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The physics of membrane protein polyhedra
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The physics of membrane protein polyhedra
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THE PHYSICS OF MEMBRANE PROTEIN POLYHEDRA by Di Li 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) May 2017 Copyright 2017 Di Li Acknowledgments The completion of this thesis would not have been possible without the guidance of my supervisor Christoph A. Haselwandter. His impressive academic insights as well as inspirations have motivated me so much in the progress of my PhD study. Besides, he has always been so helpful in personal support toward significant milestones in my life. I couldn’t be more grateful for being so lucky to have the opportunity to collaborate with him throughout those years. Very special thanks to Osman Kahraman, from whom I learned so much since the beginning of my academic research. He has been so patient in explaining complicated concepts and techniques to me and so generous in sharing fantastic research ideas. His enthusiasm in scientific research sets a lifetime example encouraging me to devote my entire life to whatever I am interested in and whatever I would like to pursue in my career. I thank James Q. Boedicker, Stephan Haas, Christoph A. Haselwandter, Noah Malm- stadt, and Paolo Zanardi for serving in the committee of both my PhD qualifying exam and thesis defense. Thank you all so much for your time and great comments on my research reports. The work in this thesis is supported by NSF Grants No. DMR-1554716 and No. DMR-1206332, an Alfred P. Sloan Research Fellowship in Physics, the James H. Zum- berge Faculty Research and Innovation Fund at the University of Southern California, ii and the USC Center for High-Performance Computing. I also acknowledge support through the Kavli Institute for Theoretical Physics, Santa Barbara, via NSF Grant No. PHY-1125915. I thank W. S. Klug, R. Phillips, D. C. Rees, M. H. B. Stowell, and H. Yin for helpful comments. Any achievement I accomplished would not have been possible without the invalu- able support from my family. Thank you mom, dad, my beloved wife Yiai, and her parents. Thank you for always standing beside me and going through every difficulty with me. I love you all. Di February 2017 iii Contents Acknowledgments ii List of Tables vi List of Figures vii Abstract xiii 1 Introduction 1 1.1 Biological membranes . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Self-assembly of lipid aggregates . . . . . . . . . . . . . . . . . . . . . 2 1.3 Membrane protein polyhedral nanoparticles (MPPNs) . . . . . . . . . . 3 1.4 Self-assembly of MPPNs . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Overview of the model of MPPN self-assembly . . . . . . . . . . . . . 7 2 Lipid bilayer elasticity theory of MPPNs 9 2.1 Protein-induced lipid bilayer deformations . . . . . . . . . . . . . . . . 9 2.2 Bilayer mechanics of MPPNs . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 MPPN midplane deformation energy . . . . . . . . . . . . . . 16 2.2.2 MPPN thickness deformation energy . . . . . . . . . . . . . . 21 2.2.3 MPPN defect energy . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.4 Steric constraints on protein separation . . . . . . . . . . . . . 26 2.3 MPPN mean-field energy . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.1 MPPN mean-field energy for perfect hydrophobic matching . . 27 2.3.2 General form of the MPPN mean-field energy . . . . . . . . . . 28 3 Minimal molecular model of MPPN organization 33 3.1 Simulated annealing Monte Carlo simulations . . . . . . . . . . . . . . 33 3.2 Polyhedral symmetry of MPPNs . . . . . . . . . . . . . . . . . . . . . 38 3.3 Polyhedral symmetry of MPPNs for a single gated ion channel . . . . . 40 iv 4 Statistical thermodynamics of MPPN self-assembly 44 4.1 Equilibrium distribution of MPPNs . . . . . . . . . . . . . . . . . . . . 44 4.2 MPPN self-assembly phase diagram . . . . . . . . . . . . . . . . . . . 46 4.2.1 MPPN self-assembly phase diagram for perfect hydrophobic matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2.2 Generalized MPPN self-assembly phase diagram . . . . . . . . 51 5 Conclusion 55 A Derivation of the zero-force boundary condition for membrane thickness deformations 58 B MscS Transmembrane Geometry 60 C Derivation of MPPN defect energy 62 C.1 First Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 C.2 Second Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 D Stretching modulus and MPPN mean-field energy 66 E Robustness of MPPN self-assembly phase diagram 71 References 75 v List of Tables 3.1 Top twenty polyhedral fits to the minimum-energy MPPN configuration in fig. 3.1 predicted by our minimal molecular model of MPPN orga- nization. We define the fit error as the sum over the squared distances between the simulated positions of protein centers and the closest fitted polyhedron vertices. In our fitting procedure we allow for the 132 poly- hedral symmetries [1] corresponding to the 5 Platonic solids (P), the 13 Archimedean solids (A) with two chiral pairs, the 13 Catalan solids (C) with two chiral pairs, and the 92 Johnson solids (J) with five chiral pairs. 39 3.2 Top twenty polyhedral fits to the minimum-energy MPPN configuration in fig. 3.3 predicted by our minimal molecular model of MPPN organi- zation, obtained as in tab. 3.1. . . . . . . . . . . . . . . . . . . . . . . . 42 vi List of Figures 1.1 Schematic of cell membranes with different types of membrane proteins embedded in lipid bilayer environments (after Ref. [2]). . . . . . . . . . 1 1.2 Stereo side view of the ribbon diagrams showing the molecular struc- ture of the mechanosensitive channel of small conductance (MscS) hep- tamer obtained from protein crystallography, viewed perpendicular to the sevenfold axis, with each subunit represented in a separate color (after Ref. [3]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Cryotomography of MscS MPPNs. (A) The Particle Estimation for Electron Tomography (PEET) isosurface derived from 162 individual particles selected from eight single-tilt tomograms. The strong align- ment bias due to the missing wedge effect in determining the MPPN symmetry is observed along the lower part of the surface, but individ- ual MscS heptamers are still discernible in the image. (B) The corre- sponding PEET isosurface, following introduction of randomized start- ing Euler angles to minimize the missing-wedge bias. (C) The use of randomized starting Euler angles results in a much-improved map with octahedral (432) symmetry that could be fit with 24 molecules of the MscS crystal structure. Isosurface renderings of the volume averages were generated using Chimera (31) (after Ref. [4]). . . . . . . . . . . . 5 vii 1.4 Illustration of membrane protein crystallisation by conceptual break- through using controlled convective-diffusive transport in microfluidic channel to boost crystallisation process (after Ref. [5]). . . . . . . . . . 6 2.1 (a) Schematic of MPPNs [4]. MscS are embedded in a lipid bilayer with the MscS cytoplasmic region outside MPPNs [4]. (b) Schematic of protein-induced lipid bilayer deformations in MPPNs. We denote the bilayer midplane radius of the protein by i and the bilayer-protein contact angle by . The transmembrane surface of the protein speci- fies boundary conditions onh andu at the bilayer-protein interface (see main text). The membrane patch radius o and boundary angle are determined by the MPPN size and the number of proteins per MPPN [6, 7]. The protein structure shown here corresponds to the closed state of MscS [3, 8] used in experiments on MPPNs [4, 5], with Protein Data Bank ID 2OAU and different colors indicating different MscS subunits [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Minimized total MPPN energyE min for MPPNs formed from MscS [4, 5] with the contributions E h and E d due to bending deformations and packing defects versusn at = 0:5 rad. The inset shows the optimal sphere coveragep(n) forn identical non-overlapping circles [10] with particularly favorable packings atn = 12 (icosahedron),n = 24 (snub cube), andn = 48. (Inset after Ref. [10].) . . . . . . . . . . . . . . . . 28 2.3 MPPN energy and MPPN stretching modulus for MPPNs formed from MscS [3, 8] at = 0:5 rad for = 0. (a) MPPN energy per protein, E min =n, obtained from eqs. (2.57), (2.38), and (2.55) vs. n andU. The magenta curves show then-states with minimalE min =n, for 10 n 80, as a function ofU. (b) MPPN stretching modulusK s obtained from viii eq. (2.56) with eqs. (2.38) and (2.55) vs. n and U. The white dashed curves indicate locations in parameter space with a discontinuous jump inK s as a function ofU. In both panels,U is changed by varyingm. . . 29 2.4 Bilayer midplane and thickness deformation energies per membrane patch,E h =n andE u =n, obtained form eqs. (2.38) and (2.55), andE 0 =n = (E h +E u )=nvs. o for MPPNs formed from MscS [3, 8] at = 0:5 rad for = 0, n = 48, and the indicated values of U, which we obtain by changing m. The positions of the optimal o = min are indicated by dots, with a two-fold degenerate global minimum of E 0 =n at U 0:175 nm for o i . The grey shaded region indicates the range o < i excluded by steric constraints. . . . . . . . . . . . . . . . . . . 31 3.1 Front and side views of the minimum-energy MPPN configuration obtained from our minimal molecular model of MPPN organization. The larger and smaller disks represent proteins and the lipid bilayer, with disk sizes corresponding to MscS [8] and diC14:0 lipids [11], respectively. The green and blue lines are obtained by connecting the centers of nearby MscS in the simulated MPPN configuration and by fitting the simulated MPPN configuration to a snub cube (dextro) using least-square mini- mization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Molecular structure of MscS. The left panel shows the closed state of MscS [3, 8] used previously in experiments on MPPNs [4, 5], with [9] Protein Data Bank ID 2OAU. The right panel shows the open state of MscS [12], with [9] Protein Data Bank ID 4HWA. Different colors in both panels indicate different MscS subunits [9]. . . . . . . . . . . . . . 40 3.3 Front and side views of the minimum-energy MPPN configuration obtained from our minimal molecular model of MPPN organization with one ix MscS, indicated by a red circle, in the open state. The larger and smaller disks represent proteins and the lipid bilayer, with disk sizes correspond- ing to MscS [8] and diC14:0 lipids [11], respectively. The green and blue lines are obtained by connecting the centers of nearby MscS in the simulated MPPN configuration and by fitting the simulated MPPN configuration to a snub cube (dextro) using least-square minimization. . 41 4.1 MPPN self-assembly phase diagram obtained from eq. (4.11) as a func- tion of protein number fraction c and bilayer-protein contact angle . The color map in the upper panel shows maximum values of(n) asso- ciated with the dominantn-states of MPPNs. The dominantn are indi- cated in each portion of the phase diagram, together with the associated MPPN symmetry [10]. Black dashed curves delineate phase boundaries. The white dashed line indicates the valuec 7:8 10 8 correspond- ing to experiments on MPPNs [4] formed from MscS [8, 13] and the -range associated with MscS. The lower panel shows(n) forn = 20, 22, 24, 27, and 30 as a function of along the white dashed line in the upper panel. We use the model parameter values i 3:2 nm [8, 13] and K b 14 k B T [14] corresponding to MPPNs formed from [4, 5] MscS and diC14:0 lipids. . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 MPPN self-assembly phase diagram obtained from eq. (4.11) as a func- tion of protein radius i and bilayer-protein contact angle. The white dashed line indicates the protein radius i 3:2 nm and-range asso- ciated with MscS [8, 13]. We use the same labeling conventions as in fig. 4.1 with the model parameter values c 7:8 10 8 [4] and K b 14k B T [14] corresponding to MPPNs formed from [4, 5] MscS and diC14:0 lipids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 x 4.3 MPPN self-assembly phase diagram obtained from eq. (4.11) withE min (n) determined by eqs. (2.57), (2.38), and (2.55) as a function of U and . The color map in the upper panel shows the maximum values of (n) associated with the dominant n-states of MPPNs. The dominant n are indicated in each portion of the phase diagram, together with the associated MPPN symmetry [10]. Black dashed curves delineate phase boundaries. The red dashed line indicates the value U 0:055 nm corresponding to the bilayer-protein hydrophobic thickness mismatch associated with [4, 5] MPPNs formed from MscS [3, 8] and diC14:0 lipids [14, 15], and the -range associated with MscS [3, 7, 8]. The lower panel shows (n) for n = 20, 22, 24, 27, and 30 as a function of along the red dashed line in the upper panel. We used the protein number fractionc 7:810 8 employed in experiments and set = 0. The orange shaded areas indicate regions for whichn = 12 is strongly penalized by large values of K s resulting [7] from o = min ! i in eq. (2.56) (see also fig. 2.3(b) of Chap. 2). In these regions of parameter space, the continuum model of MPPN bilayer mechanics may not give reliable results for the dominantn-states of MPPNs. . . . . . . . . . . . 52 4.4 MPPN radii R of dominant n-states MPPNs for n = 12, 24, and 48, as a function of bilayer-protein hydrophobic thickness mismatchU for the entire andU range considered in fig. 4.3. In each panel, the green, red, and blue curves correspond to the upper limit, lower limit, and mean value of MPPN radiiR with respect to, respectively. . . . . . . . . . 53 B.1 Molecular structure of MscS in the closed state [8] with Protein Data Bank ID 2OAU represented using Visual Molecular Dynamics [9]. The xi green beads indicate hydrophobic residues [16] at pH 7. The trun- cated cones are chosen so as to enclose the approximate transmembrane region of MscS [8, 13]. The values of quoted in each panel are calcu- lated from the respective truncated cones. . . . . . . . . . . . . . . . . 61 D.1 (a) MPPN stretching modulusK s and (b) minimized total MPPN energy E min for MPPNs formed from MscS [4] with the contributionsE h and E d due to bending deformations and packing defects versus n at = 0:5. The insets in panels (a) and (b) allow for the fulln-range 10n 80 considered here, while for the main panels we use the samen-range as in the main panel of fig. 2.2 of Chap. 2. . . . . . . . . . . . . . . . . 