 2.5 nm. In all
panels, solid lines represent calculations employing the clover-leaf MscL cross-section
shape models for MscL’s open (on) and closed (off) states described in Sec. 4.2.2, while
dashed lines incorporate a cylinder cross-section shape with MscL opened (on) and
closed (off) states possessing equivalent cross-section areas to the corresponding clover
cross-section shapes. The color legend below the panels indicates which parameters
(a, K`
b , or Kt) were assigned the temperature relations in Eqs. (4.1)–(4.3) for DOPC lipid
bilayers [116] and, by omission, which of these parameters were held constant at their
respective values at room temperature Trm = 25C. In (b,d) EcoC membranes, a was
modified using a0 = 2.45 nm in Eq. (4.1). In panels (b,d), the lightly shaded region depicts
the clover model solutions, with a 50% variation in m in Eq. (4.1), and the overlapping
darker shaded region represents solutions with variations in " by 50%.
92
Figure 4.5: Estimates of the lipid bilayer deformation contribution to MscL’s activation energy obtained using Eq. (2.3) with τ = 0, are depicted for (a,c) DOPC and (b,d) EcoC
membranes, as a function of temperature. In panels (a,b) we set Woff = Won = 3.8 nm
in Eq. (2.13), and in panels (c,d) we set Woff = 3.8 nm and Won = 2.5 nm. In all
panels, solid lines represent calculations employing the clover-leaf MscL cross-section
shape models for MscL’s open (on) and closed (off) states described in Sec. 4.2.2, while
dashed lines incorporate a cylinder cross-section shape with MscL opened (on) and
closed (off) states possessing equivalent cross-section areas to the corresponding clover
cross-section shapes. The color legend below the panels indicates which parameters
(a, Kℓ
b
, or Kt) were assigned the temperature relations in Eqs. (4.1)–(4.3) for DOPC lipid
bilayers [116] and, by omission, which of these parameters were held constant at their
respective values at room temperature Trm = 25◦C. In (b,d) EcoC membranes, a was
modified using a0 = 2.45 nm in Eq. (4.1). In panels (b,d), the lightly shaded region depicts
the clover model solutions, with a 50% variation in m in Eq. (4.1), and the overlapping
darker shaded region represents solutions with variations in ε by 50%.
decreases with increasing temperature, Gℓ,on decreases while Gℓ,off increases, yielding
a decrease in ∆Gℓ
. In DOPC, we find that ∆Gℓ decreases by about 63 kBTrm and in
EcoC, that ∆Gℓ decreases by about 55 kBTrm with increasing temperature over the range
T = 10–50◦C.
92
We performed analogous calculations using a circular cross-section model for MscL’s
closed and open states with radii 2.3 nm and 3.5 nm, respectively, which yield crosssection areas equivalent to the corresponding clover-leaf shape models we’ve already
discussed (see dashed curves in Fig. 4.5). Using circular cross-sections, we found comparable shifts in ∆Gℓ with increasing temperature over the range T = 10–50◦C. This
signifies that our results do not depend substantially on the variations in MscL’s crosssection shape. Furthermore, these findings remain consistent even when the parameters
m and ε in Eq. (4.1) and Eq. (4.2) are subject to variations, e.g., of 50%, as depicted by
the lightly shaded regions in Fig. 4.5(b,d) for m and the darker shaded regions for ε.
We estimated the activation energy in Eq. (4.5) for MscL in an EcoC membrane, considering τ = 0.01, 0.1, 1 kBTrm/nm2
, as a function of temperature. Though, ∆Gp in Eq. (4.5)
is not precisely known, MscL has been measured to have a channel opening probability
Po = 0.5 at around τ ≈ 2.7 kBTrm/nm2
[22]. So we assign ∆Gp a value that yields Po = 0.5
in Eq. (4.4 at τ ≈ 2.7 kBTrm/nm2
. For our calculations with Woff = Won = 3.8 nm, this requires we set ∆Gp = 55 kBTrm [see Fig. 4.5(a)], and for our calculations with Woff = 3.8 nm
and Won = 2.5 nm, this requires we set ∆Gp = 0 [see Fig. 4.5(c)].
In our models for MscL, ∆Gp is independent of temperature. So the changes in ∆G
due to increasing temperature in Fig. 4.5(a,c) originate from ∆Gℓ
. For our calculations
with Woff = Won = 3.8 nm, we find that the contributions due to finite τ further increase
∆Gℓ with increasing temperature. For example, at τ = 1 kBTrm/nm2 ∆Gℓ
increases by
19 kBTrm with increasing temperature over the range T = 10–50◦C [see blue curve in
Fig. 4.5(a)], which is substantial compared to an increase of 4 kBTrm in ∆Gℓ
|τ = 0 [see
93
solid red curve in Fig. 4.4(b)]. So changes in temperature can have a significant effect on
∆G due to membrane tension.
In contrast, when we consider the hydrophobic thicknesses Woff = 3.8 nm and Won =
2.5 nm for MscL’s closed and open state, respectively, we find that incorporating a finite
τ does not significantly impact the change in ∆G with increasing temperature [compare
curves in Fig. 4.5(c) with the solid red curve in Fig. 4.4(d)]. Thus, the decrease in ∆G with
increasing temperature in Fig. 4.5(c) is primarily attributed to the decrease in the bilayer’s
hydrophobic thickness, as we discussed previously when we considered τ = 0.
Using Eq. (4.4), we calculated MscL’s channel opening probability Po as a function of
τ at temperatures T = 10, 30, 50◦C. We find that MscL’s activation threshold membrane
tension¶
increases by about 0.2 kBTrm/nm2
(equivalent to about a 10% increase) with
increasing temperature over the range T = 10–50◦C, when we set Woff = Won = 3.8 nm
[see Fig. 4.6(c)]. In contrast, when we consider Woff = 3.8 nm and Won = 2.5 nm, the
activation threshold membrane tension decreases by about 2 kBTrm/nm2
(equivalent to
about a 60% decrease) with increasing temperature over the range T = 10–50◦C [see
Fig. 4.6(d)]. These findings remain consistent even when the parameters m and ε in
Eq. (4.1) and Eq. (4.2), respectively, are varied by 50%, as indicated by the lightly shaded
regions in Fig. 4.6(b,d) for m and the darker shaded regions for ε.
94
Figure 4.6: In panels (a,b), we depict estimates of MscL’s activation energy in Eq. (4.5) in
an EcoC membrane as a function of temperature, where we set the membrane tension ⌧
to the values indicated by the color legend beneath panel (c). In panels (b,d) we show the
opening channel probability in Eq. (4.4) for MscL in an EcoC membrane as a function of
⌧ , where we set the temperature T to the values indicated in the color legend underneath
panel (d), and the shaded regions. The shaded regions in panels (b,d) denote the range
of solutions for 50% variations in m in Eq. (4.1) and " in Eq. (4.2) as indicated in the
greyscale legend underneath panel (d). In panels (a,b) we set Wo↵ = Won = 3.8 nm, and
in panels (c,d) we set Wo↵ = 3.8 nm and Won = 2.5 nm in Eq. (2.13). In panels (a,b) we
set Gp = 55 kBTrm, and in panels (c,d) we set Gp = 0 in Eq. (4.5).
4.3.3 Piezo gating
We calculated the separate contributions to the activation energy of Piezo, G =
GM
` +G⌧
`,cap+Gb
`,cap+Gb
p,cap for the Piezo dome shape models discussed in Sec. 4.2.3.
95
Figure 4.6: In panels (a,b), we depict estimates of MscL’s activation energy in Eq. (4.5) in
an EcoC membrane as a function of temperature, where we set the membrane tension τ
to the values indicated by the color legend beneath panel (c). In panels (b,d) we show the
opening channel probability in Eq. (4.4) for MscL in an EcoC membrane as a function of
τ , where we set the temperature T to the values indicated in the color legend underneath
panel (d), and the shaded regions. The shaded regions in panels (b,d) denote the range
of solutions for 50% variations in m in Eq. (4.1) and ε in Eq. (4.2) as indicated in the
greyscale legend underneath panel (d). In panels (a,b) we set Woff = Won = 3.8 nm, and
in panels (c,d) we set Woff = 3.8 nm and Won = 2.5 nm in Eq. (2.13). In panels (a,b) we
set ∆Gp = 55 kBTrm, and in panels (c,d) we set ∆Gp = 0 in Eq. (4.5).
4.3.3 Piezo gating
We calculated the separate contributions to the activation energy of Piezo, ∆G =
∆GM
ℓ +∆Gτ
ℓ,cap+∆Gb
ℓ,cap+∆Gb
p,cap for the Piezo dome shape models discussed in Sec. 4.2.3.
If we assume K
p
b
(T) = 20 kBTrm and membrane tension τ is finite, we find that ∆GM
ℓ decreases approximately linearly with increasing temperature over the range T = 10–50◦C
¶Throughout this thesis, we define the activation threshold of ion channels as the value for τ at which
Po = 0.5 in Eq. (4.4).
95
Figure 4.7: Estimates of (a) the change in deformation energy associated with the lipid
bilayer surrounding the Piezo dome, GM
` , (b) the change in the energy associated with
the change in the Piezo dome’s in-plane bilayer area under membrane tension, G⌧
`,cap,
(c) the change in the bending energies associated with the lipid bilayer component of
the Piezo dome, Gb
`,cap (green curves), and the protein component of the Piezo dome,
Gb
p,cap (purple curves), and (d) the activation energy of Piezo at the membrane tension
values indicated by the color legends and as functions of temperature. For our estimates
of GM
` (T), in panel (a), we used Eq. (B.1) in the arc-length representation (see Sec. B.1
for details) with the boundary conditions in Eqs. (4.6)–(4.8), which we evaluated with
Scap = 450 nm2 and Ro↵(T) in Eq. (4.9). For our estimates of G⌧
`,cap(T), in panel (b), we
used Eq. (4.10) which we evaluated with Scap = 450 nm2 and Ro↵(T) in Eq. (4.9). For our
estimates of Gb
`,cap(T), in panel (c), we used Eq. (4.11), which we evaluated with K`
b (T)
in Eq. (4.2), Scap = 450 nm2, and Ro↵(T) in Eq. (4.9). For our estimates of Gb
p,cap(T),
in panel (c), we used Eq. (4.12), which we evaluated with the Kp
b (T) that is indicated by
the legend underneath all of the panels, Rp
0 = 10.2 nm, Scap = 450 nm2, and Ro↵(T) in
Eq. (4.9). To evaluate Ro↵(T) in Eq. (4.9), we used Rp
0 = 10.2 nm, K`
b (T) in Eq. (4.2), and
the Kp
b (T) that is indicated by the legend underneath all of the panels. For our estimates
of G(T), in panel (d), we used G(T) = GM
` (T)+G⌧
`,cap(T)+Gb
`,cap(T)+Gb
p,cap(T).
108
Figure 4.7: Estimates of (a) the change in deformation energy associated with the lipid
bilayer surrounding the Piezo dome, ∆GM
ℓ
, (b) the change in the energy associated with
the change in the Piezo dome’s in-plane bilayer area under membrane tension, ∆Gτ
ℓ,cap,
(c) the change in the bending energies associated with the lipid bilayer component of
the Piezo dome, ∆Gb
ℓ,cap (green curves), and the protein component of the Piezo dome,
∆Gb
p,cap (purple curves), and (d) the activation energy of Piezo at the membrane tension
values indicated by the color legends and as functions of temperature. For our estimates
of ∆GM
ℓ
(T), in panel (a), we used Eq. (B.1) in the arc-length representation (see Sec. B.1
for details) with the boundary conditions in Eqs. (4.6)–(4.8), which we evaluated with
Scap = 450 nm2 and Roff(T) in Eq. (4.9). For our estimates of ∆Gτ
ℓ,cap(T), in panel (b), we
used Eq. (4.10) which we evaluated with Scap = 450 nm2 and Roff(T) in Eq. (4.9). For our
estimates of ∆Gb
ℓ,cap(T), in panel (c), we used Eq. (4.11), which we evaluated with Kℓ
b
(T)
in Eq. (4.2), Scap = 450 nm2
, and Roff(T) in Eq. (4.9). For our estimates of ∆Gb
p,cap(T),
in panel (c), we used Eq. (4.12), which we evaluated with the K
p
b
(T) that is indicated by
the legend underneath all of the panels, R
p
0 = 10.2 nm, Scap = 450 nm2
, and Roff(T) in
Eq. (4.9). To evaluate Roff(T) in Eq. (4.9), we used R
p
0 = 10.2 nm, Kℓ
b
(T) in Eq. (4.2), and
the K
p
b
(T) that is indicated by the legend underneath all of the panels. For our estimates
of ∆G(T), in panel (d), we used ∆G(T) = ∆GM
ℓ
(T)+∆Gτ
ℓ,cap(T)+∆Gb
ℓ,cap(T)+∆Gb
p,cap(T).
[see dotted curves in Fig. 4.7(a)]. In particular, at τ = 1 kBTrm/nm2
, ∆GM
ℓ decreases by
about 3 kBTrm over the range T = 10–50◦C [see blue dotted curve in Fig. 4.7(a)]. Since
96
we assume in our model that Piezo’s dome flattens out in its open state, the lipid bilayer
also flattens out, so GM
ℓ,on = 0. This implies that the decrease in ∆GM
ℓ
is due to an increase in GM
ℓ,off. GM
ℓ,off increases since the Piezo dome curvature (1/Roff) increases with
increasing temperature. Since Roff in Eq. (4.9) depends linearly on the ratio Kℓ
b
/Kp
b
and
Kℓ
b decreases with increasing temperature while we assume K
p
b
remains constant, the
Piezo dome curvature increases with increasing temperature. This implies that the more
highly curved Piezo dome yields a larger GM
ℓ,off in our model. Thus, our model predicts that
∆GM
ℓ decreases due to Piezo’s dome curvature increasing with increasing temperature.
We calculated the contribution ∆Gτ
ℓ,cap in Eq. (4.10), at finite τ , as a function of temperature [see dotted curves in Fig. 4.7(b)]. We find that ∆Gτ
ℓ,cap decreases approximately
linearly with increasing temperature. In particular, at τ = 1 kBTrm/nm2 we find that ∆Gτ
ℓ,cap
decreases by about 3 kBTrm over the range T = 10–50◦C [see blue dotted curve in
Fig. 4.7(b)]. Since Piezo’s dome curvature increases with increasing temperature and
∆Acap ∼ 1/R2
off in Eq. (4.10), ∆Acap increases with increasing temperature. Thus, our
model predicts that ∆Gτ
ℓ,cap decreases with increasing temperature due to the decrease
in bilayer in-plane area that accompanies the increase in Piezo dome curvature in its off
state.
We calculated the contribution ∆Gb
ℓ,cap in Eq. (4.11) as a function of temperature [see
green dotted curve in Fig. 4.7(c)]. We find that ∆Gb
ℓ,cap decreases approximately linearly
with increasing temperature by about 1 kBTrm over the range T = 10–50◦C. Since we
assume in our model that Piezo’s dome flattens out in its open state, the lipid bilayer
also flattens out, so Gb
ℓ,cap,on = 0. This implies that the decrease in ∆Gb
ℓ,cap is due to
an increase in Gb
ℓ,cap,off. Gb
ℓ,cap,off ∼ 1/R2
off [see Eq. (4.11)], so Gb
ℓ,cap,off increases with
97
increasing Piezo dome curvature. Thus, our model predicts that ∆Gb
ℓ,cap decreases with
increasing temperature due to the increase in Piezo’s dome curvature.
We calculated the contribution ∆Gb
p,cap in Eq. (4.12) as a function of temperature [see
purple dotted curve in Fig. 4.7(c)]. We find that ∆Gb
p,cap increases approximately linearly
with increasing temperature by about 3 kBTrm over the range T = 10–50◦C. Since we
assume in our model that Piezo’s dome flattens out in its open state, the lipid bilayer
also flattens out, so Gb
p,cap,on = 0.25K
p
b Scap(2/Rp
0
)
2/2 is independent of temperature [see
Eq. (4.12)]. This implies that the increase in ∆Gb
p,cap is due to a decrease in Gb
p,off. Gb
p,off ∼
[(1/Rp
0
) − (1/Roff)]2
, with 1/Rp
0 > 1/Roff [see Eq. (4.12)]. This implies that as the Piezo
dome curvature increases, it increases towards the intrinsic curvature of Piezo (1/Rp
0
) and,
consequently, Gb
p,off decreases. Thus, our model predicts that the increase in ∆Gb
p,cap with
increasing temperature is due to the Piezo dome curvature increasing towards Piezo’s
intrinsic curvature.
These results suggest that if we assume K
p
b
(T) = 20 kBTrm the lipid bilayer contribution
biases Piezo towards its open state [∆T (∆Gℓ) = ∆T (∆GM
ℓ + ∆Gτ
ℓ,cap + ∆Gb
ℓ,cap) < 0]
||
with increasing temperature, while the contribution due to the bending of Piezo’s arms
biases Piezo towards its off state [∆T (∆Gp) = ∆T (∆b
p,cap) > 0]. At low tension (τ ⪅
0.1 kBTrm/nm2
), |∆T (∆Gℓ)| < |∆T (∆Gp)|, so increasing temperature biases Piezo towards
its off state, ∆T (∆G) > 0 [see cyan and red dotted curves in Fig. 4.7(d)]. At a sufficiently
large membrane tension (τ ⪆ 0.2 kBTrm/nm2
), |∆T (∆Gℓ)| > |∆T (∆Gp)|, so increasing
||The quantity ∆T q represents the difference between q at T = 50◦C and q at T = 10◦C for the function
q.
