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The physics of emergent membrane phenomena

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The physics of emergent membrane phenomena

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Content THE PHYSICS OF EMERGENT MEMBRANE PHENOMENA by Ahis Shrestha 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 2022 Copyright 2022 Ahis Shrestha Acknowledgements This work would not have been possible without the help and support of many people. Here I would like to take the opportunity to thank them and acknowledge their essential contributions. First and foremost, I would like to express my sincere gratitude to my advisor Prof. Christoph Haselwandter for his valued guidance and continued support throughout my years as a PhD student. His input and contributions has been paramount to the completion of my PhD thesis and research projects. I greatly appreciate all of his thoughtful insights and suggestions, which have been vital to the progress of my work. Prof. Haselwandter has been a constant source of inspiration for me in my academic and scientific pursuits. I consider myself extremely fortunate to have him as my advisor and collaborator for all these years. He has never hesitated to devote his valuable time to assist me with any of my needs and, for that, I will always be deeply grateful to him. I would like to give a huge thanks to my research collaborators Dr. Osman Kahraman and Prof. Fabien Pinaud, for their contributions, which have played a crucial role in the success of my scientific work. I am very thankful to Dr. Kahraman for always providing keen insights and helpful suggestions, and assisting me with learning new computational techniques. I also really want to thank Prof. Pinaud for his essential input and constructive feedback, and sharing his knowledge about the experimental side of research. Many thanks to Prof. Hubert Saleur, Prof. Moh El-Naggar, and Prof. James Boedicker, for their support through my academic journey at USC, and for partaking as members in my dissertation and qualification exam committee. I want to specially thank Prof. Saleur and Prof. El-Naggar for providing me the opportunity to attend their courses on soft matter physics and biological physics, from which I have learned a great deal. ii The work in this thesis was supported at USC by NSF Grants No. DMR-1554716 and No. PHY- 1806381, the James H. Zumberge Faculty Research and Innovation Fund at USC, and the USC Center for Advanced Research Computing. I would also like to thank my former mentors, professors and teachers. Special thanks to Prof. Katsushi Arisaka (University of California, Los Angeles) and Prof. Sylvio May (North Dakota State University), for their vital support and guidance in my early years as a physics student and researcher, and for inspiring me to continue pursuing scientific research. Finally, I would like to thank my family, relatives and friends. Firstly, I would like to extend my deepest gratitude to my parents Gyani Shrestha and Annurama Shrestha for always supporting me and my education throughout my life. Many thanks to my sibling Aastha Shrestha, and my relatives Mini Pradhan, Nirakar Pradhan and Asvin Pradhan, for their continued help and constant support over the years. Last but not least, I would like to thank my friends Kyle Meisch and Yatin Mondkar for engaging with me in insightful conversations and shared interests. I am truly grateful to all of them. Ahis Shrestha May 2022 iii Table of Contents Acknowledgements ii List of Tables vi List of Figures vii Abstract xv Chapter 1: Introduction 1 1.1 Emergent properties of heterogeneous membranes . . . . . . . . . . . . . . . . . . 1 1.2 Organization and regulation of lipids and proteins in cell membranes . . . . . . . . 3 1.3 Role of curved membrane domains in cell membrane plasticity . . . . . . . . . . . 5 1.4 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Local membrane organization and membrane function regulation 9 2.1 Mechanochemical coupling of membrane composition and hydrophobic thickness . 10 2.1.1 Energy of local lipid-protein domain . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Energy minimization and bilayer boundary conditions . . . . . . . . . . . 14 2.2 Lipid-protein domains in dilute membranes . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Local lipid organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Lipid-protein chemical affinity . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 Lipid chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Gating of mechanosensitive ion channels in heterogeneous bilayers . . . . . . . . . 25 Chapter 3: Crowded membranes and cooperative membrane function 30 3.1 Mean-field model of membrane protein interactions . . . . . . . . . . . . . . . . . 30 3.2 Lipid-protein domains in crowded membranes . . . . . . . . . . . . . . . . . . . . 32 3.2.1 Proteins with identical hydrophobic thickness . . . . . . . . . . . . . . . . 34 3.2.2 Proteins with distinct hydrophobic thickness . . . . . . . . . . . . . . . . 37 3.3 Membrane protein cooperativity in heterogeneous bilayers . . . . . . . . . . . . . 40 Chapter 4: Curved membrane domains and membrane plasticity 45 4.1 Energy of caveola domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Stable caveola shapes for varying membrane tension . . . . . . . . . . . . . . . . 49 4.2.1 Constant line tension at caveola domain boundary . . . . . . . . . . . . . . 51 iv 4.2.2 Caveola-shape-dependent line tension at caveola domain boundary . . . . . 53 4.3 Caveola shape distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 5: Membrane footprint of curved membrane domains 58 5.1 Caveola membrane footprint energy . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1.1 Arclength parameterization of the caveola neck . . . . . . . . . . . . . . . 59 5.1.2 Hamilton equations for the caveola neck . . . . . . . . . . . . . . . . . . . 60 5.1.3 Caveola neck boundary conditions . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Shape and deformation energy of caveolae neck . . . . . . . . . . . . . . . . . . . 63 Chapter 6: Conclusion 69 6.1 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 Outlook on future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 References 76 Appendices 86 A Derivation of bilayer energy and natural boundary conditions . . . . . . . . . . . . 87 A.1 Free energy of elastic thickness deformations . . . . . . . . . . . . . . . . 87 A.2 Free energy of lipid domain formation . . . . . . . . . . . . . . . . . . . . 89 A.3 Natural boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 90 B One-dimensional model of lipid-protein domains . . . . . . . . . . . . . . . . . . 92 C Dependence of bilayer bending rigidity and thickness deformation modulus on lipid composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 D Numerical minimization of bilayer energy . . . . . . . . . . . . . . . . . . . . . . 97 E Calculation of protein interaction potentials in heterogeneous bilayers . . . . . . . 101 F Discontinuous transitions in caveola shape at constant line tension . . . . . . . . . 102 v List of Tables 2.1 Experimentally measured values of unperturbed bilayer thicknesses 2a (in ˚ A), thickness deformation moduli K t (in k B T / ˚ A 2 ), and bending rigidities K b (in k B T ) for different lipid species. (After Refs. [1, 2].) . . . . . . . . . . . . . . . . . . . . 13 vi List of Figures 2.1 Schematic of bilayer-protein hydrophobic matching in heterogeneous bilayers. We consider a membrane protein with an approximately cylindrical hydrophobic sur- face of radius R and height 2H 0 at the center of an axisymmetric lipid bilayer patch composed of two lipid species with distinct unperturbed hydrophobic thicknesses (lipid species indicated in red and blue). The inner boundary of the lipid bilayer annulus in the membrane patch is constrained by the membrane protein at the cen- ter of the membrane patch, while we allow the outer boundary of the membrane patch to be either free or constrained by other proteins. We denote the hydropho- bic thickness of the lipid bilayer leaflets by h and the radial coordinate about the protein center by r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Illustration of the local lipid environment of a single membrane protein with (a) zero chemical potential and no chemical affinity between the membrane protein and a particular lipid species, (b) zero chemical potential with a chemical pref- erence of the membrane protein for a particular lipid species (indicated in blue), (c) a nonzero chemical potential with no chemical affinity between the membrane protein and a particular lipid species, and (d) a nonzero chemical potential with a chemical preference of the membrane protein for a particular lipid species (indi- cated in red). For ease of visualization, we took here one lipid species (indicated in red) to yield an unperturbed hydrophobic bilayer thickness that is identical to the hydrophobic thickness of the membrane protein at the center of the membrane patch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Local organization of lipid species through protein-induced lipid bilayer thick- ness deformations. (a) Thickness deformation profile h, lipid composition c (color bars), and energy density g for H 0 = 1:9 nm (upper panel) and H 0 = 2:0 nm (lower panel) versus r at t = 0. (b) Bilayer free energy G in Eq. (2.4) as a function of H 0 for a heterogeneous bilayer containing lipid species A and B and homogeneous bilayers composed of lipid species A or B for R= R c and the indicated values of t. The dashed vertical lines show H 0 = H 0 fort= 0, 1 k B T=nm 2 , 2 k B T=nm 2 , and 3 k B T=nm 2 (right to left), for which the dominant lipid composition in the bilayer patch changes from lipid species A to lipid species B. At r= R+ L we use, for all panels, the boundary conditions in Eqs. (2.7a)–(2.7c). We set L= 20 nm. . . . . . 20 vii 2.4 Lipid-protein organization for a single membrane protein with and without a lipid- protein chemical affinity. (a) Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for the fixed-value boundary conditions c(R)= c 0 with c 0 = 0 (upper panel) and c 0 = 1 (lower panel) as a function of the radial coordinate r at H 0 = 1:94 nm. (b) Total bilayer energy G in Eq. (2.4) as a function of H 0 for a heterogeneous bilayer containing lipid species A and B and homogeneous bilayers composed solely of lipid species A or B for natural boundary conditions on c at r= R or the indicated lipid-protein affinities. The dashed vertical lines show H 0 = H 0 for fixed c 0 = 0, free c 0 , and fixed c 0 = 1 (right to left), for which the dominant lipid composition in the membrane patch changes sharply from lipid species A to lipid species B. At r= R+ L we use, for all panels, the natural boundary conditions in Eqs. (2.7a)– (2.7c). We set L= 20 nm, R= R c ,m = 0, andt = 0. . . . . . . . . . . . . . . . . 22 2.5 Dependence of lipid-protein organization on the membrane patch chemical poten- tial m, in the case of a single membrane protein. Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for (a) natural boundary conditions on c at the bilayer-protein interface and H 0 = 1:7 nm with m = 10 3 k B T=nm 2 (upper panel) and m = 10 2 k B T=nm 2 (lower panel), and (c) the fixed-value boundary condition c(R)= c 0 at the bilayer- protein interface and H 0 = 1:9 nm with c 0 = 0, m = 10 2 k B T=nm 2 (upper panel) and c 0 = 1,m=10 2 k B T=nm 2 (lower panel). Total bilayer energy G in Eq. (2.4) as a function of H 0 for a heterogeneous bilayer containing lipid species A and B and homogeneous bilayers composed solely of lipid species A or B for the indi- cated values of m with (b) natural boundary conditions on c at the bilayer-protein interface and (d) the indicated fixed-value boundary conditions on c at the bilayer protein interface withm =10 2 k B T=nm 2 (m? 0) (curves corresponding to free c 0 included for reference). In panel (b), the dashed vertical lines show H 0 = H 0 for free c 0 atm = 10 3 k B T=nm 2 ,m = 0, andm =10 3 k B T=nm 2 (right to left), for which the dominant lipid composition in the membrane patch changes sharply from lipid species A to lipid species B. Similarly, the gray dashed vertical lines in panel (d) show H 0 = H 0 for c 0 = 0, free c 0 , and c 0 = 1 at m = 0 (right to left). At r= R+ L we use, for all panels, the natural boundary conditions in Eqs. (2.7a)– (2.7c). We set L= 20 nm, R= R c , andt = 0 for all panels. . . . . . . . . . . . . . 24 2.6 MscL gating probability P o in Eq. (2.9) as a function of t for heterogeneous bi- layers containing lipid species A and B, A and C, and A and D, and homogeneous bilayers containing only lipid species A, B, C, or D. All curves were calculated as in Fig. 2.2, with a= 1:7 nm, 2:2 nm, 1:8 nm, and 2:0 nm for lipid species A, B, C, and D, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 viii 2.7 Gating of single MscL in heterogeneous lipid bilayers. MscL gating probability P o in Eq. (2.9) as a function of membrane tension t for heterogeneous bilayers composed of lipid species A and B for (a)m = 0 with natural boundary conditions on c at the bilayer-MscL interface or the indicated values of c 0 in the closed (c 0 = c c 0 ) and open (c 0 = c o 0 ) conformational states of MscL, (b) m6= 0 with natural boundary conditions on c at the bilayer-MscL interface, and (c)m6= 0 with various combinations of free and fixed boundary conditions on c for the closed and open conformational states of MscL. In panel (c) we usem =10 2 k B T=nm 2 (m? 0). For reference, we show in all panels the corresponding results obtained for m = 0 with natural boundary conditions on c at the bilayer-MscL interface [3], and for homogeneous bilayers composed solely of lipid species A or B. . . . . . . . . . . 29 3.1 Schematic of the mean-field model of bilayer-mediated protein interactions in crowded membranes [3–8]. The membrane protein at the center of the membrane patch has radius R and is surrounded by a lipid bilayer annulus of thickness L. The boundary conditions at the outer rim of the bilayer annulus at r= R+L are taken to be axisymmetric about the membrane patch center, and are chosen so as to model a uniform array of membrane proteins (see Sec. 2.1.2). . . . . . . . . . . . . . . . 31 3.2 Local lipid organization and bilayer-thickness-mediated protein interactions in crowded membranes. Thickness deformation profile h, lipid composition c (color bars), and energy density g at L= 3 nm and L= 12 nm versus (r R)=L with R= R c for (a) identical proteins with H 0 = H L = 2:0 nm and (b) distinct proteins with H 0 = 2:2 nm and H L = 1:3 nm in bilayers composed of lipid species A and B. (c) Bilayer- thickness-mediated protein interaction potentials G int for identical (H 0 = H L ) and distinct (H 0 6= H L ) proteins as in panels (a) and (b) with R= R c in a heterogeneous bilayer composed of lipid species A and B, and homogeneous bilayers containing only lipid species A or B. The interaction potentials G int are obtained by subtract- ing from Eq. (2.4) the (noninteracting) large-L limit of Eq. (2.4), which is linear in L. We sett = 0 for all panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Bilayer-thickness-mediated protein interactions in heterogeneous bilayers for mem- brane proteins with identical hydrophobic thickness, H 0 = H L , using H 0 = H L = 2:0 nm. Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for L= 3 nm and L= 12 nm versus (r R)=L with R= R c for (a) fixed-value boundary conditions on c with c 0 = c L and m = 0, (b) fixed-value boundary conditions on c with c 0 6= c L and m= 0, and (c) natural boundary conditions on c andm=10 2 k B T=nm 2 (m? 0). We sett = 0 for all panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ix 3.4 Interaction potentials for proteins with identical hydrophobic thickness, H 0 = H L , using H 0 = H L = 2:0 nm. Bilayer-thickness-mediated protein interaction potentials G int (L) for (a) fixed-value boundary conditions on c with c 0 = c L and m = 0, (b) fixed-value boundary conditions on c with c 0 6= c L and m = 0, and (c) natural boundary conditions on c and m =10 2 k B T=nm 2 (m? 0). In all panels we show, for reference, also the G int (L) obtained for natural boundary conditions on c and m = 0 in heterogeneous bilayers composed of lipid species A and B (AB), as well as the G int (L) obtained for homogeneous bilayers composed solely of lipid species A or B [3]. To calculate G int (L), we subtracted from the total bilayer energy G in Eq. (2.4) the value of G obtained in the large-L, non-interacting regime (see Appendix E). We sett = 0 for all panels. . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Bilayer-thickness-mediated protein interactions in heterogeneous bilayers for mem- brane proteins with distinct hydrophobic thickness, H 0 6= H L , using H 0 = 2:2 nm and H L = 1:3 nm. Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for L= 3 nm and L= 12 nm versus(r R)=L with R= R c for (a) fixed-value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 1 atm= 0, (b) fixed-value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 0 atm= 0, and (c) natural boundary conditions on c atm =10 2 k B T=nm 2 (m? 0). We sett = 0 for all panels. . . . . . . . . . 38 3.6 Interaction potentials for proteins with distinct hydrophobic thickness, H 0 6= H L , using H 0 = 2:2 nm and H L = 1:3 nm. Bilayer-thickness-mediated protein inter- action potentials G int (L) for (a) fixed-value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 1 at m = 0, (b) fixed-value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 0 at m = 0, and (c) natural boundary conditions on c at m =10 2 k B T=nm 2 (m? 0). In all panels we show, for reference, also the G int (L) obtained for natural boundary conditions on c andm = 0 in heterogeneous bilayers composed of lipid species A and B (AB), as well as the G int (L) obtained for homogeneous bilayers composed solely of lipid species A or B [3]. To calcu- late G int (L), we subtracted from the total bilayer energy G in Eq. (2.4) the value of G obtained in the large-L, non-interacting regime (see Appendix E). We sett = 0 for all panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.7 Hydrophobic lipid-protein interactions in crowded membranes. (a) Bilayer free energy G in Eq. (2.4) as a function of H 0 for a heterogeneous bilayer composed of lipid species A and B with H L = H 0 or H L = H o and the indicated values of L. We set t = 0 and R= R c for both H L = H 0 and H L = H o . For ease of visualization we shifted the curves for H L = H o by G 0 = G min 30 k B T , where G min are the respective global minima of G(H 0 ). (b) MscL gating tension ¯ t implied by Eq. (2.9) as a function of L for a heterogeneous bilayer containing lipid species A and B and homogeneous bilayers composed of lipid species A or B with open-state MscL proteins at r= R+ L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 x 3.8 Cooperativity in the tension-dependent gating of MscL in heterogeneous bilayers. Probability that the MscL protein at the center of the membrane patch is in its open state, P o in Eq. (2.9), versus membrane tensiont for (a) identical fixed-value boundary conditions on c at all bilayer-protein interfaces with m = 0, (b) the indi- cated fixed-value boundary conditions on c at r= R and r= R+L in the closed (c c 0 ) and open (c o 0;L ) conformational states of MscL with m = 0, and (c) natural bound- ary conditions on c at all bilayer-protein interfaces with m =10 2 k B T=nm 2 (m? 0). For reference, we also show in all three panels the corresponding MscL gating curves obtained with natural boundary conditions on c at all bilayer-protein interfaces with m = 0, as well as the corresponding MscL gating curves obtained for homogeneous bilayers composed solely of lipid species A or B [3]. We use H 0 = H c and H 0 = H o for the closed and open conformational states of the MscL protein at the center of the membrane patch, and H L = H o for the proteins at the outer boundary of the membrane patch. The edge-to-edge protein separations con- sidered here, L= L c = 3 nm and L= L f = 12 nm, correspond to regimes with strong and negligible bilayer-mediated protein interactions (see also Figs. 3.3–3.6). 44 4.1 Cup-shaped caveolae. (a) Two-dimensional z-x rendering of caveolae obtained from three-dimensional superresolution fluorescence imaging of caveolin-1 at the plasma membrane for mouse embryonic fibroblast cells freely adhering to fibronectin substrates (left) and with adhesion constrained to 21010mm 2 fibronectin islands (right). Changes in the cell adhesion geometry are expected to modify the cell ad- hesion forces [9, 10], and the shape and organization of caveolae [11]. The white dotted curves indicate the approximate position of the plasma membrane for the observed caveolae. Scale bar: 100 nm. (Adapted with permission from Ref. [11].) (b) Schematic of the spherical cap model of caveola shape, with the caveola do- main indicated in red and the surrounding membrane indicated in blue. We denote the caveola surface area by S = pL 2 , the caveola radius of curvature by R, the caveola base radius by a, and the caveola invagination depth by h [12]. . . . . . . 46 4.2 Caveola energy G in Eq. (4.2) as a function of the caveola shape parameter b for g= 0:008, 0.028, and 0.048 k B T=nm 2 withs= 0 (solid curves) ands= 1 k B T=nm (dashed curves). The solid and dashed horizontal lines indicate the energy minima of the cyan solid and green dashed curves, respectively. We set L= 100 nm and C 0 = 0:04 nm 1 . The schematics in the top panels show caveolae (indicated in red) with shape parametersb = 0, 1/4, 1/2, 3/4, and 1 (left to right). We setk = 20 k B T . 50 4.3 Global minimum of G in Eq. (4.2) within the range 0b 1,b min (left axes and opaque curves) and corresponding derivative of b min with respect to membrane tension, db min =dg, for 0 0 with s = 1 k B T=nm (dashed curves). We use the same labeling scheme for b min and db min =dg. The stars along the right axes mark the maximum magnitudes of db min =dg,S max , for the s = 0 curves, using the same color scheme as for the db min =dg-curves. We setk = 20 k B T . . . . . . . . . . . . . . . . . . . . . . . . . 52 xi 4.4 Global minimum of G in Eq. (4.2) within the range 0b 1,b min (left axes and opaque curves) and corresponding derivative of b min with respect to membrane tension, db min =dg, for 0 0 and set L= 20 nm. In panel (c) we used H 0 = H L = 2:0 nm for H 0 = H L , and H 0 = 2:2 nm and H L = 1:3 nm for H 0 6= H L . In all panels we set, unless specified otherwise, m = 0 and used natural boundary conditions on h,Ñh, and c. We sett = 0 and employed a(c) in Eq. (2.5). . . . . . 96 D.1 Convergence tests for the total bilayer energy G in Eq. (2.4) for membrane proteins in heterogeneous bilayers. We consider here (a) single proteins with H 0 = 2:0 nm att = 0 (left panel) andt = 1 k B T=nm 2 (right panel), and (b) interacting proteins with identical hydrophobic thickness (H 0 = H L = 2:0 nm; left panel) and distinct hydrophobic thickness (H 0 = 2:2 nm and H L = 1:3 nm; right panel) with c 0 = 1 and c L = 0. Results obtained using the L-BFGS-B solver with the multistart method are indicated by red data points and plotted as a function of the logarithm (base 10) of the number of grid points used in the L-BFGS-B solver, log 10 (N). For comparison, we also show the corresponding results obtained by numerically solving the Euler-Lagrange equations in Eqs. (D.4) and (D.5) using the NDSolve- command in Mathematica [14] (dashed blue horizontal lines). We set here e = 100 k B T ,m = 0, R= 3 nm, and L= 5 nm for all panels and used, unless specified otherwise, natural boundary conditions on h,Ñh, and c. . . . . . . . . . . . . . . . 99 E.1 Total bilayer energy G in Eq. (2.4), protein interaction potential in Eq. (E.1), and far-field bilayer energy G f in Eq. (E.2) as a function of protein separation L for membrane proteins with identical hydrophobic thickness H 0 = H L = 2:0 nm (left panel) and distinct hydrophobic thicknesses H 0 = 2:2 nm and H L = 1:3 nm (right panel) in heterogeneous bilayers. For both panels we used natural boundary con- ditions on c at all bilayer-protein interfaces, and setm = 0 andt = 0. . . . . . . . . 101 F.1 The real solutionsb 1 ,b 2 ,b 3 , andb 4 of the quartic equation arising from the min- imization of G in Eq. (2) together with the boundary statesb = 0;1 (upper panel) and the corresponding values of G (lower panel) as a function ofg for L= 100 nm, C 0 = 0:04 nm 1 , ands = 1 k B T=nm. The gray dashed vertical line drawn across the panels indicates the critical tension g 0:028 k B T=nm 2 for which the fully budded caveola state with b = 1 becomes unstable for the parameter values used here. We setk = 20 k B T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 xiv Abstract Cell membranes are composed of a great variety of lipid and protein species that organize into distinct domains which vary in composition and structure. This thesis explores the physical mech- anisms and principles underlying the emergent properties of heterogeneous membrane domains, at scales of tens of nanometers in the context of lipid-protein organization through hydrophobic matching, and at scales of hundreds of nanometers in the context of caveola membrane domains. To achieve hydrophobic matching, the lipid bilayer tends to deform around membrane pro- teins so as to match the protein hydrophobic thickness at bilayer-protein interfaces. Such protein- induced distortions of the lipid bilayer hydrophobic thickness incur a substantial energy cost that depends critically on the bilayer-protein hydrophobic mismatch, while distinct conformational states of membrane proteins often show distinct hydrophobic thicknesses. As a result, hydrophobic interactions between membrane proteins and lipids can yield a rich interplay of lipid-protein orga- nization and transitions in protein conformational state. We combine here the membrane elasticity theory of protein-induced lipid bilayer thickness deformations with the Landau-Ginzburg theory of lipid domain formation to systematically explore the coupling between local lipid organization, lipid and protein hydrophobic thickness, and protein-induced lipid bilayer thickness deformations in membranes with heterogeneous lipid composition. We allow for a purely mechanical coupling of lipid and protein composition through the energetics of protein-induced lipid bilayer thickness deformations as well as a chemical coupling driven by preferential interactions between particular lipid and protein species. We find that the resulting lipid-protein organization can endow mem- brane proteins with diverse and controlled mechanical environments that, via protein-induced lipid bilayer thickness deformations, can strongly influence protein function. The theoretical approach xv employed here provides a general framework for the quantitative prediction of how membrane thickness deformations influence the joint organization and function of lipids and proteins in cell membranes. A striking feature of cell membranes in many mammalian cell types is the abundance of small membrane invaginations of defined lipid and protein composition, called caveolae, which flatten with increasing membrane tension. Superresolution light microscopy and electron microscopy have revealed that caveolae can take a variety of cup-like shapes. We show here that, for the range in membrane tension relevant for cell membranes, the competition between membrane tension and membrane bending yields caveolae with cup-like shapes similar to those observed experimentally. We find that the caveola shape and its sensitivity to changes in membrane tension can depend strongly on the caveola spontaneous curvature and on the size of caveola domains. Cup-shaped caveolae are also expected to deform the surrounding membrane into curved caveolae neck region. We show that the contribution of the caveola neck to the overall energy cost of caveolae depends on the membrane intrinsic curvature and the local membrane organization around caveolae. Our results suggest that heterogeneity in caveola shape produces a staggered response of caveolae to mechanical perturbations of the cell membrane, which may facilitate regulation of membrane ten- sion over the wide range of scales thought to be relevant for cell membranes. The materials in this PhD thesis are also discussed in the following publications: I. A. Shrestha, O. Kahraman, and C. A. Haselwandter. Regulation of membrane proteins through local heterogeneity in lipid bilayer thickness. Phys. Rev. E, 102(6):060401, 2020. Rapid Communication. II. A. Shrestha, O. Kahraman, and C. A. Haselwandter. Mechanochemical coupling of lipid organization and protein function through membrane thickness deformations. Submitted to Phys. Rev. E, 2021 (under review). III. A. Shrestha, F. Pinaud, and C. A. Haselwandter. Mechanics of cup-shaped caveolae. Phys. Rev. E, 104:L022401, 2021. Letter. xvi IV. A. Shrestha, F. Pinaud, and C. A. Haselwandter. The role of the caveola membrane footprint in membrane tension regulation. In preparation (expected submission in Spring 2022). xvii Chapter 1 Introduction This chapter provides an introduction to the research topics investigated in this PhD thesis. In particular, we present the background and motivation behind our research, briefly describe our key findings, and outline the contents of this thesis. In section 1.1, we begin by discussing the hetero- geneous nature of cell membranes and the emerging roles of lipid-protein interactions in regulating cellular functions. Section 1.2 focuses on lipid-protein organization in heterogeneous membranes due to hydrophobic matching and its effect on membrane protein function, while section 1.3 fo- cuses on the mechanics of curved membrane domains and its role in membrane tension regulation. Lastly, section 1.4 provides an overview of the thesis. 1.1 Emergent properties of heterogeneous membranes Cell membranes consist of a lipid bilayer which generally contains a wide variety of lipid species. The different types of lipids present in cell membranes are typically distinguished based on various features of their molecular structure, for instance the length of the hydrophobic fatty acid chain or tail region of the lipid. Much effort has been expended on the detailed cataloging of lipids in cell membranes, giving rise to the field of lipidomics [15]. In addition to different lipid species, cell membranes also contain various species of membrane proteins that are interspersed in the lipid bilayer [16]. Distinct membrane proteins of varying protein structure and orientation 1 in the bilayer are specialized in carrying out a range of different cellular functions. For example, membrane proteins such as mechanosensitive ion channels play a key role in sensing and respond- ing to mechanical perturbations in the cell [17]. Furthermore, the lipid bilayer also varies in bilayer hydrophobic thickness and bilayer mean curvature, producing local membrane shape distortions in cell membranes [16]. Ultimately, these diverse components and features of the bilayer displays a sophisticated organization of distinct domains in the membrane [18]. Hence, cell membranes are highly heterogeneous. Heterogeneity in cell membrane structure and composition allows regulation of the interaction of membrane components, and is crucial for various cellular processes [15, 18]. A wide range of experiments [15, 16, 18–25] suggest that local lipid-protein interactions provide key mechanisms for cell membrane organization and regulation of cellular functions. For instance, in experiments on lipid bilayers with heterogeneous hydrophobic thickness, it has been observed that lipid bilayer and protein composition couple through lipid-protein interactions driven by local hydrophobic matching of bilayer and protein thickness, resulting in organization of lipid and protein species according to their energetically preferred hydrophobic thickness [15, 16, 18, 23–25]. Motivated by these experimental observations, we study the general physical principles underlying lipid-protein organization due to hydrophobic matching. We quantify the hydrophobic lipid-protein interactions mediated through bilayer thickness deformations in heterogeneous bilayers and examine the effects of membrane heterogeneity on the regulation of membrane proteins that are vital to functioning of cells. Variations in the local shape of cell membranes are in part mediated by the organization of membrane domains that are enriched in particular lipid and protein species. Membrane plasticity couples to local membrane organization and responds to various mechanical stresses across the membrane, in particular, through the regulation of membrane tension which is critical to many cellular processes, such as endocytosis, exocytosis, cell adaption to mechanical perturbations, and activation of mechanosensitive ion channels [26–29]. An example of such membrane domains with a defined lipid and protein composition are small membrane invaginations called caveolae. They 2 are one the most abundant and striking features of the plasma membranes of many mammalian cell types. Caveolae alters their curved shape in response to cell swelling and cell stretching, and act as membrane area reservoirs to buffer against changes in membrane tension [30–32]. To uncover the core physical mechanisms that govern cell membrane plasticity and adaption to local mechanical perturbations, we explore here the mechanics of such a curved membrane domains and their response to modification of membrane tension in heterogeneous membranes. 