66 D.2 (a) Numerical values of the minima of the bending energy per membrane patch, b =E h (n;R)=n, obtained by minimizing eq. (D.3) with respect to o at eachn, and (b) corresponding second derivatives with respect to o , 00 b , evaluated for eachn at the o minimizing b . We used the same parameter values as in fig. 2.2 of Chap. 2. . . . . . . . . . . . . . . . . 69 E.1 Same plots as in (a,b) fig. 4.1 and (c) fig. 4.2 of Chap. 4, but using K b = 28k B T instead ofK b = 14k B T . . . . . . . . . . . . . . . . . . 72 E.2 Same plots as in (a,b) fig. 4.1 and (c) fig. 4.2 of Chap. 4, but using K b = 56k B T instead ofK b = 14k B T . . . . . . . . . . . . . . . . . . 73 xii Abstract This PhD thesis is devoted to the study of the physical mechanisms underlying self- assembly of membrane protein polyhedral nanoparticles (MPPNs). It provides analyti- cal and computational approaches that allow investigation of the roles of bilayer-protein interactions, lipid bilayer mechanical properties, topological defects in protein pack- ing, and thermal effects in MPPN self-assembly. Chapter 1 is an overall introduction to biological membranes, as well as the experimental phenomenology of MPPN self- assembly. In Chapter 2 we explore a mean-field model of the MPPN energy based on the continuum elasticity of lipid bilayer-protein interactions, which accounts for protein- induced lipid bilayer curvature deformations, lipid bilayer-protein hydrophobic thick- ness mismatch, and topological defects arising from the spherical shape of MPPNs. In Chapter 3 we justify some of the key assumptions underlying our mean-field of MPPN energy using a minimal molecular model of MPPN organization. In Chapter 4 we develop the statistical thermodynamics of MPPN self-assembly, which we then combine with our mean-field model of MPPN energy to calculate the MPPN self-assembly phase diagram. We find that, in addition to the bilayer-protein contact angle, which depends on the shape of membrane proteins, the bilayer-protein hydrophobic thickness mismatch is a key molecular control parameter for MPPN shape, with pronounced protein-induced lipid bilayer thickness deformations biasing the MPPN self-assembly phase diagram xiii towards highly symmetric and uniform MPPN shapes. Finally, in Chapter 5, we sum- merize the background, and key results of the work described in this thesis, and discuss the broader significance of our work on the physics of membrane protein polyhedra. Most of the material described in this thesis is also discussed in the following publi- cations: I. D. Li, O. Kahraman, C. A. Haselwandter. Controlling the shape of membrane protein polyhedra (submitted). II. D. Li, O. Kahraman, C. A. Haselwandter. Symmetry and size of membrane pro- tein polyhedral nanoparticles. Physical Review Letters, 117(13): 138103, 2016. arXiv:1611.00842 xiv Chapter 1 Introduction 1.1 Biological membranes The motivation of the research underlying this PhD thesis is to get a better under- standing of the physical properties and physiologically relevant functionalities of bio- logical membranes. One of the hallmarks of cellular life is the existence of membranes, which serve as envolopes separating the cellular contents from the rest of the world, and hence give spatial integrity to cells. Cell membranes are not a passive boundary, but control the flow of nutrients and other molecules critical for life, signals, and waste products between the cell and its environment. Moreover, cell membranes help main- tain differences in chemical and physical composition between the cell and the external world, and provide reaction compartments for the various fundamental processes of life. Figure 1.1: Schematic of cell membranes with different types of membrane proteins embedded in lipid bilayer environments (after Ref. [2]). 1 Fundamental constituents of cell membrane are membrane proteins and phospho- lipids (see fig. 1.1). Membrane proteins, which have a much more complicated molec- ular structure than lipids, control most of the biological functions of cell membranes. They play a central role in many essential cellular processes [17, 18] such as ion exchange, signaling, and membrane curvature regulation. Phospholipids, which in gen- eral form bilayer structures in cell membranes, act like a “fluid glue” between membrane proteins, and are essential for the structural integrity of cell membranes. 1.2 Self-assembly of lipid aggregates The phospholipids found in cell membranes have a head group domain and a bulky tail which contains a pair of hydrocarbon chains. The non-polar hydrocarbon chain is water-avoiding, or “hydrophobic” [19]. The carboxyl head group is polar and water- loving, or “hydrophilic”. These two important characteristics combine to make the lipid molecules amphiphilic. When amphiphilic molecules are dispersed in an aqueous envi- ronment, such as water, the hydrophobicity of the lipid tails means that it is energetically favorable for the lipid tails to be shielded from the aqueous environment, while the head groups of lipids are attracted by water due to hydrophilic interactions [20]. The result of these two competing effects gives rise to the self-assembly of lipid aggregates. Self- assembly of lipid aggregates competes with dispersal of lipids in the aqueous solution, which is favorable from an entropic perspective. In practice, a variety of different types of lipid aggregates can be formed in solution, depending on many factors such as the length and unsaturation of double C=C bonds in the hydrocarbon chain, the ionic environment, and temperature [21]. For example, some single-chained lipids with a large head group area can have a conical shape and hence tend to form spherical micelles. In contrast, double-chained lipids with a large 2 head group area can have a truncated conical shape, and therefore prefer to form bilayer structures. Note that phosphocholine lipids lie in the latter lipid category, and cell mem- branes are vesicles formed by such lipid bilayer structures. Regardless of which specific type of aggregate lipids form, self-assembled lipid aggregates typically have character- istic sizes that are much greater than single molecules [20]. 1.3 Membrane protein polyhedral nanoparticles (M- PPNs) Membrane proteins play a central role in many essential cellular processes. Determi- nation of membrane protein structure (see, e.g., fig. 1.2 for an example of a membrane protein structure) is therefore a subject of immense scientific interest. The biologically Figure 1.2: Stereo side view of the ribbon diagrams showing the molecular structure of the mechanosensitive channel of small conductance (MscS) heptamer obtained from protein crystallography, viewed perpendicular to the sevenfold axis, with each subunit represented in a separate color (after Ref. [3]). 3 relevant structures, and resulting functions, of many membrane proteins depend criti- cally [13, 18] on their lipid bilayer environment, and on chemical and voltage gradients across the cell membrane. Yet, traditional approaches for determination of membrane protein structure require removal of membrane proteins from their lipid bilayer envi- ronment. As a result, it is difficult to access different in vivo conformational states of a membrane protein, which are often stabilized by the lipid bilayer environment in cell membranes together with chemical or voltage gradients across the cell membrane. Recent experiments on membrane protein polyhedral nanoparticles (MPPNs) [4] (see fig. 1.3) present an exciting step towards overcoming this challenge. In these experi- ments [4], lipids and proteins were observed to self-assemble in an aqueous environment into MPPNs—closed lipid bilayer vesicles with a uniform size and polyhedral arrange- ment of membrane proteins. In particular,Escherichiacoli mechanosensitive channel of small conductance (MscS 1 ) was observed [4, 5] to predominantly yield MPPNs with the symmetry of a snub cube, with one MscS located at each of its 24 vertices, and a char- acteristic overall radius 20 nm. MPPNs, therefore, constitute a novel form of ordered lipid-protein assembly intermediate between single particles and large crystalline struc- tures. Conditions for generating MPPNs resembled those for other types of 2D-ordered bilayer arrangements of membrane proteins, particularly 2D crystals, in that membrane proteins are mixed with a specific phospholipid at a defined ratio, followed by removing the solubilizing detergent [22]. The main distinction here is that because of the poly- hedral shape of MPPNs, conditions are tuned in such a way that stabilizes the highly curved surfaces of polyhedra rather than the planar specimens desired for 2D crystals. 1 MscS is a stretch-activated heptameric channel with 21 transmembrane helices (3 from each subunit) and a large cytoplasmic domain with overall dimensions (in the closed state) of8 nm (parallel to the membrane plane)12 nm (perpendicular to the membrane plane) [3, 8]. 4 Figure 1.3: Cryotomography of MscS MPPNs. (A) The Particle Estimation for Elec- tron Tomography (PEET) isosurface derived from 162 individual particles selected from eight single-tilt tomograms. The strong alignment bias due to the missing wedge effect in determining the MPPN symmetry is observed along the lower part of the surface, but individual MscS heptamers are still discernible in the image. (B) The corresponding PEET isosurface, following introduction of randomized starting Euler angles to mini- mize the missing-wedge bias. (C) The use of randomized starting Euler angles results in a much-improved map with octahedral (432) symmetry that could be fit with 24 molecules of the MscS crystal structure. Isosurface renderings of the volume averages were generated using Chimera (31) (after Ref. [4]). The resulting well-defined symmetry and size of MPPNs may [4], in addition to poten- tial applications as novel drug delivery carriers with precisely controlled release mecha- nisms, offer a new route for the structure determination of membrane proteins, with the membrane proteins embedded in a lipid bilayer environment and the closed surfaces of MPPNs supporting physiologically relevant transmembrane gradients. 1.4 Self-assembly of MPPNs The standard method to reconstitute membrane proteins in a native lipid bilayer environment is to use dialysis with solubilized lipids and proteins, a buffer solution, and a dialysis membrane in the middle. However, this method is inconvenient and inefficient — requiring one or two weeks to generate membrane protein nanoparticles [5] — as it is driven by slow diffusion process removing higher concentrations of detergents from the 5 Figure 1.4: Illustration of membrane protein crystallisation by conceptual breakthrough using controlled convective-diffusive transport in microfluidic channel to boost crystalli- sation process (after Ref. [5]). protein/lipid/detergent complex sample to a lower concentration of the buffer solution. The pores of the dialysis membrane are designed to only allow small molecules such as detergents to pass through, while large molecules such as membrane proteins remain in the sample. As detergents are fully removed by the diffusion process, membrane proteins aggregate with lipid bilayers to form MPPNs [5]. In recent experiments on MPPNs [4, 5] formed from MscS, an innovative microflu- idic device (see fig. 1.4) was implemented to rapidly form MPPNs over time scales of seconds. The experimental samples are mixtures of E. coli MscS, the detergents Fos- Choline (FC-14), and the lipids Phosphocholine (PC-14). A higher concentration of protein/lipid/detergent complex was injected as a stream into the center of the main channel of the device, merged by flows with equal speed from two side buffer solu- tions [5]. The interface between the center and two side buffers created a concentration gradient so that diffusion occurs from central complex stream to the buffer solutions. 6 The flow rate needs to be adjusted in a precise way so as to achieve removal of detergent micelles from the protein/detergent complex, and to guarantee formation of lipid bilayer during the self-assembly of MPPNs. In particular, the high concentration of micelle detergents (FC-14) encapsulating MscS decreases quickly along the mixing channel, due to the centre-to-side convective-diffusive transport [5]. When the concen- tration drops below the critical micelle concentration (CMC) [19, 20] of FC-14, the micelle detergents become monomers. The monomers are then removed to the side buffer stream by diffusion. In the meantime, the lipids phosphocholine (PC-14), which has a much smaller CMC than FC-14, are transformed into bilayer structures and simul- taneously aggregate with MscS to form MPPNs. 1.5 Overview of the model of MPPN self-assembly In a previous publication [7], we introduce a simple mean-field model of MPPNs inspired by previous work on membrane budding [6, 23, 24] and viral capsid self- assembly [25], where the observed symmetry and size of MPPNs [4] can be under- stood based on the interplay of protein-induced lipid bilayer curvature deformations [6, 23, 24] arising [13] from the conical shape of MscS [3, 8] (see Chap. 2 Sec. 2.1), and topological defects in protein packing necessitated by the spherical shape of MPPNs [25] (see Chap. 2 Sec. 2.2). We confirm some of the key assumptions underlying our mean-field model by carrying out Simulated Annealing Monte Carlo simulations of a minimal molecular model of MPPN organization (see Chap. 3 Sec. 3.1), which we for- mulate following previous work on viral capsid symmetry [26]. Finally, we use our mean-field model of MPPNs to calculate [20, 25, 27] the MPPN self-assembly phase diagram, based on thermal fluctuations, as a function of protein concentration, bilayer- protein contact angle, and protein size (see Chap. 4 Sec. 4.2). We show that our model 7 correctly predicts, with all model parameters determined directly from experiments, the observed [4] symmetry and size of MPPNs formed from MscS. On the other hand, realization of MPPNs as a novel method for membrane protein structural analysis, as well as targeted drug delivery, requires [4] control over MPPN symmetry and size. Current experimental approaches, as well as our aforementioned analytic results, however, yield a distribution of different MPPN shapes [4, 7], which limits the resolution of MPPN-based structural studies [4] and potential applications of MPPNs as novel drug delivery carriers. To explore strategies for controlling and opti- mizing MPPN shape, we generalize our mean-field model of MPPN self-assembly [7] to account for the effects of a hydrophobic thickness mismatch between membrane pro- tein and the unperturbed lipid bilayer. We provide general analytic solutions for the dependence of the MPPN energy on bilayer-protein hydrophobic thickness mismatch (see Chap. 2 Sec. 2.1 and Sec. 2.2) and, on this basis, calculate the generalized MPPN self-assembly phase diagram (see Chap. 4 Sec. 4.2). Our results suggest that, in addi- tion to the bilayer-protein contact angle [7], the bilayer-protein hydrophobic thickness mismatch is a key molecular control parameter for MPPN shape. In particular, we find that modification of the lipid bilayer composition, or protein hydrophobic thickness, so as to produce pronounced protein-induced lipid bilayer thickness deformations biases the MPPN self-assembly phase diagram towards highly symmetric and uniform MPPN shapes. Our results provide general insights into the roles of bilayer-protein interactions and lipid bilayer mechanical properties in MPPN self-assembly, and suggest strategies for controlling MPPN shape in experiments. 