98
temperature biases Piezo towards its on state, ∆T (∆G) < 0 [see blue dotted curve in
Fig. 4.7(d)].
Piezo’s activation threshold was previously measured [111] to be about
τ ≈ 0.4 kBTrm/nm2
. We find that as temperature increases, this threshold decreases
by about 0.05 kBTrm/nm2
(roughly equivalent to about a 10% decrease) over the range
T = 10–50◦C [see Fig. 4.8(a)]. So our model suggests changes in temperature have a
slight effect on Piezo’s gating threshold if K
p
b
does not depend on temperature. In our
model, this effect is primarily attributed to the increase in Piezo dome curvature (1/Roff)
[see Eq. (4.9)] with increasing temperature.
If we consider, instead, that K
p
b
(T) = Kℓ
b
(T), the Piezo dome curvature is independent
of temperature, with 1/Roff = 0.25/Rp
0
[see Eq. (4.9)]. Assuming K
p
b
(T) = Kℓ
b
(T), we
calculated ∆GM
ℓ as function of temperature [see solid curves in Fig. 4.7(a)]. We find
that ∆GM
ℓ
increases approximately linearly with increasing temperature. In particulate, at
τ = 1 kBTrm/nm2
, we find ∆GM
ℓ
increases by about 1 kBTrm over the temperature range
T = 10–50◦C [see blue solid curve in Fig. 4.7(a)]. Since our model assumes the Piezo
dome is flat in its open state, we have GM
ℓ,on = 0. This implies the increase in ∆GM
ℓ
is
due to a decrease in GM
ℓ,off. GM
ℓ,off ∼ Kℓ
b
[see Eq. (B.1)], so GM
ℓ,off decreases with increasing
temperature since Kℓ
b decreases with increasing temperature [see Eq. (4.2)]. In essence,
the lipid bilayer becomes easier to bend. Thus, our model predicts that the increase in
∆GM
ℓ with increasing temperature is due to the bilayer becoming easier to bend.
We calculated ∆Gτ
ℓ,cap as function of temperature [see solid curves in Fig. 4.7(b)].
We find that ∆Gτ
ℓ,cap is independent of temperature. This result is attributed to the Piezo
dome curvature not changing with temperature. Since the Piezo dome curvature does not
99
11
FIG. 5. Estimate of Piezo channel opening probability Po curves calculated using Eq. (14) at the indicated values of T as a
function of membrane tension ⌧ for the Piezo-membrane system in Fig. 4, with (a) Kp = Kb|T =Trm , (b) Kp = Kb(Flipped(T)),
and (c) Kp = Kb(T), where Kb(T) is taken from Eq. (4).
4 kBTrm (approximately 47%) under elevated membrane temperature conditions over the temperature range
T = 10–50C (as indicated by the blue dotted curve in
Fig. 4(d)).
When analyzing Piezo’s channel opening probability
Po as a function of membrane tension ⌧ (Eq. (14)), we observe that the transition of increasing temperature biasing Piezo towards its closed state to biasing Piezo towards
its opened state occurs at a membrane tension smaller
than Piezo’s gating tension. Furthermore, the shifts in
the isotherms are negligible at low membrane tension
(⌧ . 0.1 kBTrm/nm2) (Fig. 5(a)). However, as membrane tension increases, the spacing between isotherms
widens due to the escalating work required against membrane tension to form the Piezo dome in its closed (o↵)
state. This bias pushes Piezo towards its opened (on)
state at higher temperatures. These observations align
with the previous findings that variations in G are
more pronounced at greater membrane tension (as seen
in the dotted curves in Fig. 4(d)). To illustrate the robustness of our findings, we included shaded regions in
Fig. 5 to represent the gating curve variations at each
indicated temperature, resulting from a 50% variation in
" = 7 ⇥ 1021 J in Eq. 4.
Conversely, if the ratio of rigidities K`
b /Kp
b increases
with temperature, warmer temperatures would tend to
strain Piezo towards a larger radius of curvature (Ro↵)
away from its intrinsic radius of curvature (Rp
0). Assuming that the rigidity of Piezo’s arms decreases with temperature at twice the rate of the lipid bilayer (Kp
b (T) =
K`
b (2" ! ", T)), the increase in K`
b /Kp
b with warmer
temperature results in variations in the opposite direction compared to the scenario with Kp
b (T) = 20 kBTrm
(dashed curves in Fig. 4); notably, these variations are
more significant with temperature changes. In contrast, the shifts between Piezo’s isotherm gating curves
Figure 4.8: Estimates of channel opening probability in Eq. (4.4) for Piezo as a function of
membrane tension and at the indicated values of temperature T, assuming (a) K
p
b
(T) =
20 kBTrm, (b) K
p
b
(T) = Kℓ
b
(T), and (c) K
p
b
(T) = Kℓ
b
(2ε → ε, T), with Kb(T) calculated by
Eq. (4.2). To evaluate ∆G = ∆GM
ℓ + ∆Gτ
ℓ,cap + ∆Gb
ℓ,cap + ∆Gb
p,cap in Eq. (4.4) we followed
the caption of Fig. 4.7 to calculate all of its various contributions. The shaded regions
denote the range of solutions that include 50% variations in ε about ε = 7 × 10−21 J at the
temperatures indicated by the color legend in each panel.
100
change with temperature ∆Ap in Eq. (4.10) does not change with temperature. Thus our
model predicts that ∆Gτ
ℓ,cap is independent of temperature since the Piezo dome curvature
does not change with temperature when K
p
b
(T) = Kℓ
b
(T).
We calculated ∆Gb
ℓ,cap as a function of temperature [see solid green curve in Fig. 4.7(c)].
We find that ∆Gb
ℓ,cap increases approximately linearly with increasing temperature by
about 2 kBTrm over the range T = 10–50◦C. Since our model assumes the Piezo dome
flattens out when it gates open, we have Gb
ℓ,cap,on = 0. This implies the increase in ∆Gb
ℓ,cap
is due to a decrease in Gb
ℓ,cap,off. Gb
ℓ,cap,off ∼ Kℓ
b
[see Eq. (4.11)], so Gb
ℓ,cap,off decreases with
increasing temperature since Kℓ
b decreases with increasing temperature [see Eq. (4.2)].
This amounts to the lipid bilayer becoming easier to bend with increasing temperature.
Thus, our model predicts that the increase in ∆Gb
ℓ,cap with increasing temperature is due
to the bilayer becoming easier to bend.
We calculated the contribution ∆Gb
p,cap in Eq. (4.12) as a function of temperature
[see purple solid curve in Fig. 4.7(c)]. We find that ∆Gb
p,cap decreases approximately
linearly with increasing temperature by about 4 kBTrm over the range T = 10–50◦C. Since
Piezo’s dome curvature is independent of temperature, the dependence on temperature
of ∆Gb
p,cap in encapsulated entirely in K
p
b
(T) (∆Gb
p,cap ∼ K
p
b
)[see Eq. (4.12)]. Since it is
always the case that 1/Rp
0 > 1/Rp
0 − 1/Roff for finite Roff, in our model it is always true that
∆Gb
p,cap > 0. Thus, our model predicts that ∆Gb
p,cap decreases with increasing temperature
due to Piezo’s arms becoming easier to bend.
If we assume K
p
b
(T) = Kℓ
b
(T), our model results suggest that over the membrane
tension range τ = 0–1 kBTrm, ∆G decreases with increasing temperature [see Fig. 4.7(d)].
Collecting our results for the effect of increasing temperature on the many contributions to
101
∆G, we find that the lipid bilayer contribution (∆Gℓ = ∆M
ℓ + ∆Gb
ℓ,cap) biases Piezo towards
its off state with increasing temperature while the protein contribution ∆Gp = ∆Gb
p,cap
biases Piezo towards its on state. Over the membrane tension range τ = 0–1 kBTrm, we
have |∆T (∆Gℓ)| < |∆T (∆Gp)|, thus ∆G decreases with increasing temperature.
We calculated the channel opening probability in Eq. (4.4) of Piezo [see Fig. 4.8(b)].
We find that the activation threshold decreases by about 0.07 kBTrm/nm2
(roughly equivalent to about a 20% decrease) with increasing temperature over the range T = 10–50◦C.
Thus, if K
p
b
(T) = Kℓ
b
(T), our model predicts changes in temperature can produce shifts in
Piezo’s activation threshold membrane tension that are on the same order of magnitude
as the activation threshold.
We also explore the possibility that K
p
b
decreases with increasing temperature at faster
rate than Kℓ
b
. In particular, we assume K
p
b = Kℓ
b
(2ε → ε, T) (see Sec. 4.2.3). We calculated ∆GM
ℓ as function of temperature [see dashed curves in Fig. 4.7(a)]. We find
that ∆GM
ℓ
increases approximately linearly with increasing temperature. In particulate, at
τ = 1 kBTrm/nm2
, we find ∆GM
ℓ
increases by about 5 kBTrm over the temperature range
T = 10–50◦C [see blue dashed curve in Fig. 4.7(a)]. Since our model assumes the Piezo
dome is flat in its open state, we have GM
ℓ,on = 0. This implies the increase in ∆GM
ℓ
is due
to a decrease in GM
ℓ,off. GM
ℓ,off decreases since the Piezo dome curvature (1/Roff) decreases
with increasing temperature. Since Roff in Eq. (4.9) depends linearly on the ratio Kℓ
b
/Kp
b
and K
p
b
decreases at a faster rate with increasing temperature than Kℓ
b
, Roff increases with
increasing temperature. Thus, the Piezo dome curvature decreases with increasing temperature. This implies that a less curved Piezo dome yields a smaller GM
ℓ,off in our model.
102
Thus, our model predicts that ∆GM
ℓ
increases due to Piezo’s dome curvature decreasing
with increasing temperature.
We calculated the contribution ∆Gτ
ℓ,cap in Eq. (4.10), at finite τ , as a function of temperature [see dashed curves in Fig. 4.7(b)]. We find that ∆Gτ
ℓ,cap increases approximately
linearly with increasing temperature. In particular, at τ = 1 kBTrm/nm2 we find that ∆Gτ
ℓ,cap
increases by about 3 kBTrm over the range T = 10–50◦C [see blue dashed curve in
Fig. 4.7(b)]. Since Piezo’s dome curvature decreases with increasing temperature and
∆Acap ∼ 1/R2
off in Eq. (4.10), ∆Acap decreases with increasing temperature. Thus, our
model predicts that ∆Gτ
ℓ,cap increases with increasing temperature due to the increase in
bilayer in-plane area that accompanies the decrease in Piezo dome curvature in its off
state.
We calculated the contribution ∆Gb
ℓ,cap in Eq. (4.11) as a function of temperature [see
green dashed curve in Fig. 4.7(c)]. We find that ∆Gb
ℓ,cap increases approximately linearly
with increasing temperature by about 4 kBTrm over the range T = 10–50◦C. Since we
assume in our model that Piezo’s dome flattens out when it gates open, the lipid bilayer
also flattens out, so Gb
ℓ,cap,on = 0. This implies that the increase in ∆Gb
ℓ,cap is due to
an decrease in Gb
ℓ,cap,off. Gb
ℓ,cap,off ∼ 1/R2
off [see Eq. (4.11)], so Gb
ℓ,cap,off decreases with
decreasing Piezo dome curvature. Thus, our model predicts that ∆Gb
ℓ,cap increases with
increasing temperature due to the decrease in Piezo’s dome curvature.
We calculated the contribution ∆Gb
p,cap in Eq. (4.12) as a function of temperature [see
purple dashed curve in Fig. 4.7(c)]. We find that ∆Gb
p,cap decreases approximately linearly
with increasing temperature by about 11 kBTrm over the range T = 10–50◦C. Since we
assume in our model that Piezo’s dome flattens out when it gates open, the lipid bilayer
103
also flattens out, so Gb
p,cap,on = 0.25K
p
b Scap(2/Rp
0
)
2/2 is independent of temperature [see
Eq. (4.12)]. This implies that the decrease in ∆Gb
p,cap is due to an increase in Gb
p,off. Gb
p,off ∼
[(1/Rp
0
) − (1/Roff)]2
, with 1/Rp
0 > 1/Roff [see Eq. (4.12)]. From this, it follows that as the
Piezo dome curvature decreases, it decreases away from the intrinsic curvature of Piezo
(1/Rp
0
) and, consequently, Gb
p,off increases. Thus, our model predicts that the decrease in
∆Gb
p,cap with increasing temperature is due to the Piezo dome curvature decreasing away
from Piezo’s intrinsic curvature.
These results suggest that if we assume K
p
b = Kℓ
b
(2ε → ε, T), the lipid bilayer contribution biases Piezo towards its closed state [∆T (∆Gℓ) = ∆T (∆GM
ℓ + ∆Gτ
ℓ,cap + ∆Gb
ℓ,cap) > 0]
with increasing temperature, while the contribution due to the bending of Piezo’s arms
biases Piezo towards its on state [∆T (∆Gp) = ∆T (∆b
p,cap) > 0].
At low tension (τ ≲ 0.1 kBTrm/nm2
), |∆T (∆Gℓ)| < |∆T (∆Gp)|, so increasing temperature biases Piezo towards its on state, ∆T (∆G) < 0 [see cyan and red dashed
curves in Fig. 4.7(d)]. At a sufficiently large membrane tension (τ ⪆ 1 kBTrm/nm2
),
|∆T (∆Gℓ)| > |∆T (∆Gp)|, so increasing temperature biases Piezo towards its off state,
∆T (∆G) > 0 [see blue dashed curve in Fig. 4.7(d)].
Piezo’s activation threshold was previously measured [111] to be about
τ ≈ 0.4 kBTrm/nm2
, at τ ∼ 0.4 kBTrm/nm2
, our model give that increasing temperature
biases Piezo towards it on state. We find that as temperature increases, Piezo’s activation threshold decreases by about 0.13 kBTrm/nm2
(roughly equivalent to about a 30%
decrease) over the range T = 10–50◦C [see Fig. 4.8(a)]. So our model suggests that
changes in temperature can have a substantial effect on Piezo’s activation threshold if
104
K
p
b = Kℓ
b
(2ε → ε, T). In our model, this effect is primarily attributed to the decrease in
Piezo dome curvature (1/Roff) [see Eq. (4.9)] with increasing temperature.
The shaded regions in Fig. 4.8 denote the range of solutions for Po, at the indicated
temperatures, over which ε Eq. (4.2) was varied by 50% about the value 7 × 10−21 J. The
shaded regions demonstrate that our results for the shifts in Piezo’s activation curves with
increasing temperature, in all the models we considered, are robust to variations in ε by
as much as, e.g., 50%.
4.4 Connection to experiments
4.4.1 Chemoreceptors
Microorganisms have also long been recognized for their impressive capability to detect temperature variations and orient their movement in response to temperature gradients [13, 62–69, 219–223]. This ability to measure temperature gradients is vital for
microorganisms to navigate towards optimal growth enviroments. While the molecular
mechanisms underlying chemotaxis have been fairly well characterized and understood,
the molecular mechanisms underpinning the thermosensing behavior remained elusive
until experiments in recent decades began to shed light on the subject [13, 63–65, 68,
69, 222, 224]. Surprisingly, these investigations revealed that bacteria, including E. coli,
repurpose some of the very same chemoreceptors they use for chemotaxis to carry out
thermotaxis. Still, the physical mechanisms by which the chemoreceptors are activated
by temperature have remained a mystery.
105
In this context, our model provides quantitative insights into the thermosensing capabilities of E. coli chemoreceptors within the framework of temperature-induced changes
in lipid bilayer mechanics. Our findings indicate that alterations in bilayer properties,
driven by temperature fluctuations, can exert a significant influence on the energetics of
chemoreceptor activation. Over a biologically relevant temperature range of 10–50◦C, our
predictions suggest that moderate changes in temperature can induce variations of several kBTrm in the energetic contribution from protein-induced deformations in the E. coli
cytoplasmic membrane to the chemoreceptor activation energy largely due to changes
in membrane thickness. Importantly, this magnitude of variation is comparable to the
changes in activation energy typically associated with chemoreceptor methylation, which
are about 1 kBTrm per methylation site [209]. These findings suggest that regions characterized by locally elevated temperatures could act as areas with shorter bacterial run
times or function as “sinks" for bacterial diffusion. This phenomenon might lead to a
bias in bacterial motion toward higher temperatures. In essence, our model suggests
that bacteria might be drawn to warmer regions due to the reduced activation energy
for chemoreceptor-mediated responses that comes from a decrease in the contribution
from the lipid bilayer deformations with increasing temperature. This aligns intriguingly
with microfluid device experiments [13], which have indicated that bacteria tend to exhibit
thermophilic behavior in response to moderate temperature gradients through the use of
chemoreceptors.