1.2 Organization and regulation of lipids and proteins in cell membranes Lipids and membrane proteins in cell membranes show an intricate, submicron organization into membrane domains that vary in molecular composition and hydrophobic thickness [15, 16, 18, 19]. The resulting heterogeneity in the mechanical and chemical properties of cell membranes permits the local regulation of membrane protein function through lipid-protein interactions and plays a critical role in a variety of cellular processes [15, 18, 19, 33]. Experiments suggest that cell membrane organization is influenced by preferential chemical interactions between particular lipid and protein species as well as nonspecific hydrophobic lipid-protein interactions [15,16,18–22,33]. For instance, some ion channels are thought to aggregate specific lipid species in their vicinity, thereby creating a local bilayer environment with controlled mechanical properties [34–36]. The resulting membrane domains may serve to bias ion channels towards closed or open conformational states, and hence modulate ion channel function. The preferred hydrophobic thickness of lipid bilayers depends strongly on the lipid compo- sition [1, 15, 16, 18], with a variation in bilayer hydrophobic thickness between roughly 3.4 nm and 4.4 nm for the lipid tail lengths typically found in cell membranes. Similarly, the hydropho- bic thickness of membrane proteins varies substantially between different membrane proteins, and even among distinct conformational states of the same membrane protein. For instance, the closed and open conformational states of the mechanosensitive channel of large conductance (MscL) 3 have approximate hydrophobic thicknesses of 3.8 nm and 2.6 nm [37, 38], respectively. Differ- ences in the preferred hydrophobic thickness of lipids and membrane proteins generally result in deformations of the lipid bilayer thickness in the vicinity of membrane proteins so as to achieve hydrophobic matching at the bilayer-protein interfaces [13, 39–41]. Such protein-induced lipid bi- layer thickness deformations tend to incur a substantial energy cost> 10 k B T [13,39–41]. Protein- induced lipid bilayer thickness deformations yield, on the one hand, a coupling between protein conformational state and lipid bilayer thickness [39, 42–46]. On the other hand, protein-induced lipid bilayer thickness deformations give rise to bilayer-mediated interactions between membrane proteins [4, 40, 47–50]. Such bilayer-thickness-mediated protein interactions tend to be strongly favorable at close enough separations for proteins with identical hydrophobic thickness, and can yield self-assembly of membrane protein clusters [51–58]. The energy cost of protein-induced lipid bilayer thickness deformations depends critically on the lipid tail length [13, 39–41, 59]. In membranes containing lipids with distinct tail lengths, one therefore expects a coupling between local lipid composition and membrane protein hydrophobic thickness. In the following chapters, we systematically explore the mechanochemical coupling of local lipid organization, protein and lipid hydrophobic thickness, and protein-induced lipid bi- layer thickness deformations in bilayers with heterogeneous lipid composition and hydrophobic thickness. Our approach builds on previous work on lipid phase separation in bilayers with het- erogeneous hydrophobic thickness [60, 61], and on the coupling between bilayer mean curvature and lipid composition [62–64]. We describe how lipid-protein organization in bilayers with het- erogeneous hydrophobic thickness can be captured quantitatively by combining the elasticity the- ory of lipid bilayer thickness deformations [2, 4, 39–41, 47–50] with the Landau-Ginzburg (LG) theory of lipid domain formation [62, 65–73]. We illustrate our theoretical approach for MscL, which provides a paradigm for the coupling of local lipid composition and membrane protein func- tion [34, 44, 45, 74]. In agreement with a wide range of experimental studies [15, 16, 18, 23–25], we find that membrane hydrophobic thickness provides a key control parameter for lipid-protein organization and regulation in membranes with heterogeneous lipid composition. 4 1.3 Role of curved membrane domains in cell membrane plas- ticity More than 60 years ago, curved membrane domains known as caveolae were discovered by some of the early pioneers of electron microscopy in cell biology [75, 76]. Caveolae are small membrane invaginations with a size of the order of 100 nm that are enriched in caveolin-1 pro- teins as well as a number of other proteins and lipids [77, 78]. Caveolae are highly abundant in the plasma membranes of certain mammalian cell types, such as muscle and endothelial cells, in which they can encompass half of the total membrane area [78, 79]. Caveolae respond to changes in membrane tension by altering their curved shape [30, 75, 77–80]. In particular, caveolae flatten out and, ultimately, (partially) disassemble in response to cell swelling and cell stretching, and thereby provide a reservoir of (in-plane) membrane area and a buffer against changes in membrane tension [30–32]. Caveolae are thought to have biologically important roles in mechanosensing, mechanotransduction, membrane area and membrane tension homeostasis, plasma membrane or- ganization and signaling, and lipid regulation [75–79, 81]. While early electron microcopy (EM) experiments indicated that caveola membrane domains either occur in approximately flat or budded spherical shapes, more recent EM as well as superres- olution light microscopy (SRM) experiments have demonstrated that caveolae can take a variety of cup-like shapes resembling spherical caps [11, 76–78, 82–86]. For instance, three-dimensional superresolution fluorescence imaging of caveolin-1 in mouse embryonic fibroblast cells [11] shows that caveolae can take cup-like shapes with variable invagination depths, areas, and radii of curva- ture (see Fig. 2.1). Motivated by these experimental observations, we employ here the theoretical framework developed in Ref. [87] to study the mechanics of cup-shaped caveolae. Previously it was found [30,88] that the competition between membrane tension and line tension along the cave- ola domain boundary yields fully flattened or fully budded (spherical) caveola shapes. We show that, as the membrane tension is changed, the preferred spontaneous curvature of caveola domains can yield a continuous spectrum of cup-like caveola shapes intermediate between fully flattened 5 and fully budded caveola states [89]. For the range in membrane tension relevant for cell mem- branes and the spontaneous curvature associated with caveola domains, our calculations predict that caveolae adopt cup-like shapes similar to those observed experimentally [11, 76–78, 82–86] [Fig. 2.1(a)]. We find that the caveola shape and its response to changes in membrane tension can depend strongly on the caveola spontaneous curvature and on the size of caveola domains. Our cal- culations suggest that caveolae show a staggered response to changes in membrane tension, which may facilitate regulation of membrane tension over the wide range of scales thought to be relevant for cell membranes [13, 90, 91]. Cup-shaped caveolae are expected to deform the surrounding membrane [92]. Curvature- sensing or curvature-generating lipids or proteins may be enriched in the curved neck region of caveolae, and thus modify stable caveola shapes [93, 94]. We use membrane elasticity theory to quantify the contribution of the caveola neck to the spectrum of minimum-energy caveolae shapes, and how this contribution depends on the membrane intrinsic curvature and the local membrane organization around caveolae. Furthermore, we consider the effect of constraints on membrane shape provided by, for instance, membrane-cytoskeletal interactions [95–98], on the shape and energetics of caveolae and the caveola neck. We find that the mechanical properties of the cave- ola neck may serve to prime caveolae to changes in membrane tension, and hence allow cells to modify the regulation of in-plane membrane area through caveolae. 1.4 Overview of the thesis The prevailing theme throughout this thesis is the study of emergent features of heterogeneous membranes. In particular, we focus on developing theoretical models that aim to provide a ro- bust understanding of key physical principles underlying biologically important cell membrane phenomena. Chapter 2 develops the general theory of protein-induced lipid bilayer thickness de- formations in bilayers with heterogeneous hydrophobic thickness. On the one hand, we thereby consider a purely mechanical, nonspecific coupling of lipid and protein composition through the 6 energetics of protein-induced lipid bilayer thickness deformations [3]. On the other hand, we allow for a chemical coupling between the local lipid and protein compositions driven by pref- erential interactions between particular lipid and protein species [99–106]. We first consider the most straightforward scenario of a single membrane protein in a heterogeneous lipid bilayer com- posed of two lipid species with distinct lipid tail lengths. We determine how the local lipid com- position around the membrane protein depends on the protein hydrophobic thickness, the lipid- protein chemical affinity, and the local lipid chemical potential. Using MscL as a model system, we show that lipid heterogeneity can induce transitions in the protein conformational state. In chapter 3, we then consider the crowded membrane protein environments typical for cell mem- branes [13, 15, 16, 18, 19, 33]. We find that heterogeneity in the lipid composition of membranes can expand the repertoire and range of bilayer-thickness-mediated protein interactions, can yield colocalization of lipids and membrane proteins according to their preferred hydrophobic thickness, and can strongly affect bilayer-mediated protein cooperativity in cell membranes. In addition to the emerging roles of membrane heterogeneity in local membrane organization and function, this thesis also explores the emergent properties of curved membrane domains in cell membranes, in particular, through the phenomenology of small membrane invaginations known as caveolae. In chapter 4, we develop here a physical model describing cup-like shapes of caveolae. We show that, for the range in mechanical parameters relevant for caveolae in cell membranes, the competition between membrane tension and membrane bending yields caveolae with cup-like shapes similar to those observed experimentally [11,76–78,82–86]. We find that the caveola shape can depend strongly on the caveola spontaneous curvature and on the size of caveola domains, with larger caveola domains responding more sensitively to changes in membrane tension [89]. Furthermore, chapter 5 extends the caveola model to incorporate the contribution of the curved neck region surrounding caveolae. We quantify this contribution as an additional membrane foot- print energy, and explore how its dependence on the membrane intrinsic curvature and the local membrane organization around caveolae affects caveolae shape. Overall, our work suggest that 7 heterogeneity in caveola shape produces a staggered response of caveolae to mechanical pertur- bations of the cell membrane, which may facilitate regulation of membrane tension over the wide range of scales thought to be relevant for cell membranes. Finally, we conclude this thesis with chapter 6, which includes a summary and discussion of our key results. 8 Chapter 2 Local membrane organization and membrane function regulation This chapter explores local lipid-protein organization mediated through membrane thickness deformations and the regulation of membrane protein function in heterogeneous bilayers. In sec- tion 2.1, we present our physical model of lipid-protein domains in heterogeneous lipid bilayers, where the local lipid organization surrounding a membrane protein is coupled with protein-induced bilayer thickness deformations. Section 2.2 examines the different scenarios of this mechanochem- ical coupling for local domains containing a single membrane protein in dilute membranes. In sec- tion 2.3, we then study the effects of bilayer heterogeneity on the gating behavior of mechanosen- sitive ion channels. Overall, this chapter lays out our theoretical framework and main results of the coupling between local lipid organization, lipid and protein hydrophobic thickness, and protein- induced lipid bilayer thickness deformations in membranes with heterogeneous lipid composition. 9 2.1 Mechanochemical coupling of membrane composition and hydrophobic thickness We aim here to explore generic features of the interplay between lipid-protein organization and membrane thickness deformations that are independent of most molecular details [3]. We therefore consider an idealized bilayer-protein system, in which a cylindrical membrane protein of radius R is located at the center of a circular membrane patch representing the local lipid environment of the membrane protein (see Fig. 2.1). We take the local lipid environment of the membrane protein to be approximately axisymmetric about the protein center. Furthermore, we assume up-down symmetry in the hydrophobic thickness and composition of the upper and lower lipid bilayer leaflets. Models assuming such an idealized, symmetric lipid bilayer environment have been employed successfully to describe the coupling of membrane protein conformational state and lipid bilayer thickness in homogeneous bilayers [39, 42–46] as well as bilayer-thickness-mediated protein interactions in homogeneous bilayers [4, 40, 47–58]. Further simplification of the modeling approach employed here could be achieved through an effectively one-dimensional version of this model, which we explore in Appendix B. 2.1.1 Energy of local lipid-protein domain Protein-induced lipid bilayer thickness deformations tend to decay rapidly away from the lipid- protein boundary, with a decay length 1 nm [13, 74], tend to be small compared to the unper- turbed lipid bilayer thickness, and tend to show small gradients [2, 43, 45, 50] (Fig. 2.1). As a result, it is useful to describe protein-induced lipid bilayer thickness deformations by expanding the Monge representation of the elastic bilayer deformation energy to leading order in(h a) and its derivatives [4, 5, 40, 50], where h is the hydrophobic thickness of the lipid bilayer leaflet at the Cartesian coordinates (x;y) and a is the unperturbed hydrophobic thickness of the lipid bilayer 10 Figure 2.1: Schematic of bilayer-protein hydrophobic matching in heterogeneous bilayers. We consider a membrane protein with an approximately cylindrical hydrophobic surface of radius R and height 2H 0 at the center of an axisymmetric lipid bilayer patch composed of two lipid species with distinct unperturbed hydrophobic thicknesses (lipid species indicated in red and blue). The inner boundary of the lipid bilayer annulus in the membrane patch is constrained by the membrane protein at the center of the membrane patch, while we allow the outer boundary of the membrane patch to be either free or constrained by other proteins. We denote the hydrophobic thickness of the lipid bilayer leaflets by h and the radial coordinate about the protein center by r. leaflet, which depends on the lipid tail length (Fig. 2.1). The elastic thickness deformation energy of the lipid bilayer can then be written in the form [2, 4, 39–41, 45, 47–50, 74] G h = Z dxdy ( K b 2 (Ñ 2 h) 2 + K t 2 h a a 2 + t 2 (Ñh) 2 + 2 h a a + t 2 2K t ) ; (2.1) where the integral runs over the (in-plane) lipid bilayer surface, K b is the lipid bilayer bending rigidity, K t is the bilayer thickness deformation modulus, and t is the (lateral) membrane tension (see Appendix A.1). For a given lipid composition, the values of the parameters K b , K t , and a in Eq. (2.1) can be measured directly in experiments [1, 13, 41]. We have added the constant term t 2 =2K t in the integrand in Eq. (2.1) such that G h = 0 for all extremal functions of G h corresponding to a constant h(x;y) [50]. 