8 Chapter 2 Lipid bilayer elasticity theory of MPPNs This chapter discusses the essential concepts and techniques that we use here to derive the MPPN energetics. The outline of this chapter is as follows. In Sec. 2.1, we derive the protein-induced lipid bilayer deformation energies as a functional of midplane curvature and bilayer thickness deformation profiles, based on the continuum elasticity theory of lipid bilayer membranes. We then construct Euler-Lagrange equations that can be used for solving for the minimum-energy membrane deformation profiles, and hence membrane deformation energies. In Sec. 2.2, we apply the equations obtained in Sec. 2.1 to MPPNs and formulate the MPPN bilayer mechanics taking into account the protein packing defect energy originating from topological constraints. In Sec. 2.3, we derive the analytic expressions for the MPPN energy. 2.1 Protein-induced lipid bilayer deformations As membrane proteins are generally found to be rigid compared to lipid bilayers, the protein-induced membrane curvature deformations [13, 28–30] largely depend on the transmembrane geometry of membrane proteins, together with the closed spheri- cal shape of MPPNs. In the simplest case, there are only protein-induced curvature deformations in MPPNs induced by approximately conical membrane inclusions such as 9 Figure 2.1: (a) Schematic of MPPNs [4]. MscS are embedded in a lipid bilayer with the MscS cytoplasmic region outside MPPNs [4]. (b) Schematic of protein-induced lipid bilayer deformations in MPPNs. We denote the bilayer midplane radius of the protein by i and the bilayer-protein contact angle by . The transmembrane surface of the protein specifies boundary conditions on h and u at the bilayer-protein interface (see main text). The membrane patch radius o and boundary angle are determined by the MPPN size and the number of proteins per MPPN [6, 7]. The protein structure shown here corresponds to the closed state of MscS [3, 8] used in experiments on MPPNs [4, 5], with Protein Data Bank ID 2OAU and different colors indicating different MscS subunits [9]. MscS [3, 8]. In addition, there may also occur protein-induced bilayer thickness defor- mations in the direction perpendicular to the surface of bilayer membranes of MPPNs, due to a hydrophobic thickness mismatch at the bilayer-protein contact interface. As a result, the lipid tail length is correspondingly adjusted (stretched if the protein hydropho- bic thickness is larger than that of unperturbed lipid bilayers, or compressed otherwise) in order to match the protein hydrophobic thickness. The continuum elasticity of protein-induced lipid bilayer deformations has been employed successfully for a variety of different lipid-protein systems, including gram- icidin channels [31, 32] and mechanosensitive channels of large conductance (MscL) [13, 33–35]. Following these studies based on the continuum elasticity theory of lipid bilayers [13, 31, 33, 36–40], bilayer-protein interactions are captured by two coupled 10 scalar fields h + and h (see fig. 2.1) that specify the positions of the hydrophilic- hydrophobic interface at the outer and inner lipid bilayer leaflets with respect to a planar reference surface. We also introduce the excess hydrophobic thickness u + and u of the outer and inner lipid bilayer leaflets, respectively, defined [41] as its hydrophobic thickness along the normal axis with respect to its hydrophobic-hydrophilic interface subtracted by its unperturbed hydrophobic thickness, denoted bym (see fig. 2.1). For the diC14:0 lipids [15] used for MPPNs formed from MscS [4, 5], we have [14, 33] m 1:76 nm. The membrane-spanning region of MscS [3, 8] has an approximately conical shape [13] with radius i 3:2 nm in the lipid bilayer midplane and bilayer- protein contact angle 0:46–0:54 rad (see fig. 2.1 and App. B). In the Monge representation, the bending deformation energyE b of a lipid bilayer is captured by the Helfrich-Canham-Evans free energy of bending [36–38] E b = Z dA K b 2 (C 1 +C 2 ) 2 ; (2.1) whereK b is the lipid bilayer bending rigidity, andC 1 andC 2 are the two principal cur- vatures pointwise on the membrane. Here we have neglected the constant contribution due to the Gaussian curvature [6, 42], and set the lipid spontaneous curvature equal to zero. In the almost planar membrane (Monge) approximation, we have, to leading order, the bending deformation energy as a functional ofh + andh G b [h + ;h ] = K b 4 Z dA (r 2 h + ) 2 + (r 2 h ) 2 ; (2.2) where we assume that the bending rigidity associated with individual membrane leaflet is half of the bilayer bending rigidity,K b . As discussed previously, a hydrophobic thickness mismatch at the bilayer-protein interface will induce bilayer thickness deformations, i.e. u + and u , which are equal 11 to zero for the unperturbed lipid bilayer. A harmonic spring model that describes the mechanism of lipid bilayer leaflets is developed [32, 33, 40] to characterize the energy cost associated with bilayer thickness deformations G t [u + ;u ] = K t 2 Z dA u + +u 2m 2 ; (2.3) whereK t is the stiffness associated with lipid bilayer thickness deformations. Note that, following the same convention as we have for the potential energy of a harmonic spring, we include the prefactor 1=2 in eq. (2.3). To simplify our notation, it is mathematically convenient to recasth in terms of the midplane deformation field (see fig. 2.1), h = h + +h 2 ; andu in terms of the average thickness deformation field of the lipid bilayer leaflets (see fig. 2.1), u = u + +u 2 : Note that the height difference between the hydrophobic-hydrophilic interfaces of the two leaflets should, in principle, be equal to the projected bilayer thickness, which, to second order, yields the following geometrical constraint [41] h + h = (u + +m) 1 (rh + ) 2 2 + (u +m) 1 (rh ) 2 2 ; (2.4) which, to leading order [40, 41], becomes u = h + h 2m 2 ; (2.5) 12 With the notations above, eq. (2.3) can be written as G t [u] = K t 2 Z dA u m 2 ; (2.6) while, to leading order, eq. (2.2) becomes G b [h;u] = K b 4 Z dA n r 2 (h +u +m) 2 + r 2 (hum) 2 o (2.7) = K b 4 Z dA n r 2 (h +u) 2 + r 2 (hu) 2 o (2.8) = K b 2 Z dA (r 2 h) 2 + (r 2 u) 2 : (2.9) Furthermore, we allow for lipid bilayer perturbations due to externally applied ten- sion [43], which penalizes the deviation of the membrane area from the unperturbed projected membrane area. In the Monge representation, the leading-order elastic energy cost due to tension is given by G [h;u] = 2 Z dA (rh + ) 2 2 + (rh ) 2 2 + u + m + u m (2.10) = 2 Z dA (rh) 2 + (ru) 2 + 2u m : (2.11) For generality we consider in eq. (2.11) the term 2u=m, which accounts for stretch- ing deformations tangential to the leaflet surfaces [20, 35, 44], as well as the term (rh) 2 +(ru) 2 , which accounts for changes in the projection of the bilayer area onto the reference plane used in the Monge representation ofh [19, 33, 45]. For diC14:0 lipids [15], we have [14, 33]K b 14k B T , andK t 56:5k B T=nm 2 . 13 Therefore, to leading order inh andu, the total membrane deformation energy of the bilayer-protein system, denoted byG def [h;u], can be written as the sum of two decou- pled energies associated with midplane curvature deformations and bilayer thickness deformations, i.e., G def [h;u] =G b [h;u] +G t [u] +G [h;u]G h [h] +G u [u]; (2.12) where G h [h] = 1 2 Z dA K b (r 2 h) 2 +(rh) 2 ; (2.13) G u [u] = 1 2 Z dA K b (r 2 u) 2 +K t u m 2 +(ru) 2 + 2 u m : (2.14) In order to obtain the equilibrium deformation profiles h and u that minimize eqs. (2.13) and (2.14) with respect to h and u and their higher order derivatives, we recall the procedure for minimizing the functionalL =L(f;rf;r 2 f) (f can beh or u): 0 = Z dAL(f;rf;r 2 f) (2.15) = Z dA @L @f f + @L @(rf) (rf) + @L @(r 2 f) (r 2 f) (2.16) = Z dA @L @f fr @L @(rf) f +r 2 @L @(r 2 f) f (2.17) = Z dAf @L @f r @L @(rf) +r 2 @L @(r 2 f) ; (2.18) 14 where we have integrated by parts and eliminated the boundary terms because they are either fixed or optimized to minimize the deformation energy. Sincef is arbitrary, we must have @L @f r @L @(rf) +r 2 @L @(r 2 f) = 0: (2.19) Substituting eqs. (2.13) and (2.14) into eq. (2.19) we obtain K b r 4 hr 2 h = 0; (2.20) K b r 4 ur 2 u + K t m 2 u + m = 0: (2.21) The solutions to eqs. (2.20) and (2.21) depend on the boundary conditions at the bilayer- protein interface, which are subject to the molecular properties of lipids and proteins considered in our model. 2.2 Bilayer mechanics of MPPNs For further convenience, we use here the Monge representation of h and assume rotational symmetry in h and u about the protein center, i.e. h = h(), u = u(), and dA = 2 d, with denoting the distance from the protein center. The protein- induced lipid bilayer deformations in eqs. (2.13) and (2.14) yield [13, 39, 42, 46– 48] bilayer-mediated interactions between membrane proteins in MPPNs. For the case of rotationally-symmetric membrane inclusions considered here, G h and G u are both expected to favor hexagonal protein arrangements [6, 23, 24, 40, 49–51]. The resulting contributions to the MPPN energy can be calculated [7] from eqs. (2.13) and (2.14), at the mean-field level, by approximating the hexagonal unit cell by a circle [6, 23, 24, 40, 49] of radius o (see fig. 2.1). The membrane patch radius o depends on the number of proteins per MPPN,n, and the MPPN bilayer midplane radius,R, via 15 o =R sin [6, 7], where the membrane patch angle = arccos[(n 2)=n] so that the total area covered by membrane patches is equal to the total MPPN area. Through min- imization ofG h andG u in eqs. (2.13) and (2.14) with respect toh() andu() in each membrane patch we derive, below, general analytic expressions for the MPPN midplane and thickness deformation energies,E h (n;R) andE u (n;R) [see eqs. (2.38) and (2.55)]. During MPPN self-assembly [4, 5], MPPNs are not expected to be able to support pres- sure gradients, suggesting that = 0. For completeness, however, we provide, below, general analytic solutions for arbitrary. These solutions could be used, for instance, to determine the shape of MPPNs for finite at fixedn andR. 2.2.1 MPPN midplane deformation energy The Euler-Lagrange equation in eq. (2.20) can be written as r 4 h = 2 r 2 h; (2.22) where = p =K b is the inverse decay length of midplane deformations [33], with the general solution [42] h() =A h I 0 () +B h K 0 () +C h +D h ln; (2.23) whereI 0 andK 0 are the zeroth-order modified Bessel functions of the first and second kind, respectively. The constantsA h , B h , C h , andD h in eq. (2.23) are determined by the boundary conditions along the bilayer-protein interface and the outer boundary of the membrane patch. In particular, the slope of the lipid bilayer at the bilayer-protein interface is given by h 0 ( i ) a = tan, while the slope at the outer boundary is 16 given byh 0 ( o ) b = tan. Furthermore, we impose [6, 7] a zero-force boundary condition [42] at = o , @ @ h() 2 h() =o = 0; (2.24) and fix the (arbitrary) reference point ofh viah( i ) = 0. These four boundary condi- tions, together with eq. (2.23), imply that A h = [bK 1 ( i )aK 1 ( o )]=F; (2.25) B h = [bI 1 ( i )aI 1 ( o )]=F; (2.26) C h = [aK 0 ( i )I 1 ( o ) +aI 0 ( i )K 1 ( o )b=( i )]=F; (2.27) D h = 0; (2.28) where F =[K 1 ( i )I 1 ( o )I 1 ( i )K 1 ( o )]; (2.29) andI 1 andK 1 are the first-order modified Bessel functions of the first and second kind, respectively. Integration of eq. (2.13) with eq. (2.23) from = i to = o for alln membrane patches thus results in the MPPN midplane deformation energy,E h (n;R). For mathematical simplicity, we apply here a similar method for calculating the energy as provided by Weikl et al. [42] (i.e., integrate by parts so that the results only depend on boundary terms), and writeh as h() =f() +g(); (2.30) 17 withf() A h I 0 () +B h K 0 () satisfying the nonhomogeneous equationr 2 f = 2 f and g() C h +D h ln satisfying the homogeneous equationr 2 g = 0. The midplane deformation energy is then E h (n;R) =n Z o i d[K b (r 2 f +r 2 g) 2 +(rf +rg) 2 ]; (2.31) which, expanding the quadratic terms, can be written as E h (n;R) =n Z o i d 8 < : [K b (r 2 f) 2 +(rf) 2 ] | {z } T 1 + 2[K b r 2 fr 2 g +rfrg] | {z } T 2 + [K b (r 2 g) 2 +(rg) 2 ] | {z } T 3 9 = ; ; (2.32) 18 where T 1 = n Z o i d[K b ( 2 f)(r 2 f) +(rf) 2 ] = n Z o i d[fr 2 f + (rf) 2 ] = n Z o i dr (frf) = n o f @f @ o i f @f @ i ! ; (2.33) T 2 = 2n Z o i d[K b r 2 fr 2 g +r (frg)fr 2 g] = 2n Z o i d(K b r 2 ff)r 2 g + 2n o f @g @ o i f @g @ i ! = 2n o f @g @ o i f @g @ i ! = 2nD h [f( o )f( i )] = 0; (2.34) T 3 = n Z o i d[K b (r 2 g) 2 +(rg) 2 ] = n Z o i d(rg) 2 = nD 2 h Z o i d= = 0: (2.35) 19 Therefore we have that E h (n;R) = T 1 +T 2 +T 3 = T 1 = n o f @f @ o i f @f @ i ! : (2.36) At the bilayer-protein interface we have a @h @ i = @(f +g) @ i = @f @ i + D h i = @f @ i ; (2.37) and similarly @f @ o =b at the outer patch boundary. The midplane deformation energy now becomes E h (n;R) = n [b o f( o )a i f( i )] = nfb o [A h I 0 ( o ) +B h K 0 ( o )]a i [A h I 0 ( i ) +B h K 0 ( i )]g : (2.38) Finally, by plugging eqs. (2.25 – 2.28) into eq. (2.38) we obtain the midplane curvature deformation energy: E h (n;R) = nK b I 1 ( o )K 1 ( i )I 1 ( i )K 1 ( o ) (2ab +[b 2 o (I 1 ( i )K 0 ( o ) +I 0 ( o )K 1 ( i )) +a 2 i (I 1 ( o )K 0 ( i ) +I 0 ( i )K 1 ( o ))]): (2.39) For o !1 andb! 0 [6], eq. (2.38) yields the minimum of the midplane deformation energy in eq. (2.13) for a single conical inclusion (n = 1) in an infinite, asymptotically flat lipid bilayer membrane [33, 42], while for! 0 we recover the results in Refs. [6, 7]. 