106
4.4.2 MscL
Regarding MscL’s temperature-dependent gating threshold in prokaryotic membranes,
there is a paucity of available data. However, it is worth noting that over a decade ago,
experimental evidence indicated that MscL’s activation threshold decreases as temperature decreases when reconstituted in mammalian cells [225]. This suggests, at least in
this particular investigation, that lower temperatures tend to bias MscL towards its open
state. We found here that bilayer deformations bias MscL towards its open state at lower
temperatures under two conditions: (1) MscL exhibits negligible change in hydrophobic
thickness upon activation and (2) MscL’s hydrophobic thickness exceeds that of the lipid
bilayer. Currently, to our knowledge, there is a lack of adequate data regarding the thickness of mammalian cell membranes, preventing us from drawing any definite conclusions
about the hydrophobic thickness in the reported study in mammalian cells [225]. The
observed temperature dependence of MscL’s activation threshold is intriguing, given the
expectation that MscL exhibits a decrease in thickness when gated open [27, 42, 59,
91, 214], a point we wil return to later. Yet, we acknowledge that it is possible for tight
binding lipids or peptides to act as structural co-factors that can effectively reduce the
change in MscL’s hydrophobic thickness when activated [22]. Assuming mammalian cell
membranes are thinner than MscL, our findings suggest that MscL’s effective hydrophobic
thickness remains relatively constant during activation.
In contrast, if we assume that MscL’s effective hydrophobic thickness decreases to a
value thinner than the lipid bilayer when it gates open [27, 42, 59, 91, 214], our model predicts that MscL’s activation threshold decreases with increasing temperature. In particular,
107
our model predicts MscL’s activation threshold tension decreases by a couple kBTrm/nm2
over the range T = 10–50◦C. While this result does not align with the experimental observations of MscL’s activation threshold increasing with increasing temperature in mammalian cells [225], it is not unsupported. Since membrane hydrophobic thickness has
been observed to decrease with increasing temperature [15, 116], predictions of MscL’s
activation threshold decreasing with increasing temperature align with the observations
of MscL’s activation threshold decreasing with decreasing membrane thickness [14, 22].
Moreover, there are also experiments that indicate that cell membrane rupture tension
decreases with increasing temperature, albeit with limited available data [117]. These observations could be of biological significance if MscL’s activation threshold was found to
consistently remain just below that of the cell membrane rupture tension over the range of
biologically relevant temperatures. This would be beneficial for bacteria given the potential for damage inflicted on the cell from the premature activation of MscL and, conversely,
if MscL’s activation was not accessible when needed.
Clearly, more data is needed to decipher MscL’s functional dependence on temperature, but available experimental data and our calculations suggest that temperature may
have a significant effect. Given the general principles on which our model is built, similar
conclusions are expected to apply more broadly to other protein sensors.
108
4.4.3 Piezo
Recent electrophysiological experiments indicate that Piezo’s activation is inhibited by
cold temperatures and increased lipid bilayer rigidity [108, 226]. Since cold temperatures are expected to increase lipid bilayer rigidity, this suggests that Piezo’s functional
dependence on temperature is intricately linked to changes in lipid bilayer rigidity. Our
model predicts that decreasing temperature (and increasing lipid bilayer rigidity) yields an
increase in Piezo’s activation threshold in qualitative agreement with experimental observations [108, 226]. This result follows regardless of whether we assumed the bending
rigidity of Piezo’s arms increases in colder temperatures or remains constant.
The shifts in Piezo’s activation threshold with increasing temperature over the range
T = 10–50◦C, predicted by our model in terms of membrane tension, are on the same order of magnitude as Piezo’s gating tension at room temperature (∆τ ∼ −0.1 kBTrm/nm2
).
Furthermore, our calculations indicate that these shifts are anticipated to be more substantial if the bending rigidity of Piezo’s arms exhibits a faster rate of decrease with increasing temperature. These findings suggest that temperature can have a significant
impact on the functionality of Piezo ion channels. Moreover our results indicate that the
lipid bilayer rigidity may provide a general physical mechanism by which transmembrane
mechanosensors can sense changes in temperature.
109
Chapter 5
Physical mechanism for the self-assembly of emerin
nanodomains at the inner nuclear membrane
This chapter proposes and develops a physical mechanism for the self-assembly and
stabilization of emerin nanodomains observed in experiments [12]. The chapter is organized as follows. In Sec. 5.1 we develop the general mathematical form of the reactiondiffusion equations we employ to model emerin nanodomains, and present a linear stability analysis to obtain the general mathematical conditions on the reaction rate constants and diffusion coefficients for which they yield a Turing instability. In Sec. 5.2 we
introduce our physical model, based on a Turing mechanism, for the self-assembly and
stabilization of emerin nanodomains. In Sec. 5.2.1 we demonstrate how the molecular
reactions relevant for emerin nanodomains can be expressed mathematically in terms of
our reaction-diffusion equations. In Sec. 5.3 we employ our reaction-diffusion model to
predict key properties of emerin nanodomains observed in experiments. We summarize
our conclusions in Sec. 6.4.
110
5.1 Reaction-diffusion equations and linear stability analysis
Here we present the mathematical framework of our reaction-diffusion system, where
I (x, y, t) and A (x, y, t) represent dynamic molecular concentration fields along the INM
governed by the reaction-diffusion equations
∂I
∂t = F (I, A) + νI
(1 − A) ∇2
I + I∇2A
(5.1)
and
∂A
∂t = G (I, A) + νA
(1 − I) ∇2A + A∇2
I
, (5.2)
which involve the cubic polynomials F and G describing the reaction dynamics of the
I and A molecule populations (see Sec. 5.2.1). The diffusion coefficients, νI and νA,
dictate the dispersion rates of these molecular complexes. Notably, we impose here the
constraint 0 ≤ I + A ≤ 1, which accounts for the finite sizes [227] of I and A complexes
in the confined INM area, on all reaction and diffusion processes; we thereby scale I and
A to represent the local fractional coverage of INM area. This constraint produces the
non-linear modifications to the standard diffusion terms νI∇2
I and νA∇2A in Eqs. (5.1)
and (5.2).
A Turing instability, also known as a diffusion-driven instability, was theorized by Alan
Turing [140, 141], as a generic mechanism for nonequilibrium pattern formation from random initial conditions. For our model to support a Turing instability [9, 11, 120–122, 140–
111
144], Eqs. (5.1) and (5.2) must exhibit a non-trivial homogeneous fixed point, represented
as (I, A) =
¯I, A¯
. For a non-trivial homogeneous fixed point, both ¯I and A¯ should not
equal 0 or 1. We have
F
¯I, A¯
= 0, G
¯I, A¯
= 0 , (5.3)
In the absence of diffusion, random perturbations of this fixed point decay over time.
Our perturbation can be represented as a planar wave with small amplitude,
⃗δf(x, y, t) =
δI
δA
=
ˆδI
ˆδA
e
λte
i·⃗κ·⃗r
, (5.4)
where λ characterizes whether the perturbation grows or decays, ⃗κ = (κx, κy)
T
is the
perturbation’s spatial wavevector, and ⃗r = (x, y)
T
. Introducing random perturbations of
the I and A concentration fields about I = ¯I and A = A¯,
⃗δf and setting νI = νA = 0, we
have from Eqs. (5.1)–(5.3) that
∂
⃗δf (x, y, t)
∂t = J¯ ⃗δf (x, y, t) , (5.5)
with the homogeneous stability matrix
J¯ =
I11 I12
A21 A22
≡
∂F
∂I
(I,A)=(I,¯ A¯)
∂F
∂A
(I,A)=(I,¯ A¯)
∂G
∂I
(I,A)=(I,¯ A¯)
∂G
∂A
(I,A)=(I,¯ A¯)
, (5.6)
to leading order in ⃗δf. Substitution of this representation of the perturbation ⃗δf into
Eq. (5.5) leads to the eigenvector equation
J¯ ·
⃗δf (t) = λ ⃗δf (t) , (5.7)
For the perturbation ⃗δf to decay with time, Eq. (5.7) must yield two negative eigenvalues
(λ
±
0 < 0). One finds that the condition
Tr[J¯] = I11 + A22 < 0 (5.8)
guarantees one negative eigenvalue, while the condition
Det[J¯] = I11A22 − I12A21 > 0 (5.9)
ensures that both eigenvalues share the same sign. Consequently, Eqs. (5.8) and (5.9)
ensure the stability of the reaction-only system under (spatially homogeneous) perturbations.
The subsequent stage in our analysis involves introducing diffusion into our system
and investigating the joint conditions on the reaction-diffusion processes in the system
leading to a Turing instability. In the presence of diffusion with νI ̸= νA, random perturbations of the fixed point (I, A) =
¯I, A¯
trigger pattern formation via a Turing instability.
11
Introducing random perturbations of the I and A concentration fields about I = ¯I and
A = A¯,
⃗δf, we have from Eqs. (5.1)–(5.3) that
∂
⃗δf (x, y, t)
∂t =
J¯ + D¯ · ∇2
⃗δf (x, y, t) , (5.10)
with the coefficients of the diffusive terms in matrix
D¯ =
νI
1 − A¯
¯I
A ν ¯
A
1 − ¯I
, (5.11)
to leading order in ⃗δf. Substitution of this representation of the perturbation ⃗δf into
Eq. (5.10) leads to the eigenvector equation
J¯ − κ
2D¯
⃗δf (t) = λ ⃗δf (t) . (5.12)
For a Turing instability, the perturbation ⃗δf must not decay with time. The perturbation ⃗δf
decays with time if Tr
J¯ − κ
2D¯
< 0 and Det
J¯ − κ
2D¯
> 0, so, for their to be a Turing
instability, one or both of these conditions must not be satisfied.
The terms within D¯ in Eq. (5.11) are positive, since (I, A) =
¯I, A¯
is a non-trivial homogeneous fixed point and 0 ≤ I + A ≤ 1. Thus Tr
J¯ − κ
2D¯
= I11+
A22 − κ
2
νI
1 − A¯
+ νA
1 − ¯I
< 0. This implies, for a Turing instability, we require
Det
J¯ − κ
2D¯
< 0.
Det
J¯ − κ
2D¯
= Aκ
4 − Bκ
2 + C < 0, with A = 4νIνA
1 − ¯I − A¯
,
B = νI
A22
1 − A¯
− A21 ¯I
+ νA
I11
1 − ¯I
− I12A¯
, and C = Det
J¯
. So Det
J¯ − κ
is a quadratic function in κ
2
. Note that A > 0 since (I, A) =
¯I, A¯
is a non-trivial homogeneous fixed point and 0 ≤ I + A ≤ 1. Therefore, Det
J¯ − κ
2D¯
is a convex parabolic
function in κ
2
, with its vertex at κ
2 = B/2A. To have Det
J¯ − κ
2D¯
< 0, the vertex of
Det
J¯ − κ
2D¯
must lie below the κ
2
-axis. Since C = Det[J¯] > 0 [see Eq. (5.9)], this can
only be true if B
2 − 4AC > 0, and thus we have the condition
νI
A22
1 − A¯
− A21 ¯I
+ νA
I11
1 − ¯I
− I12A¯
2
− 4νIνA
1 − ¯I − A¯
Det
J¯
> 0 .
(5.13)
The perturbation ⃗δf has a finite spatial frequency κ, so for a Turing instability, κ must
be real where Det
J¯ − κ
2D¯
< 0. This is the case if the intercepts κ
2
± > 0. κ
2
± > 0 if
B > 0, and consequently we have the condition
νI
A22
1 − A¯
− A21 ¯I
+ νA
I11
1 − ¯I
− I12A¯
> 0 . (5.14)
The range
κ
2
−, κ2
+
signifies a band of unstable perturbation modes that give rise to a
Turing instability. An estimate of the characteristic scale arising from the Turing instability
can be calculated from the midpoint of this band, denoted as κ
2
m, which corresponds
to the vertex of Det
J¯ − κ
2D¯
, κm =
p
B/2A [121]. The characteristic length scale is
determined by
ℓc =
2π
κm
= 2π
s
2νIνA(1 − ¯I − A¯)
νI [A22(1 − A¯) − A21 ¯I] + νA[I11(1 − ¯I) − I12A¯]
. (5.1
5.2 Physical model of emerin nanodomains
Here we develop a physical model of the self-assembly of emerin nanodomains at the
INM. Our model serves three related purposes: (a) to explain how wild-type emerin nanodomains form when no force is applied, (b) to predict how wild-type emerin nanodomain
properties change under force application based on experimental data on changes in
emerin diffusion under force application, and (c) to trace observed changes in emerin organization in mutated forms of emerin to changes in key reaction or diffusion processes.
Experiments have revealed two distinct populations of fast and slowly diffusing emerin
at the INM [12]. On this basis, we consider in our model two distinct types of emerin-NBP
complexes, fast and slowly diffusing emerin-NBP complexes. We assume that there are
fast diffusing emerin that on their own cannot assemble molecular complexes, but can do
so by interacting with other emerin or NBPs. We refer to these emerin and NBP molecular
components, collectively, as ∅. On the one hand, emerin can interact with other emerin
or NBPs to form molecular complexes that can transiently assemble into higher-order
structures, resulting in a local increase in the molecule concentration at the INM. Such
molecular complexes therefore activate increased molecule concentrations at the INM,
and we denote them by A. We assume that the diffusion of A complexes is slowed down
substantially by their interactions with other emerin and with molecular binding partners,
and therefore associate A complexes with the slowly diffusing emerin populations seen in
experiments [12]. We set their diffusion coefficient to νA = νslow (see Sec. 5.3 for data).
116
On the other hand, we assume that emerin can also form transient complexes that only
show weak interactions with potential molecular binding partners and do not form higherorder structures*, thus locally inhibiting increased molecule concentrations at the INM
through steric constraints. We denote these molecular complexes by I. We assume that,
compared to A complexes, the diffusion of I complexes is less affected by interactions with
potential molecular binding partners, and therefore associate I complexes with the more
rapidly diffusing emerin populations seen in experiments. We set their diffusion coefficient
to νI = νfast (see Sec. 5.3 for data). Note that I complexes may contain a single emerin
molecule or multiple emerin molecules. For instance, the EDMD-associated P183H mutation yields spontaneous formation of emerin dimers that are long-lived enough to be
tracked in microscopy experiments, and seem unable to form higher-order structures [12,
127]. The diffusion properties of these P183H dimers are relatively similar to those found
for emerin monomers that are not part of higher-order structures [12].
Given experimental uncertainties, and to keep our model as general as possible, we
do not specify the exact stoichiometry of A and I complexes in terms of emerin molecules
or NBPs. A and I complexes may therefore correspond to a range of molecular structures, and are defined by their reaction and diffusion properties rather than their detailed
molecular composition. The precise diffusion properties of A and I complexes are likely
to depend on their detailed molecular composition. A more detailed model of emerin nanodomain self-assembly would therefore allow for more than just two types of molecular
complexes. We also note that, since experiments suggest that the regime of slow emerin
*Possibly, because emerin or NBPs are present in these complexes at stoichiometries or in molecular
conformations that are inconsistent with the assembly of higher-order structures.
117
diffusion applies predominantly to emerin inside emerin nanodomains, we implicitly make
here the assumption that emerin nanodomains are dominated by A, rather than I, complexes. Our model calculations are consistent with this assumption (see Sec. 5.3).
To understand the spontaneous formation of emerin nanodomains within the framework of our model, consider a local fluctuation in the molecule concentrations at the INM
that produces a local excess of the activating molecule species. Due to the reaction properties of the activating molecule species, the molecule concentrations of both activators
and inhibitors will subsequently be increased at that membrane location. Since it is assumed that νI > νA, the inhibiting molecule species diffuse faster out of this membrane
region than the activating molecule species, producing a positive feedback in which local
molecule concentrations of both the activating and inhibiting molecule species are locally
increased. Finally, a steady state is reached when enough of the inhibiting molecules
diffuse into the membrane regions from the surrounding membrane so that the local concentrations of activating and inhibiting molecule species balance each other.
At the most basic level, I and A complexes may both spontaneously assemble or
disassemble. Since, at least in wild-type systems, the self-assembly of emerin into nanodomains seems to rely on interactions with A complexes (see below), we assume that
the disassembly of I and A is much faster than their spontaneous assembly. For simplicity, we therefore set the spontaneous assembly rates of both I and A to be equal to zero
118
for wild-type emerin (but see Sec. 5.3.4.1) while allowing for the spontaneous dissociation
reactions
I
f1 −→ ∅ , (5.16)
A
g1 −→ ∅ , (5.17)
where f1 and g1 denote decay rate constants. The rate constants f1 and g1 fix the time
scales associated with the reaction and diffusion properties of the I and A populations in
our model of emerin nanodomain self-assembly.
In the absence of direct experimental measurements of f1 and g1, we fix f1 and g1
through a simple physical argument inspired by the mathematics of Turing instabilities
[140]. Note that, in the absence of reactions increasing the I and A concentrations, f1
and g1 in Eqs. (5.16) and (5.17) yield estimates for the root-mean-square displacements
(RMSDs) of the I and A populations over their lifetime at the membrane, 2
p
νI/f1 and
2
p
νA/g1. We assume that the I population can globally explore the system and, hence,
diffuse over the characteristic scale of emerin nanodomains (∼ 20 nm), while the A population stays localized over molecular scales. We therefore set f1 = 30 s−1 and g1 = 30f1
resulting, for wild-type systems under no mechanical stress, in RMSDs of about 16 nm
and 1 nm for the I and A populations. While we use these values of f1 and g1 throughout,
we find that f1 or g1 can be changed by as much as 50%, or even more, to obtain emerin
nanodomains with similar properties.