11 We consider here protein-induced lipid bilayer thickness deformations in heterogeneous bilayer membranes that effectively consist of two lipid species with distinct unperturbed hydrophobic thicknesses, in a regime in which the two lipid species can form distinct domains with rapid lateral diffusion of lipids in the plane of the membrane [15, 16, 18–21, 33, 61, 70, 71]. At the mean-field level, lipid domain formation in such a binary system is successfully described by the LG free energy [60, 61, 67, 70–72] G c = Z dxdy h e 2 (Ñc) 2 +f(c)mc+m 0 i ; (2.2) where the integral runs over the (in-plane) lipid bilayer surface, c(x;y) is a continuous function describing the (mean) lipid composition at the Cartesian coordinates (x;y) such that c= 0 and c= 1 correspond to the two lipid species under consideration, the parameter e specifies the en- ergy penalty associated with lipid domain boundaries, the function f(c) is the (even) mean-field potential associated with lipid phase separation, and m is the (local) lipid chemical potential (see Appendix A.2). In analogy to Eq. (2.1), we include in the integrand in Eq. (2.2) the constant term m 0 =(jmj+m)=2 such that G c 0 for all extremal functions of G c corresponding to a constant lipid composition c(x;y)= 0 or c(x;y)= 1. The chemical potentialm in Eq. (2.2) allows for a breaking of the symmetry between c= 0 and c= 1 in Eq. (2.2), and locally biases the lipid composition of the membrane patch under consideration towards a particular lipid species. Form = 0 in Eq. (2.2) we recover the scenario we considered previously [3], which corresponds to situations in which lipids of either species can freely diffuse into and out of the membrane patch without any external constraints on the lipid composition of the membrane patch. Following previous work [70,107], we consider for the mean-field potentialf(c) in Eq. (2.2) a fourth-order polynomial in c centered at c= 1=2: f(c)= b 0 b 1 2 c 1 2 2 + b 2 4 c 1 2 4 : (2.3) 12 Lipid 2a ( ˚ A) K t (k B T / ˚ A 2 ) K b (k B T ) diC13:0 34.1 0.5 0.576 0.03 14 2 diC14:0 35.2 0.6 0.565 0.05 14 2 C18:0/1 40.7 0.6 0.568 0.03 21 2 diC18:1c9 36.9 0.4 0.638 0.04 20 2 diC18:2 34.9 0.3 0.596 0.05 10 2 diC18:3 34.3 0.6 0.588 0.08 9.3 1 diC20:4 34.4 0.7 0.603 0.02 10 1 diC22:1 43.7 0.5 0.634 0.02 29 3 Table 2.1: Experimentally measured values of unperturbed bilayer thicknesses 2a (in ˚ A), thickness deformation moduli K t (in k B T / ˚ A 2 ), and bending rigidities K b (in k B T ) for different lipid species. (After Refs. [1, 2].) From the LG theory of lipid domain formation [70] we have b 1 = 4n 0 k B T=3 and b 2 = 16n 0 k B T=3, where n 0 is the mean lipid number per unit area, so that the mean-field potential in Eq. (2.2) has minima at c= 0 and c= 1. Furthermore, we set b 0 = n 0 k B T=12 so that the mean-field potential in Eq. (2.2) is zero at these two minima. The two key physical parameters entering Eq. (2.2) are thus e and n 0 . Following Refs. [61, 70] we use here, unless specified otherwise, the values e = 1 k B T and n 0 = 1 nm 2 , which were found previously to successfully describe lipid domain formation in heterogeneous bilayers composed of distinct lipid species [60, 71, 72]. We obtain the total bilayer (free) energy of the membrane patch, G, by adding Eqs. (2.1) and (2.2): G= G h + G c : (2.4) The fields describing the lipid leaflet thickness, h(x;y), and the lipid bilayer composition, c(x;y), in Eq. (2.4) are, in principle, coupled [60, 61] via the elastic parameters K b , K t , and a in Eq. (2.1), which all depend on the lipid composition [1, 13, 41]. A variety of experiments on membranes with heterogeneous hydrophobic thickness [15, 16, 18, 23–25] indicate that lipid bilayer and protein composition couple through hydrophobic lipid- protein interactions driven by differences in the energetically preferred hydrophobic thickness of lipids and membrane proteins, suggesting that a in Eq. (2.1) provides the dominant coupling of 13 h and c. Based on the experimental data on the dependence of bilayer elastic properties on lipid composition compiled in Ref. [1] (see Table 2.1), we explore in Appendix C the relative importance of the dependence of K b , K t , and a in Eq. (2.1) on lipid composition [13, 41]. For all the scenarios considered here we find that the dependence of a on c dominates over the dependence of K b and K t on c. We therefore focus on the coupling of h and c through a(c) [60,61]. Notably, experiments show that a depends crucially on the lipid chain length [1]. A major lipid component of cell membranes is provided by phospholipids, for which a roughly varies from a 1:7 nm to a 2:2 nm for the approximate range in lipid chain length relevant for cell membranes [1, 15, 16, 18]. Following Refs. [1–3, 45] we approximate a(c) by a linear function and identify the two lipid species in the bilayer with a(c = 0) = 1:7 nm (lipid species A) and a(c = 1) = 2:2 nm (lipid species B). We thus set a(c)a 1 c+a 0 ; (2.5) where a 1 = 0:5 nm and a 0 = 1:7 nm [1–3, 45]. Unless stated otherwise, we use the fixed values K b = 20 k B T and K t = 60 k B T=nm 2 typical for phospholipid bilayer membranes [1, 13]. Ap- pendix C considers expressions analogous to Eq. (2.5) for K b and K t . 2.1.2 Energy minimization and bilayer boundary conditions We assume that the dominant lipid leaflet thickness field h(x;y) and the dominant lipid bilayer composition field c(x;y) minimize G in Eq. (2.4) with Eqs. (2.1)–(2.3) subject to suitable boundary conditions. To determine the minimal h(x;y) and c(x;y) of G, it is convenient to write the total energy in Eq. (2.4) in polar coordinates such that the fields h and c only depend on the radial coordinate r = p x 2 + y 2 . The integrands in Eq. (2.1) and (2.2) are easily transformed to polar coordinates by noting that, for rotationally symmetric systems such as considered here,Ñh= dh dr ˆ r and Ñc= dc dr ˆ r, where ˆ r is the radial unit vector, and Ñ 2 h= 1 r d dr r dh dr . We minimize G in the 14 domain R r R+L (Fig. 2.1), where L is the width of the lipid bilayer annulus in the membrane patch. To this end, it is useful to write G in the form G= 2p Z R+L R g(r)rdr; (2.6) where g(r) is the lipid bilayer energy density. We numerically minimize G in Eq. (2.6) using the L- BFGS-B solver [108, 109]. Appendix D provides a detailed discussion of the numerical approach employed here, and compares the results obtained using the L-BFGS-B solver to the corresponding results obtained by numerical solution of the Euler-Lagrange equations associated with Eq. (2.4). To calculate protein interaction potentials G int (L) in heterogeneous lipid bilayers, we subtract from the total bilayer energy G in Eq. (2.4) the G obtained in the large-L, non-interacting regime (see Appendix E). The mathematical form of the bilayer boundary conditions associated with G in Eq. (2.4) fol- lows from the calculus of variations [110, 111] (see Appendix A.3). In particular, we can have natural (free) or fixed-value boundary conditions on h(r),Ñh(r), and c(r) at r= R and r= R+ L. At a boundary r= r b , with r b = R or r b = R+ L, the natural boundary conditions on h(r),Ñh(r), and c(r) are given by 1 r d dr r dh dr r=r b = 0; (2.7a) d dr th K b 1 r d dr r dh dr r=r b = 0; (2.7b) dc dr r=r b = 0; (2.7c) respectively. Conversely, the fixed-value boundary conditions on h(r),Ñh(r), and c(r) at a bound- ary r= r b are given by h(r b )= H b ; (2.8a) 15 dh dr r=r b = s b ; (2.8b) c(r b )= c b ; (2.8c) respectively, where H b , s b , and c b take given, fixed values. In principle, one can have any combination of the natural and fixed-value boundary conditions on h(r),Ñh(r), and c(r) in Eqs. (2.7a)–(2.7c) and Eqs. (2.8a)–(2.8c) at r b = R and r b = R+L. The specific form of the boundary conditions used for a given scenario encodes some of the key physical properties of the particular system at hand. We first note that it is energetically very unfavorable to expose the hydrophobic surfaces of lipids or membrane proteins to water, while membrane proteins are considerably more rigid than lipid bilayers [2, 13, 39, 41–43, 45, 74]. As a result, one expects that membrane proteins impose a fixed bilayer leaflet thickness H b at bilayer-protein boundaries. Throughout this thesis, we therefore use Eq. (2.8a) at bilayer-protein interfaces. We employ here MscL as a model system to explore lipid-protein organization and regulation through membrane thickness deformations. The closed and open conformational states of MscL yield the approximate hydrophobic thicknesses H c = 1:9 nm and H o = 1:3 nm, respectively, with the approximate protein radii R c = 2:5 nm and R o = 3:5 nm in the closed and open conformational states of MscL [2, 37, 38, 45, 74], which we use as reference values for H b in Eq. (2.8a) and R. The choice of the boundary condition onÑh at bilayer-protein interfaces has been a matter of debate, with some studies suggesting the natural boundary condition in Eq. (2.7b) and other stud- ies suggesting the fixed-value boundary condition in Eq. (2.8b) with s 0 = 0 [2, 39, 43, 45, 74, 112]. Following Refs. [2, 39, 45, 50, 74] we use here Eq. (2.8b) with s 0 = 0 at bilayer-protein interfaces. We consider both the natural and fixed-value boundary conditions on c in Eqs. (2.7c) and (2.8c) at bilayer-protein interfaces, respectively, with c b = 0 or c b = 1. The former boundary condition cor- responds to nonspecific lipid-protein interactions, while the latter boundary condition corresponds to a lipid-protein chemical affinity driven by preferential interactions between particular lipid and 16 protein species. Finally, for scenarios where the (outer) boundary of the membrane patch under consideration is not constrained by membrane proteins, we use the natural boundary conditions in Eqs. (2.7a)–(2.7c) for h,Ñh, and c. 2.2 Lipid-protein domains in dilute membranes This section focuses on lipid-protein organization and regulation through membrane thickness deformations for a single membrane protein in a heterogeneous lipid bilayer composed of two lipid species with distinct unperturbed hydrophobic thicknesses. It is useful to distinguish here between four basic scenarios, which are illustrated in Fig. 2.2. First, we consider the case of zero chemical potential in Eq. (2.2) with no chemical affinity between the membrane protein at the center of the membrane patch and a particular lipid species [see Fig. 2.2(a)], for which lipid- protein organization is driven solely by the energetics of lipid bilayer thickness deformations [3]. Second, we consider the case of zero chemical potential in Eq. (2.2) with a chemical preference of the membrane protein at the center of the membrane patch for a particular lipid species [see Fig. 2.2(b)]. If the preferred lipid species yields an unperturbed bilayer thickness that is distinct from the hydrophobic thickness of the membrane protein, such a chemical affinity frustrates hy- drophobic lipid-protein interactions. Third, we consider the case of a nonzero chemical potential in Eq. (2.2) with no chemical affinity between the membrane protein at the center of the membrane patch and a particular lipid species [see Fig. 2.2(c)]. Similarly to Fig. 2.2(a), the energetics of protein-induced lipid bilayer thickness deformations can then yield, driven by local matching of bilayer and protein hydrophobic thickness, local segregation of a particular lipid species around the membrane protein. Finally, we allow for an interplay of bilayer thickness deformations with a nonzero chemical potential in Eq. (2.2) and a chemical affinity between particular lipid and pro- tein species [see Fig. 2.2(d)], which can perturb lipid-protein organization due to local matching of bilayer and protein hydrophobic thickness. Since we focus here on a single membrane protein in a heterogeneous lipid bilayer membrane we use, throughout this section, the natural boundary 17 Figure 2.2: Illustration of the local lipid environment of a single membrane protein with (a) zero chemical potential and no chemical affinity between the membrane protein and a particular lipid species, (b) zero chemical potential with a chemical preference of the membrane protein for a par- ticular lipid species (indicated in blue), (c) a nonzero chemical potential with no chemical affinity between the membrane protein and a particular lipid species, and (d) a nonzero chemical poten- tial with a chemical preference of the membrane protein for a particular lipid species (indicated in red). For ease of visualization, we took here one lipid species (indicated in red) to yield an unperturbed hydrophobic bilayer thickness that is identical to the hydrophobic thickness of the membrane protein at the center of the membrane patch. conditions in Eqs. (2.7a)–(2.7c) at the (outer) membrane patch boundary r b = R+ L. Further- more, we choose membrane patch sizes large enough so that g(R+ L) 0 in Eq. (2.6). We first consider how lipid-protein organization and the energy of protein-induced lipid bilayer thickness deformations depend on the membrane protein hydrophobic thickness, for a purely mechanical coupling through a(c) without a lipid specific chemical affinity and at zero chemical potential in Eq. (2.2) (see Sec. 2.2.1). We then introduce a chemical affinity between the membrane protein and a particular lipid species and compare how this affects the former scenario, at zero chemical potential in Eq. (2.2) and at zero membrane tension in Eq. (2.1) (see Sec. 2.2.2). After that, we allow for a nonzero chemical potential in Eq. (2.2), which effectively biases the composition of the lipid bilayer patch towards a particular lipid species, and explore the effects of such a bias on lipid-protein organization and the energy of protein-induced lipid bilayer thickness deformations, 18 again at zero membrane tension in Eq. (2.1) (see Sec. 2.2.3). Finally, we consider the effects of a nonzero membrane tension in Eq. (2.1) on protein-induced lipid bilayer thickness deformations in heterogeneous bilayers (see Sec. 2.3). In particular, using MscL as a model system, we study how heterogeneity in the preferred bilayer hydrophobic thickness affects the tension-dependent gating of ion channels. We find that, even at fixed membrane tension, changes in lipid heterogeneity can induce MscL gating. 