20 2.2.2 MPPN thickness deformation energy The Euler-Lagrange equation (eq. (2.21)) associated with lipid bilayer thickness deformations can be written as (r 2 + )(r 2 ) u = 0; (2.40) where u() =u() + m Kt and = 1 2K b r 2 4K b K t m 2 ! : (2.41) Equation (2.40) has the solution [31, 48, 52] u() =A + u K 0 ( p + ) +A u K 0 ( p ) +B + u I 0 ( p + ) +B u I 0 ( p ); (2.42) where the constantsA u andB u are fixed by the boundary conditions onu() along the bilayer-protein interface and the outer boundary of the membrane patch, respectively. To determine the boundary conditions onu() we first note [39, 49] that, by symme- try (i.e. periodic boundary condition),u 0 ( o ) = 0 in our mean-field model of MPPNs. We also set u 0 ( i ) = 0, which is consistent with experiments on gramicidin channels [31, 53, 54] and the mechanosensitive channel of large conductance [33, 35, 44, 45], but other choices for this boundary condition could be implemented [32, 39, 41, 49, 55– 61]. Furthermore, we assume [13, 28, 31, 39, 49] that the lipid bilayer deforms along the bilayer-protein interface so as to match the hydrophobic thickness of the membrane protein, yieldingu( i ) = U, with the hydrophobic thickness mismatchU = 1 2 Wm, whereW 3:63 nm for MscS [3, 8] so thatU 0:055 nm for MPPNs formed from MscS and diC14:0 lipids [4, 5, 11, 15]. 21 Finally, a fourth boundary condition is obtained (see App. A for greater detail) by lettingu( o ) vary so as to minimize the thickness deformation energy, which amounts [6, 42] to a zero-force boundary condition at = o analogous to eq. (2.24): @ @ r 2 u() 2 u() =o = 0; (2.43) where, as in eq. (2.24), = p =K b . Note that the zero-force boundary condition in eq. (2.43) does not explicitly depend on the terms in eq. (2.14) that only involveu, and not its derivatives. Together with eq. (2.42), the above boundary conditions imply that A u = I 1o (Q 1o1i Q 1i1o ) U + m Kt S ; (2.44) B u = K 1o (Q 1o1i Q 1i1o ) U + m Kt S ; (2.45) where S =I 1o (P + 0i1i P ++ 0i1i )I + 1o + (Q ++ 1i0i +Q + 0i1i )K + 1o +K 1o (Q +++ 1i1o Q +++ 1o1i )I 0i (Q ++ 1o0i +Q ++ 0i1o )I 1i ; (2.46) and we define, forj = 0; 1,l = 0; 1, =i;o, and =i;o, P jl p K j K l ; (2.47) Q jl p I j K l ; (2.48) 22 andK j K j ( p ) andI j I j ( p ). Integration of eq. (2.14) with eq. (2.42) from = i to = o for alln membrane patches thus results in the MPPN thickness deformation energy,E u (n;R). First note that the integrand in eq. (2.14) can be rearranged as follows: = K b (r 2 u) 2 +K t u m Kt m 2 +(r u) 2 + 2 u m Kt m = K b (r 2 u) 2 +K t u m 2 2 u m + 2 K t +(r u) 2 + 2 u m 2 2 K t = K b (r 2 u) 2 +K t u m 2 +(r u) 2 2 K t : (2.49) From eq. (2.21), we know that Kt m 2 u =r 2 uK b r 4 u, and hence = K b (r 2 u) 2 +(r u) 2 + ur 2 uK b ur 4 u 2 K t = r [K b (r u)r 2 uK b ur 3 u + ur u] 2 K t : (2.50) Therefore, the bilayer thickness deformation energy can be expressed as E u (n;R) =n Z o i d 0 B B @ r [K b (r u)r 2 uK b ur 3 u + ur u] | {z } T 1 2 K t |{z} T 2 1 C C A ; (2.51) where T 2 = n 2 K t Z o i d = n 2 2K t ( 2 o 2 i ): (2.52) 23 ForT 1 , we can again use the divergence theorem to transform the surface integral to a line integral: T 1 = n o K b @ u @ r 2 uK b u @ @ r 2 u + u @ u @ o n i K b @ u @ r 2 uK b u @ @ r 2 u + u @ u @ i : (2.53) Note that @ u @ o = 0 and @ u @ i = 0 from the boundary conditions on the slope of bilayer thickness deformations, and hence we have T 1 =nK b i u @ @ r 2 u i o u @ @ r 2 u o ! : (2.54) The bilayer thickness deformation energy in eq. (2.51) therefore becomes E u (n;R) = T 1 T 2 = nK b i u @ @ r 2 u i o u @ @ r 2 u o ! + n 2 2K t ( 2 i 2 o ) = nK b i U + m K t @ @ r 2 u i + n 2 2K t ( 2 i 2 o ) = n K b i U + m K t 3=2 + B + u I + 1i A + u K + 1i + 3=2 (B u I 1i A u K 1i ) + 2 2K t 2 i 2 o : (2.55) 24 For o !1, eq. (2.55) reproduces previous results [31, 33, 62] on the minimum of the thickness deformation energy in eq. (2.14) for a single cylindrical inclusion (n = 1) in an infinite, asymptotically flat lipid bilayer membrane. 2.2.3 MPPN defect energy The spherical shape of MPPNs necessitates topological defects in the preferred hexagonal packing of membrane proteins which, in analogy to viral capsids [25, 26], yields [7] an energy penalty characteristic ofn. This energy penalty can be quantified [7, 25], at the mean-field level, by approximating the spring network associated with the preferred hexagonal protein arrangements [6] by a uniform elastic sheet [34, 63] with stretching modulus K s = p 3 24n @ 2 E 0 @ o 2 o= min ; (2.56) where E 0 = E h +E u and min yields the global minimum of E 0 for o i . We quantify, at the mean-field level, the deviation from the preferred hexagonal packing of membrane proteins due to the spherical shape of MPPNs through [7, 25] the fraction of the surface of a sphere enclosed byn identical non-overlapping circles at closest packing [10],p(n), resulting in the MPPN defect energy (see App. C) E d (n;R) = 2K s R 2 p max p(n) p max 2 ; (2.57) where p max = =2 p 3 corresponds to uniform hexagonal protein arrangements. We calculate the MPPN energyE min (n) by minimizing the sum ofE h ,E u , andE d at each n with respect toR. 25 2.2.4 Steric constraints on protein separation To account for steric constraints on lipid and protein size we only allow [7] for membrane patch sizes> i + l when calculatingE min (n), where the lipid radius l 0:45 nm for the diC14:0 lipids [11, 15] used for MPPNs formed from MscS [4, 5]. In particular, to determine the steric constraint on protein separation in our mean-field model of MPPNs we consider neighboring non-overlapping circular membrane patches. We then require that the sum of the areas of the spherical caps enclosed by these circles must be equal to the overall MPPN area multiplied by p(n). Denoting the radius of the non-overlapping circular membrane patches by 0 o 1 and the patch boundary angle by 0 , we have sin 0 = 0 o =R. The solid angle enclosed by each patch is given by 2 (1 cos 0 ). Upon equating the area of then spherical caps to the total MPPN area multiplied byp(n) we find n 2R 2 2 4 1 s 1 0 o R 2 3 5 = 4R 2 p(n): (2.58) The steric constraint on protein separation in our mean-field model is then given by 0 o i + l , with the membrane patch size 0 o defined by eq. (2.58). 2.3 MPPN mean-field energy To calculate the MPPN energy E min (n) we minimize the sum of E d in eq. (2.57), E h in eq. (2.55), and E u in eq. (2.55) at each n with respect to R, with all parame- ters determined directly by the molecular properties of the lipids and proteins forming MPPNs [4, 5]. Below, we first present the results for a special case where no thickness 1 The variable o (> 0 o ) (see fig. 2.1) corresponds to overlapping membrane patches [6, 24] and, as such, is less convenient for defining a suitable steric constraint on protein separation. 26 deformations are taken into account, i.e., the MPPN energyE = E h +E d , which can be achieved in experiments by matching the hydrophobic thickness of lipid bilayers and proteins, and then consider a more generalized MPPN energy whereE =E h +E u +E d . 2.3.1 MPPN mean-field energy for perfect hydrophobic matching Following Sec. 2.2 we estimate the MPPN bending energy E b using a formalism developed in the context of membrane budding [6, 23, 24], where minimization of the Helfrich-Canham-Evans bending energy [19, 36–38] with respect to the bilayer mid- plane height field yields E h (n;R) = 2nK b (b o a i ) 2 2 o 2 i : (2.59) Equation (2.59) yields [6] a preferred unit cell size of hexagonally packed proteins, withE h = 0, which can be achieved if>, corresponding to close-packed catenoidal bilayer deformation profiles. Based on eq. (2.59) only (i.e. E u = 0), the stretching modulus given by eq. (2.56) based on eq. (2.59) becomes K s = K b 2 p 3 min (a 4 ;b 4 ) ja 2 b 2 j 2 i ; (2.60) which depends onn viab. We minimize the total MPPN energyE given by eqs. (2.59), (2.60), and (2.57), at each n, with respect to R, which yields the minimum MPPN energy E min (n) with all parameters determined directly by the molecular properties of the lipids and proteins forming MPPNs (see fig. 2.2). We find that MPPNs with n < n 0 , where n 0 20 for MPPNs formed from MscS [4, 5] with 0:5 rad, are strongly penalized by the MPPN bending energy, which cannot be minimized to zero in this regime. Furthermore, MPPNs with n < n 0 also tend to be penalized by the MPPN defect energy because 27 Figure 2.2: Minimized total MPPN energyE min for MPPNs formed from MscS [4, 5] with the contributions E h and E d due to bending deformations and packing defects versus n at = 0:5 rad. The inset shows the optimal sphere coverage p(n) for n identical non-overlapping circles [10] with particularly favorable packings at n = 12 (icosahedron),n = 24 (snub cube), andn = 48. (Inset after Ref. [10].) K s in eq. (2.60) can be large for smalln (see App. D). Forn > n 0 , in which case also >, we find a range of favorablen corresponding to locally optimal protein packings. However, forn>n 0 the MPPN energies associated with distinctn fall within just a few k B T of each other and, as we discuss further below, thermal effects are therefore crucial in this regime. Finally we note that, forn which allowE h = 0 in fig. 2.2, the preferred protein separation (and, hence, MPPN size) is set, within 0:5%, byE h in eq. (2.59). 2.3.2 General form of the MPPN mean-field energy To calculate the general MPPN energy E min (n) we minimize the sum of E d in eq. (2.57),E h in eq. (2.38), andE u in eq. (2.55) at eachn with respect toR, from which we obtain the MPPN energy per protein, E min (n)=n, with all parameters determined directly by the molecular properties of the lipids and proteins forming MPPNs [see fig. 2.3(a)]. We varym to produce values ofU betweenU0:5 nm andU 0:5 nm. 28 Figure 2.3: MPPN energy and MPPN stretching modulus for MPPNs formed from MscS [3, 8] at = 0:5 rad for = 0. (a) MPPN energy per protein,E min =n, obtained from eqs. (2.57), (2.38), and (2.55) vs. n and U. The magenta curves show the n-states with minimal E min =n, for 10 n 80, as a function of U. (b) MPPN stretching modulusK s obtained from eq. (2.56) with eqs. (2.38) and (2.55)vs.n andU. The white dashed curves indicate locations in parameter space with a discontinuous jump inK s as a function ofU. In both panels,U is changed by varyingm. 29 Such a U-range could potentially be realized in experiments on MPPNs formed from MscS [4, 5] by using lipids with different acyl-chain lengths [14, 33, 45, 64]. Note that, ifU is changed by varyingm, E min =n is not invariant underU!U becauseG u in eq. (2.14) explicitly depends onm. We find that the magnitude of the MPPN energy tends to increase with increasing jUj, becauseE u increases withjUj [33] (see fig. 2.3(a)). Similarly as for the caseU = 0 [7], the contributionE d toE min yields, forU6= 0, a series of local minima inE min =n at locally optimal protein packing states [7, 10]. We find that, at smalljUj, n = 48 provides the minimum of E min =n in the range 10 n 80, with several competing n yielding E min =n within a fraction of k B T of n = 48. AsjUj is increased, we find MPPNs with snub cube symmetry,n = 24, as well as icosahedral symmetry,n = 12, as the minima ofE min =n in the range 10 n 80. This can be understood by noting that bilayer-thickness-mediated interactions between integral membrane proteins favor close packing of membrane proteins [49, 51]. Particularly favorable protein packing states such as the icosahedron and the snub cube therefore become dominant asjUj is increased, with the icosahedron providing [10] the optimal protein packing for 10 n 80. Note that the transition fromn = 24 ton = 12 as the minimum ofE min =n with increasingjUj only occurs forU > 0 in fig. 2.3(a). This can be understood by noting that, for U > 0, m is smaller than for U < 0, yielding a larger magnitude of G u in eq. (2.14). The MPPN stretching modulus K s entering the MPPN defect energy in eq. (2.57) tends to increase with increasingjUj (see fig. 2.3(b)). Similarly as forE min ,K s is not invariant under U !U if, as in fig. 2.3, U is changed by varying m, because G u in eq. (2.14) explicitly depends on m. K s takes particularly large values at n 16 in fig. 2.3(b) because [7] the membrane patch radius o = min yielding the global minimum ofE 0 in eq. (2.56) for o i approaches i forn 16 (see App. D). The 30 Figure 2.4: Bilayer midplane and thickness deformation energies per membrane patch, E h =n andE u =n, obtained form eqs. (2.38) and (2.55), andE 0 =n = (E h +E u )=nvs. o for MPPNs formed from MscS [3, 8] at = 0:5 rad for = 0, n = 48, and the indicated values of U, which we obtain by changing m. The positions of the optimal o = min are indicated by dots, with a two-fold degenerate global minimum ofE 0 =n at U 0:175 nm for o i . The grey shaded region indicates the range o < i excluded by steric constraints. continuum model of MPPN bilayer mechanics used here may not give reliable results in this regime. We also find that, for certainn,K s is a discontinuous function ofU in fig. 2.3(b). This can be understood by noting that min is set by the competition between E h , which yields short-range repulsion and long-range attraction [6] between membrane proteins in MPPNs, andE u , which favors the smallest o allowed by steric constraints, but also yields a local energy minimum at intermediate o [39, 48, 49, 65] (see fig. 2.4). Forn < 40 withU > 0 (n < 41 withU < 0) in fig. 2.3(b), min always lies within the small- o regime ofE u . But, forn 40 withU > 0 (n 41 withU < 0) and small jUj in fig. 2.3(b), min falls into the intermediate- o regime ofE u . AsjUj is increased, the magnitude of E u increases while E h remains constant (fig. 2.4). As a result, for n 40 withU > 0 (n 41 withU < 0) and large enoughjUj in fig. 2.3(b), we find a transition in the position of min , from the intermediate- o regime ofE u to the small- o regime ofE u (fig. 2.4), resulting in a discontinuous jump in min and, hence,K s . The 31 discontinuity inK s in fig. 2.3(b) with increasingjUj is accompanied, for a givenn, by a discontinuous decrease in the preferred MPPN radius. 