119
Based on the experimental observation that emerin can form higher-order structures,
we assume that the presence of A at a particular membrane location increases the likelihood that additional emerin complexes assemble at that membrane location. This could,
for instance, result from the preferential recruitment of NBPs to membrane regions with
elevated A concentrations, from a direct binding of A to emerin monomers, or from a local
slowing down of emerin diffusion in membrane regions with elevated A concentrations.
We allow here for both the possibility that an A complex can facilitate the assembly of
another A complex,
A + ∅
g2 −→ 2A , (5.18)
and, oppositely, for the possibility that a newly formed emerin complex is unable to form
higher-order structures,
A + ∅
f2 −→ A + I . (5.19)
In the case of emerin complexes that can form higher-order structures, we also allow for
the possibility that two emerin complexes present in a particular membrane region can
interact to facilitate the formation of a third emerin complex,
2A + ∅
f3 −→ 3A . (5.20)
120
Note that, mathematically, the above reaction provides the only fundamental distinction
between I and A, together with our assumption that A can locally increase the concentration of I, but not vice versa. Equations (5.18)–(5.20) thus encapsulate our model assumptions about the key reaction properties of A and, by extension, I complexes underlying
emerin nanodomain self-assembly.
To achieve a non-trivial steady-state distribution of emerin, decay and creation rates
across multiple emerin complexes must take comparable magnitudes. In particular, due
to the slow diffusion rates of A complexes, the decay and creation rates of different A
complexes must be approximately equal to each other so that a non-trivial steady state
can be achieved. Furthermore, as mentioned above, we expect from experiments that
emerin nanodomains are predominantly composed of A, rather than I, complexes. We
therefore expect f2 in Eq. (5.19) to be smaller than f1 in Eq. (5.16). For the calculations
described here we set f2 = f1/2, but other choices f2 ≲ f1 give similar results. In the most
basic picture, we expect g1 and g2 in Eqs. (5.17) and (5.18) to take similar magnitudes,
and we therefore set g2 = g1. Since the reactions in Eqs. (5.18) and (5.19) correspond
to first-order reactions, while the reaction in Eq. (5.20) corresponds to a second-order
reaction, we generally expect g3 to be substantially smaller than g1 and g2. We set here
g3 = g2/15 = 2f1 for wild-type emerin under no mechanical stress. Again, we find that
other choices for g1 ≈ g2 and g3 ≪ g2 give similar results.
In all reaction (and diffusion) processes that locally increase the molecule concentration at the INM, we include steric factors inhibiting the formation of emerin complexes in
crowded membrane regions (or their diffusion to such membrane regions) such that the
sum of the dimensionless I and A concentration fields, (I + A) with I = I (x, y, t) and
121
A = A (x, y, t), cannot exceed 1 anywhere in the system (see Sec. 5.1). Thus, I (x, y, t)
and A (x, y, t) represent the fraction of the free membrane area covered by I and A complexes.
5.2.1 Reaction kinetics of emerin nanodomains
Equations (5.16), (5.17), (5.19), and (5.20) provide the key reactions entering our
model of emerin nanodomain self-assembly. In particular, the reactions I → ∅ and
A → ∅ in Eqs. (5.16) and (5.17) represent the spontaneous disassembly of I and A
complexes. Conversely, the reactions A + ∅ → A + I, A + ∅ → 2A, and 2A + ∅ → 3A
in Eqs. (5.19), (5.18), and (5.20) represent the assembly of I and A complexes, which we
take to be catalyzed by A complexes. We mathematically account for these reactions in
the reaction-diffusion equations in Eqs. (5.1) and (5.2) through the formalism of chemical
dynamics [140–144],
F (I, A) = −f1I + f2 (1 − I − A) A (5.21)
and
G (I, A) = −g1A + g2 (1 − I − A) A + g3
1 − I − A
1 − ¯I − A¯
A2
A¯
(5.22)
with
A¯ =
f1 (g2 − g1 + g3)
f1g2 + f2 (g1 − g3)
,
¯I =
f2
f1
g1 − g3
g2
A . ¯ (5.23)
For instance, the decay of I and A complexes into emerin and NBPs is characterized by
the terms −f1I and −g1A in F and G. We consider steric effects by imposing the constraint 0 ≤ I + A ≤ 1, leading to a reduction in the relevant reaction rates by a factor of
122
(1 − I − A). The reaction A + ∅ → A + I, represented by the term +f2 (1 − I − A) A
in F, and the reactions A + ∅ → 2A and 2A + ∅ → 3A, represented by the terms
+g2 (1 − I − A) A and +g3 [A2
(1 − I − A)] /
1 − ¯I − A¯
A¯
in G, model the activation of
elevated concentrations of I complexes by A complexes and the self-activation of A complexes.
As explained in Section 5.2, our Turing model assumes that the I complexes act as
inhibitors and A complexes as activators, leading to I11 < 0 and A22 > 0 in Eq. (5.6),
respectively. Note that the evaluation of I11 in Eq. (5.6), utilizing Eq. (5.21), results in the
condition
I11 = −f1 − f2A <¯ 0 , (5.24)
Since our model does not have any reactions for which I complexes stabilize other I
complexes, we have that I complexes have the property inhibiting other I complexes
through steric repulsion, I11 < 0, due to their finite size. Utilizing Eq. (5.22) to assess A22
in Eq. (5.6) yields the following condition for a Turing instability,
A22 = −g1 + g2
1 − ¯I − 2A¯
+ g3
2 −
A¯
1 − ¯I − A¯
> 0 . (5.25)
The reaction 2A + ∅ → 3A represents the minimal reaction capable of yielding A22 > 0 in
our model. We note here that the reaction A+∅ → 2A is insufficient for the self-activation
of A complexes since it would fail to give A22 > 0 in our model.
Consider that, as per the conditions outlined in Sec. 5.1, we must have I11A22 −
I12A21 > 0 [refer to Eq. (5.9)]. Consequently, it follows that I12 and A21 must possess
1
opposite signs. In our model, we do not have any reaction in which I complexes stabilize
A complexes. Thus, we find, by evaluating A21 in Eq. (5.6) and utilizing Eq. (5.22), that
A21 = −g2A¯ − g3A/¯ (1 − ¯I − A¯) < 0 . (5.26)
Notably, in our model, the reaction A+∅ → A+I embodies the simplest scenario wherein
I complexes are stabilized by A complexes. Evaluating I12 in Eq. (5.6), using Eq. (5.21),
yields the following condition for a Turing instability,
I12 = f2
1 − ¯I − 2A¯
> 0 . (5.27)
The reaction terms in F and G are such that the reaction rate constants f1, f2, g1, g2,
and g3 are all expressed in units of s
−1
. For emerin nanodomain self-assembly to occur
through a Turing instability, these reaction rates, for a given, observed set of values of
the I and A diffusion coefficients νI and νA, must satisfy the mathematical constraints in
Eqs. (5.8), (5.9), (5.13), (5.14), and (5.24)–(5.27).
5.2.2 Numerical implementation
The reaction and diffusion processes considered in our model imply, via the Turing
mechanism for nonequilibrium pattern formation [140], a characteristic length scale ℓc,
which can be estimated through a linear stability analysis (see Sec. 5.1) [121] and which
we corroborate through numerical solutions of our reaction-diffusion equations. The corresponding characteristic diameter of emerin nanodomains, ℓΦ, is given by ℓΦ ≈ ℓc/2.
12
Similarly, the characteristic time scale for the self-assembly of emerin nanodomains, τ ,
can be estimated as τ ≈ ℓ
2
c
/νA, where νA < νI is the diffusion coefficient of the slower
molecule species (A complexes) considered in our model. However, due to uncertainties in our estimates of f1 and g1, which fix the time scales associated with I and A (see
Sec. 5.2), as well as the mean-field character of the formulation of our model [11], we
only attach limited physical significance to the values of τ obtained from our model calculations, and show τ for completeness.
For our numerical solutions of the reaction-diffusion Eqs. (5.1) and (5.2), we employed
the DifferentialEquations library in Julia [228, 229]. We compared numerical solutions
obtained with a range of solvers—including BS3, Tsit5, Runge-Kutta, and GMRES—and
found similar results. The numerical solutions shown here were obtained using the GMRES solver. We used periodic boundary conditions with a system size 400 × 400 nm2
,
which is significantly larger than the size of emerin nanodomains observed in experiments [12]. Smaller system sizes approaching the size of emerin nanodomains can yield
finite-size artifacts in the emerin patterns generated by our model.
The numerical solutions presented in (Sec. 5.3) were obtained from initial conditions
that were perturbed randomly about the homogeneous steady state of the system, (I, A) =
¯I, A¯
, setting the diffusion coefficients νI = νfast and νA = νslow (see Fig. 5.1 for data of
νfast and νslow for each system considered here), and utilizing the reaction rate constants
in Sec. 5.3 for each individual system discussed. Random perturbations were drawn from
a uniform distribution over [−0.0005, 0.0005]. Note that the reaction rates in our model fix
the homogeneous steady state (I, A) =
¯I, A¯
of the system (see Sec. 5.2.1) and were
chosen for each system considered here to satisfy the constraints for a Turing instability
1
3
FIG. 1. A-I reaction scheme in our model of the self-assembly of emerin nanodomains in the INM. In the left column, we
show the reactions that change the dimensionless I concentration (fractional area covered by I), which include a first-order
spontaneous decay reaction and a first-order creation reaction activated by A. In the right column, we show the reactions that
change the dimensionless A concentration (fractional area covered by A), which include first-order reactions analogous to those
in the left column (spontaneous decay and self-activation of A), as well as a second-order self-activation reaction. All reactions
locally increasing I or A include the steric repulsion factor (1 I A). Analogous steric factors are included in the di↵usion
terms for I and A. The constant concentrations (I,¯ A¯) are determined by the reaction rates and correspond to the homogeneous
steady state of the system, (I,A)=(I,¯ A¯).
molecule concentration at the INM, we include steric factors to prevent complex formation in crowded membrane
regions. This ensures that the sum of the dimensionless
I and A concentration fields, (I + A) with I = I(x, y, t)
and A = A(x, y, t), never exceeds 1 anywhere in the system (see Fig. S3). Thus, I(x, y, t) and A(x, y, t) represent
the fraction of free membrane area covered by I and A
complexes.
In Sec. SII, we present analytical predictions of
emerin concentrations (⇢slow) and relative densities
(hNslowi/hNfasti) in nanodomains compared to the surrounding areas, based on measured di↵usion properties
and experimental data. In Sec. SIII, we introduce the
reaction scheme (see Fig. S1) for our reaction-di↵usion
model, along with the estimated reaction rate constants
used for various emerin systems. Details and predictions
of these models applied to di↵erent emerin systems, including wild-type and mutant forms under di↵erent conditions, are summarized in Fig. ??, with corresponding
numerical solution snapshots shown in Fig. 2. Sec. SIV
outlines our numerical implementation approach, while
the remaining portions of this letter discuss our model’s
predictions of emerin nanodomain properties, namely
nanodomain diameter (`) and fractional area coverage
(F), adapting the model to explain observed changes
under force application and mutation.
III. RESULTS
A. Wild-type system under no mechanical stress
In unperturbed wild-type (“WT”) systems, our model
spontaneously self-assembles emerin nanodomains from
random initial conditions. These nanodomains have a
characteristic diameter ` ⇡ 22 nm in the steady state,
closely matching experimental observations (22 ± 11 nm
[1]). Our model also predicts that the fraction of available
emerin nanodomain area covered by I and A complexes
is F ⇡ 0.15. This is comparable to the mean fractional
area of emerin nanodomains covered by emerin, approximately 3–15%, suggested by previously published data [1]
and further experimental estimates (see Sec. SI). As expected, our numerical solutions yield elevated concentrations of A complexes compared to I complexes in emerin
nanodomains [Fig. ??(WT)].
Though our model’s output aligns closely with experimental findings in several ways, our model exhibits some
disparities with experiments which can be potentially explained. For instance, our model predicts F ⇡ 0.15
which lies within experimental margins for emerin area
coverage of nanodomains, but is on the high side. Since
we impose steric constraints so that 0 I + A 1,
if I and A were to cover the entire free INM area in
emerin nanodomains, we would have F = 1. Assuming the steric constraints on I and A are set by emerin
rather than other components, F = f corresponds to an
average fractional area coverage f of emerin in the nanodomain. Our numerical solutions yield f ⇡ 15% inside
Figure 5.1: Table comparing experimental data on emerin [12] (orange) and predictions
of our reaction-diffusion model (emerin nanodomain diameter, ℓΦ, and ratio of the fraction
of emerin nanodomain area covered by I and A complexes relative to that of the wild
type system under no mechanical stress, FΦ/F WT
Φ ) (red) for the various emerin systems
in Fig. 5.2 and the ∆95-99 system under no mechanical stress. For ∆95-99 systems
under no mechanical stress, emerin nanodomains were not observed to self-assemble
in experiments [12] and were not predicted to self-assemble by our model. So, for the
∆95-99 system under no mechanical stress, we specify “null" for FΦ/F WT
Φ and ℓΦ.
in Eqs. (5.8), (5.9), (5.13), (5.14), (5.24)–(5.27), with the exception of our solutions for
the ∆95-99 emerin system under no mechanical stress. Our model for the ∆95-99 emerin
system under no mechanical stress does not yield a Turing pattern due to not satisfying
the constraints for a Turing instability. In Sec. 5.2.1 we express the reaction scheme used
to obtain our numerical solutions in Sec. 5.3 in a form that, for mathematical convenience,
explicitly involves the homogeneous steady state (I, A) =
¯I, A¯
, with all reaction rates
having units of s
−1
. We used a 150 × 150 grid for all numerical solutions, and checked
that a finer grid produced similar results. The steady states solutions provided in Sec. 5.3
correspond to times t = 100τ in our numerical solutions.
To calculate the average joint concentration of (I + A) inside emerin nanodomains,
FΦ, we first find, for a given nanodomain, the grid point associated with the maximum of
(I + A) in the steady state of the system. We then average (I + A) over all grid points
12
within a radius ℓΦ/2, rounded to the nearest multiple of the grid spacing, about this (local)
maximum of (I + A). We repeat this algorithm for four additional emerin nanodomains
and average over five nanodomains to obtain FΦ. We found that this last step was, strictly
speaking, not necessary, with FΦ evaluated over a single nanodomain and FΦ evaluated
over multiple nanodomains yielding similar results.
5.3 Results
5.3.1 Wild-type system under no mechanical stress
For wild-type (“WT") systems under no mechanical stress, we find that our model
yields spontaneous self-assembly of emerin nanodomains from random initial conditions.
In the steady state of the system, we find that the emerin nanodomains obtained from our
model have a characteristic diameter ℓΦ ≈ 22 nm, in good agreement with 22 ± 11 nm in
diameter observed in experiments [12] [see Figs. 5.1 and 5.2(a)]. Thus, our model yields
spontaneous self-assembly of emerin nanodomains with a size that is consistent with
experimental observations. Our model also predicts that the fraction of available emerin
nanodomain area covered by I and A complexes is FΦ ≈ 0.15. This is comparable to
the mean fractional area of emerin nanodomains covered by emerin, approximately 3–
15%, suggested by previously published data [12] and further unpublished experimental
estimates [230]. As expected, our numerical solutions yield elevated concentrations of A
complexes compared to I complexes in emerin nanodomains [Fig. 5.2(a)].
127
Figure 5.2: Colormaps depict numerical solutions for A, I, and (A + I) (overlay) calculated from our emerin nanodomain self-assembly model (see Sec. 5.2.2 for numerical
implementation), where I and A denote fields for the fraction of the INM area locally
covered by I and A complexes, respectively. The colorbar scale indicates the values of
the fields A, I, or (A + I). We show model solutions for wild-type systems (a) with and
(b) without mechanical stress, (c) Q133H systems without mechanical stress, (d) P183H
systems without mechanical stress, and (e) ∆95-99 systems under mechanical stress.
Diffusion coefficients νI and νA are as indicated in Fig. 5.1 with νA = νslow and νI = νfast.
The reaction rates f1, f2, g1, g2, and g3 utilized in our calculations are discussed in (a)
Sec. 5.2, (b) Sec. 5.3.2, (c) Sec. 5.3.3, (d) Sec. 5.3.4, and (e) Sec. 5.3.6. The colormaps
depict numerical solutions at the corresponding time t = 100 τ (see Sec. 5.2.2), with (a)
τ ≈ 6 s, (b) τ ≈ 17 s, (c) τ ≈ 3 s, (d) τ ≈ 50 s, (e) τ ≈ 74 s.
128
Though our model’s output aligns closely with experimental findings in several ways,
our model exhibits some disparities with experiments which can be potentially explained.