2.2.1 Local lipid organization We begin by considering a purely mechanical, non-specific coupling between protein hy- drophobic thickness and the local lipid organization mediated by the energetics of protein-induced bilayer thickness deformations. We find that, even if the protein at r= R does not show a chemical preference for a particular lipid species and has no chemical bias towards a particular lipid compo- sition, the coupling of local lipid and protein composition through a(c) [1,15,16,18,60,61] yields, depending on the value of H 0 , accumulation of lipid species A or B around the membrane protein [Fig. 2.3(a)]. As the value of H 0 is changed through a critical value H 0 , the dominant composition of the bilayer patch shows a sharp change from lipid species A to lipid species B. We find that the bilayer energy G in Eq. (2.4) has two minima as a function of protein hydrophobic thickness, at H 0 = a(0) and H 0 = a(1) [see Fig. 2.3(b)]. These two minima are associated with lipid species A and B, with a sharp change in G at H 0 = H 0 that mirrors the sharp change in local lipid composition at H 0 = H 0 . As the membrane tension t is increased, the preferred hydrophobic thickness of the lipid bilayer is reduced, thus shifting H 0 to smaller values. For a membrane protein with H 0 H 0 , a change in membrane tension can therefore induce a change in the local lipid environment of the membrane protein, and hence alter local membrane composition. 2.2.2 Lipid-protein chemical affinity Figure 2.4 contrasts lipid-protein organization and the energy of protein-induced lipid bilayer thickness deformations for scenarios without and with a chemical affinity between particular lipid 19 Figure 2.3: Local organization of lipid species through protein-induced lipid bilayer thickness deformations. (a) Thickness deformation profile h, lipid composition c (color bars), and energy density g for H 0 = 1:9 nm (upper panel) and H 0 = 2:0 nm (lower panel) versus r at t = 0. (b) Bilayer free energy G in Eq. (2.4) as a function of H 0 for a heterogeneous bilayer containing lipid species A and B and homogeneous bilayers composed of lipid species A or B for R= R c and the indicated values of t. The dashed vertical lines show H 0 = H 0 for t = 0, 1 k B T=nm 2 , 2 k B T=nm 2 , and 3 k B T=nm 2 (right to left), for which the dominant lipid composition in the bilayer patch changes from lipid species A to lipid species B. At r = R+ L we use, for all panels, the boundary conditions in Eqs. (2.7a)–(2.7c). We set L= 20 nm. and protein species [Figs. 2.2(a) and 2.2(b)]. Throughout this section we assume zero membrane tension in Eq. (2.1) and zero chemical potential in Eq. (2.2). For general values of the hydrophobic thickness of the membrane protein at the center of the membrane patch, H b = H 0 , we find that a chemical affinity between the membrane protein and a particular lipid species only has a minor effect on the overall lipid composition of the bilayer patch surrounding the membrane protein, with the average lipid composition of the bilayer patch being driven primarily by the interplay of lipid and protein hydrophobic thickness [3]. However, we also find that, if the hydrophobic thickness 20 of the membrane protein (or of the lipids) is tuned to lie within a certain (narrow) range, modi- fication of the lipid-protein chemical affinity can have a dramatic effect on the lipid composition of the membrane patch. For instance, for the parameter values used for Fig. 2.4 we find that two proteins with similar hydrophobic thicknesses within the range 1:91 nm/ H 0 / 1:98 nm but dif- ferent chemical affinities—one for lipid species A (c b = c 0 = 0) and the other for lipid species B (c b = c 0 = 1)—can yield markedly different lipid compositions of the membrane patch, with the membrane patch composition being strongly biased towards either lipid species A or B [see Fig. 2.4(a)]. As mentioned in Sec. 2.2.1, for membrane proteins without a chemical affinity for a particular lipid species, changes in the protein hydrophobic thickness H 0 across a critical value H 0 yield a sharp change in the lipid composition of the bilayer patch [3] [see Fig. 2.4(b)]. A similar transition in lipid composition is obtained for membrane proteins that have a chemical affinity for a particular lipid species, but at different values of H 0 [Fig. 2.4(b)]. For instance, for the parameter values used for Fig. 2.4, we have H 0 = 1:95 nm for natural boundary conditions on c at the bilayer-protein interface, but H 0 = 1:98 nm and H 0 = 1:91 nm for the fixed-value boundary conditions c 0 = 0 and c 0 = 1, respectively. Furthermore, a chemical affinity between particular lipid and protein species substantially affects the energy landscape of lipid-protein interactions [Fig. 2.4(b)]. Notably, G shows minima at both H 0 = a(0) and H 0 = a(1) for natural boundary conditions on c at the bilayer- protein interface. In contrast, fixed-value boundary conditions on c at the bilayer-protein interface yield a unique minimum of G at H 0 = a(c 0 ). The foregoing results can be understood intuitively by noting that, in the case of a (strong) chemical affinity between particular lipid and protein species, the lipids at the bilayer-protein boundary deviate from their preferred hydrophobic thickness if H 0 6= a(c 0 ), yielding a frustrated bilayer-protein configuration. However, since gradients in c are energetically unfavorable, the membrane patch maintains an approximately uniform lipid composition with c c 0 , as long as H 0 a(c 0 ). But, if H 0 deviates strongly enough from a(c 0 ), the energetics of bilayer thickness deformations can make it favorable for the lipid bilayer composition to rapidly transition around 21 Figure 2.4: Lipid-protein organization for a single membrane protein with and without a lipid- protein chemical affinity. (a) Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for the fixed-value boundary conditions c(R)= c 0 with c 0 = 0 (upper panel) and c 0 = 1 (lower panel) as a function of the radial coordinate r at H 0 = 1:94 nm. (b) Total bilayer energy G in Eq. (2.4) as a function of H 0 for a heteroge- neous bilayer containing lipid species A and B and homogeneous bilayers composed solely of lipid species A or B for natural boundary conditions on c at r= R or the indicated lipid-protein affinities. The dashed vertical lines show H 0 = H 0 for fixed c 0 = 0, free c 0 , and fixed c 0 = 1 (right to left), for which the dominant lipid composition in the membrane patch changes sharply from lipid species A to lipid species B. At r= R+ L we use, for all panels, the natural boundary conditions in Eqs. (2.7a)–(2.7c). We set L= 20 nm, R= R c ,m = 0, andt = 0. the membrane protein to a composition c6= c 0 , yielding a dominant lipid composition of the mem- brane patch that is distinct from c 0 , with large bilayer thickness deformations and largejÑcj in the immediate vicinity of the membrane protein. 2.2.3 Lipid chemical potential A nonzero chemical potential m introduces an external bias to the lipid composition of the membrane patch. Such an external bias may arise through interactions of the membrane patch 22 with the surrounding membrane in which, for instance, one lipid species may be dominant overall. A nonzero chemical potential m competes with bilayer thickness deformations and/or a chemical affinity between particular lipid and protein species to drive lipid-protein organization [Figs. 2.2(c) and 2.2(d)]. For example, for a membrane protein with a hydrophobic thickness H 0 = 1:7 nm matching the unperturbed bilayer leaflet thickness associated with lipid species A and no lipid- protein chemical affinity, the membrane patch only contains lipids of species A for m = 0. How- ever, asm is increased from zero, the lipid bilayer composition is increasingly biased towards lipid species B, eventually resulting in the localization of lipid species A in the immediate vicinity of the membrane protein, with the overall lipid composition of the membrane patch being dominated by lipid species B [see Fig. 2.5(a)]. For moderate values ofjmj, the total bilayer energy of the mem- brane patch, G in Eq. (2.4), shows two minima as a function of H 0 corresponding to lipid species A and B, respectively, with the minimum associated with lipid species A providing the global energy minimum for m < 0, and vice versa [see Fig. 2.5(b)]. Asjmj is increased, the energy landscape of the membrane patch approaches that associated with a membrane patch containing only lipids of species A or B, yielding a unique energy minimum at H 0 a(0) or H 0 a(1) [Fig. 2.5(b)]. The competition between a nonzero chemical potential, bilayer thickness deformations, and a chemical affinity between particular lipid and protein species can produce a variety of scenarios for lipid-protein organization [see Figs. 2.5(c) and 2.5(d)]. For instance, the external chemical potential may compete with lipid-protein affinity, thereby frustrating lipid-protein organization. In this case, the lipid composition of the membrane patch in the immediate vicinity of the membrane protein is determined by the chemical preference of the membrane protein for a particular lipid species, while the lipid composition of the membrane patch away from the membrane protein is dominated by the competition between the external chemical potential and bilayer thickness deformations [Fig. 2.5(c)]. Examining G in Eq. (2.4) as a function of H 0 we find a unique energy minimum for large enoughjmj that neither corresponds to the unperturbed hydrophobic thickness associated with lipid species A nor to the unperturbed hydrophobic thickness associated with lipid species B [Fig. 2.5(d)]. However, this minimum in G(H 0 ) can be rather shallow, suggesting strong 23 Figure 2.5: Dependence of lipid-protein organization on the membrane patch chemical potential m, in the case of a single membrane protein. Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for (a) natural boundary conditions on c at the bilayer-protein interface and H 0 = 1:7 nm with m = 10 3 k B T=nm 2 (upper panel) andm= 10 2 k B T=nm 2 (lower panel), and (c) the fixed-value boundary condition c(R)= c 0 at the bilayer-protein interface and H 0 = 1:9 nm with c 0 = 0,m= 10 2 k B T=nm 2 (upper panel) and c 0 = 1,m=10 2 k B T=nm 2 (lower panel). Total bilayer energy G in Eq. (2.4) as a function of H 0 for a heterogeneous bilayer containing lipid species A and B and homogeneous bilayers composed solely of lipid species A or B for the indicated values of m with (b) natural boundary conditions on c at the bilayer-protein interface and (d) the indicated fixed-value boundary conditions on c at the bilayer protein interface with m =10 2 k B T=nm 2 (m? 0) (curves corresponding to free c 0 included for reference). In panel (b), the dashed vertical lines show H 0 = H 0 for free c 0 at m = 10 3 k B T=nm 2 , m = 0, and m =10 3 k B T=nm 2 (right to left), for which the dominant lipid composition in the membrane patch changes sharply from lipid species A to lipid species B. Similarly, the gray dashed vertical lines in panel (d) show H 0 = H 0 for c 0 = 0, free c 0 , and c 0 = 1 at m = 0 (right to left). At r= R+ L we use, for all panels, the natural boundary conditions in Eqs. (2.7a)–(2.7c). We set L= 20 nm, R= R c , andt = 0 for all panels. fluctuations in the lipid-protein organization in this regime, with no sharp transition in the lipid composition of the membrane patch as H 0 is varied. 24 2.3 Gating of mechanosensitive ion channels in heterogeneous bilayers The results in Figs. 2.3–2.5 show that heterogeneity in the unperturbed lipid bilayer thickness can have a pronounced effect on the energy of protein-induced lipid bilayer thickness deformations. For conformational transitions in membrane proteins that involve a change in protein hydrophobic thickness [13,39,41–46], lipid heterogeneity is therefore expected to bias the protein conformation towards particular states. We illustrate here the regulation of protein function through hetero- geneity in the unperturbed lipid bilayer thickness using the tension-dependent gating of MscL as a model system. The experimental phenomenology of MscL is captured by a two-state model, in which MscL is assumed to be either in closed or open conformational states with gating probability P o = 1 1+ e (DGtDA)=k B T ; (2.9) where DG= G o G c is the energy difference between the open and closed states of MscL and DA=p[(R o ) 2 (R c ) 2 ] is the change in (in-plane) membrane area. The gating energyDG involves contributions due to internal protein conformational changes as well as bilayer-protein interactions. Remarkably, the basic experimental phenomenology of MscL gating in homogeneous bilayer membranes can already be captured based solely on contributions toDG arising from bilayer thick- ness deformations [2,44,45,74]. We consider here MscL gating in heterogeneous lipid bilayers [3], and calculateDG from Eq. (2.4) for the open and closed states of MscL, using the boundary con- ditions described in Sec. 2.1.2. To quantify MscL gating, we define the MscL gating tension ¯ t as the smallest membrane tension t with P o 1=2. Experiments on MscL gating in E. coli giant spheroplasts have yielded an estimate ¯ t 2:5 k B T=nm 2 [74, 113], but ¯ t is known to depend on the lipid bilayer composition [2, 34, 44, 45]. While there is, to our knowledge, currently no exper- imental evidence demonstrating a strong chemical affinity between (closed- or open-state) MscL and a particular lipid species, MscL could be modified to synthetically engineer such an affinity 25 Figure 2.6: MscL gating probability P o in Eq. (2.9) as a function of t for heterogeneous bilayers containing lipid species A and B, A and C, and A and D, and homogeneous bilayers containing only lipid species A, B, C, or D. All curves were calculated as in Fig. 2.2, with a= 1:7 nm, 2:2 nm, 1:8 nm, and 2:0 nm for lipid species A, B, C, and D, respectively. and, more generally, a great variety of other membrane proteins have been found to show chemical affinities for particular lipid species [102, 103, 105, 106]. As suggested by Fig. 2.3(b), for zero chemical potential and no lipid-protein chemical affinity, introduction of lipid species A into a bilayer containing originally only lipid species B energet- ically biases MscL towards its open state, thus decreasing ¯ t, and vice versa (see Fig. 2.6). The magnitudes and signs of the predicted shifts in ¯ t depend non-monotonically on the unperturbed hydrophobic thickness of the lipids under consideration. For instance, introduction of lipid species C or D in Fig. 2.6, which both have smaller a than lipid species B, into a bilayer containing origi- nally only lipid species A can result in a smaller as well as larger ¯ t than obtained with lipid species A and B. Figure 2.6 shows that, for lipids typically found in cell membranes [1, 15], the coupling of local lipid and protein composition through a(c) [1, 15, 16, 18, 60, 61] can produce shifts in ¯ t that are a substantial fraction of the bilayer rupture tension t r 3 k B T=nm 2 [1, 74], and can thus strongly affect MscL gating [34, 44]. In contrast to the purely mechanical scenario, we find that introducing a lipid-protein chemical affinity [see Fig. 2.7(a)], a nonzero chemical potential [see Fig. 2.7(b)], or a combination of both effects [see Fig. 2.7(c)] can further modify MscL gating, with a complex dependence of ¯ t on the lipid bilayer properties. For instance, Fig. 2.7(a) suggests that, if MscL shows an affinity 26 for a particular lipid species in its closed state but not in its open state, the gating tension of MscL is generally lowered, and vice versa. This can be understood intuitively by noting that a lipid-protein affinity increases the constraints on the lipid-protein system, and thus generally increases the energy of protein-induced lipid bilayer thickness deformations. An exception to this rule occurs if the fixed-value boundary condition on c at the bilayer-protein interface is chosen so as to coincide with the value of c implied by the corresponding natural boundary condition on c. Interestingly, Fig. 2.7(a) shows that, for certain types of lipid-protein affinity, MscL gating is effectively suppressed in the physically relevant tension range 0tt r , thus stabilizing MscL in its closed conformational state. This behavior can arise if the chemical affinity of open-state MscL for a particular lipid species yields a lipid composition in the vicinity of MscL that, due to MscL-induced lipid bilayer thickness deformations, is highly unfavorable from an energetic perspective. In addition to a chemical affinity between MscL and particular lipid species, a nonzero chemical potential can also substantially affect MscL gating [Fig. 2.7(b)]. In particular, for the parameter values used for Fig. 2.7(b),m > 0 tends to increase ¯ t whilem < 0 tends to decrease ¯ t. This can be understood intuitively by noting that H (c) > H (o) for MscL [2,37,38,45,74], while lipid species A has a smaller unperturbed hydrophobic thickness than lipid species B. As a result, a bias of the lipid bilayer composition towards lipid species B over lipid species A favors the closed state of MscL over the open state of MscL, and vice versa. Note, however, that the predicted shifts in ¯ t are not symmetric about m = 0, because the thickness deformation energy in Eq. (2.1) explicitly depends on a(c). Combining a chemical affinity of MscL for a particular lipid species with a nonzero chemical potential we obtain a complex dependence of ¯ t on the values of c 0 andm [see Fig. 2.7(c)]. For instance, with m > 0 we find substantial differences in ¯ t for scenarios in which MscL has an affinity for lipid species A in both its open and closed states [green curve in Fig. 2.7(c)], and in which MscL has an affinity for lipid species A in its open state and an affinity for lipid species B in its closed state [indigo curve in Fig. 2.7(c)]. In contrast, the MscL gating curves corresponding to these two scenarios are almost identical if the sign of m is flipped so that m < 0 [blue and red 27 curves in Fig. 2.7(c)]. Collectively, we find in Fig. 2.7 that heterogeneity in the lipid composition of bilayer membranes can, through protein-induced bilayer thickness deformations, lipid-protein chemical affinity, and the local lipid chemical potential, strongly affect the competition between distinct conformational states of membrane proteins. 