32 Chapter 3 Minimal molecular model of MPPN organization This chapter is organized as follows. In Sec. 3.1, we discuss our minimal molec- ular model of MPPN organization, which employs simulated annealing Monte Carlo simulations of protein-protein, lipid-protein, and lipid-lipid pair interactions potentials. In Sec. 3.2, we introduce a method to determine the polyhedral symmetry of MPPNs obtained from simulated annealing Monte Carlo simulations. In Sec. 3.3, we explore how the preferred symmetry of MPPNs varies if one of the MscS in MPPNs has gated and is stabilized in its open state. 3.1 Simulated annealing Monte Carlo simulations Our mean-field model of MPPNs assumes [6, 23–25] that, for a givenn, the protein arrangement in MPPNs is determined by close packing of circular membrane patches, each with a protein at its center. Following previous work on viral capsid symmetry [26], we test these assumptions through Monte Carlo simulations of a minimal molec- ular (particle-based) model of MPPN organization, which focuses on short-range inter- actions between lipids and proteins. In this model, we represent the lipid bilayer and membrane proteins by disks lying on the surface of a sphere and assume that lipids inter- act with other lipids and proteins via Lennard-Jones potentials [26] with, for simplicity, hardcore steric repulsion between proteins. These interactions can be parameterized as 33 described below, based on experiments and previous calculations [8, 11, 19, 66], but our simulation results are not sensitive to the particular interactions used. We employed sim- ulated annealing Monte Carlo simulations [67] with linear cooling to numerically deter- mine the minimum-energy configuration of lipids and proteins in our minimal molecular model of MPPN organization. Following experiments on MPPNs formed from MscS [4] we focused in our simulations on MPPNs withn = 24 and a total of 1700 lipids. Our mean-field model of MPPNs focuses on long-range (bilayer-mediated) inter- actions between proteins, and the interplay between the conical protein shape and the spherical geometry of MPPNs. In our minimal molecular model of MPPN organiza- tion we focus on the complementary scenario of short-range interactions between lipids and proteins, for which we consider a generic (Lennard-Jones) model capturing steric and hydrophobic effects. In particular, we used for our simulated annealing Monte Carlo simulations of our minimal molecular model of MPPN organization protein and lipid parameter values consistent with experiments on MPPNs formed from MscS and diC14:0 lipids [4, 5]. We represent the lipid bilayer and membrane proteins by disks lying on the surface of a sphere and assume that lipids interact with other lipids and proteins via Lennard-Jones potentials [26], V i (r) = i r i r 12 2 r i r 6 ; (3.1) withi = 1; 2 corresponding to lipid-lipid and lipid-protein interactions, wherer is the center-to-center particle distance. For simplicity, we use hardcore steric repulsion for the protein-protein interactions. The r i in eq. (3.1) are the minima of V i (r) and are determined by the lipid and protein disk sizes. The interaction strengths i in eq. (3.1) can be viewed as the energy penalties for exposing lipids or membrane proteins to an aqueous environment. As discussed below, experiments and previous calculations [19, 34 66] suggest 1 10k B T and 2 20k B T for diC14:0 lipids and MscS in eq. (3.1). To check for robustness of our simulation results we repeated our simulated annealing Monte Carlo simulations for 2 = 1 = 1–10, and also allowed for unfavorable long-range interactions [42, 46] between MscS, which we calculated analytically for general protein separations using the formalism developed in Refs. [47, 68]. We find that our results regarding the minimum-energy symmetry of MPPNs are not sensitive to the particular value of 2 = 1 and protein-protein interactions used. In our simulated annealing Monte Carlo simulations we took the MPPN sphere radius in our minimal molecular model of MPPN organization to correspond to the hydrophobic-hydrophilic interface in the outer membrane leaflet of MPPNs. The val- ues of the model parameters entering our simulations can then be estimated as follows. We first note that MPPNs formed from diC14:0 lipids and MscS have a lipid-protein ratio 70 [4], which yields a total number of lipids 1700 for MPPNs withn = 24. Experiments suggest [69] that, for lipid bilayer vesicles formed from diC14:0 lipids [4], 72:5% of the lipids forming the vesicle are located in the outer lipid bilayer leaflet. Assuming that the ratio of lipids in the outer and inner lipid bilayer leaflets in MPPNs [4] is the same as for lipid bilayer vesicles [69], we estimate that the lipid number in the outer MPPN bilayer leaflets 1200. Figure 3.1 shows the minimum-energy MPPN configuration found in our simula- tions, obtained from our minimal molecular model of MPPN organization for 24 MscS and 1200 lipids in the outer membrane leaflet. The results in fig. 3.1 suggest that, in the ground state of the system, MscS are arranged in the form of a snub cube, in agree- ment with the corresponding optimal protein packing assumed in the mean-field model of MPPNs. We checked for systematic bias by repeating the simulated annealing Monte Carlo simulations in fig. 3.1 with different random seeds. We also repeated our simula- tions for a range of initial temperatures in the simulated annealing procedure, between 35 Figure 3.1: Front and side views of the minimum-energy MPPN configuration obtained from our minimal molecular model of MPPN organization. The larger and smaller disks represent proteins and the lipid bilayer, with disk sizes corresponding to MscS [8] and diC14:0 lipids [11], respectively. The green and blue lines are obtained by connecting the centers of nearby MscS in the simulated MPPN configuration and by fitting the simulated MPPN configuration to a snub cube (dextro) using least-square minimization. five and twenty-five times the room temperature, with fig. 3.1 corresponding to an initial temperature of approximately ten times the room temperature. We used a protein disk radius 4:0 nm corresponding to [8] the outer membrane leaflet of MPPNs and a lipid disk radius l 0:45 nm for diC14:0 lipids [11]. As initial conditions for our simu- lated annealing Monte Carlo simulations we used non-overlapping but otherwise ran- dom configurations of protein disks and random configurations of lipid disks located on the surface of a sphere. We repeated our simulations for sphere radii between 11 nm and 13 nm with step size 0:1 nm. The minimum-energy MPPN configuration in fig. 3.1 was obtained with a sphere radius 12:3 nm which, within the limits of the various approxi- mations made here, is consistent with the corresponding MPPN size obtained from our mean-field model and observed [4] in experiments on MPPNs formed from MscS and diC14:0 lipids. To provide simple estimates of the approximate values of 1 and 2 in our minimal molecular model of MPPN organization we note that membrane proteins and lipids both 36 have substantial hydrophobic domains, which makes it energetically unfavorable for membrane proteins and lipids to be dissolved in water. We denote byE lipid the energy required to create the new water-hydrocarbon interface when lipids are dissolved in water. The value ofE lipid can be estimated [19] based on the water-hydrocarbon surface tension together with the effective area of the contact interface. For simplicity, we approximate [19] the hydrophobic regions of lipids as cylinders of radiusR c and length n cc l cc , wheren cc is the number of carbon atoms along the lipid hydrocarbon chain and l cc = 0:126 nm [19] is the average C-C bond length along the chain. Therefore, the area of the lipid hydrophobic region 2n cc R c l cc , which yields E lipid 2n cc R c l cc : (3.2) For the double-chained diC14:0 lipids used in experiments on MPPNs [4] there are n cc = 14 carbon atoms per hydrocarbon chain, with an effective cross-sectional radius R c = 0:3 nm [19]. Using a water-hydrocarbon surface tension 0:02 J=m 2 [19, 70], we thus findE lipid 30k B T for the two leaflets of a diC14:0 lipid bilayer. Assuming an approximately hexagonal packing of lipids in the lipid bilayer, each lipid has six nearest- neighbor lipids, which yields an effective interaction strength per lipid-lipid bond of 1 1 3 E lipid 10k B T . To estimate the strength of lipid-protein interactions we assume [66] that the transfer energy of protein residues between polar and hydrophobic media can be estimated based on the water-hydrocarbon surface tension . For proteins such as MscS which are much larger than lipids, one protein can be regarded as substituting for two nearest-neighbor lipids in the approximately hexagonal packing of lipids in the lipid bilayer. This yields an effective interaction strength per lipid-protein bond of 2 2 3 E lipid 20k B T . 37 3.2 Polyhedral symmetry of MPPNs To quantify the quality of the polyhedral fit in fig. 3.1 we proceeded as in exper- iments on MPPNs [4] and used least-square minimization to calculate the minimum fit error for 132 convex polyhedra [1]: the Platonic, Archimedean, Catalan, and John- son solids. We define [4] the minimization function as the sum over the squared dis- tances between the simulated positions of protein centers in MPPNs and the closest fitted polyhedron vertices. This corresponds to a many-to-one or one-to-many mapping for polyhedra which have fewer or more than 24 vertices, respectively. We used sim- ulated annealing Monte Carlo simulations to minimize the fit error with respect to the following variables: the position of the polyhedron center, the polyhedron size, and the Euler angles. The best fits among the Platonic, Archimedean, Catalan, and Johnson solids obtained via this minimization procedure are summarized in tab. 3.1. We find that the snub cube (dextro) provides overall the best fit to the simulated minimum-energy MPPN configuration in fig. 3.1. We note that in our minimal molecular model of MPPN organization the preference of the minimum-energy configuration for one chiral poly- hedral symmetry, such as the snub cube (dextro), rather than its mirror-symmetric con- figuration, such as the snub cube (levo), only results from the particular (random) initial conditions used. Indeed, repeating the simulations of our minimal molecular model of MPPN organization in fig. 3.1 using the mirror symmetric initial protein configuration we find the snub cube (levo), rather than the snub cube (dextro), as the best fit. The optimal sphere coveragep(n) [10], plotted in fig. 2.2(inset) of Chap. 2, suggests a simple intuitive explanation for our result in fig. 3.1 and tab. 3.1 that, for n = 24, the snub cube yields the minimum-energy MPPN configuration in our minimal molec- ular model of MPPN organization. From an intuitive perspective it is to be expected that the minimum-energy configuration of lipids and proteins in our minimal molecular model corresponds to a configuration in which proteins are surrounded by lipids so as to 38 Type Polyhedron Fit error Å 2 A Snub cuboctahedron (dextro) 420.2 A Truncated cuboctahedron 3724.6 C Pentagonal hexecontahedron (levo) 5848.8 C Pentagonal hexecontahedron (dextro) 6414.5 J Gyroelongated square bicupola M2 7743.8 A Rhombicuboctahedron 7749.3 J Metabigyrate rhombicosidodecahedron 8254.4 J Trigyrate rhombicosidodecahedron 8841.9 J Gyroelongated square bicupola M1 8877.0 C Disdyakistriacontahedron 8947.2 A Snub icosidodecahedron (levo) 9107.8 J Parabigyrate rhombicosidodecahedron 9181.1 C Pentagonal icositetrahedron (dextro) 9289.2 J Gyrate rhombicosidodecahedron 9309.3 C Trapezoidal hexecontahedron 9425.9 A Snub cuboctahedron (levo) 9459.6 J Elongated square gyrobicupola 9867.1 A Rhombicosidodecahedron 12085.8 C Pentagonal icositetrahedron (levo) 12361.2 J Bigyrate diminished rhombicosidodecahedron 12455.7 Table 3.1: Top twenty polyhedral fits to the minimum-energy MPPN configuration in fig. 3.1 predicted by our minimal molecular model of MPPN organization. We define the fit error as the sum over the squared distances between the simulated positions of protein centers and the closest fitted polyhedron vertices. In our fitting procedure we allow for the 132 polyhedral symmetries [1] corresponding to the 5 Platonic solids (P), the 13 Archimedean solids (A) with two chiral pairs, the 13 Catalan solids (C) with two chiral pairs, and the 92 Johnson solids (J) with five chiral pairs. maximize favorable lipid-protein interactions, with dense packing of lipids in the inter- vening space between proteins. One therefore expects that the minimum-energy MPPN configuration in our minimal molecular model of MPPN organization corresponds to an arrangement in which proteins are at the centers of (non-overlapping) circles of maxi- mal radius. This is the protein arrangement implied byp(n) in fig. 2.2(inset) of Chap. 2 [10] yielding, for n = 24, a snub cube symmetry of protein centers, as also found in 39 our simulated annealing Monte Carlo simulations of our minimal molecular model of MPPN organization. 3.3 Polyhedral symmetry of MPPNs for a single gated ion channel Note that in experiments [4, 5] on the self-assembly of MPPNs formed from MscS, the mechanosensitive channels are all in their closed state. However, in a native cell environment, mechanosensitive channels may open as a response to changes in trans- membrane gradients, and may also be possible [4] to realize such transmembrane gra- dients for MPPNs. Figure 3.2 shows the geometric difference in the transmembrane region of the closed (left panel) and open (right panel) state of MscS [3, 8, 9, 12]. As far as our minimal molecular model is concerned, proteins in these two conformational states exhibit different sizes, which correspond to two protein disk radii. We consider Figure 3.2: Molecular structure of MscS. The left panel shows the closed state of MscS [3, 8] used previously in experiments on MPPNs [4, 5], with [9] Protein Data Bank ID 2OAU. The right panel shows the open state of MscS [12], with [9] Protein Data Bank ID 4HWA. Different colors in both panels indicate different MscS subunits [9]. 