For instance, our model predicts FΦ ≈ 0.15 which lies within experimental margins for
emerin area coverage of nanodomains, but is on the high side. Since we impose steric
constraints so that 0 ≤ I + A ≤ 1, if I and A were to cover the entire free INM area in
emerin nanodomains, we would have FΦ = 1. Assuming the steric constraints on I and
A are set by emerin rather than other components, FΦ = f corresponds to an average
fractional area coverage f of emerin in the nanodomain. Our numerical solutions yield
f ≈ 15% inside emerin nanodomains, while experiments suggest a range 3–15%, referring
to the fractional area coverage of emerin alone. This implies that steric constraints on I
and A in our model arise not only from emerin but also from other interacting molecule
species, supporting our assumption that I and A complexes involve both emerin and
NBPs.
Additionally, the model generates closely spaced emerin nanodomains, approximately
10 nm apart and consistently 22 nm in diameter. In contrast, experimental emerin nanodomains exhibit variations in size, shape, and spacing, often exceeding 10 nm between
them [12]. However, this discrepancy is unsurprising, since, by definition, our meanfield model inherently lacks the molecular noise present in actual molecular reactions and
diffusion [231]. In the INM, variations in nanodomain properties likely arise from heterogeneous INM composition involving molecule species other than emerin and NBPs and
fluctuations in NBP properties. We note that the I domains of our model have a broader
profile than the A domains they overlap, which is a characteristic of the Turing mechanism,
where the outward diffusing flow of the fast inhibitor I complexes balances the surge of
129
emerin complexes at the center of nanodomains and stabilizes the nanodomains. Due to
the close spacing between emerin nanodomains, the I domains exhibit linkage.
Lastly, our numerical solutions yield an area coverage of (I + A) of about 4% outside
emerin nanodomains. This is substantially higher than the expected mean local fractional
area covered by emerin outside emerin nanodomains (≲ 2%) [12, 230]. Here, again, we
note that our model generates closely spaced emerin nanodomains with an inter-domain
distance of around 10 nm, in contrast to experiments which often reveal larger spacings
between nanodomains, likely due to disruptions in the noisy INM environment [12]. So,
when comparing our model’s results on (I + A) outside emerin nanodomains to experimental findings, it’s important to focus on closely spaced nanodomains. Experimentally
observed emerin density maps, albeit somewhat blurred by rendering effects, suggest that
emerin density is notably higher between closely spaced emerin nanodomains. Together
with our model’s more stringent steric constraints on I and A, which tend to overestimate
the local emerin density, this could explain why our model shows somewhat elevated values of (I + A) outside emerin nanodomains compared to experimentally observed mean
emerin concentrations outside these nanodomains.
5.3.1.1 Crowding effects due to emerin monomers
Emerin monomers cover ∼ 0.1% of the INM [230]. To test the possible effect of
steric constraints due to emerin monomers on our model predictions, we accounted for a
“background” concentration of emerin monomers by including a modified steric repulsion
term (1 − I − A − m) in the reaction-diffusion dynamics considered here, with a constant
fractional area m = 0.002 = 0.2% of emerin monomers. Inserting m = 0.002 into our
130
model and leaving all other model parameters unchanged resulted in slightly smaller nanodomains, of diameter ℓΦ ≈ 21 nm as opposed to the value ℓΦ ≈ 22 nm obtained with
m = 0, and a slight increase in the nanodomain coverage by I and A complexes, from
FΦ ≈ 15% to FΦ ≈ 17%. Thus, as expected, steric effects due to emerin monomers seem
to have little impact on the self-assembly of emerin nanodomains in our model.
For completeness, we tested whether steric effects due to emerin monomers could be
effectively compensated by decreasing the creation rate of I (“crowder”) complexes, from
f2 = f1/2 to f2 = 0.44f1. These adjustments yielded results that were nearly identical to
those obtained with the corresponding system with m = 0 [see Fig. 5.1 and Fig. 5.2(a)].
Using values of m smaller or not much greater than m = 0.002 also yielded similar results
as in Fig. 5.2(a). By increasing m to values substantially greater than m = 0.002 (e.g.,
m = 0.004), which effectively leads to a crowding out of I and A complexes, a transition
to stripe patterns can be obtained in our model.
5.3.2 Wild-type system under mechanical stress
Wild-type systems under mechanical stress (“WT; force") were observed to exhibit
emerin nanodomains with an increased diameter 60 ± 12 nm, but a density of emerin
inside nanodomains that was reduced by about 40% compared to unperturbed wild-type
systems [12]. Interestingly, it was thus found that while the density of emerin in emerin
nanodomains is decreased by mechanical stress, the overall number of emerin molecules
in nanodomains is increased. Our model of emerin nanodomain self-assembly provides
a simple explanation of these observations.
131
Experiments suggest that, possibly due to a disruption in the interactions between
emerin and NBPs by mechanical force, the diffusion coefficients νA and νI are approximately doubled in the perturbed system compared to the unperturbed system [12] [see
Fig. 5.1]. In our model, this adjustment results in an increase in the diameter of emerin
nanodomains by about a factor of √
2 (∼ 40%)
†
, while the fractional area of emerin nanodomains occupied by I and A complexes remains unchanged. This suggests that the
observed changes in νI and νA can partially, but not completely, account for the observed
changes in emerin nanodomains under mechanical stress.
As noted above, experiments show that mechanical stress tends to decrease the
emerin density in nanodomains. A physical interpretation of this observation is that mechanical stress diminishes interactions between emerin and NBPs necessary for the formation of higher-order emerin structures. To test whether such a modification of the
reaction dynamics can explain the observed changes in nanodomain size, we decreased
the reaction rate g3, which parameterizes the formation of higher-order emerin structures
in our model, 2A + ∅ → 3A, by a factor of 1/2, to g3 = f1. As shown in Fig. 5.1 and
Fig. 5.2(b), this perturbation resulted in an increase in the characteristic emerin nanodomain diameter to ℓΦ ≈ 50 nm. Importantly, with this decrease in g3 we found that the
fractional area of emerin nanodomains occupied by I and A complexes was FΦ ≈ 0.1,
which amounts to FΦ/F WT
Φ ≈ 0.7
‡
. Thus, the observed increases in the emerin diffusion
coefficients together with a reduction in the effective rate for the formation of higher-order
†This can be understood by noting that ℓc in Eq. (5.15) scales with the square roots of νI and νA [121].
‡Using g3 = 0.9f1 yields ℓΦ ≈ 56 nm and FΦ/F WT
Φ ≈ 0.6, further aligning model outcomes with experimental observations.
132
emerin structures seem to underlie the observed changes in emerin nanodomains under
mechanical stress.
5.3.3 Q133H mutation
Experiments indicate that the EDMD-associated Q133H mutation of emerin decreases
emerin’s interactions with nuclear actin, resulting in an increased mobility of emerin as
well as increased formation of higher-order emerin structures [12, 136, 232–234]. It was
found experimentally that, under no mechanical stress, the Q133H mutation yields emerin
nanodomains of statistically similar size to the wild-type system under no mechanical
stress and leads to an increase in the density of emerin in nanodomains by approximately
50% compared to the wild-type system under no mechanical stress [12]. Our model can
account for these observations.
It was found in experiments that, for Q133H emerin, the rapidly- and slowly-diffusing
emerin populations have diffusion coefficients of approximately νfast = 3 × 10−3 µm2/s
and νslow = 4 × 10−4 µm2/s, respectively [12]. Proceeding as for wild-type emerin, we
thus set νA = νslow and νI = νfast for Q133H emerin. We find that these changes in the
diffusion coefficients of I and A complexes result in a slight increase in the diameter of
emerin nanodomains, by approximately§ 10%, with no appreciable change in the fractional
area of emerin nanodomains occupied by I and A complexes. Thus, the changes in the
diffusion properties observed for Q133H emerin do not account for the observed changes
in the overall properties of emerin nanodomains.
§Again, this can be understood from the scaling of ℓc in Eq. (5.15) with the diffusion coefficients νI and
νA [121].
133
We hypothesized that, possibly due to its decreased interactions with nuclear actin,
Q133H emerin has an increased propensity to interact with other Q133H emerin as well
as with NBPs to form higher-order structures. In our model, A complexes capture interactions between emerin and NBPs that can yield higher-order emerin structures, while I
complexes passively inhibit the formation of higher-order emerin structures through steric
constraints. We therefore assume that, compared to A complexes formed from wild-type
emerin, all reaction rates associated with A complexes formed from Q133H emerin are increased, including the reaction A+∅ → A+I, while reactions solely involving I complexes
are not affected. In the specific reaction scheme considered here (see Sec. 5.2), the reaction I → ∅ with rate f1 is the only reaction not directly involving A complexes. Thus, we
take f1 to be unaffected by the Q133H mutation while, for simplicity, we increase all other
reaction rates in the model—f2, g1, g2, and g3—by the same factor. Figs. 5.1 and 5.2(c)
shows the resulting model results obtained with an increase by 30% in the reaction rates
f2, g1, g2, and g3 compared to the wild-type system under no mechanical stress.
With the above assumptions, our model yields Q133H emerin nanodomains with ℓΦ ≈
18 nm [see Figs. 5.1 and Fig. 5.2(c)]. Furthermore, we find that FΦ ≈ 0.23, which amounts
to FΦ/F WT
Φ ≈ 1.5. These model results are in good agreement with the aforementioned
experiments on the Q133H emerin system [12].
5.3.4 P183H mutation
The EDMD-associated P183H mutation of emerin is thought to yield more pronounced
interactions between emerin and NBPs than wild-type emerin [131, 135, 136], which are
134
believed to slow down the diffusion of P183H emerin compared to wild-type emerin and
disrupt the formation of higher-order emerin complexes—possibly due to incorrect stoichiometries or arrangements of emerin and NBPs in supramolecular complexes. Experiments show that P183H emerin can form dimers that diffuse significantly faster than other
P183H oligomers [12]. P183H dimers seem to be unable to efficiently form higher-order
emerin structures [12, 127]. It was found experimentally that, under no mechanical stress,
the P183H mutation yields larger emerin nanodomains than the wild-type system under
no mechanical stress, with an emerin nanodomain diameter of approximately 35 ± 12 nm
(compared to approximately 22 ± 11 nm for the wild-type system under no mechanical
stress), and a reduction in the density of emerin in nanodomains by approximately 70%
compared to the wild-type system under no mechanical stress [12]. Thus, contrary to
the Q133H mutation, the P183H mutation of emerin increases the nanodomain size while
decreasing FΦ. Combining insights gleaned from the wild-type system under mechanical
stress (see Sec. 5.3.2) and Q133H emerin (see Sec. 5.3.3), our model is able to account
for these observations.
It was found in experiments that, for P183H emerin, the rapidly- and slowly-diffusing
emerin populations have diffusion coefficients of approximately νfast ≈ 1.5 × 10−3 µm2/s
and νslow ≈ 1×10−4 µm2/s, respectively [12]. On this basis, we set here νI = νfast and νA =
νslow. These changes to our model produced, compared to the wild-type system under
no mechanical stress, a decrease in ℓΦ by approximately¶ 50% with an increase in FΦ by
about|| 30% compared to the wild-type system under no mechanical stress. Thus, similarly
¶Again, this can be understood from the scaling of ℓc in Eq. (5.15) with the diffusion coefficients νI and
νA [121].
||This can be understood by noting that, compared to the wild-type system, the ratio νA/νI is decreased
in the P183H system, thus facilitating larger concentrations of A and, hence, I inside emerin nanodomains.
135
as for Q133H emerin (see Sec. 5.3.3), the changes in the diffusion properties observed
for P183H emerin do not account for the observed changes in the overall properties of
emerin nanodomains.
As mentioned above, experiments suggest that P183H emerin shows, compared to
wild-type emerin, a reduced ability to self-assemble into higher-order structures. Following similar reasoning as for Q133H emerin, we therefore decreased the rates associated
with all “activating” reactions in our model. In particular, we now decreased (rather than
increased) the rates f2, g1, g2, and g3 by 30% compared to the wild-type system under
no mechanical stress. This adjustment to our model produced emerin nanodomains that
were about 18 nm in diameter, and reduced FΦ by about 20% compared to the wild-type
system under no mechanical stress. Thus, such a global reduction in all rates associated
with reactions involving A complexes decreased, consistent with experimental observations, ⟨I + A⟩, but also decreased (rather than increased) the nanodomain size.
In analogy to the wild-type system under mechanical stress (see Sec. 5.3.2), we hypothesized that the P183H mutation has a more pronounced effect on the formation of
higher-order emerin structures. Decreasing g3 to g3 = f1/2, we found that the nanodomain diameter increased to ℓΦ ≈ 35 nm and FΦ/F WT
Φ decreased to about 0.4 [see
Figs. 5.1 and 5.2(d)]**. Thus, with the values of νI and νA measured for P183H emerin,
we are able to account for the aforementioned experiments on the P183H emerin system [12] via a reduction in the rates of all reactions involving A complexes, with a more
**If we use g3 = 0.6f1 rather than g3 = f1/2 and decrease f2, g1, and g2 by 35% rather than 30%
compared to the wild-type system under no mechanical stress, our model produces P183H emerin nanodomains with ℓΦ ≈ 32 nm and FΦ/F WT
Φ ≈ 0.3, further improving the agreement between model results
and experimental observations.
136
pronounced reduction in the reaction rate associated with the formation of higher-order
emerin complexes.
5.3.4.1 Spontaneous dimerization
P183H emerin has a higher propensity to dimerize than wild-type emerin. In particular,
P183H emerin can dimerize even at the outer nuclear membrane in the absence of emerin
nanodomains [12]. This observation suggests that P183H dimers, which are represented
by I complexes in our model, can form spontaneously in the absence of A complexes. To
test to what extent our model predictions change if one allows for the spontaneous formation of dimers, we extended our reaction scheme in Sec. 5.2.1 to allow for the reaction
∅ → I with reaction rate f0 in units of s
−1
. This modifies F to
F (I, A) = f0
¯I
1 − I − A
1 − ¯I − A¯
− f1I + f2 (1 − I − A) A (5.28)
with
A¯ =
(f1 − f0) (g2 − g1 + g3)
g2 (f1 − f0) + f2 (g1 − g3)
,
¯I =
f2 (g1 − g3)
g2 (f1 − f0)
A . ¯ (5.29)
We found that such a modified reaction scheme can yield similar results for P183H emerin
as described above. For instance, setting f0 = 0.1f1, decreasing f2 from f2 = 0.35f1 to
f2 = 0.25f1 to compensate for the additional creation of I complexes through the reaction
∅ → I, and setting g3 = 0.6f1 rather than g3 = f1/2 produces results similar to those in
Figs. 5.1 and 5.2(d).
137
5.3.5 ∆95-99 mutant system under no mechanical stress
Experiments indicate that the EDMD-associated ∆95-99 mutation of emerin yields an
approximately random emerin distribution across the INM at no mechanical stress, with
little or no domain formation and ∆95-99 emerin covering a fractional area ≲ 2% of the
INM [12]. ∆95-99 emerin is thought to exhibit reduced interactions with some NBPs (e.g.,
lamin A/C and/or nuclear actin) [128, 131, 135]. This conclusion is also supported by
experiments on wild-type emerin, which show that depletion of NBPs results in impaired
formation of higher-order emerin structures in the INM [12]. Based on our results on
P183H emerin (see Sec 5.3.4), our model is able to account for these observations.
It was found in experiments that the mobility of ∆95-99 emerin on the INM is reduced by
approximately a factor of two compared to wild-type emerin under no mechanical stress,
νfast ≈ 1 × 10−3 µm2/s and νslow ≈ 1.5 × 10−4 µm2/s [12] (see Fig. 5.1). We thus set
νI = νfast and νA = νslow in our model. This adjustment to our model results in a decrease
in the diameter of emerin nanodomains by about†† 30%, while FΦ is decreased by about
20% compared to the wild-type system. Thus, the observed decrease in νA and νI is
not sufficient to account for the severely impaired self-assembly of emerin nanodomains
observed for the ∆95-99 mutation of emerin.
As mentioned above, experiments suggest that ∆95-99 emerin has a decreased propensity to self-assemble into higher-order structures. Following similar reasoning as for
P183H emerin, we therefore decreased the rates associated with all “activating” reactions
in our model. Interestingly we find that, for the diffusion coefficients measured for ∆95-99
††Again, this can be understood from the scaling of ℓc in Eq. (5.15) with the diffusion coefficients νI and
νA [121].
138
emerin, a decrease in the rates f2, g1, g2, and g3 by as little as 20% compared to the unperturbed wild-type emerin system leads to a destabilization of emerin nanodomains and
results, in our mean-field model, in a homogeneous steady state. In particular, proceeding as for P183H emerin and decreasing f2, g1, and g2 by 30% while decreasing g3 from
g3 = 2f1 to g3 = f1/2 (see Sec 5.3.4) resulted in a homogeneous steady state of the system with ⟨I + A⟩ ≈ 2%. As discussed above, we expect the value of ⟨I + A⟩ obtained from
our model to yield an upper bound on the fractional area coverage of emerin, because I
and A involve not only emerin but also NBPs. Our model results for ∆95-99 emerin under
no mechanical stress are therefore consistent with experimental observations [12].