28 Figure 2.7: Gating of single MscL in heterogeneous lipid bilayers. MscL gating probability P o in Eq. (2.9) as a function of membrane tension t for heterogeneous bilayers composed of lipid species A and B for (a)m = 0 with natural boundary conditions on c at the bilayer-MscL interface or the indicated values of c 0 in the closed (c 0 = c c 0 ) and open (c 0 = c o 0 ) conformational states of MscL, (b) m6= 0 with natural boundary conditions on c at the bilayer-MscL interface, and (c) m6= 0 with various combinations of free and fixed boundary conditions on c for the closed and open conformational states of MscL. In panel (c) we use m =10 2 k B T=nm 2 (m? 0). For reference, we show in all panels the corresponding results obtained form= 0 with natural boundary conditions on c at the bilayer-MscL interface [3], and for homogeneous bilayers composed solely of lipid species A or B. 29 Chapter 3 Crowded membranes and cooperative membrane function In this chapter, we extend the framework used in chapter 2 to study bilayer-mediated protein in- teractions in heterogeneous membranes that are crowded with membrane proteins. Section 3.1 mo- tivates our mean-field approach for modeling the interaction between membrane proteins through bilayer thickness deformations. Section 3.2 explores a variety of different scenarios for interacting membrane proteins in heterogeneous bilayers. Then in section 3.3, we investigate cooperative gat- ing of mechanosensitive ion channels in heterogeneous lipid environments. Altogether, this chapter expands upon the previous chapter to incorporate the effects of bilayer heterogeneity in bilayer- thickness-mediated protein-protein interactions and membrane protein cooperativity in crowded membranes. 3.1 Mean-field model of membrane protein interactions Cell membranes are crowded with membrane proteins, with the mean protein separation and size being of the same order of magnitude [13, 19]. An overlap in the bilayer thickness de- formations induced by neighboring membrane proteins gives rise to bilayer-thickness-mediated 30 Figure 3.1: Schematic of the mean-field model of bilayer-mediated protein interactions in crowded membranes [3–8]. The membrane protein at the center of the membrane patch has radius R and is surrounded by a lipid bilayer annulus of thickness L. The boundary conditions at the outer rim of the bilayer annulus at r= R+ L are taken to be axisymmetric about the membrane patch center, and are chosen so as to model a uniform array of membrane proteins (see Sec. 2.1.2). protein interactions, which can yield self-assembly of supramolecular membrane protein assem- blies [4, 40, 47–58]. Interestingly, it follows from the mechanics of bilayer thickness deforma- tions [2, 4, 39–41, 45, 47–50, 74] that protein-induced bilayer thickness deformations are localized over a scale of approximately 4 nm about each membrane protein, which corresponds to roughly one-half the typical protein-protein separation in cell membranes [13, 19]. Bilayer-thickness- mediated protein interactions may therefore have a broad impact on the local organization and cooperative function of cell membranes. To ascertain the role of bilayer-thickness-mediated pro- tein interactions in cell membrane organization and function, it is important to understand how bilayer-thickness-mediated protein interactions are modified by the heterogeneous lipid bilayer compositions typically found in cell membranes [1, 15, 16, 18]. 31 We employ a mean-field approach [3–8], and consider a membrane protein at the center of a membrane patch surrounded by a uniform array of identical proteins (see Fig. 3.1). This mean- field approach cannot capture the effect of protein shape on bilayer-thickness-mediated protein interactions [48–50, 57] but, based on previous work on bilayer-thickness-mediated protein inter- actions in homogeneous bilayers [4], is expected to correctly capture the sign, approximate range, and order of magnitude of bilayer-mediated protein interactions in heterogeneous bilayers. We first consider bilayer-thickness-mediated protein interactions for an array of membrane proteins with a hydrophobic thickness that is identical to or distinct from the hydrophobic thickness of the protein at the center of the membrane patch, at zero chemical potential in Eq. (2.2) and without a lipid-protein chemical affinity (see Sec. 3.2). Following that, we focus on the effects of a nonzero chemical potential in Eq. (2.2) and a chemical affinity between particular lipid and protein species on bilayer-thickness-mediated protein interactions in heterogeneous lipid bilayers (see Sec. 3.2.1 and 3.2.2). On this basis, we then investigate protein cooperativity in heterogeneous bilayer mem- branes (see Sec. 3.3). We find that heterogeneity in the lipid tail length can expand the repertoire and range of bilayer-thickness-mediated protein interactions, can yield colocalization of lipids and membrane proteins according to their preferred hydrophobic thickness, and can have a pronounced effect on bilayer-mediated protein cooperativity. 3.2 Lipid-protein domains in crowded membranes We first explore the influence of lipid heterogeneity on bilayer-thickness-mediated protein interactions without lipid-protein chemical affinities and at zero chemical potential in Eq. (2.2) [3]. We find that the local lipid organization due to protein-induced lipid bilayer thickness de- formations not only depends crucially on the preferred lipid and protein hydrophobic thickness [15,16,18,23–25] but also on the (edge-to-edge) protein separation, L [see Figs. 3.2(a) and 3.2(b)]. In particular, proteins with identical hydrophobic thickness locally yield separate lipid domains for L' 10 nm, which merge into a single domain at small L [Fig. 3.2(a)]. In contrast, proteins with 32 Figure 3.2: Local lipid organization and bilayer-thickness-mediated protein interactions in crowded membranes. Thickness deformation profile h, lipid composition c (color bars), and en- ergy density g at L= 3 nm and L= 12 nm versus(r R)=L with R= R c for (a) identical proteins with H 0 = H L = 2:0 nm and (b) distinct proteins with H 0 = 2:2 nm and H L = 1:3 nm in bilayers composed of lipid species A and B. (c) Bilayer-thickness-mediated protein interaction potentials G int for identical (H 0 = H L ) and distinct (H 0 6= H L ) proteins as in panels (a) and (b) with R= R c in a heterogeneous bilayer composed of lipid species A and B, and homogeneous bilayers containing only lipid species A or B. The interaction potentials G int are obtained by subtracting from Eq. (2.4) the (noninteracting) large-L limit of Eq. (2.4), which is linear in L. We sett = 0 for all panels. distinct hydrophobic thickness locally induce, at large L, lipid domains with distinct composition, but these domains tend to be dispersed at small L [Fig. 3.2(b)]. In bilayers with homogeneous lipid composition, membrane proteins with identical (distinct) hydrophobic thickness generally show favorable (unfavorable) bilayer-thickness-mediated interac- tions at small L [4, 13, 40, 47, 50]. We find that, depending on the lipid and protein hydropho- bic thickness considered, heterogeneous lipid bilayers can yield similar bilayer-mediated pro- tein interactions, or favorable (unfavorable) bilayer-mediated interactions between proteins with 33 identical (distinct) hydrophobic thickness that are longer in range and smaller (greater) in magni- tude than in homogeneous bilayers [see Fig. 3.2(c)]. Lipid heterogeneity thus expands the reper- toire of bilayer-thickness-mediated protein interactions [4, 13, 40, 47, 50]. Experiments on a wide range of membrane proteins suggest that bilayer-thickness-mediated protein interactions provide a general mechanism for protein clustering [51–54, 56, 58]. Figure 3.2 predicts that such clus- tering of membrane proteins is accompanied by changes in local lipid organization, leading to co-localization of lipids and membrane proteins according to their preferred hydrophobic thick- ness [15, 16, 18, 23–25, 51–54, 56, 58]. 3.2.1 Proteins with identical hydrophobic thickness In this section we explore further scenarios in which the protein at the center of the membrane patch has a hydrophobic thickness H 0 that is identical to the hydrophobic thickness of the sur- rounding membrane proteins, H L = H 0 (see Fig. 3.3). We set here H 0 = H L = 2:0 nm, which lies in between the unperturbed bilayer leaflet thicknesses a= 1:7 nm and a= 2:2 nm associated with lipids of species A and B, respectively. We first consider situations in which the protein at the center of the membrane patch and its neighboring membrane proteins have chemical affinities for the same [see Fig. 3.3(a)] or distinct [see Fig. 3.3(b)] lipid species with zero chemical potential in Eq. (2.2). Subsequently, we consider bilayer-thickness-mediated protein interactions for situ- ations in which the protein at the center of the membrane patch and its neighboring membrane proteins have natural boundary conditions on c at the bilayer-protein interfaces but m6= 0 with, again, H L = H 0 [see Fig. 3.3(c)]. Comparison of the results in Figs. 3.4(a), 3.4(b), and 3.4(c) with the corresponding results obtained for natural boundary conditions on c at the bilayer-protein in- terfaces withm = 0 [3] shows that a chemical preference of the membrane proteins for a particular lipid species or a non-zero chemical potential can produce substantial shifts in bilayer-thickness- mediated protein interactions. For c 0 = c L = 0, m = 0, and large L, we find that lipid domains with c 0 form near the protein boundaries at r= R and r= R+ L [Fig. 3.3(a)]. As L is decreased, these domains merge, 34 Figure 3.3: Bilayer-thickness-mediated protein interactions in heterogeneous bilayers for mem- brane proteins with identical hydrophobic thickness, H 0 = H L , using H 0 = H L = 2:0 nm. Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for L= 3 nm and L= 12 nm versus(rR)=L with R= R c for (a) fixed-value boundary conditions on c with c 0 = c L and m = 0, (b) fixed-value boundary conditions on c with c 0 6= c L and m = 0, and (c) natural boundary conditions on c and m =10 2 k B T=nm 2 (m? 0). We sett = 0 for all panels. as also found for membrane proteins with natural boundary conditions on c [3], but the resulting bilayer-thickness-mediated protein interactions G int (L) are not as favorable as in the case of natural boundary conditions on c [Fig. 3.4(a)]. This can be understood intuitively by noting that H 0 = H L = 2:0 nm and c 0 = c L = 0 produce substantial bilayer thickness deformations in the vicinity of the proteins, which incurs an energy penalty and produces less favorable interactions at small L. As expected, we find similar, but less pronounced, effects of lipid-protein chemical affinity on lipid organization and bilayer-thickness-mediated protein interactions with H 0 = H L = 2:0 nm and c 0 = c L = 1, for which the bilayer thickness is deformed less strongly in the vicinity of the proteins than for c 0 = c L = 0 [Fig. 3.3(a)]. For situations with c 0 6= c L and m = 0, lipid-protein chemical 35 Figure 3.4: Interaction potentials for proteins with identical hydrophobic thickness, H 0 = H L , using H 0 = H L = 2:0 nm. Bilayer-thickness-mediated protein interaction potentials G int (L) for (a) fixed- value boundary conditions on c with c 0 = c L and m = 0, (b) fixed-value boundary conditions on c with c 0 6= c L and m = 0, and (c) natural boundary conditions on c and m =10 2 k B T=nm 2 (m ? 0). In all panels we show, for reference, also the G int (L) obtained for natural boundary conditions on c and m = 0 in heterogeneous bilayers composed of lipid species A and B (AB), as well as the G int (L) obtained for homogeneous bilayers composed solely of lipid species A or B [3]. To calculate G int (L), we subtracted from the total bilayer energy G in Eq. (2.4) the value of G obtained in the large-L, non-interacting regime (see Appendix E). We sett = 0 for all panels. affinity can further reduce the strength of energetically favorable interactions, and even render bilayer-thickness-mediated protein interactions unfavorable for all protein separations considered in Fig. 3.4(b). This can be understood intuitively by noting that, in this case, distinct lipid-protein affinities at r= R and r= R+ L induce distinct lipid environments at r= R and r= R+ L, which prevents a merging of lipid domains at small L [Fig. 3.3(b)]. Furthermore, we find that, compared 36 to situations with natural boundary conditions on c, fixed-value boundary conditions on c at the bilayer-protein interfaces can substantially increase the range of (unfavorable) bilayer-thickness- mediated protein interactions [Figs. 3.4(a) and 3.4(b)]. In Fig. 3.3(c) we explore bilayer-thickness-mediated protein interactions for situations in which m< 0 orm> 0 in Eq. (2.2), with H 0 = H L and natural boundary conditions on c. A positive chem- ical potential tends to bias the lipid composition of the membrane patch towards lipids of species B, while a negative chemical potential tends to bias the lipid composition of the membrane patch towards lipids of species A. We find that, depending on the interplay of protein hydrophobic thick- ness, the sign and magnitude of the chemical potential, and the unperturbed bilayer thicknesses associated with the particular lipid species under consideration, a nonzero chemical potential can strongly affect bilayer-thickness-mediated protein interactions [Fig. 3.4(c)]. For instance, with H 0 = H L = 2:0 nm we find thatm = 0 andm > 0 give similar interaction potentials, but thatm < 0 gives bilayer-thickness-mediated protein interactions that are much more favorable at intermediate L, and have a somewhat greater range, than obtained with m = 0 or m > 0 [Fig. 3.4(c)]. The latter result can be understood intuitively by noting that m < 0 and m > 0 (or m = 0) yield a similar lipid-protein organization at small L, but drastically different lipid bilayer compositions at large L that are dominated by lipid species A and B, respectively [Fig. 3.3(c)]. 