40 here the simplest scenario, where 1 out of 24 MscS is in the open state with a protein disk radius 4.5 nm, and the remaining 23 MscS are in the closed state with a protein radius 4.0 nm, both corresponding to [8] the outer membrane leaflet of MPPNs. We repeated our simulated annealing Monte Carlo simulations to determine the symmetry of MPPNs in our minimal molecular model of MPPN organization. Since the locations of the 24 vertices on a snub cube are identical, it is therefore arbitrary for the selec- tion of the open-state protein, provided that the randomness of initial configurations and simulated annealing process is checked for systematic bias. Note that in the limit where the size of open-state protein goes to infinity, the optimal sphere size goes to infinity as well so that all the other particles (including closed-state proteins and lipids) become infinitely small. As a result, the original snub cube arrange- ment of proteins will be largely distorted and in fact no polyhedral symmetry is guaran- teed for such MPPNs. On the other hand, we have shown in Sec. 3.2 that MPPNs with Figure 3.3: Front and side views of the minimum-energy MPPN configuration obtained from our minimal molecular model of MPPN organization with one MscS, indicated by a red circle, in the open state. The larger and smaller disks represent proteins and the lipid bilayer, with disk sizes corresponding to MscS [8] and diC14:0 lipids [11], respec- tively. The green and blue lines are obtained by connecting the centers of nearby MscS in the simulated MPPN configuration and by fitting the simulated MPPN configuration to a snub cube (dextro) using least-square minimization. 41 Type Polyhedron Fit error Å 2 A Snub cuboctahedron (dextro) 1349.0 A Snub cuboctahedron (levo) 6320.1 C Pentagonal hexecontahedron (levo) 6456.7 C Pentagonal hexecontahedron (dextro) 7594.9 J Gyroelongated square bicupola M1 7929.2 C Trapezoidal hexecontahedron 8073.0 C Pentagonal icositetrahedron (dextro) 8336.2 J Gyroelongated square bicupola M2 8438.6 A Truncated cuboctahedron 8458.5 A Rhombicuboctahedron 8572.7 J Trigyrate rhombicosidodecahedron 8891.1 J Metabigyrate rhombicosidodecahedron 8960.9 J Parabigyrate rhombicosidodecahedron 9845.1 J Elongated square gyrobicupola 9966.2 A Snub icosidodecahedron (levo) 10290.0 J Gyrate rhombicosidodecahedron 10544.1 C Pentagonal icositetrahedron (levo) 10598.7 A Snub icosidodecahedron (Dextro) 10649.8 C Disdyakistriacontahedron 11475.4 C Pentakisdodecahedron 11491.4 Table 3.2: Top twenty polyhedral fits to the minimum-energy MPPN configuration in fig. 3.3 predicted by our minimal molecular model of MPPN organization, obtained as in tab. 3.1. a uniform protein disk radius 4.0 nm have a very well-defined snub cube symmetry. Therefore, if it is the case that MPPNs with 1 open-state and 23 closed-state MscS still have snub cube symmetry with an open-state protein radius 4.5 nm, we conclude that also for a smaller open-state protein radius the snub cube symmetry of MPPNs will still hold. Figure 3.3 shows the minimum-energy MPPN configuration found in our simula- tions, obtained from our minimal molecular model of MPPN organization for 23 closed- state MscS, 1 open-state MscS, and 1200 lipids in the outer membrane leaflet. The results in fig. 3.3 suggest that, in the ground state of the system, MscS are still arranged 42 in the form of a snub cube (dextro) [4], with a best fit error 1300 Å 2 , while the second- and third-best fits are provided by the snub cube (levo) and pentagonal hexecontahedron (levo) with the substantially larger fit errors 6300 Å 2 and 6500 Å 2 , respectively (see tab. 3.2). 43 Chapter 4 Statistical thermodynamics of MPPN self-assembly This chapter discusses the statistical thermodynamics of MPPN self-assembly. In Sec. 4.1, we discuss the equilibrium distribution of MPPNs in an aqueous environment. In Sec. 4.2, we obtain MPPN self-assembly phase diagrams in different sections of parameter space. 4.1 Equilibrium distribution of MPPNs Following the formalism developed in Refs. [19–21] in the context of self-assembly of amphiphile aggregates, we note that for a mixture of N w water molecules and a variety of different types of coexisting MPPNs with N M n MPPNs in each MPPN type characterized byn, the system entropy is [20] S =k B ln =k B ln (N w + P n N M n )! N w ! Q n N M n ! : (4.1) Using Stirling’s approximation, i.e., ln N!N lnNN forN 1, we have 44 S=k B (N w + X n N M n ) ln(N w + X n N M n ) (N w + X n N M n ) (N w lnN w N w ) ( X n N M n lnN M n X n N M n ) (4.2) = (N w + X n N M n ) ln(N w + X n N M n )N w lnN w X n N M n lnN M n (4.3) =N w ln 1 P n N M n N w + P n N M n X n N M n ln N M n N w + P n N M n : (4.4) In the dilute limit of MPPNs, which is the regime relevant for experiments on MPPNs formed from MscS [4, 5], we have N w P n N M n , so we can replace the denominator (N w + P n N M n ) byN w . We can further simplify the first logarithmic term in eq. (4.4) via ln(1 +) for small, resulting in S=k B X n N M n ln N M n N w 1 : (4.5) The free energy of the system is thus F =ETS = X n N M n +k B T ln N M n N w 1 ; (4.6) where is the MPPN internal energy, which depends onn. If different types of MPPNs contain the same type of membrane proteins 1 , the system free energy becomes F = X n N n n +k B T ln N n =n N w 1 ; (4.7) 1 For MPPNs withn proteins embedded in each MPPN, if the number of lipids in each MPPN is deter- mined by minimizing the MPPN internal energy, then we have a unique one-to-one mapping regarding the number of enclosed proteins and lipids in each MPPN. Therefore, either the protein number, denoted byn, or the lipid number suffices to define the MPPN state. 45 whereN n is the total number of proteins bound in MPPNs withn proteins each (denoted as n-state MPPNs), and N M n = N n =n. For experiments [4] on the self-assembly of MPPNs formed from MscS, the total supply of proteins is restricted to their overall (fixed) concentration in the solution. We hence minimize the free energy under the contraint X n N n =N total ; (4.8) whereN total denotes the total number of proteins in the solution. If we define the number fraction of n-state MPPNs with respect to water by (n) = N n =nN w , and note that =E min (n) whereE min (n) is the minimized internal energy of n-state MPPNs from our mean-field model in Chap. 2 Sec. 2.3, we obtain from eq. (4.7) F =N w X n (n) [E min (n) +k B T (ln (n) 1)] : (4.9) The aforementioned molecule number constraint is thus X n n (n) = N total N w : (4.10) 4.2 MPPN self-assembly phase diagram To calculate the MPPN self-assembly phase diagram we note [7] that for MPPNs formed from MscS the protein concentration 1 mg/mL [4], with the molecular mass 2:2 10 5 g/mol for MscS [71], yielding the protein number fractionc = P n N n =N w 7:810 8 . In the dilute limitc 1 with no interactions between MPPNs, minimization of the system free energy given by equation (4.9) with respect to the MPPN number fraction (n) [20, 25, 27] yields (n) =e [nE min (n)] ; (4.11) 46 where = 1=k B T , and the chemical potential is fixed by the constraint P n n(n) = c. For simplicity, we restrictn to the range 10n 80, yielding the MPPN equilibrium distribution(n) = (n)= P 80 n=10 (n). 4.2.1 MPPN self-assembly phase diagram for perfect hydrophobic matching Figure 4.1 shows the MPPN self-assembly phase diagram as a function of protein number fractionc and bilayer-protein contact angle for the region of parameter space relevant for MPPNs formed from MscS [4, 5]. In agreement with experiments [4, 5] our model predicts that MPPNs withn = 24 are dominant for MPPNs formed from MscS. As observed experimentally [4] the MPPNs in fig. 4.1 withn = 24 have the symmetry of a snub cube with MscS located at the polyhedron vertices. We note that MPPNs with n = 12 and icosahedral symmetry, which exhibit the closest packing of MscS for 10 n 80 [fig. 2.2(inset) of Chap. 2], yield(12) 1 in fig. 4.1 due to the relatively small [6] of MscS [3, 8, 13], which results in a largeE h forn = 12. Our mean-field model of MPPNs predicts, with all parameters fixed directly by experiments [3, 4, 8, 13, 14], that MPPNs withn = 24 have a bilayer midplane radiusR 10 nm for MPPNs formed from MscS [4, 5]. Adjusting for the length of the MscS cytoplasmic region 10 nm [8] (fig. 2.1 of Chap. 2), the size of the dominant MPPNs predicted by our model is in quantitative agreement with the total MPPN radius 20 nm measured in experiments [4] forn = 24. Apart from the dominant MPPNs withn = 24, experiments also suggest [4] that lipids and MscS can self-assemble into MPPNs with a smaller average radius, but the symmetry of these MPPNs is unclear. The observed sub-dominant MPPNs [4] may correspond to the low-symmetry structures competing with MPPNs withn = 24 in fig. 4.1. In particular, fig. 4.1 predicts that the most abundant low-symmetry MPPNs 47 Figure 4.1: MPPN self-assembly phase diagram obtained from eq. (4.11) as a function of protein number fractionc and bilayer-protein contact angle. The color map in the upper panel shows maximum values of(n) associated with the dominantn-states of MPPNs. The dominantn are indicated in each portion of the phase diagram, together with the associated MPPN symmetry [10]. Black dashed curves delineate phase bound- aries. The white dashed line indicates the valuec 7:8 10 8 corresponding to exper- iments on MPPNs [4] formed from MscS [8, 13] and the-range associated with MscS. The lower panel shows(n) forn = 20, 22, 24, 27, and 30 as a function of along the white dashed line in the upper panel. We use the model parameter values i 3:2 nm [8, 13] andK b 14k B T [14] corresponding to MPPNs formed from [4, 5] MscS and diC14:0 lipids. correspond to MPPNs withn = 20,D 3h symmetry, and a radius at the bilayer midplane which is reduced by 1 nm compared to MPPNs withn = 24. 48 Figure 4.1 shows that the dominant MPPN symmetry and size only weakly depend on c. This suggests that even if not all proteins in the system are incorporated into MPPNs, and the effective value ofc is smaller thanc 7:8 10 8 [4], the key model predictions discussed above remain unchanged—indeed, smallerc tend to increase the dominance of MPPNs withn = 24 andn = 20 (fig. 4.1) while leaving the MPPN radius unchanged. In contrast, fig. 4.1 suggests that is a key parameter setting the preferred symmetry and size of MPPNs. Calculating the MPPN self-assembly phase diagram as a function of and the protein radius i (see fig. 4.2), we find that the dominant MPPN symmetry is more sensitive to variations in than i . With the exception of n = 18, which is almost as closely packed as the locally optimal packing staten = 17 (fig. 2.2(inset) of Chap. 2), all of the dominant MPPN symmetries in fig. 4.2 correspond to locally optimal protein packings, with n = 24 yielding the largest (n). Finally, we note that the bilayer bending rigidity K b 14 k B T [14] of the lipids used for MPPNs [4] is small compared to other lipids [13, 14], and that it has also been suggested Figure 4.2: MPPN self-assembly phase diagram obtained from eq. (4.11) as a function of protein radius i and bilayer-protein contact angle. The white dashed line indicates the protein radius i 3:2 nm and-range associated with MscS [8, 13]. We use the same labeling conventions as in fig. 4.1 with the model parameter valuesc 7:810 8 [4] and K b 14 k B T [14] corresponding to MPPNs formed from [4, 5] MscS and diC14:0 lipids. 49 [55, 61, 72] thatK b may be increased in the vicinity of membrane proteins. We find (see App. E) that, asK b is being increased, the dominance of MPPNs withn = 24 becomes increasingly pronounced for MPPNs formed from MscS [4, 5]. We finally note that, as far as the use of MPPNs for structural studies is concerned, it is desirable to produce MPPNs that have a uniform polyhedral symmetry and size. As shown in figs. 4.1 and 4.2 as well as figs. E.1 and E.2 of App. E, small tend to yield large n. In the large-n regime, different n-states of MPPNs do not show sub- stantial differences in the optimal sphere coverage p(n) (fig. 2.2(inset) of Chap. 2). This generally leads to non-uniform MPPN symmetries, i.e., broad distributions ofn- states, which is expected to decrease the resolution of MPPN-based approaches for the structural analysis of membrane proteins. Our model suggests possible strategies for overcoming this practical limitation. In particular, our model suggests that MPPNs with smaller n are produced if the effective is increased. This could be achieved, for instance, through addition of suitable toxins [13] that localize to the lipid bilayer- protein interface and amplify protein-induced lipid bilayer bending deformations. Fur- thermore, to explore strategies for controlling and optimizing MPPN shape, we present below a generalized MPPN self-assembly phase diagram showing how bilayer-protein hydrophobic thickness mismatch acts as a key molecular control parameter for MPPN shape, with pronounced protein-induced lipid bilayer thickness deformations biasing the MPPN self-assembly phase diagram towards highly symmetric and uniform MPPN shapes. Through appropriate tuning of the competition between protein-induced lipid bilayer bending and thickness deformations (via, for instance, changes in the lipid tail length) it may be possible to stabilize MPPNs with highly symmetric and uniform shape, which may thus help to improve the structural studies of membrane proteins and control targeted drug delivery. 50 4.2.2 Generalized MPPN self-assembly phase diagram Figure 4.3 shows the MPPN self-assembly phase diagram as a function of bilayer- protein hydrophobic thickness mismatchU and bilayer-protein contact angle for the protein number fractionc 7:8 10 8 used in experiments on MPPNs formed from MscS [4, 5]. The lower panel in fig. 4.3 provides the MPPN fractions(n) for then- states dominant in the region of parameter space associated with [4, 5] MPPNs formed from MscS [3, 8] and diC14:0 lipids [11, 14, 15], which is indicated by a dashed hori- zontal line in the upper panel in fig. 4.3. In agreement with experiments [4, 5] and our previous results forU = 0 [7] (see Subsec. 