5.3.6 ∆95-99 mutant system under mechanical stress
Experiments indicate that application of mechanical stress to the ∆95-99 emerin system (“∆95-99; force") results in the self-assembly of emerin nanodomains with approximate diameter 75 ± 20 nm and emerin densities about 25% compared to the unperturbed
wild-type emerin system [12]. Based on the wild-type emerin system under mechanical
stress (see Sec. 5.3.2) we modified our model of the unperturbed ∆95-99 emerin system
(see Sec. 5.3.5) to capture the mechanically stressed state of the ∆95-99 emerin system.
From experimental observations of the perturbed state of the wild-type emerin system, we expect the mobility of ∆95-99 emerin to increase when the system is put under
mechanical stress [12]. However, it was initially not clear how exactly mechanical perturbation of the ∆95-99 system impacts the values of νI and νA. Assuming that A complexes formed from ∆95-99 emerin respond similarly to mechanical stress as A complexes
139
formed from wild-type emerin, we double νA to νA = 3 × 10−4 µm2/s. Furthermore, we
hypothesize that the diffusion rate of I complexes may respond more strongly to mechanical stress than the diffusion rate of A complexes‡‡—this could, for instance, be the case if
the combined effects of the ∆95-99 mutation and application of mechanical stress largely
decouple I complexes from their potential binding partners—as indicated by the large diameter of the nanodomains observed (75±20 nm), since ℓc depends on √
νI . We find that
with, for instance, νI = 6 × 10−3 µm2/s, ∆95-99 emerin self-assemble into nanodomains
with ℓΦ ≈ 53 nm and FΦ/F WT
Φ ≈ 0.5.
These results are broadly consistent with published experimental data on the ∆95-99
emerin system under mechanical stress [12]. Closer agreement between model results
and experimental observations can be obtained by assuming, in analogy to the wild-type
emerin system under mechanical stress (see Sec. 5.3.2), that mechanical stress also
modifies the reaction rate for 2A + ∅ → 3A in the ∆95-99 system. For instance, if we
decrease g3 from g3 = f1/2 to g3 = 0.35f1 we find ∆95-99 emerin nanodomains with
ℓΦ ≈ 74 nm and FΦ/F WT
Φ ≈ 0.4 [see Figs. 5.1 and 5.2(e)]§§
.
Interestingly, the diffusion coefficients of the ∆95-99 emerin system have recently been
measured [230] after we had made our initial predictions, and found them to be νfast ≈
6 × 10−3 µm2/s and νslow ≈ 4 × 10−4 µm2/s, which agree quite close with our predictions.
We found if we change νA = 3 × 10−4 µm2/s to νA = 4 × 10−4 µm2/s to better align with
these new measurements, and do not decrease g3 to g3 = 0.35f1, we get similar results
‡‡Simply doubling νI to νI = 2 × 10−3 µm2/s in our model, yields a homogeneous steady state of I
and A complexes, which does not align with experimental observations that mechanical perturbation of the
∆95-99 emerin system evokes the formation of nanodomains [12].
§§Decreasing g3 to g3 = 0.34f1, rather than to g3 = 0.35f1, yields nanodomains with ℓΦ ≈ 77 nm and
FΦ/F WT
Φ ≈ 0.3, further improving the agreement between model results and experimental observations.
140
as described above, with ℓΦ ≈ 71 nm and FΦ/F WT
Φ ≈ 0.4. Since our initial prediction for νI
was based on the nanodomain diameter, these results further support a direct correlation
between emerin nanodomain properties and emerin reaction-diffusion characteristics.
141
Chapter 6
Overview and conclusions
This chapter offers summaries and conclusions for each of Chapters 2 through 5 and
discusses potential future research directions suggested by our findings.
6.1 Overview and conclusions of Chapter 2
In Chapter 2 we introduced a novel BVM for lipid bilayer deformations based on the
elasticity theory of protein-induced lipid bilayer thickness deformations [21–25, 27–34,
36–40]. In Sec. 2.1, we discussed the bilayer thickness deformation energy, in Eq. 2.3
and demonstrated how we incorporate the hydrophobic shape parameters of proteins in
the boundary conditions in Eqs. 2.12–2.14 to fix the lipid bilayer thickness deformation
energy. In Sec. 2.2 we described in detail our BVM for bilayer thickness deformations,
tested it against FEM solutions, and discussed how the BVM can be used to calculate
protein-induced lipid bilayer thickness deformations and their associated elastic energy
for general protein shapes. We also introduced a measure of accuracy [see Eq. (2.25)
142
in Sec. 2.2.1], and APDs to optimize the efficiency and accuracy [see Sec. 2.2.2] of our
BVM.
Our BVM permits the construction of analytic series solutions of protein-induced lipid
bilayer deformations for arbitrarily large deviations from a circular protein cross section,
albeit can become computationally limited by the available floating point precision of numbers utilized (see Appendix A.2). In addition to the membrane protein cross section, our
BVM allows for a breaking of rotational symmetry about the protein center through angular
variations in the boundary conditions along the bilayer-protein interface—in particular, for
the scenarios considered in Chapters 2 and 3, in the protein hydrophobic thickness and in
the bilayer-protein contact slope along the bilayer-protein boundary. Our BVM reproduces
available analytic solutions for membrane proteins with circular cross section [23, 27, 30,
58, 59, 118] and yields, for membrane proteins with non-circular cross section, excellent
agreement with numerical, finite element solutions.
A limitation of the BVM arises for protein shapes that show extreme deviations from
circular symmetry, in which case BVM solutions tend to involve a large number of terms
and, hence, become increasingly intractable. In such cases it may be advisable to modify the APD method for the distribution of boundary points employed in Chapters 2–4,
so as to reduce the number of terms required in the lipid bilayer thickness deformation
field in Eq. (2.20) with Eq. (2.21). While we have focused here on bilayer thickness deformations, it would be interesting to use a BVM approach analogous to that employed
in Chapters 2–4 to construct analytic series solutions for other modes of protein-induced
lipid bilayer deformations such as, for instance, bilayer midplane or lipid tilt deformations
[22, 32–38, 54–56, 107, 153, 154, 158–161, 163, 186, 187]. On this basis one could,
143
for instance, further investigate how anisotropic membrane protein shapes can give rise
to anisotropic membrane elastic properties [54, 55]. Furthermore, it would be interesting
to construct BVM solutions for membrane proteins embedded in bilayers with heterogeneous lipid composition [162, 235–239]. Notably, for calculating long-ranged lipid bilayer
deformations due to proteins of non-circular cross-section shape, the BVM may offer a
convenient and computationally efficient approach. In particular, as membrane tension
approaches zero, leading to theoretically infinite decay lengths for midplane deformations, numerical techniques like FDM and FEM can become computationally infeasible,
whereas the BVM remains viable.
6.2 Overview and conclusions of Chapter 3
In Chapter 3 we investigated the significance of protein shape in protein-induced lipid
bilayer thickness deformations. Based on the BVM solutions, in Sec. 3.1, we formulated
a simple analytic approximation of the lipid bilayer thickness deformation energy associated with general protein shapes [see Eq. (3.1) with Eq. (3.2)]. Through our BVM and
analytic approximation of the lipid bilayer thickness deformation energy, we surveyed the
dependence of protein-induced lipid bilayer thickness deformations on protein shape in
Sec. 3.2. We then applied our BVM and analytic approximation to investigate the impact
of protein shape in the assembly of transmembrane protein oligomers [see Sec. 3.3.1]
and transitions in transmembrane protein conformational states [see Sec. 3.3.2]. Here
we provide the conclusions of studies in regards to the impact of protein shape on lipid
bilayer deformations.
144
We find that, for modest deviations from rotational symmetry, our analytic approximation of the lipid bilayer thickness deformation energy is in good agreement with BVM solutions. These results suggest that, to a first approximation, the effect of membrane protein
shape on the energy of bilayer thickness deformations can be understood based on the
length of the circumference of non-circular protein cross sections. Moreover, our survey of
the dependence of protein-induced lipid bilayer thickness deformations on protein shape
reveals that protein shape tends to have a large effect on the energy of protein-induced
lipid bilayer thickness deformations, typically shifting the bilayer deformation energy by
more than 10 kBT.
In the case of non-circular protein cross sections, we find that protein self-interactions
provide an important motif for the energy of protein-induced lipid bilayer thickness deformations. Such self-interactions arise for invaginations in the protein cross section, from
overlaps in the bilayer deformations induced at different sections of the bilayer-protein
interface. The basic phenomenology of membrane protein self-interactions can be understood by drawing analogies with bilayer-thickness-mediated interactions between proteins
[22, 24, 25, 30, 41, 57, 58, 60, 61, 157, 161, 178–183]. In particular, membrane protein
self-interactions can effectively lower the energy cost of protein-induced lipid bilayer thickness deformations for proteins with constant bilayer-protein hydrophobic mismatch and
zero bilayer-protein contact slope. For non-zero bilayer-protein contact slopes, or for variations in the bilayer-protein hydrophobic mismatch or in the bilayer-protein contact slope
along the bilayer-protein interface, protein self-interactions can yield dramatic shifts in the
bilayer thickness deformation energy. Thus, the interplay between the cross-sectional
shape of membrane proteins, protein hydrophobic thickness, and bilayer-protein contact
145
slope yields a rich energy landscape of protein-induced lipid bilayer thickness deformations. Interestingly, the hydrophobic thickness or bilayer-protein contact slope of membrane proteins may be modified in cells through, for instance, protein mutations, changes
in lipid composition, or the binding of peptides at the bilayer-protein interface, while protein
oligomerization and transitions in protein conformational state tend to change the crosssectional shape of membrane proteins. The results described here therefore suggest
general physical mechanisms for how protein shape couples to the function, regulation,
and organization of membrane proteins.
6.3 Overview and conclusions of Chapter 4
In Chapter 4 we introduced a simple model of the effect of temperature changes
on protein-induced elastic bilayer deformations to explore the intricate relationship between temperature, membrane mechanics, and the activation energies of key sensory
proteins. In Sec. 4.1 we described, based on experimental measurements, how temperature changes modify key bilayer mechanical properties. In Sec. 4.2 we developed, based
on previous work [27, 57, 59, 102, 107, 110, 111], membrane-mechanical models of transitions in the conformational states of bacterial chemoreceptor trimers, MscL ion channels, and Piezo ion channels. In Sec. 4.3 we combined the methodologies developed in
Secs. 3.1 and 4.2 to quantify the effect of temperature changes on the conformational
states of chemroeceptors, MscL, and Piezo. In Sec. 4.4 we discussed the implications of
our finding for chemoreceptor thermotaxis and tempertaure-dependent shifts in the gating
thresholds of MscL and Piezo.
146
Temperature, being a fundamental environmental factor, significantly influences cellular physiology, impacting decision-making processes in microorganisms and the response
of multicellular organisms. Our research sought to understand how alterations in membrane mechanical properties induced by temperature changes can impact the activation
energies of pivotal proteins involved in cellular perception. Our findings highlight several
critical aspects of the interplay between temperature and membrane protein conformational state. Most notably, we found that temperature variations can induce substantial
changes in the energy cost of protein-induced lipid bilayer deformations, with magnitudes
on the order of several kBTrm to tens of kBTrm. These results suggest a role for membrane
elastic properties in bacterial thermosensation.
Our findings suggest that the effect of increasing temperature on lipid bilayer deformations may be sufficient for the activation of chemoreceptors in E. coli, which aligns with
their observed thermophilic response [13]. We also found changes in temperature may
produce significant shifts in the activation energy of MscL through changes in membrane
thickness. These results are based on measurements of the temperature dependence
of the membrane hydrophobic thickness of DOPC lipid bilayers, representing synthetic
and pure lipid compositions [116]. However, biological membranes generally have highly
heterogeneous compositions. Interestingly, measurements of the Bacillus subtilis membrane thickness dependence on temperature have revealed a decrease rate in membrane
thickness that is three-fold faster than that used in our model based on measurements of
DOPC bilayers [15, 116]. This observation suggests that our calculations for estimating
the effect of changes in temperature on chemoreceptors and MscL in E. coli cytoplasmic
membranes are conservative. Thus we may expect the functionality of chemoreceptors
147
and MscL to have a stronger dependence on temperature through membrane mechanics
than that suggested by our results.
We have predicted that Piezo’s gating tension increases with decreasing temperature,
which we can trace back to the observed increase in lipid bilayer bending rigidity with
decreasing temperature. Interestingly, these results align with experimental observations
[108, 226] on the dependence of Piezo gating on temperature and bilayer bending rigidity, suggesting that membrane rigidity may provide a membrane property through which
proteins can sense variations in temperature.
The results of our investigation in Chapter 4 indicates that certain transmembrane
proteins can detect variations in temperature through alterations in membrane thickness.
Interestingly, there is substantial evidence suggesting that DesK, a transmembrane protein known for its temperature-sensing ability, gauges temperature changes by monitoring
variations in membrane thickness [15, 43, 44, 46, 47, 70–79, 94, 95, 98–100, 240, 241].
Thus, it would be intriguing to conduct a quantitative analysis, akin to our investigations
on chemoreceptors and MscL, to assess the influence of temperature-induced membrane
thickness variations on DesK’s activation energy.
Beyond our focus on MscL and Piezo, the realm of thermosensation encompasses
other mechanosensors with intriguing temperature-sensitive attributes. Among these,
TREK/TRAK channels have emerged as notable examples. These mammalian channels are known to be gated by membrane tension and several have been shown to be
activated either through increases in temperature or through decreases in temperature
[48, 83, 84, 96, 97, 242–244]. These observations highlight the multifaceted nature of
mechanosensors, which can also double as thermosensors, suggesting that the interplay
148
between membrane mechanics and temperature sensitivity extends beyond the proteins
we explored in this study [48].
Another well-known group of thermosensors is the Transient Receptor Potential (TRP)
channels which also double as chemoreceptors in mammalian cells [16, 52, 80–82]. In
recent years, mounting evidence has supported the hypothesis that TRPs are intrinsically
activated by temperature, owing to a temperature-dependent protein molar heat capacity mechanism and, perhaps, also contributions due to bilayer-protein interactions [52,
195, 196]. Structural studies suggest significant conformational changes in TRPs upon
activation [51, 53, 195–206, 245, 246], aligning with the principles of thermodynamics,
which predict that substantial changes in protein molar heat capacities are associated
with substantial structural changes.
In addition to temperature-sensing through thermodynamic properties intrinsic to TRP
proteins, there is evidence suggesting that membrane mechanics may play a secondary
role in temperature sensing in TRP channels. For instance, experiments have shown that
alterations in membrane composition can influence the temperature activation thresholds
of TRPs [196, 245]. Notably, enriching the membrane with cholesterol leads to significant
increases in the activation temperature in TRVP1, causing shifts of several degrees Celsius [196]. In these experiments, it was reported that the cholesterol enrichment of the cell
membrane resulted in an increase in membrane stiffness and a decrease in membrane
fluidity. So the observed increase in the activation temperature in TRVP1 due to cholesterol enrichment may be attributed to an increase in membrane rigidity and a decrease
in membrane fluidity. In this context and in light of the results described in Chapter 4,
149
it would be interesting to see experiments measuring the effect of changing membrane
thickness on TRP activation.
We acknowledge the complex nature of temperature sensing mechanisms in cells—in
particular, experimental difficulties associated with isolating temperature sensing mechanisms from the effects of temperature on other physiological parameters, and the need
to devise experimental techniques for this purpose. Perhaps, in analogy to studies on
mechanosensitive ion channels [14, 22], one potential experimental approach for assessing whether bilayer mechanics plays a role in thermosensing could involve compensating
for temperature-induced changes in bilayer mechanical properties by altering the lipid
composition. For example, according to the results obtained in Chapter 4, chemoreceptor
trimers are increasingly biased towards the “on" state with increasing temperature. This
prediction critically depends on the observed decrease in lipid bilayer thickness with increasing temperature. It might thus be possible to test the role of bilayer mechanics in the
thermosensitive behavior of chemoreceptors by counterbalancing temperature-induced
variations in lipid bilayer thickness through adjustments in lipid tail length.
Overall, we find here that changes in membrane mechanical properties—in particular, lipid bilayer thickness and rigidity—can substantially impact the activation energies of
various protein sensors. Our calculations, which utilized a simple temperature-dependent
membrane mechanical model, underscore the potential significance of membrane mechanical properties as crucial contributors to the temperature sensing abilities of key sensory proteins, including chemoreceptors, MscL, and Piezo. Thus, our work suggests that
the elastic coupling of lipid bilayer properties and membrane protein conformational state
150
may provide a generic physical mechanism for temperature sensing through membrane
mechanics.
6.4 Overview and conclusions of Chapter 5
In Chapter 5, we investigated the mechanisms governing nuclear adaptation to mechanical stress, with a specific focus on the critical role played by emerin proteins and
protein mutations associated with EDMD. We also examined the implications of EDMD
mutations on this process. Our work led to the development of a simple framework for
understanding the self-assembly and stabilization of emerin nanodomains at the INM. In
Sec. 5.1 we introduced of the general mathematical structure of the reaction-diffusion
equations used here, and carried out a linear stability analysis of these equations to identify the conditions leading to Turing patterns in our model. In Sec. 5.2, we developed
in detail our physical model of emerin nanodomain self-assembly. In Sec. 5.3, we combined experimental measurements and simple estimates of the reaction and diffusion
parameters in our model to predict emerin nanodomain formation, and compared these
predictions to experimental observations.