3.2.2 Proteins with distinct hydrophobic thickness We consider in this section further scenarios in which the protein at the center of the mem- brane patch has a hydrophobic thickness H 0 that is different from the hydrophobic thickness of the surrounding membrane proteins, H L 6= H 0 (see Fig. 3.5). In particular, we set H 0 = 2:2 nm and H L = 1:3 nm. These values of H 0 and H L should be contrasted with the unperturbed bilayer leaflet thicknesses a= 1:7 nm and a= 2:2 nm associated with lipid species A and B, respectively. We thus have perfect hydrophobic matching for the protein at r = R and lipid species B, and a preference of the proteins at r= R+ L for lipid species A over lipid species B. 37 Figure 3.5: Bilayer-thickness-mediated protein interactions in heterogeneous bilayers for mem- brane proteins with distinct hydrophobic thickness, H 0 6= H L , using H 0 = 2:2 nm and H L = 1:3 nm. Bilayer leaflet thickness profile h (left axes), lipid composition c (color bars), and energy density g in Eq. (2.6) (right axes) for L= 3 nm and L= 12 nm versus(r R)=L with R= R c for (a) fixed- value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 1 atm= 0, (b) fixed-value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 0 atm = 0, and (c) natural boundary conditions on c atm =10 2 k B T=nm 2 (m? 0). We sett = 0 for all panels. Focusing first on situations withm= 0 we find that, if c L = 1 so that that lipids at r= R+L are substantially expanded in hydrophobic thickness, bilayer-thickness-mediated protein interactions are more unfavorable and longer in range than in the case of natural boundary conditions on c, irrespective of whether c 0 = 0 or c 0 = 1 [see Fig. 3.6(a)]. This can be understood intuitively by noting that, to reduce the energy cost of bilayer thickness deformations, the bilayer composition (rapidly) changes here from lipid species B towards lipid species A as r is decreased from r= R+L for both c 0 = 0 and c 0 = 1 [see Fig. 3.5(a)], which results in lipid-protein configurations that are increasingly unfavorable as L is decreased. Conversely, if c L = 0 the lipids at r= R+ L are not as expanded in hydrophobic thickness as for c L = 1, and we find bilayer-thickness-mediated protein 38 Figure 3.6: Interaction potentials for proteins with distinct hydrophobic thickness, H 0 6= H L , using H 0 = 2:2 nm and H L = 1:3 nm. Bilayer-thickness-mediated protein interaction potentials G int (L) for (a) fixed-value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 1 at m = 0, (b) fixed-value boundary conditions on c with c 0 = 0 or c 0 = 1 and c L = 0 at m = 0, and (c) natural boundary conditions on c at m =10 2 k B T=nm 2 (m? 0). In all panels we show, for reference, also the G int (L) obtained for natural boundary conditions on c andm= 0 in heterogeneous bilayers composed of lipid species A and B (AB), as well as the G int (L) obtained for homogeneous bilayers composed solely of lipid species A or B [3]. To calculate G int (L), we subtracted from the total bilayer energy G in Eq. (2.4) the value of G obtained in the large-L, non-interacting regime (see Appendix E). We sett = 0 for all panels. interactions that are similar to those obtained with natural boundary conditions on c, irrespective of whether c 0 = 0 or c 0 = 1 [see Fig. 3.6(b)]. In Fig. 3.5(c) we consider bilayer-thickness-mediated protein interactions for H 0 = 2:2 nm and H L = 1:3 nm with natural boundary conditions on c for m < 0 and m > 0 in Eq. (2.2). Simi- larly as in Fig. 3.3(c), we find that a nonzero chemical potential can have a pronounced effect on 39 bilayer-mediated protein interactions. In particular, m < 0 results, at intermediate L, in a regime with strongly favorable bilayer-mediated protein interactions [Fig. 3.6(c)]. In contrast, m = 0 and m> 0 both yield unfavorable bilayer-mediated protein interactions for all L in Fig. 3.6(c). This can be understood intuitively by noting that homogeneous bilayers composed solely of lipid species A yield an energetically favorable interaction regime in Fig. 3.6(c). Since a negative lipid chemical potential biases the lipid bilayer composition towards lipids of species A,m < 0 can therefore also yield an energetically favorable interaction regime. Interestingly, for m < 0 the energetically fa- vorable regime of bilayer-thickness-mediated protein interactions in Fig. 3.6(c) has a longer range than for homogeneous bilayers composed solely of lipid species A. 3.3 Membrane protein cooperativity in heterogeneous bilayers We first consider the cooperative effects of membrane crowding on membrane protein function in heterogeneous bilayers without lipid-protein chemical affinities and at zero chemical potential in Eq. (2.2) [3]. Our calculations predict that protein crowding affects the energy landscape of hydrophobic lipid-protein interactions in heterogeneous lipid bilayers [see Fig. 3.7(a)]. For mem- brane proteins with identical hydrophobic thickness we find that the interaction of lipid domains smoothens the transition in lipid patch composition at the critical H 0 = H L = H [solid curves in Fig. 3.7(a)]. Calculating G as a function of H 0 for fixed H L , we find at large L a global minimum of G corresponding to H 0 = H L and a local minimum with H 0 6= H L for which, as illustrated in Fig. 3.2(b), the two proteins have distinct local lipid environments [dashed curves in Fig. 3.7(a)]. As L is decreased, the lipid domains induced by proteins with H 0 6= H L tend to be dispersed, and the corresponding local minimum in G(H L ) is suppressed. The interplay of local lipid organization and protein crowding can have intricate effects on membrane protein cooperativity, which we illustrate in Fig. 3.7(b) for MscL. As in Fig. 2.6 we 40 Figure 3.7: Hydrophobic lipid-protein interactions in crowded membranes. (a) Bilayer free energy G in Eq. (2.4) as a function of H 0 for a heterogeneous bilayer composed of lipid species A and B with H L = H 0 or H L = H o and the indicated values of L. We set t = 0 and R= R c for both H L = H 0 and H L = H o . For ease of visualization we shifted the curves for H L = H o by G 0 = G min 30 k B T , where G min are the respective global minima of G(H 0 ). (b) MscL gating tension ¯ t implied by Eq. (2.9) as a function of L for a heterogeneous bilayer containing lipid species A and B and homogeneous bilayers composed of lipid species A or B with open-state MscL proteins at r= R+ L. thereby follow Refs. [2, 45, 47, 74] and assume that contributions due to bilayer thickness defor- mations dominate the MscL gating energy. Figure 3.7(b) implies that, compared to dilute mem- brane protein environments, the presence of open-state MscL proteins with L/ 2:0 nm leads to a decrease in the MscL gating tension ¯ t byD ¯ t AB 2:4 k B T=nm 2 for heterogeneous bilayers com- posed of lipid species A and B, but byD ¯ t A 2:0 k B T=nm 2 for homogeneous bilayers composed of lipid species A. In contrast, Fig. 3.7(b) suggests that bilayers composed of lipid species B only yield an MscL gating tension smaller than the membrane rupture tension, ¯ t <t r , for L/ 2:5 nm or after introduction of some other lipid species, such as lipid species A, into the membrane. Thus, we find that even the pure mechanical coupling of local lipid and protein composition through 41 a(c) [1, 15, 16, 18, 60, 61] can strongly affect bilayer-mediated protein cooperativity in crowded membranes. Moreover, the results in Figs. 3.3–3.6 show that introducing a nonzero chemical potential in Eq. (2.2) and a chemical affinity between particular lipid and protein species can also have a pro- nounced effect on bilayer-thickness-mediated protein interactions in heterogeneous lipid bilayers. This, in turn, suggests that lipid heterogeneity modifies protein cooperativity in the crowded pro- tein environments provided by cell membranes [13, 15, 16, 18, 19, 33]. In analogy to Sec. 2.3, we explore here the effect of lipid heterogeneity on protein cooperativity using MscL as a model system. For MscL embedded in lipid bilayers with homogeneous lipid composition, it has been predicted [47] and observed experimentally [54] that MscL gating is affected by protein crowding. As for Figs. 3.3–3.6, we use the mean-field model in Fig. 3.1 to explore protein cooperativity in heterogeneous bilayers. Specifically, we take the protein at the center of the membrane patch to correspond to the closed or open state of MscL, and take the neighboring proteins to impose bilayer boundary conditions corresponding to open-state MscL proteins. Employing Eq. (2.9) with differ- ent L we then calculate, as a function of membrane tension, the probability that the MscL protein at the center of the membrane patch is in its open conformational state, P o , and thus quantify protein cooperativity (see Fig. 3.8). We denote the protein edge-to-edge separations in the far-field (non- interacting) and closed-packed (strongly interacting) regimes by L= L f and L= L c , respectively. We use here L f = 12 nm and L c = 3 nm. We denote the corresponding MscL gating tensions by ¯ t f and ¯ t c , respectively, which are the smallest values of the membrane tension for which P o 1=2. We explore here how a lipid-MscL chemical affinity affects cooperative gating of MscL in heterogeneous bilayers, at zero chemical potential and assuming that MscL shows the same lipid affinity in its closed and open states [see Fig. 3.8(a)]. We first note [3] that, for heterogeneous bilayers with natural boundary conditions on c at the bilayer-protein interfaces, the MscL gating tension decreases as L is decreased, ¯ t c < ¯ t f , with amplified MscL cooperativity as compared to homogeneous bilayers. Figure 3.8(a) shows that a lipid-MscL chemical affinity can also yield amplified MscL cooperativity with ¯ t c < ¯ t f , and can result in substantial shifts in the MscL gating 42 tension as L is decreased. Interestingly, we find in Fig. 3.8(a) that, if MscL has an affinity for lipid species B in its closed and open states, the (hypothetical) MscL gating tension is greater than the approximate bilayer rupture tensiont r 3 k B T=nm 2 [74] for L= L f , ¯ t f >t r , but ¯ t c <t r for L= L c . Thus, for these boundary conditions on c, MscL gating relies on cooperative effects. Allowing for distinct lipid-MscL affinities in the closed and open conformational states of MscL further broadens the cooperative response of MscL to changes in membrane tension [see Fig. 3.8(b)], and can further amplify cooperative shifts in the MscL gating tension. In addition to lipid-protein chemical affinity, a nonzero chemical potential can also substantially affect protein cooperativity in heterogeneous bilayers [see Fig. 3.8(c)]. For instance, using natural boundary conditions on c in the closed and open states of MscL, cooperativity in MscL gating is markedly amplified for m > 0 in Fig. 3.8(c) as compared to m = 0. This can be understood intuitively by noting thatm> 0 biases the lipid bilayer composition towards lipid species B, which tends to produce a strongly increased bilayer thickness deformation energy for the open state of MscL, thus increasing the significance of cooperative effects with respect to scenarios withm = 0. In contrast, m < 0 only produces comparatively modest shifts in the MscL gating cooperativity in Fig. 3.8(c) with respect to m = 0. Similarly as in Figs. 3.8(a) and 3.8(b), cooperative effects are needed in Fig. 3.8(c) for m > 0 to shift the MscL gating tension to values smaller than the bilayer rupture tension t r . Taken together, the results in Fig. 3.8 thus show that a lipid-protein chemical affinity or a nonzero lipid chemical potential can substantially affect protein cooperativity in heterogeneous lipid bilayers and, in particular, amplify cooperative effects. 43 Figure 3.8: Cooperativity in the tension-dependent gating of MscL in heterogeneous bilayers. Probability that the MscL protein at the center of the membrane patch is in its open state, P o in Eq. (2.9), versus membrane tensiont for (a) identical fixed-value boundary conditions on c at all bilayer-protein interfaces with m = 0, (b) the indicated fixed-value boundary conditions on c at r= R and r= R+ L in the closed (c c 0 ) and open (c o 0;L ) conformational states of MscL with m = 0, and (c) natural boundary conditions on c at all bilayer-protein interfaces withm=10 2 k B T=nm 2 (m? 0). For reference, we also show in all three panels the corresponding MscL gating curves obtained with natural boundary conditions on c at all bilayer-protein interfaces withm= 0, as well as the corresponding MscL gating curves obtained for homogeneous bilayers composed solely of lipid species A or B [3]. We use H 0 = H c and H 0 = H o for the closed and open conformational states of the MscL protein at the center of the membrane patch, and H L = H o for the proteins at the outer boundary of the membrane patch. The edge-to-edge protein separations considered here, L= L c = 3 nm and L= L f = 12 nm, correspond to regimes with strong and negligible bilayer- mediated protein interactions (see also Figs. 3.3–3.6). 44 Chapter 4 Curved membrane domains and membrane plasticity This chapter focuses on the general physical principles underlying the stability of curved cave- ola domains and their response to mechanical perturbations in cell membranes. In section 4.1, we introduce a spherical cap model for cup-shaped caveola. Section 4.2 explores the mechanics of cup-shaped caveolae and their response to changes in membrane tension, with varying caveolae spontaneous curvature and domain size, for a variety of different scenarios for the caveola line tension. Section 4.2 discussed the distribution of stable caveola shapes under different membrane tension regimes that are relevant for caveolae in cell membranes. Overall, we present in this chap- ter a mechanical model of caveolae that aims to capture the physical mechanisms underlying the experimentally observed cup-like caveola shapes. 4.1 Energy of caveola domain Based on SRM and EM observations [11, 76–78, 82–86] and previous theoretical models [30, 88, 92] we describe caveolae as membrane domains that take the shape of spherical caps [12] with radius of curvature R and fixed surface area S=pL 2 , where L is the in-plane radius of the flattened caveola (Fig. 4.1). While the assumption of a fixed caveola area is consistent with experiments 45 Figure 4.1: Cup-shaped caveolae. (a) Two-dimensional z-x rendering of caveolae obtained from three-dimensional superresolution fluorescence imaging of caveolin-1 at the plasma membrane for mouse embryonic fibroblast cells freely adhering to fibronectin substrates (left) and with adhesion constrained to 210 10 mm 2 fibronectin islands (right). Changes in the cell adhesion geometry are expected to modify the cell adhesion forces [9,10], and the shape and organization of caveolae [11]. The white dotted curves indicate the approximate position of the plasma membrane for the observed caveolae. Scale bar: 100 nm. (Adapted with permission from Ref. [11].) (b) Schematic of the spherical cap model of caveola shape, with the caveola domain indicated in red and the surrounding membrane indicated in blue. We denote the caveola surface area by S =pL 2 , the caveola radius of curvature by R, the caveola base radius by a, and the caveola invagination depth by h [12]. showing that the number of caveolin-1 proteins in curved caveolae is approximately constant with only a small pool of caveolin-1 outside caveolae [114, 115], we also note that flattened caveolae may disassemble [31], which may provide a mechanism for plasticity in caveola size. The caveola shape is conveniently specified by the fraction of the surface of a sphere of radius R covered by the caveola, b = L 2 =4R 2 , with b = 0 for completely flattened caveolae, b = 1 for fully budded caveolae, and 0 0, permitting exact analytic solutions. However, for the scenarios with a (linear) dependence ofs onb considered here, mini- mization of Eq. (4.2) leads to a polynomial equation of degree greater than four, in which case an 48 analytic solution is not feasible [124]. To treat all the choices for s considered here on the same footing, we numerically determineb min throughout this chapter by minimizing G in Eq. (4.2) us- ing the numpy.argmin function implemented in the NumPy package of the programming language Python [125]. We checked our numerical solution procedure against the exact analytic solutions available for constants. For constant s and g = 0, the caveola energy in Eq. (4.2) reduces to the energy studied in Ref. [87] in the context of membrane budding driven by line tension. For constants and C 0 = 0, Eq. (4.2) reduces to the energy introduced in Ref. [30] to describe bistable caveolae with b 0 or b 1. An energy similar to Eq. (4.2) with constant s was also recently employed [88] in the context of a two-state model of caveolae withb 0 orb 1. We show here that, for the values of g, L,s, and C 0 relevant for caveolae in cell membranes, Eq. (4.2) can yield stable cup-like caveola shapes with 0 0, Eq. (4.2) yields stable cup-like 49 Figure 4.2: Caveola energy G in Eq. (4.2) as a function of the caveola shape parameter b for g = 0:008, 0.028, and 0.048 k B T=nm 2 with s = 0 (solid curves) and s = 1 k B T=nm (dashed curves). The solid and dashed horizontal lines indicate the energy minima of the cyan solid and green dashed curves, respectively. We set L= 100 nm and C 0 = 0:04 nm 1 . The schematics in the top panels show caveolae (indicated in red) with shape parametersb = 0, 1/4, 1/2, 3/4, and 1 (left to right). We setk = 20 k B T . caveola shapes with 0 0 with s = 1 k B T=nm (dashed curves). We use the same labeling scheme for b min and db min =dg. The stars along the right axes mark the maximum magnitudes of db min =dg,S max , for the s = 0 curves, using the same color scheme as for the db min =dg-curves. We setk = 20 k B T . predicts that the (fully budded) caveola shape is independent of membrane tension. In contrast, for g >g each value ofb min corresponds, for given L and C 0 , to a unique value ofg. For a constant line tensions > 0 in Eq. (4.2), cup-like caveola shapes tend to be ruled out by a sharp transition fromb min = 1 tob min =b < 1 with increasing membrane tension. In particular, at a critical membrane tensiong caveola states withb = 1 andb =b have equal energy (see the 52 green dashed curve in Fig. 4.2), yielding a discontinuous transition in caveola shape fromb min = 1 tob =b with increasing membrane tension (see the dashed curves in Fig. 4.3). The value ofg and the magnitude of this discontinuous jump in caveola shape depend on the values of L, C 0 and s. In particular, for a given L and s such that g 0, a decrease in C 0 results in a smaller g and a greater discontinuous jump length 1b . As C 0 ! 