4.2.1), we find that MPPNs with snub cube symmetry,n = 24, are dominant for MPPNs formed from MscS. Figure 4.3 shows that, compared to the case U = 0 [7], MscS-induced lipid bilayer thickness deformations enhance the dominance of MPPNs withn = 24. Apart from the dominant MPPNs with n = 24, we also find subdominant MPPNs withn = 20,D 3h symmetry, and a MPPN radius that is reduced by 1 nm compared to MPPNs withn = 24. Again, these results are consistent with experiments [4] as well as our previous results forU = 0 [7]. Figure 4.3 suggests that, in addition to [7], the bilayer-protein hydrophobic thick- ness mismatchU is a key molecular parameter controlling MPPN shape. We find that, as the magnitude ofU is being increased and contributions due to protein-induced lipid bilayer thickness deformations come to dominate the MPPN energy, highly symmet- ric protein packings such as n = 12, n = 24, and n = 48 [10] become increasingly dominant over large portions of the phase diagram. This can be understood by noting that bilayer-thickness-mediated interactions between integral membrane proteins favor close packing of membrane proteins [49, 51], making MPPN states with large pack- ing fractionsp(n) [10] strongly favorable, from an energetic perspective, for largejUj (see also fig. 2.3(a) of Chap. 2). Indeed, the icosahedron,n = 12, provides the largest value ofp(n) for then-range considered here [10]. We find that the MPPN radiiR of 51 Figure 4.3: MPPN self-assembly phase diagram obtained from eq. (4.11) withE min (n) determined by eqs. (2.57), (2.38), and (2.55) as a function ofU and. The color map in the upper panel shows the maximum values of (n) associated with the dominant n-states of MPPNs. The dominantn are indicated in each portion of the phase diagram, together with the associated MPPN symmetry [10]. Black dashed curves delineate phase boundaries. The red dashed line indicates the valueU 0:055 nm corresponding to the bilayer-protein hydrophobic thickness mismatch associated with [4, 5] MPPNs formed from MscS [3, 8] and diC14:0 lipids [14, 15], and the -range associated with MscS [3, 7, 8]. The lower panel shows(n) forn = 20, 22, 24, 27, and 30 as a function of along the red dashed line in the upper panel. We used the protein number fraction c 7:8 10 8 employed in experiments and set = 0. The orange shaded areas indicate regions for whichn = 12 is strongly penalized by large values ofK s resulting [7] from o = min ! i in eq. (2.56) (see also fig. 2.3(b) of Chap. 2). In these regions of parameter space, the continuum model of MPPN bilayer mechanics may not give reliable results for the dominantn-states of MPPNs. 52 Figure 4.4: MPPN radiiR of dominantn-states MPPNs forn = 12, 24, and 48, as a function of bilayer-protein hydrophobic thickness mismatch U for the entire and U range considered in fig. 4.3. In each panel, the green, red, and blue curves correspond to the upper limit, lower limit, and mean value of MPPN radii R with respect to , respectively. 53 the dominant MPPNs with n = 12, 24, and 48 in fig. 4.3 only show small variations with respect to U and for the parameter range considered in fig. 4.3 (see fig. 4.4). Figures 4.3 and 4.4 thus suggest that protein-induced lipid bilayer thickness deforma- tions tend to bias MPPN self-assembly towards highly symmetric and uniform MPPN shapes. In particular, we find, with all model parameters determined directly by exper- iments [3, 4, 8, 13, 14],R 7 nm, 10 nm, 14 nm for the regions in the phase diagram in fig. 4.3 for whichn = 12, 24, and 48 are dominant, respectively. As already noted in Subsec. 4.2.1, this implies that, adjusting [7] for the length of the MscS cytoplasmic region 10 nm [8], the value ofR predicted by our model of MPPN self-assembly for n = 24 is in quantitative agreement [7] with the MPPN size observed experimentally [4] forn = 24. 54 Chapter 5 Conclusion MPPNs constitute a novel form of ordered lipid-protein assembly intermediate between single particles and large crystalline structures [4, 5]. To aid the utilization of MPPNs for structural studies of membrane proteins in the presence of physiologi- cally relevant transmembrane gradients [4, 5], and for targeted drug delivery with pre- cisely controlled release mechanisms [4, 5], we have developed a physical description of MPPNs which connects the symmetry and size of MPPNs to key molecular proper- ties of the lipids and proteins forming MPPNs. Our model accounts for the energy cost of protein-induced lipid bilayer midplane curvature deformations [6, 23, 24], protein- induced lipid bilayer thickness deformations [33, 40, 41], topological defects in protein packing in MPPNs [25, 26], and the statistical thermodynamics [20, 25–27] of MPPN self-assembly. With all model parameters determined directly from experiments, our model correctly predicts the observed [4] symmetry and size of MPPNs formed from MscS. Realization of MPPNs as a novel method for membrane protein structural analysis [4, 5], and targeted drug delivery [4, 5], requires [4] control over MPPN shape. Our results suggest that, in addition to the bilayer-protein contact angle [7], the bilayer- protein hydrophobic thickness mismatch U is a key molecular control parameter for MPPN shape. It has been proposed [13, 73] that can be perturbed through addition of peptide toxins that localize to the bilayer-protein interface. Our results suggest [7] that, in general, small effective yield MPPNs with largen, and vice versa. However, it may be experimentally challenging to tune the effective associated with a given integral 55 membrane protein of unknown structure with sufficient precision so as to produce a particular MPPN symmetry. In contrast, a range ofU can be generated experimentally [13, 28–30], for a given integral membrane protein, via systematic changes in the lipid acyl-chain length [14, 64]. Moreover, U can also be modified experimentally through repositioning of amphipathic protein residues [74]. The general analytic expression of the MPPN energy, and corresponding self-assembly phase diagram, obtained here show that pronounced protein-induced lipid bilayer thickness deformations favor highly symmetric and uniform MPPN shapes. Our results suggest strategies for producing highly symmetric and uniform MPPNs in experiments, and may thus help to optimize MPPN shape for structural studies of membrane proteins and targeted drug delivery. The results on MPPN self-assembly described here suggest several directions for future work: In our analytic calculations we have been using a mean-field model in which we approximate hexagonally packed unit cells of membrane patches as circles. It would be interesting to lift this mean-field assumption, and to directly account for the polyhedral arrangement of proteins on MPPNs suggested by our minimal molecular model of MPPN organization (see Chap. 3). In particular, our mean- field model does not allow isolation of the possible role of many-body interactions in setting the MPPN symmetry and size (see also App. D), with the MPPN energy being additive in terms of the membrane patch energy. Moreover, a more precise representation of membrane protein geometry (e.g. the seven subunits of MscS [3, 8] form an overall heptameric shape in the MscS transmembrane region) could be applied as the boundary condition when calculating MPPN energy [75, 76], provided that we are able to construct a global analytical or numerical expression for the MPPN energy. 56 Utilization of MPPNs as a novel approach for targeted drug delivery with precisely controlled release mechanisms could possibly be realized through MPPNs with an open state of mechanosensitive ion channels. As a potential future research topic, it would therefore be worthwhile to consider the effect of opening one or more membrane proteins on MPPNs from both analytical and computational perspec- tives. Presumably, the size of open-state mechanosensitive channels, which can be directly measured from experiments, as well as the number fraction of open-state proteins, may perturb or even destroy the polyhedral arrangement of membrane proteins forn that correspond to locally optimal packing states [7, 10], and hence alter the thermodynamic competition between differentn-states of MPPNs. We considered here in our minimal molecular model of MPPN organization a simple scenario in which only one channel was assumed to be in the open state and found that, for physically reasonable protein sizes, the snub cube symmetry of MPPNs was preserved. However, if more than one channel in MPPNs is in the open state, the simulated annealing Monte Carlo simulations of our minimal molec- ular model of MPPN organization become considerably more involved and, in particular, it becomes more difficult to avoid biasing the simulations through the initial conditions used for the protein arrangement. Therefore, sampling of many different random initial conditions becomes necessary in order to obtain accurate outcomes. As a result, depending on whether the original polyhedral symmetry of MPPNs formed from all closed-state proteins is destroyed, we might need to revisit the validity of the assumptions underlying our analytical model for the case of multiple channels in MPPNs being in the open state. 57 Appendix A Derivation of the zero-force boundary condition for membrane thickness deformations To derive the zero force boundary condition for membrane thickness deformations, we study a perturbation [42] v(;) =u() +u() (A.1) of the equilibrium bilayer thickness deformation u() on an annulus S: i o around a conical inclusion. The perturbation is restricted by the boundary conditions of the membrane patch, i.e., uj o = u, uj i = 0,u j o @u @ o = 0, andu j i @u @ i = 0. The thickness deformation energy from eq. (2.14) is G u [v] = 1 2 Z dA K b (r 2 v) 2 +K t v m 2 +(rv) 2 + 2 v m =n Z o i " K b @ 2 v @ 2 + 1 @v @ 2 + @v @ 2 +K t v m 2 + 2 v m # d =n Z o i f(v;v ;v ;) d; (A.2) 58 where f(v;v ;v ;) K b (v +v =) 2 +v 2 +K t v m 2 + 2 v m : (A.3) In equilibrium,G u [v] is minimal, indicating that dG d =0 =n Z o i @f @v dv d + @f @v dv d + @f @v dv d =0 d = 0: (A.4) Via integration by parts we obtain 1 n dG d = Z o i @f @v @ @ @f @v + @ 2 @ 2 @f @v dv d d+ @f @v dv d @ @ @f @v dv d + @f @v dv d o i : (A.5) Inserting the definition off in eq. (A.3), we are led to 1 2n dG d = Z o i K b r 4 vr 2 v + K t m 2 v + m dv d d+ @ @ (vK b r 2 v) dv d +K b r 2 v dv d o i : (A.6) The equilibrium thickness deformation u() fulfills the Euler-Lagrange equation eq. (2.21). So the above integral vanishes at = 0. Since dv d =0 = u and dv d =0 =u , we must have, at = 0, @ @ (uK b r 2 u) o = 0: (A.7) 59 Appendix B MscS Transmembrane Geometry Structural studies suggest [3, 8, 13] that the membrane-spanning region of MscS has the shape of a truncated cone. To estimate the cross-sectional radius i of MscS in the lipid bilayer midplane and the lipid bilayer-MscS contact angle we rotate the known MscS structure [8] about its symmetry axis normal to the membrane, and fit truncated cones to the MscS transmembrane region (see fig. B.1 for representative snapshots). This yields the estimates i 3:2 nm and 0:46–0:54 rad, respectively. We simi- larly estimate that the length of the MscS cytoplasmic region 10 nm. Following an analogous procedure we estimate the radius of the transmembrane region of open MscS. 60 Figure B.1: Molecular structure of MscS in the closed state [8] with Protein Data Bank ID 2OAU represented using Visual Molecular Dynamics [9]. The green beads indicate hydrophobic residues [16] at pH 7. The truncated cones are chosen so as to enclose the approximate transmembrane region of MscS [8, 13]. The values of quoted in each panel are calculated from the respective truncated cones. 61 Appendix C Derivation of MPPN defect energy C.1 First Approach The continuum limit of the stretching energy of a hexagonal network of harmonic springs is given by [63] H = K 2 Z d 2 x(rr) 2 ; (C.1) where (rr) 2 = (@r=@x 1 ) 2 +(@r=@x 2 ) 2 , in whichx 1 andx 2 are internal coordinates and r = r(x 1 ;x 2 ) denotes the external coordinate specifying the location of the surface in the (three-dimensional) embedding space. The continuum force constantK in eq. (C.1) is related to the discrete force constantK 0 of the harmonic springs viaK = p 3K 0 [63]. For a uniform hexagonal lattice of MscS, the spring constant [6] K 0 = 1 12n @ 2 E 0 @ o 2 o= min : (C.2) To relate eq. (C.1) to the standard stretching energy of a uniform elastic sheet [34] formulated in terms of the areal strain A=A we consider the position of a point in a flat rectangular patch of an elastic sheet. We denote the width and height of the rectan- gular patch by L 1 and L 2 , respectively. Now assume that L 1 and L 2 are extended (or compressed) by L 1 and L 2 , respectively. A point in the patchr 0 (x 1 ;x 2 ) then moves to a new location r(x 1 ;x 2 ) =r 0 + L 1 L 1 x 1 i + L 2 L 2 x 2 j; (C.3) 62 with i and j denoting two orthogonal unit vectors. We therefore have that (rr) 2 = (@r=@x 1 ) 2 + (@r=@x 2 ) 2 = (L 1 =L 1 ) 2 + (L 2 =L 2 ) 2 . For uniform strain we also have that L 1 =L 1 = L 2 =L 2 , which yields (rr) 2 = 2(L 1 =L 1 ) 2 . Furthermore, to leading order the area change due to L 1 and L 2 is given by A = (L 1 + L 1 )(L 2 + L 2 )L 1 L 2 L 1 L 2 +L 2 L 1 ; (C.4) which, again to leading order, yields the squared areal strain A A 2 = L 1 L 1 + L 2 L 2 2 = 4 L 1 L 1 2 = 2(rr) 2 : (C.5) Thus, equation (C.1) can be expressed in terms of areal strain [34] as H = p 3K 0 4 Z d 2 x A A 2 : (C.6) For uniform areal strain, (A=A) 2 is a constant and hence [34] can be pulled out of the integral in eq. (C.6). This yields H = K s 2 A A A 2 ; (C.7) where the stretching modulus K s = p 3K 0 2 (C.8) as in eq. (2.56) of Chap. 2. The MPPN defect energy is therefore given by E d (n;R) = K s 2 A A A 2 ; (C.9) 63 whereA = 4R 2 . Similarly as in previous work on viral capsid self-assembly [25], we approximate, at the mean-field level, the areal strain by A A = p max p(n) p max ; (C.10) wherep(n) denotes the fraction of the surface of a sphere enclosed byn identical non- overlapping circles at closest packing [10] and the optimal coverage p max = =2 p 3 corresponds to hexagonal packing of circular membrane patches. Equation (C.2) with eqs. (C.8)–(C.10) result in the defect energyE d (n;R) in eq. (2.57) of Chap. 2, with the stretching modulus in eq. (2.56) of Chap. 2. C.2 Second Approach In our first approach above for deriving the MPPN defect energy in eq. (2.57) of Chap. 2 we used the result in Ref. [63] thatK = p 3K 0 , and then derived the stretch- ing energy in eq. (C.7) from eq. (C.1). To complement this derivation, we obtain here the stretching energy in eq. (C.7) directly from a discrete hexagonal lattice of harmonic springs. In the ground state, we take each triangular element of the lattice, composed of three proteins in the hexagonal spring network, to have side lengthr with each side cor- responding to a harmonic bond. If all three sides of the triangular element are stretched by r, the elastic energy associated with each side is given by 1 2 K 0 (r) 2 . We have three of these bonds per triangle, with each bond shared by two adjacent triangles. The overall energy associated with this area deformation is therefore given by E discrete = 1 2 3 1 2 K 0 (r) 2 : (C.11) 64 To determine the continuum stretching energy associated with eq. (C.11) we note that the area of an equilateral triangle of side lengthr is given byA = p 3r 2 =4. To leading order, this implies an area change A = p 3rr=2 due to the stretching deformation r!r + r. We therefore have that (A) 2 A = p 3(r) 2 (C.12) to leading order. The continuum stretching energy associated with uniform stretching deformations [34] can then be written as E continuum = K s 2 (A) 2 A = p 3K s 2 (r) 2 : (C.13) Setting E continuum = E discrete yields K s = p 3 2 K 0 as in eq. (C.8), resulting in the stretching modulus in eq. (2.56) of Chap. 2 and the defect energyE d (n;R) in eq. (2.57) of Chap. 2. 