Our model of emerin nanodomain self-assembly links emerin’s diffusion and reaction
characteristics to key physical attributes of emerin nanodomains, such as their size and
fractional area coverd by emerin. We were able to show that our model accurately predicts
the emerin nandomain size and emerin fractional area coverage for a variety of experimental conditions corresponding to mutations and changes in mechanical stress based
on observations of changes in emerin reaction and diffusion properties. Furthermore, we
151
were also able to show that our model accurately predicts the diffusion coefficients of the
∆95-99 mutant emerin system under mechanical stress based on the nanodomain diameter and the wild-type emerin system’s response to mechanical stress. This model also
explained how these properties change in response to mechanical stress and emerin mutations. Our model suggests that emerin nanodomain self-assembly is rooted in a Turing
instability exhibited by the two distinct emerin populations observed at INM: a rapidlydiffusing emerin population that locally inhibits increases in the emerin concentration at
the membrane through steric repulsion, and slowly-diffusing emerin population locally
binds emerin and NBPs to further increase the emerin concentration at memrbane locations with elevated emerin concentration. Our results emphasize the critical role of
rapidly-diffusing emerin in the self-assembly of stable emerin nanodomains. The work
described in Chapter 5 thus helps to illuminate the fundamental mechanisms underlying
the formation and physical characteristics of emerin nanodomains, offering insight into
nuclear adaptation to mechanical perturbations and key features of EDMD-associated
mutations or emerin.
152
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177
Appendices
A Supplemental material for Chapter 2
A.1 Computational implementation of the boundary value method
The general solution for the bilayer thickness deformation field in Eq. (2.20) with
Eq. (2.21) involves modified Bessel functions of the second kind, Kn, of any order n [163].
Kn (x) can vary rapidly with x, leading to numerical overflow and large round-off (floating
point) errors [173]. These numerical issues are compounded by error propagation in the
arithmetic operations necessary for solving the linear system of equations imposing the
bilayer-protein boundary conditions in the BVM [173, 174]. In particular, if the values of
the matrix A in Eq. (2.24) vary over many orders of magnitude, which is typically the case
for the scenarios considered here, the resulting propagation of floating point errors can be
catastrophic. These problems are ameliorated through the APD method, which effectively
reduces the number of terms required in the series in Eq. (2.20) with Eq. (2.21), as well
as LU decomposition with partial pivoting of A [173, 174], which reduces the pairing of
matrix elements that differ over many orders of magnitude.
178
We solved the linear system of equations in Eq. (2.24) in C++ using F. Johansson’s
arbitrary precision library for C/C++, Arb [175], which includes built-in functions for LU
decomposition with partial pivoting. Importantly, Arb also includes Bessel functions with
support for complex arguments. The linear system of equations in Eq. (2.24) encompasses 4N + 2 independent equations. As N is increased in Eq. (2.20) with Eq. (2.21),
solving Eq. (2.24) therefore becomes increasingly intensive from a computational perspective. To improve the computational efficiency of our calculations, we use OpenMP
multi-threading [177] to spread computations across multiple CPU cores.
As discussed in Sec. 2.2, the APD method involves the gap factor Ω in Eq. (2.28),
which we optimized so that the boundary error ηb
′ ≤ 0.1% in Eq. (2.25) and we obtained
changes in G and ηb
′ of no more than 10−5% as the numerical precision was increased.
A suitable choice for Ω thus allows construction of accurate solutions through Eq. (2.20)
with Eq. (2.21) at lower orders N, thus improving the numerical performance of the BVM.
For example, Fig. A.1 shows ηb
′ as a function of Ω for clover-leaf protein shapes with
symmetries s = 1 [see Fig. A.1(a)] and s = 3 [see Fig. A.1(b)], with constant U and
U
′ along the bilayer-protein interface. The solutions in Fig. A.1 were computed at the
indicated orders N in Eq. (2.20) with Eq. (2.21). Figure A.1 illustrates how the optimal
gap factor Ω converges with increasing N. For clover-leaf protein shapes we generally
find that the optimal Ω increases with increasing ϵ. For the scenarios considered here we
also find that, for a given N in Eq. (2.20) with Eq. (2.21) and shape of the protein cross
section, the optimal Ω changes only weakly if one allows for variations in U or U
′ along
the bilayer-protein interface.
179
20
FIG. 18. Boundary error ⌘b0 in Eq. (25) in BVM calculations
(see Sec. III) for clover-leaf protein shapes with (a) s = 1 and
✏ = 0.54 in Eq. (17) and (b) s = 3 and ✏ = 0.38 in Eq. (17) as
a function of the gap factor ⌦ in Eq. (28). We set R ⇡ 2.3 nm,
U = 0.3 nm, and U0 = 0. For ease of comparison we used the
indicated, fixed values of N in Eq. (20) with Eq. (21).
meet the error tolerances on ⌘b0 in Eq. (25). A suitable
choice for ⌦ allows construction of accurate solutions in
Eq. (20) with Eq. (21) at lower orders N, thus improving the numerical performance of the BVM. For example,
Fig. 18 shows ⌘b0 as a function of ⌦ for clover-leaf protein shapes with symmetries s = 1 [see Fig. 18(a)] and
s = 3 [see Fig. 18(b)], with constant U and U0 along the
bilayer-protein interface. The solutions in Fig. 18 were
computed at the indicated orders N in Eq. (20) with
Eq. (21). Figure 18 illustrates how the optimal gap factor ⌦ converges with increasing N. We generally find
for clover-leaf protein shapes that the optimal choice for
⌦ increases with increasing ✏. When U and U0 are allowed to vary along the protein-bilayer interface, we find
that the optimal choice of ⌦ and the corresponding ⌘b0
for a given N in Eq. (20) and protein cross-section shape
C(✓) in Eq. (17) or Eq. (18) with Eq. (19) has a weak
dependence on the assigned profiles of U(✓) and U0
(✓).
20
FIG. 19: (a) Bilayer thickness deformation energy G in
Eq. (26) calculated using the BVM (see Sec. III) and (b)
corresponding boundary error ⌘b0 in Eq. (25) for clover-leaf
protein shapes as a function of the bit precision used in the
numerical computations. We used the indicated values of s
and ✏ in Eq. (17), and R ⇡ 2.3 nm, U = 0.3 nm, and U0 = 0.
For ease of comparison we employed for s = 1 the fixed values
N = 20 in Eq. (20) with Eq. (21) and ⌦ = 0.726 in Eq. (28),
N = 48 and ⌦ = 0.62 for s = 2, N = 72 and ⌦ = 0.52 for
s = 3, N = 90 and ⌦ = 0.552 For s = 4, and N = 125 and
⌦ = 0.45 for s = 5.
same symmetries s with larger ✏ would require greater
numerical precision than double precision in the computational implementation of the BVM used here.
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FIG. 19. (a) Bilayer thickness deformation energy G in
Eq. (26) calculated using the BVM (see Sec. III) and (b)
corresponding boundary error ⌘b0 in Eq. (25) for clover-leaf
protein shapes as a function of the bit precision used in the
numerical computations. We used the indicated values of s
and ✏ in Eq. (17), and R ⇡ 2.3 nm, U = 0.3 nm, and U0 = 0.
For ease of comparison we employed for s = 1 the fixed values
N = 20 in Eq. (20) with Eq. (21) and ⌦ = 0.726 in Eq. (28),
N = 48 and ⌦ = 0.62 for s = 2, N = 72 and ⌦ = 0.52 for
s = 3, N = 90 and ⌦ = 0.552 For s = 4, and N = 125 and
⌦ = 0.45 for s = 5.
APPENDIX B: Protein-induced deformation energy
calculations in double precision
For some of the numerical calculations of the bilayer
thickness deformation energy G in Eq. (26) presented
in this work, we used numbers with substantially more
precision than double precision (64 bits) to meet the
numerical precision criteria of the boundary error ⌘b0 in
Eq. (25) described in Sec. III, with changes in G and
⌘b0 of no more than 105% as the numerical precision
of numbers is increased. However, many computing
languages do not have built in support for numerical
precision beyond double precision. In Fig. 19, we plotted
G in panel (a) and ⌘b0 in panel (b) as functions of
bit precision for several clover symmetries s and the
indicated values of ✏ to illustrate rough upper bounds on
✏ for these symmetries s, beyond which clover protein
cross-section shapes of the same symmetries s with
larger ✏ would require greater numerical precision than
double precision in the computational implementation
of the BVM used here. In Fig. 19, the G-curves in panel
(a) and ⌘b0 -curves in panel (b) were calculated with
Figure A.1: Boundary error ηb
′ in Eq. (2.25) in BVM calculations (see Sec. 2.2) for cloverleaf protein shapes with (a) s = 1 and ϵ = 0.54 in Eq. (2.17) and (b) s = 3 and ϵ = 0.38 in
Eq. (2.17) as a function of the gap factor Ω in Eq. (2.28). We set R ≈ 2.3 nm, U = 0.3 nm,
and U
′ = 0. For ease of comparison we used, for each curve, the indicated, fixed values
of N in Eq. (2.20) with Eq. (2.21).
A.2 Numerical precision
For the numerical calculations of the lipid bilayer thickness deformation energy G in
Eq. (2.26) presented in this thesis, we generally used numbers with precision (substantially) greater than double precision (64 bits) [175], so as to meet the numerical precision
criteria described in Sec. 2.2 with the boundary error ηb
′ ≤ 0.1% in Eq. (2.25) and changes
180
in G and ηb
′ of no more than 10−5% as the numerical precision is increased (see also Appendix A.1). However, many programming languages do not have built-in support for
numerical precision greater than double precision. To illustrate the extent to which double precision calculations could be used to approximate the BVM results described here,
we plot in Fig. A.2 the bilayer thickness deformation energy G [see Fig. A.2(a)] and the
corresponding boundary error ηb
′ [see Fig. A.2(b)] versus bit precision for several cloverleaf protein symmetries s and the indicated values of ϵ. As described in Appendix A.1,
the results in Fig. A.2 were obtained with F. Johansson’s arbitrary precision library for
C/C++, Arb [175]. We have ηb
′ ≤ 0.1% in Fig. A.2 as the floating point precision is increased beyond double precision, with changes in G and ηb
′ of no more than 10−5%. For
the clover-leaf protein shapes considered in this thesis, we generally find that numerical
precision greater than double precision is required for large s or large ϵ. For the polygonal
protein shapes considered in this thesis, we find that a numerical precision greater than
double precision is generally required to meet the numerical precision criteria described
in Sec. 2.2.
181
20
APPENDIX A: Computational implementation of
the boundary value method
The general solution for the bilayer thickness deformation field in Eq. (20) with Eq. (21) involves modified
Bessel functions of the second kind, Kn, of any order
n [43]. Kn(x) can vary rapidly with x, leading to numerical overflow and large round-o↵ (floating point) errors [53]. These numerical issues are further compounded
by error propagation in the arithmetic operations necessary for solving the linear systems of equations relevant
for the BVM [53, 54]. If the values of the matrix A
in Eq. (24) vary over many orders of magnitude, which
is typically the case for the scenarios considered here,
the resulting propagation of floating point errors can be
catastrophic. These problems are ameliorated through
the APD method, which e↵ectively reduces the number
of terms required in the series in Eq. (20) with Eq. (21),
as well as LU decomposition with partial pivoting of A
[53, 54], which reduces the pairing of matrix elements
that di↵er over many orders of magnitude.
We solved the linear system of equations in Eq. (24)
in C++ using F. Johansson’s arbitrary precision library
for C/C++, Arb [55], which includes built-in functions
for LU decomposition with partial pivoting. Importantly,
Arb also includes Bessel functions with support for complex arguments. The linear system in Eq. (24) encompasses Q = 4N + 2 independent equations. When Q is
large, solving Eq. (24) is computationally intensive. To
improve the computational eciency of our calculations,
we use OpenMP multi-threading [57] to spread computations across multiple CPU cores.
As discussed in Sec. III, the APD method involves the
gap factor ⌦ in Eq. (28), which we optimized so as to
meet the error tolerances on ⌘b0 in Eq. (25). A suitable
choice for ⌦ allows construction of accurate solutions in
Eq. (20) with Eq. (21) at lower orders N, thus improving the numerical performance of the BVM. For example,
Fig. 18 shows ⌘b0 as a function of ⌦ for clover-leaf protein shapes with symmetries s = 1 [see Fig. 18(a)] and
s = 3 [see Fig. 18(b)], with constant U and U0 along the
bilayer-protein interface. The solutions in Fig. 18 were
computed at the indicated orders N in Eq. (20) with
Eq. (21). Figure 18 illustrates how the optimal gap factor ⌦ converges with increasing N. We generally find
for clover-leaf protein shapes that the optimal choice for
⌦ increases with increasing ✏. When U and U0 are allowed to vary along the protein-bilayer interface, we find
that the optimal choice of ⌦ and the corresponding ⌘b0
for a given N in Eq. (20) and protein cross-section shape
C(✓) in Eq. (17) or Eq. (18) with Eq. (19) has a weak
dependence on the assigned profiles of U(✓) and U0
(✓).
FIG. 19. (a) Bilayer thickness deformation energy G in
Eq. (26) calculated using the BVM (see Sec. III) and (b)
corresponding boundary error ⌘b0 in Eq. (25) for clover-leaf
protein shapes as a function of the bit precision used in the
numerical computations. We used the indicated values of s
and ✏ in Eq. (17), and R ⇡ 2.3 nm, U = 0.3 nm, and U0 = 0.
For ease of comparison we employed for s = 1 the fixed values
N = 20 in Eq. (20) with Eq. (21) and ⌦ = 0.726 in Eq. (28),
N = 48 and ⌦ = 0.62 for s = 2, N = 72 and ⌦ = 0.52 for
s = 3, N = 90 and ⌦ = 0.552 For s = 4, and N = 125 and
⌦ = 0.45 for s = 5.
APPENDIX B: Protein-induced deformation energy
calculations in double precision
For some of the numerical calculations of the bilayer
thickness deformation energy G in Eq. (26) presented
in this work, we used numbers with substantially more
precision than double precision (64 bits) to meet the
numerical precision criteria of the boundary error ⌘b0 in
Eq. (25) described in Sec. III, with changes in G and
⌘b0 of no more than 105% as the numerical precision
of numbers is increased. However, many computing
languages do not have built in support for numerical
precision beyond double precision. In Fig. 19, we plotted
G in panel (a) and ⌘b0 in panel (b) as functions of
bit precision for several clover symmetries s and the
indicated values of ✏ to illustrate rough upper bounds on
✏ for these symmetries s, beyond which clover protein
cross-section shapes of the same symmetries s with
Figure A.2: (a) Lipid bilayer thickness deformation energy G in Eq. (2.26) calculated using the BVM (see Sec. 2.2) and (b) corresponding boundary error ηb
′ in Eq. (2.25) for
clover-leaf protein shapes as a function of the bit precision employed in the numerical
computations. We used the indicated values of s and ϵ in Eq. (2.17), and R ≈ 2.3 nm,
U = 0.3 nm, and U
′ = 0. For ease of comparison we used for s = 1 the fixed values
N = 20 in Eq. (2.20) with Eq. (2.21) and Ω = 0.726 in Eq. (2.28), N = 48 and Ω = 0.62 for
s = 2, N = 72 and Ω = 0.52 for s = 3, N = 90 and Ω = 0.552 for s = 4, and N = 125 and
Ω = 0.45 for s = 5.
182
B Supplemental material for Chapter 4
B.1 Axisymmetric bilayer midplane deformations
We assume that, for given bilayer-protein boundary conditions and membrane mechanical properties, the minimum of the bilayer midplane deformation energy [32–34]
G =
Kb
2
Z
dA (c1 + c2)
2 − τ∆A , (B.1)
with Kb as the lipid bilayer bending rigidity, determines the dominant lipid bilayer shape.
We thereby, we assume the bilayer to be asymptotically flat at a large distance away from
the bilayer-protein boundary. Equation (B.1) considers the (mean curvature) energy due
to bending of the bilayer midplane and the change in its projected in-plane area ∆A under
membrane tension τ , where c1 and c2 represent the local principal curvatures of the bilayer
midplane surface.
Solving the nonlinear shape (Euler-Lagrange) equations for bilayer midplane deformations is, in general, a very challenging mathematical problem. However, if the deformations are axisymmetric, we can reformulate the problem using the arc-length parametrization of surfaces and accurately calculate G in Eq. (B.1). We follow here Refs. [107, 110,
111, 247]. The axial symmetry allows us to parameterize the bilayer midplane shape
solely as a function of the arc-length s measured along the midplane deformation profile.
The bilayer midplane deformation field h (s) represents the vertical coordinate parallel to
the axial line of symmetry (the h-axis), r (s) denotes the radial coordinate perpendicular to
the h-axis, and ψ (s) is the angle between the tangent to the bilayer midplane surface and
183
the r-axis. The assumption of a smooth membrane surface at s = 0 yields the following
boundary conditions [107, 110, 111, 247]:
ψ (0) ≡ α , (B.2)
r (0) ≡ r0 , (B.3)
and
h (0) ≡ h0 , (B.4)
where r0, h0, and α are determined by the shape of the bilayer-protein boundary. The
Hamiltonian formalism allows us to derive a set of first-order differential equations whose
solutions yield the stationary lipid bilayer shapes. We numerically solved these equations
using Mathematica’s FindRoot command with a shooting method as described in greater
detail in a previous work [247, 248].
184
C Supplemental material for Chapter 5
C.1 Molecule distributions in emerin nanodomains from spatially
heterogeneous diffusion coefficients
Experimental data on emerin systems under various conditions, including mutations
and mechanical stress [12], suggests that emerin cluster at the INM to form stable nanodomains that coincide with regions in diffusion maps with slowed-down diffusion (diffusion coefficient νslow ≈ 3 × 10−4µm2/s in wild-type systems under no mechanical stress),
while membrane regions outside emerin nanodomains show faster diffusion of emerin (diffusion coefficient νfast ≈ 2 × 10−3µm2/s in wild-type systems under no mechanical stress).
In particular, for wild type emerin systems under no mechanical stress, 56% of the emerin
population at the INM were shown to be fast diffusers. Moreover, local cluster maps of the
emerin revealed that emerin nanodomains roughly cover 15% of the INM area and the relative density of emerin inside nanodomains was shown to be about 6. In this appendix we
show that, treating the INM as a two-dimensional medium with the observed differences
in diffusion coefficients [227], one finds steady state molecular concentrations of emerin
inside nanodomains, ρslow, and their relative densities inside nanodomains, ⟨Nslow⟩/⟨Nfast⟩,
at the INM that roughly agree with experiments, at least in the case of mechanically unperturbed wild-type systems.
185
C.1.1 Free diffusion
If no steric constraints are considered and if the particles do not interact with each
other, our system corresponds to standard, “free” diffusion with spatially heterogeneous
diffusion coefficients [227]. For now, we will not directly incorporate effects connected to
molecular crowding, and focus on the free diffusion of emerin molecules. Starting with the
(stochastic) master equation (ME) describing emerin diffusion, we obtained exact analytic
solutions of the steady-state distributions of emerin molecules. Solutions of the ME at
steady state correspond to zero net molecular fluxes across the nanodomain boundaries.
The steady-state solution of our model necessarily corresponds to uniform (average) distributions of molecules inside and outside nanodomains. In particular, our exact analytic
solution of the ME shows that, in the steady state of the system, the fraction of emerin
molecules inside nanodomains is given by
ρslow =
1 +
Γfast
Γslow −1
, (C.1)
where, in free diffusion, Γslow and Γfast are the characteristic times randomly diffusing
emerin molecules spend inside and outside emerin nanodomains, respectively. Thus,
for freely diffusing molecules we have Γ = A/ν, where A is the area of the membrane
region characterized by the diffusion coefficient ν. Thus, for our free-diffusion emerin
system we have ρslow = [1 + (Afast/Aslow) / (νfast/νslow)]−1 with Aslow and Afast as the total
areas of the membrane regions inside and outside nanodomains, respectively. Assuming nanodomains cover 15% of the availabile INM area and the diffusion coefficients are
νslow = 3 × 10−4 µm2/s and νfast = 2 × 10−3 µm2/s, for wild-type emerin systems under
186
no mechanical stress, Eq. (C.1) yields the steady state fraction of freely diffusing emerin
inside nanodomains at the INM ρslow ≈ 54%. This aligns well with the experimental measurement of 56% [12].
The density of emerin molecules inside nanodomains is ⟨Nslow⟩ = ρslowM/Aslow, where
M is the total number of emerin molecules at the INM. Likewise, the molecular density of
emerin outside nanodomains is ⟨Nfast⟩ = ρfastM/Afast, with ρfast = 1−ρslow. So, the relative
molecular density of emerin inside nanodomains is
⟨Nslow⟩
⟨Nfast⟩
=
ρslow
ρfast
Afast
Aslow
, (C.2)
Inserting, Eq. (C.1) into Eq. (C.2) and simplifying yields ⟨Nslow⟩/⟨Nfast⟩ = νfast/νslow, and so
the ME predicts that the relative molecular density of emerin inside emerin nanodomains
is governed by the ratio of emerin diffusion coefficients outside to inside emerin nanodomains. For wild type emerin systems under no mechanical stress, this ratio evaluates
to about 7 which aligns with the corresponding value of this ratio, ≈ 6, implied by experiments [12].
We also calculated ρslow and ⟨Nslow⟩/⟨Nfast⟩, using Eqs. (C.1)–(C.2), for wild type emerin
systems under mechanical stress, Q133H emerin systems under no mechanical stress,
P183H emerin systems under no mechanical stress, and ∆95-99 emerin systems under
no mechanical stress and mechanical stress, assuming their emerin nanodomains cover
15% of the INM, like wild type emerin systems under no mechanical stress, and using the
diffusion coefficients for νslow and νfast measured in experiments [12] and summarized in
Fig. C.1. We also summarize the results of our free-diffusion model in Fig. C.1.
187
Figure C.1: Table comparing experimental data [12] (orange), estimated rate constants
(blue), free diffusion model predictions [fraction of emerin molecules that are slow diffusers ⇢slow and relative emerin molecule density inside nanodomains hNslowi/hNfasti]
(green), and predictions of our reaction-diffusion model [emerin nanodomain diameter,
`, the fraction of emerin nanodomain area covered by I and A complexes, F, of its
ratio to that of the wild type system under no mechanical stress, F/F WT
] (red) for the
various emerin systems in Fig. 5.2 and the 95–99 system under no mechanical stress.
For 95–99 systems under no mechanical stress, emerin nanodomains were essentially
non-existent in experiments [12] and the predicted steady state from our reaction-diffusion
model, so we indicate “null" to denote there are no values to report for their nanodomain
characteristics.
For ⇢slow, our predictions roughly agree with experiments for all cases considered. For
hNslowi/hNfasti, our predictions roughly agree with experiments for wild-type and Q133H
emerin systems under no mechanical stress, but our predictions generally fail to describe
experiments if mechanical force is applied to the system or if emerin is mutated so as to
impair emerin’s ability to cluster [12]. when the system in subjected to mechanical stress
or mutations which greatly impair emerin’s clustering potential [12]. Our results suggest
161
Figure C.1: Table comparing experimental data [12] (orange), estimated parameters
(blue), free diffusion model predictions (fraction of emerin molecules that are slow diffusers, ρslow, and relative emerin molecule density inside nanodomains, ⟨Nslow⟩/⟨Nfast⟩)
(green) wild-type emerin systems under no mechanical stress (“WT") and mechanical
stress (“WT; force"), Q133H and P183H mutant emerin systems under no mechanical
stress, and ∆95-99 mutant emerin systems under no mechanical stress (“∆95-99") and
mechanical stress (“∆95-99; force").
For ρslow, our predictions roughly agree with experiments for all cases considered. For
⟨Nslow⟩/⟨Nfast⟩, our predictions roughly agree with experiments for wild-type and Q133H
emerin systems under no mechanical stress, but our predictions generally fail to describe
experiments if mechanical force is applied to the system or if emerin is mutated so as to
impair emerin’s ability to cluster [12]. when the system in subjected to mechanical stress
or mutations which greatly impair emerin’s clustering potential [12]. Our results suggest
that the deficient clustering potential of these systems is greatly hindered by the alteration
in interactions with nuclear binding partners imparted by force and mutations.
Equation (C.1) suggests that the steady-state fractions of emerin molecules concentrated inside nanodomains depend only on the fraction of available INM area covered by
nanodomains and on the relative diffusion coefficients inside and outside nanodomains,
and are independent of the detailed arrangement and shape of nanodomains [227]. By
188
construction, our simple model is unable to to predict the self-assembly or size of emerin
nandomains, which we address through the reaction-diffusion model described in Chapter 5.
C.1.2 Diffusion in crowded membranes
Our model is readily extended to directly account for steric repulsion arising from the
finite size of emerin molecules, in which case the effective diffusion rates also depend on
the number of emerin molecules occupying the “target” sites of randomly diffusing emerin
molecules. In particular, steric constraints due to the finite size of emerin molecules imply
that, locally, the membrane area can only accommodate some finite number of emerin
molecules, which is expected to modify the results in Sec. C.1.1. To model emerin steric
repulsion, we assume that the rates of diffusion processes locally increasing the molecule
number are ∝ (1 − N), where the field N (x, y, t) is the local fractional INM area covered
by emerin molecules. We thereby take N (x, y, t) to be normalized so that 0 ≤ N ≤ 1
[227].
Similarly as in Sec. C.1.1 we directly solved the ME defining our model of diffusion
in crowded membranes in the steady state to obtain the steady-state fractions of emerin
molecules inside emerin nanodomains, yielding Γslow = Aslow/µslow and Γfast = Afast/µfast
[227], where
µfast =
−b + (b
2 − 4ac)
1/2
2a
(C.3)
and
µslow =
1 +
νfast
νslow
1 − µfast
µfast −1
, (C.4)
189
with
a = −Aslow
νfast − νslow
νfastνslow
, b =
Aslow
νslow
+
Afast
νfast
+ ⟨N⟩
νfast − νslow
νfastνslow
, c = −
⟨N⟩
νslow
. (C.5)
We find, as in Sec. C.1.1, that the direct solution of the ME for our system with steric constraints depends on the ratio of membrane area inside and outside emerin nanodomains
and on the ratio of the diffusion coefficients measured inside and outside emerin nanodomains, and is independent of detailed emerin nanodomain properties such as the
shape or number of emerin nanodomains. Furthermore, as in Sec. C.1.1, the ME describing diffusion in crowded membranes implies, in the steady state, uniform (average)
concentrations of emerin inside and outside nanodomains. Importantly, and contrary to
the case of free diffusion, the steady-state fractions of emerin inside emerin nanodomains
and, consequently, the relative densities of emerin inside emerin nanodomains now depend on the fractional INM area covered by emerin molecules ⟨N⟩.
In wild-type emerin systems under no mechanical stress, increasing the fraction of the
INM area covered by emerin molecules, in our model, leads to a decrease in the steadystate fractions and relative molecular densities inside emerin nanodomains [Fig. C.2(a,b)].
For example, when ⟨N⟩ = 0.02, an upper bound suggested by counting experiments [230],
ρslow ≈ 53% in the steady state, corresponding to ⟨Nslow⟩/⟨Nfast⟩ ≈ 6. We found similar
minor decreases in mutant and mechanically stressed emerin systems.
While incorporation of steric constraints yields somewhat improved agreement between model predictions and experimental observations (see Fig. C.1), we also find that
steric constraints only tend to have a minor effect on the emerin distributions at the INM.
190
Figure C.2: Diffusion-only models applied to wild-type emerin systems under no mechanical stress. Steady-state (a) fractions of emerin molecules inside emerin nanodomains, ⇢slow, and (b) relative densities of emerin molecules inside emerin nanodomains, hNslowi/hNfasti, as a function of the global fractional INM area covered by
emerin molecules, hNi, assuming free diffusion (green curves) and diffusion with steric
constraints linear in the local fractional INM area covered by emerin molecules N (purple
curves). All results were obtained through direct solution of the ME (see Ref. [227]), with
the analytical solutions shown in (a) Eq. (C.1) and (b) Eq. (C.2).
166
Figure C.2: Diffusion-only models applied to wild-type emerin systems under no mechanical stress. Steady-state (a) fractions of emerin molecules inside emerin nanodomains, ρslow, and (b) relative densities of emerin molecules inside emerin nanodomains, ⟨Nslow⟩/⟨Nfast⟩, as a function of the global fractional INM area covered by
emerin molecules, ⟨N⟩, assuming free diffusion (green curves) and diffusion with steric
constraints linear in the local fractional INM area covered by emerin molecules N (purple
curves). All results were obtained through direct solution of the ME (see Ref. [227]), with
the analytical solutions shown in (a) Eq. (C.1) and (b) Eq. (C.2).
In contrast, our reaction-diffusion model requires steric repulsion for the self-assembly
and stabilization of emerin nanodomains through a Turing mechanism (see Chapter 5).
We also note that, according to the results discussed here, these steric effects alone
cannot explain the relatively low molecular densities of emerin within nanodomains seen
in experiments of various mutant emerin type systems [12]. This suggests that the observed relative densities of emerin inside and outside emerin nanodomains depend on the
interactions of emerin with its nuclear binding partners at the INM. Additionally, emerin’s
diffusion properties alone cannot explain the characteristic shape, size, and self-assembly
of emerin nanodomains, highlighting the necessity of reactions for a more accurate description of these properties.
191
Abstract (if available)
Abstract
Over recent years, a diverse range of experiments have provided much quantitative data on the role of membrane proteins in cellular signal transduction and the adaptation of cells to dynamic environments. Membrane proteins exhibit diverse molecular mechanisms for sensing stimuli, initiating signaling pathways through structural changes, and engaging in collective signaling activities. In particular, protein clustering into domains dedicated to specialized functions provides important mechanisms for cell membrane organization. Furthermore, the perturbation of the lipid bilayer by membrane proteins is thought to play an important role in membrane protein function. This thesis comprises a set of interconnected studies that employ theoretical physics to investigate fundamental aspects of the intricate coupling between membrane proteins and cellular responses to stimuli. Chapter 1 provides a general introduction to cell membranes, membrane proteins, and the diverse functions they serve in cell membranes, and as well as the theory of membrane mechanics. Chapter 2 introduces a novel boundary value method (BVM) that bridges structural biology with membrane elasticity theory, enabling the analytic determination of protein-induced lipid bilayer deformations, even for non-circular protein cross-sections, in excellent agreement with finite element solutions. Inspired by our BVM, Chapter 3 formulates a simple analytic approximation of the bilayer thickness deformation energy associated with general protein shapes and shows that, for modest deviations from rotational symmetry, this analytic approximation is in good agreement with BVM solutions. The BVM and analytical approximation are utilized to explore how variations in protein shape influence elastic bilayer thickness deformations. Our findings reveal that alterations in protein shape induce changes to the lipid bilayer deformation energy exceeding 10 k_{B}T, which may have important implications for protein conformational changes and protein oligomerization processes. Chapter 4 examines the interplay between membrane mechanics and thermosensing, revealing how, mediated by lipid bilayer properties such as hydrophobic thickness and bending rigidity, temperature changes influence the conformational transitions of membrane proteins. We thus investigate the fundamental principles underlying the coordination of thermosensing and mechanosensing in living systems. Chapter 5 explores the physical principles underlying the self-assembly of emerin nanodomains at the inner nuclear membrane, which may shed new light on the role of emerin nanodomains in mechanotransduction. By employing a comprehensive modeling approach rooted in the Turing mechanism of nonequilibrium pattern formation, we develop a simple model quantifying the intricate reaction-diffusion properties of proteins and their nuclear binding partners. On this basis, we provide insight into the wild-type properties of emerin nanodomains and their response to applied forces, as well as the mechanisms underlying the observed defects in the self-assembly of emerin nanodomains for mutated forms of emerin associated with Emery-Dreifuss muscular dystrophy. Chapter 6 provides an overview and conclusions from our studies, and suggests potential future directions of research inspired by our findings. By integrating these diverse research strands, our work contributes to a deeper understanding of the fundamental principles governing membrane mechanics and pattern formation, with implications for both physics and biology.
Most of the material described in this thesis is/will be discussed in the following publications:
1. C. D. Alas and C. A. Haselwandter. Dependence of protein-induced lipid bilayer deformations on protein shape. Phys. Rev. E, 107:024403, 2023.
2. C. D. Alas, O. Kahraman, and C. A. Haselwandter. Thermosensing through membrane mechanics, (expected submission in early 2024).
3. C. D. Alas, F. Pinaud, and C. A. Haselwandter. Physical mechanism for the self-assembly of emerin nanodomains at the inner nuclear membrane, (expected submission in 2023).
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Asset Metadata
Creator
Alas, Carlos Daniel
(filename)
Core Title
Physical principles of membrane mechanics, membrane domain formation, and cellular signal transduction
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Degree Conferral Date
2023-12
Publication Date
12/11/2023
Defense Date
11/27/2023
Publisher
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(original),
University of Southern California
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Tag
bilayer deformations,boundary value method,cell membrane organization,cellular signal transduction,conformational transitions,emerin nanodomains,lipid bilayer perturbation,mechanotransduction,membrane elasticity theory,membrane mechanics,membrane proteins,molecular mechanisms,OAI-PMH Harvest,pattern formation,protein clustering,protein shapes,rotational symmetry breaking,stimuli sensing,structural biology,thermosensing,Turing mechanism
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), Boedicker, James (
committee member
), Foster, Peter (
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), Nakano, Aiichiro (
committee member
), Pinaud, Fabien (
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)
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Tags
bilayer deformations
boundary value method
cell membrane organization
cellular signal transduction
conformational transitions
emerin nanodomains
lipid bilayer perturbation
mechanotransduction
membrane elasticity theory
membrane mechanics
membrane proteins
molecular mechanisms
pattern formation
protein clustering
protein shapes
rotational symmetry breaking
stimuli sensing
structural biology
thermosensing
Turing mechanism