65 Appendix D Stretching modulus and MPPN mean-field energy Figure D.1(a) shows the stretching modulusK s in eq. (2.60) of Chap. 2 as a function ofn. Asn becomes large the MPPN “background curvature” is reduced, which leads to smallerb and hence smallerK s —indeed, in the planar limit, lipid bilayer curvature- mediated interactions are expected to be repulsive [6, 42, 46]. The maximum in K s is reached for n such that a b. Although the MPPN defect energy in eq. (2.57) of Chap. 2 is proportional to the overall MPPN area, which tends to increase withn, K s decreases sufficiently rapidly withn so that the MPPN defect energy tends to decrease with n in the large-n regime considered here. As in fig. 2.2 of Chap. 2, fig. D.1(b) Figure D.1: (a) MPPN stretching modulusK s and (b) minimized total MPPN energy E min for MPPNs formed from MscS [4] with the contributionsE h andE d due to bending deformations and packing defects versusn at = 0:5. The insets in panels (a) and (b) allow for the fulln-range 10 n 80 considered here, while for the main panels we use the samen-range as in the main panel of fig. 2.2 of Chap. 2. 66 shows the minimized total MPPN energyE min for MPPNs formed from MscS [4] with the contributionsE h andE d due to bending deformations and packing defects, but over the full n-range considered here. The local maximum of E min at n = 16 occurs as a result of the maximum ofK s atn = 16 for 0:5 (see fig. D.1(a; inset)). Based on Ref. [6], the divergence in K s in eq. (2.60) of Chap. 2 at a = b can be understood as follows. As in Ref. [6], we calculateK s by differentiatingE h in eq. (2.59) of Chap. 2 twice with respect to o , and then evaluating this second derivative ofE h at the minimum of E h . For any a6= b, E h diverges at o = i (this state is unphysical because it corresponds to a vanishing width of the lipid bilayer annulus around each protein; please see the discussion below for further details). Asa! b, the minimum in E h is obtained with o ! + i . As a! b, K s is therefore evaluated at o which come infinitesimally close to a o yielding a divergingE h , resulting in a divergingK s . However, as discussed below, this divergence does not affect the central results and predictions of our model. Below, we provide a more detailed discussion of the physical significance and mathematical properties ofE h andK s asa!b. We first note that, in each circular membrane patch, the protein-induced lipid bilayer curvature (bending) deformationsh(r), wherer is the radial coordinate, are given by [6] h(r) = (r 2 2 i )(b o a i ) + 2 o i (a o b i ) ln (r= i ) 2 ( 2 o 2 i ) ; (D.1) which, upon substitution into the Helfrich-Canham-Evans bending energy, E h [h] = K b 2 Z dxdy(r 2 h) 2 ; (D.2) yields, forn membrane patches, the MPPN bending energy in eq. (2.59) of Chap. 2: E h (n;R) = 2nK b (b o a i ) 2 2 o 2 i : (D.3) 67 For all physically relevant scenarios we must have i < o . To calculateK s in eq. (2.60) of Chap. 2, we minimize eq. (D.3), at eachn, with respect to the membrane patch radius o [77]. We thus find that eq. (D.3) exhibits two extrema: (1) o = a b i ; (2) o = b a i : (D.4) The solution (1) o yieldsE h = 0, while the solution (2) o yields a finiteE h . Depending on the relative values ofa andb, we can then distinguish between three cases: Forjaj>jbj (i.e.,> [78]), the solution (1) o satisfies (1) o > i and is therefore physically relevant, allowing states of minimum bending energy with E h = 0. From eq. (D.1) we see that these states correspond to purely logarithmic radial deformation profiles, resulting in catenoidal lipid bilayer deformations in each membrane patch. Forjaj <jbj (i.e., < ), we have that (1) o < i while (2) o > i . The solution (1) o is therefore physically irrelevant, the solution (2) o gives the states of minimum bending energy, and we necessarily have E h > 0. From eq. (D.1) we see that these states correspond to purely quadratic radial deformation profiles, yielding membrane patch deformations that are no longer catenoidal. For a = b (i.e., = ), one can still have E h = 0 as the minimum bending energy, but only for the (unphysical) case of a vanishing width of the lipid bilayer annulus, i = o . Figure D.2 shows the numerical values of the minima of the bending energy per membrane patch, b =E h (n;R)=n, as a function ofn 1 , together with the corresponding 1 Orb =tanfarccos[(n2)=n]g, which increases monotonically withn for the range10n 80 we focus on here. 68 Figure D.2: (a) Numerical values of the minima of the bending energy per membrane patch, b =E h (n;R)=n, obtained by minimizing eq. (D.3) with respect to o at eachn, and (b) corresponding second derivatives with respect to o , 00 b , evaluated for eachn at the o minimizing b . We used the same parameter values as in fig. 2.2 of Chap. 2. values of its second derivative with respect to o , 00 b , which is proportional to K s in eq. (2.60) of Chap. 2. We find that the divergence in 00 b and, hence,K s occurs atn 16, at the transition between (1) o and (2) o as the physically relevant states of minimum bending energy. Note from eq. (D.3) that, for anya6=b,E h diverges at o = i . Since at the transition between the (1) o -branch and the (2) o -branch of the minima of eq. (D.3) we have o ! + i , o -states with a finiteE h come infinitesimally close to a o -state with E h !1, yielding a diverging second derivative ofE h with respect to o . A few remarks are in order here. First we note that, to obtain a “true” divergence ofK s , one needsa = b. Sincen is an integer and, hence, andb are both discretized, K s !1 would require a fine-tuning of. In practice, one therefore only hasa b, which results in a large, but finite,K s when transitioning between the (1) o -branch and the (2) o -branch of the minima of eq. (D.3) (see fig. D.2). Furthermore, even ifa is fine- tuned to allowa =b, this unphysical state is effectively ruled out in our model because it would require an infinite energy. Finally, when calculating the total MPPN energy in fig. 2.2 of Chap. 2, we account for steric constraints on lipid and protein size by only allowing for membrane patch sizes> i + l , where l is the lipid radius. This means that, in the small-n regimen<n 0 where one can haveab, withn 0 20 for MPPNs 69 formed from MscS [4, 5] with 0:5, the contribution to the total MPPN energy due to bending deformations is large compared to the thermal energy scalek B T (see fig. 2.2 of Chap. 2 and fig. D.1(b)). Thus, the MPPN bending energy already effectively rules out states withn < n 0 . The divergence inK s in eq. (2.60) of Chap. 2 fora = b does therefore not affect the central results and predictions of our model. The protein-induced lipid bilayer curvature deformations considered here are thought [40, 50, 79] to induce non-pairwise-additive bilayer-mediated interactions between proteins. The many-body character of curvature-mediated interactions can potentially have a variety of interesting consequences [50, 80–82], such as stabiliza- tion of certain cluster geometries. The mean-field approach we employ here retains, in analogy to planar crystalline lattices [40], some features of the many-body charac- ter of curvature-mediated interactions, but does not allow isolation of the possible role of many-body interactions in setting the MPPN symmetry and size. Our mean-field model is motivated by the experimental phenomenology of MPPNs [4], which shows that MPPNs have a spherical shape, and that proteins have an approximately uniform distribution on MPPNs. It was found previously [6] that the mean-field model of MPPN bending energy employed here, which assumes a spherical shape of MPPNs and an approximately uniform distribution of proteins on MPPNs, yields good agreement with computer simulations of budding through interacting conical membrane inclusions or adsorbed, curvature-inducing particles [83, 84]. However, many-body effects not cap- tured by our mean-field approach may, for instance, have interesting consequences for MPPN symmetry and size in situations where the composition of MPPNs is modified to produce departures from approximately uniform protein distributions, such as in cases where some channels are trapped in the open state. 70 Appendix E Robustness of MPPN self-assembly phase diagram It has been suggested [55, 61, 72] that the lipid bilayer bending rigidity may be increased in the vicinity of membrane proteins. More generally, the value of the lipid bilayer bending rigidity K b 14 k B T reported for the diC14:0 lipids [14] used for MPPNs [4] lies at the lower end of the range of values of K b typically measured in experiments on general lipids [13, 14]. We therefore repeated our calculation of the MPPN self-assembly phase diagram in fig. 4.1 and fig. 4.2 of Chap. 4 for increased values of K b (see figs. E.1 and E.2). We confirmed that for the dominant MPPNs in fig. 4.1 and fig. 4.2 of Chap. 4 and figs. E.1 and E.2 we always have that, consistent with the Helfrich-Canham-Evans model [36–38],jrhj < 1, whereh(r) is the (rotationally- symmetric) height field describing the membrane deformation profile around each pro- tein andr is the radial coordinate [6]. Figures E.1(a,b) and E.2(a,b) demonstrate that, as the bending rigidityK b is being increased, the snub cube becomes more and more dominant within the parameter range relevant for MPPNs formed from MscS [4, 5]. More generally, figs. E.1(c) and E.2(c) indicate that a largerK b yields a stronger preference for close packing of proteins. This can be understood by noting that, for smallK b , the energy differences between different n-states of MPPNs are small in the large-n regime dominated by the MPPN defect energy (see, e.g., fig. 2.2 of Chap. 2), which allows strong thermal effects. In contrast, 71 Figure E.1: Same plots as in (a,b) fig. 4.1 and (c) fig. 4.2 of Chap. 4, but usingK b = 28k B T instead ofK b = 14k B T . 72 Figure E.2: Same plots as in (a,b) fig. 4.1 and (c) fig. 4.2 of Chap. 4, but usingK b = 56k B T instead ofK b = 14k B T . 73 large K b tend to produce large magnitudes of the MPPN defect energy in the large- n regime, increasingly biasing the MPPN self-assembly phase diagram towards states with (locally) optimal packing of proteins. Finally we note that the MPPNs withn = 24 obtained along the white dashed lines in figs. E.1 and E.2, which correspond to experiments on MPPNs formed from MscS [4, 5], have the same approximate size as the corresponding MPPNs withn = 24 in fig. 4.1 and fig. 4.2 of Chap. 4. 74 References [1] G. W. Hart. The encyclopedia of polyhedra. Available at www.georgehart.com/virtual-polyhedra/vp.html. Accessed Nov. 12, 2015., 2000. [2] M. R. Villarreal. Available athttps://commons.wikimedia.org/wiki/File:Cell_mem- brane_detailed_diagram_en.svg. Accessed Feb. 24, 2017. [3] R. B. Bass, P. Strop, M. Barclay, and D. C. Rees. Crystal structure of escherichia coli mscs, a voltage-modulated and mechanosensitive channel. Sci- ence, 298(5598):1582–1587, 2002. [4] T. Basta, H.-J. Wu, M. K. Morphew, J. Lee, N. Ghosh, J. Lai, J. M. Heumann, K. Wang, Y . C. Lee, D. C. Rees, and M. H. B. Stowell. Self-assembled lipid and membrane protein polyhedral nanoparticles. Proc. Natl. Acad. Sci. U.S.A., 111(2):670–674, 2014. [5] H.-J. Wu, T. Basta, M. Morphew, D. C. Rees, M. H. B. Stowell, and Y . C. Lee. 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Abstract (if available)
Abstract
This PhD thesis is devoted to the study of the physical mechanisms underlying self-assembly of membrane protein polyhedral nanoparticles (MPPNs). It provides analytical and computational approaches that allow investigation of the roles of bilayer-protein interactions, lipid bilayer mechanical properties, topological defects in protein packing, and thermal effects in MPPN self-assembly. Chapter 1 is an overall introduction to biological membranes, as well as the experimental phenomenology of MPPN self-assembly. In Chapter 2 we explore a mean-field model of the MPPN energy based on the continuum elasticity of lipid bilayer-protein interactions, which accounts for protein-induced lipid bilayer curvature deformations, lipid bilayer-protein hydrophobic thickness mismatch, and topological defects arising from the spherical shape of MPPNs. In Chapter 3 we justify some of the key assumptions underlying our mean-field of MPPN energy using a minimal molecular model of MPPN organization. In Chapter 4 we develop the statistical thermodynamics of MPPN self-assembly, which we then combine with our mean-field model of MPPN energy to calculate the MPPN self-assembly phase diagram. We find that, in addition to the bilayer-protein contact angle, which depends on the shape of membrane proteins, the bilayer-protein hydrophobic thickness mismatch is a key molecular control parameter for MPPN shape, with pronounced protein-induced lipid bilayer thickness deformations biasing the MPPN self-assembly phase diagram towards highly symmetric and uniform MPPN shapes. Finally, in Chapter 5, we summarize the background, and key results of the work described in this thesis, and discuss the broader significance of our work on the physics of membrane protein polyhedra.
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Li, Di
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The physics of membrane protein polyhedra
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Physics
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04/11/2017
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