0 with s > 0, we recover the bistable system described in Ref. [30], which rules out cup-like caveola shapes. Indeed, if G s dominates over G k and C 0 ! 0 in Eq. (4.2) the only two stable caveola shapes correspond to b 0 and b 1 at large and small membrane tension, respectively [30]. In contrast, for small enoughs, the discontinuity in the dependence of caveola shape ong disappears, yielding a continuous spectrum of stable cup-like caveola shapes (see Appendix F). Figure 4.3 shows that the sensitivity of the stable caveola shape to changes in membrane tension depends crucially on the size and spontaneous curvature of caveolae. For 0 0 in Fig. 4.3, we find thatS max increases (weakly) with L as well as C 0 for the parameter ranges relevant for caveolae. 4.2.2 Caveola-shape-dependent line tension at caveola domain boundary As noted above, cup-shaped caveolae are expected to deform the surrounding membrane [92], leaving a membrane footprint. Such caveola-induced membrane shape deformations may incur an energy cost that depends on b. Most straightforwardly, membrane footprints with smaller curvatures may be more favorable from an energetic perspective, thus increasing the stability of caveola states with smallb. This scenario corresponds to a composition of the membrane footprint 53 with zero spontaneous curvature [128]. We phenomenologically account for such situations by allowing fors =s + (b) withs + (b)=b k B T=nm. Alternatively, curvature-sensing or curvature- generating lipids or proteins may be enriched in the curved membrane footprint of caveolae, and thus assist caveola budding [93, 94]. Such situations correspond to a caveola membrane footprint with non-zero spontaneous curvature. A simple phenomenological description of this scenario is obtained by setting s =s (b) with s (b)=(1b) k B T=nm. Effectively, with s =s , the tendency of a finite line tension s > 0 to stabilize curved membrane domains [30, 87, 88] is thus further amplified. In analogy to Fig. 4.3, Fig. 4.4 shows the energetically preferred caveola shapeb min as a func- tion of membrane tension for different values of C 0 and L, using the b-dependent line tensions s + (b) and s (b). We find that, compared to s = 1 k B T=nm, s + yields a sharper transition in caveola shape from b = 1 to b < 1 as the membrane tension is increased from zero, with this transition occurring at smaller values ofg (see the dashed curves in Figs. 4.3 and 4.4). This can be understood by noting that, compared tos = 1 k B T=nm,s + biases the caveola shape towards more flattened states and thus decreases the energy cost of caveola flattening. In contrast, s yields a greater range of stable caveola shapes with 0 0,S max increases with increasing C 0 fors (see the stars along the right axes in Fig. 4.4) and with decreasing C 0 fors + . For boths ands + , largerS max are obtained for larger L in Fig. 4.4. 55 4.3 Caveola shape distribution Thermal fluctuations are expected to perturb the caveola shape about the energetically most favorable state b =b min . A simple way to account for such thermal effects is to assume that the caveola shape is governed by a Boltzmann distribution. The probability of caveola states with b 1 b b 2 is then given by P(b 1 ;b 2 )= 1 Z Z b 2 b 1 dbe G=k B T ; (4.5) where the normalization constant Z is chosen such that P(0;1)= 1, G is given by Eq. (4.2), k B is Boltzmann’s constant, and T denotes the temperature of the system. We employ here Eq. (4.5) to predict the probability of finding cup-like caveola shapes and, in particular, focus on b 1 = 1=4 and b 2 = 3=4 in Eq. (4.5). We numerically evaluate P(b 1 ;b 2 ) in Eq. (4.5) using Python (version 2.7.12). Figures 4.5(a) and 4.5(b) show P(1=4;3=4) in Eq. (4.5) as a function of membrane tension for s = 0 and s =s (b)=(1b) k B T=nm in Eq. (4.2), respectively. We focus in Fig. 4.5 on these two choices for s because the foregoing results show that a wide range of cup-like caveola shapes can be obtained with zero (or small)s ands that decrease withb. Furthermore, we focus in Fig. 4.5 on the range in spontaneous curvature 0.04 nm 1 / C 0 / 0:08 nm 1 most relevant for caveolae [76–78, 82–84, 121]. We find in Fig. 4.5 that, for s = 0 as well as s , larger values of C 0 and smaller values of L tend to yield a larger range in membrane tension for which caveola shapes with 1=4b 3=4 are dominant. For instance, for s = 0 and C 0 = 0:08 nm 1 we find P(1=4;3=4)> 0:5 in the membrane tension range 0:01/g/ 0:07 k B T=nm 2 with L= 50 nm and 0:02/g/ 0:05 k B T=nm 2 with L= 100 nm [see the solid and dashed purple curves in Fig. 4.5(a)]. For s =s (b) and C 0 = 0:08 nm 1 we find P(1=4;3=4)> 0:5 in the membrane tension range 0:04/g/ 0:12 k B T=nm 2 with L= 50 nm and 0:04/g/ 0:07 k B T=nm 2 with L= 100 nm [see the solid and dashed purple curves in Fig. 4.5(b)]. Figure 4.5 illustrates that, within the ranges of C 0 and L relevant for caveolae, different values of C 0 and L tend to yield distinct caveola shapes in 56 Figure 4.5: Probability of caveola states with 1=4b 3=4, P(1=4;3=4) in Eq. (4.5) with the caveola energy in Eq. (4.2), as a function of membrane tension for (a)s = 0 and (b)s =s (b)= (1b) k B T=nm using C 0 = 0:04, 0.06, and 0.08 nm 1 and L= 50 nm (solid curves) and 100 nm (dashed curves). We setk = 20 k B T . distinct membrane tension regimes. Combined with the results in Figs. 4.2–4.4, this suggests that cells may use heterogeneity in the values of C 0 and L to produce a staggered response of caveola shape to changes in membrane tension. 57 Chapter 5 Membrane footprint of curved membrane domains In this chapter, we extend the caveola model described in chapter 4 to incorporate the effects of the curved neck region surrounding caveolae. Section 5.1 describes the membrane footprint model of the caveola neck, where the energy contribution of caveola-induced membrane deformation is quantified using an arclength parameterization of curved surfaces. In section 5.2, we explore the minimum-energy shapes of the caveola neck and their dependence on the membrane intrinsic curvature and the local membrane organization around caveolae. This chapter thus expands upon the previous chapter to study how a curved caveola neck, through the mechanics of caveola-induced membrane shape deformations, affects the stable shapes of caveolae. 5.1 Caveola membrane footprint energy We build upon the spherical cap model of caveolae described in section 4.1 of chapter 4, to account for the effects of the curved membrane neck surrounding caveolae domains. In particular, we consider an axisymmetric membrane patch composed of two concentric membrane domains: a cup-like caveola membrane domain taking the shape of a spherical cap, and the membrane footprint (caveola neck) associated with this caveola shape. As in the previous chapter, we take the spherical 58 Figure 5.1: Schematic of a cup- or cap-shaped caveola (red) and its associated membrane footprint or caveola neck (green). We denote the caveola radius of curvature and area by R and S, yielding a contact anglea at the caveola-neck interface. We denote the coordinates parallel and perpendicular to the axis of symmetry by z(s) and r(s) , where s is the arclength along the caveola profile, and the angle between the r-axis and the tangent to the neck profile byy(s). cap domain to have radius of curvature R and area S=pL 2 , where L is the caveola size. Following chapter 4, we use b = L 2 =4R 2 to characterize the caveola (cap) shape, with shapes ranging from flattened to budded caveolae corresponding to the range 0 < b < 1. We note that the contact anglea at the caveola-neck interface can be expressed in terms of the caveola shape parameter as a= cos 1 (1b) (see Fig. 5.1). Based on the experimental phenomenology of caveolae and other curved membrane domains [93, 94, 129], we allow for distinct molecular compositions of caveola membrane domains and caveola necks. We thus allow for the distinct spontaneous curvatures C (cap) 0 and C (neck) 0 in caveola and caveola neck regions, respectively. 5.1.1 Arclength parameterization of the caveola neck Caveola necks can be highly curved, which necessitates the consideration of arbitrarily large membrane shape deformations in the membrane region surrounding caveolae. To this end, we employ the arclength parameterization of membrane shape deformations [128, 130–136]. We thus consider the caveola neck energy G neck = 2p Z s b 0 k 2 ˙ y+ siny r C (neck) 0 2 +g(1 cosy) rds; (5.1) 59 where s denotes the arclength along the contour of the neck profile with 0 s s b , r(s) denotes the coordinate perpendicular to the axis of symmetry, and y(s) denotes the angle between the r-axis and the tangent to the neck profile (see Fig. 5.1). The first term in Eq. (5.1) arises from the Helfrich bending energy of a membrane with bending rigidity k and a spontaneous curvature C (neck) 0 . The second term in Eq. (5.1) corresponds to the work required to form the caveola membrane footprint against a lateral membrane tensiong. We note that this parameterization constraints r(s) and z(s) withy(s) through the geometric relations ˙ r= cosy; (5.2) ˙ z= siny; (5.3) where the dot notation represents the first derivative of the coordinates with respect to s. The two principal curvatures of the membrane footprint are given by ˙ y and siny=r in Eq. (5.1). This can be understood by noting from differential geometry [137, 138] that the planes of the two principal curvatures are perpendicular to each other, and that the radius of curvature in the plane perpendic- ular to the r-z plane is given by r=siny. From Eq. (5.2), it can be seen that the r cosy term in Eq. (5.1) yields the in-plane area of the membrane footprint, with the flat membrane conformation y = 0 as the reference state. 5.1.2 Hamilton equations for the caveola neck We can express Eq. (5.1) in terms of a LagrangianL that captures the geometric constraints of Eqs. (5.2) and (5.3), as G neck =pk Z s b 0 L(y; ˙ y;r; ˙ r;z; ˙ z)ds; (5.4) with the Lagrangian L = r ˙ y+ siny r C (neck) 0 2 + 2 l 2 (1 cosy) +l r (˙ r cosy)+l z (˙ z siny); (5.5) 60 where l r (s) and l z (s) are the Lagrange multipliers for the constraints on the coordinates r(s) and z(s) Eqs. (5.2) and (5.3), respectively, and the characteristic length scale l = p k=g. The minimum-energy caveola neck shape can be obtained through the Euler-Lagrange equations as- sociated with Eq. (5.4) with Eq. (5.5). Alternatively, one can minimize the membrane footprint energy and find the corresponding stable caveola neck shape by solving the Hamilton equations associated with Eq. (5.4) with Eq. (5.5). From a numerical perspective, the set Hamilton equations tend to be more tractable as they only contain first-order derivatives, in contrast to the Euler- Lagrange equations that typically contains second-order derivatives [132, 139]. Thus, in this work we follow the Hamiltonian approach. Following previous works [130, 132, 133], we note that the corresponding Hamiltonian associ- ated with the Lagrangian in Eq. (5.5) is given by, H = p 2 y 4r p y siny r + p y C (neck) 0 2 l 2 (1 cosy)+ p r cosy+ p z siny; (5.6) where the generalized momenta associated with the parameterized coordinates are given by, p y = ¶L ¶ ˙ y = 2r ˙ y+ siny r C (neck) 0 ; (5.7) p r = ¶L ¶ ˙ r =l r ; (5.8) p z = ¶L ¶ ˙ z =l z : (5.9) The minimum-energy state of the caveola neck follows from the Hamilton equations, ˙ y = p y 2r siny r +C (neck) 0 ; (5.10) ˙ r= cosy; (5.11) ˙ z= siny; (5.12) ˙ p y = p y r p z cosy+ 2r l 2 + p r siny; (5.13) 61 ˙ p r = p 2 y 4r 2 p y siny r 2 + 2 l 2 (1 cosy); (5.14) ˙ p z = 0: (5.15) The solutions to Eqs. (5.10)–(5.15) subject to the boundary conditions on the caveola neck provides the minimum-energy caveola neck shapes associated with Eq. (5.4) with Eq. (5.5). 5.1.3 Caveola neck boundary conditions We determine the minimum energy for G neck in Eq. (5.1) by solving the Hamilton equations in Eqs. (5.10)–(5.15), subject to the boundary condition that the caveola shape and the neck shape connect smoothly at the caveola-neck interface at s= 0. Thus, we constrain these equations to the following fixed boundary conditions, expressed in terms ofb: y(0)=a = cos 1 (1 2b); (5.16) r(0)= Rsina = L p 1b; (5.17) z(0)=Rcosa = L(2b 1) 2 p b ; (5.18) At the outer boundary of the caveola neck, s= s b , we consider the coordinates r(s) and z(s) to be free boundaries satisfying natural boundary conditions, which amounts to ¶L ¶ ˙ r s=s b = p r (s b )= 0; (5.19) ¶L ¶ ˙ z s=s b = p z (s b )= 0: (5.20) Whereas fory(s), we consider both the natural (free) boundary condition at s= s b given by ¶L ¶ ˙ y s=s b = p y (s b )= 0; (5.21) 62 and the fixed boundary condition corresponding to a flat membrane shape at s= s b , with y(s b )= 0: (5.22) Moreover, we also consider different values of s b representing the different finite size constraints on the membrane footprint. From Eq. (5.15) and Eq. (5.20), it follows that p z (s)= 0 for 0 s s b . The remaining unknowns are p r (0) are p y (0). Following previous works [130, 140], we solve this set of Hamilton equation numerically using a shooting method, in which a set of solutions are first generated for different values of (p r (0), p y (0)) and then, among these solutions, the particular solution that simultaneously satisfy the boundary conditions p r (s b )= 0 and p y (s b )= 0 (ory(s b )= 0) is selected. We numerically solve Eqs. (5.10)–(5.15) with the conditions Eqs. (5.16)–(5.15) imposed through the NDSolve-command, and determine p r (0) and p y (0) from Eq. (5.19) and Eq. (5.21) (or Eq. (5.22)) through the FindRoot-command in Mathematica [14]. 5.2 Shape and deformation energy of caveolae neck In this section, we examine the stable caveola neck shapes corresponding to the minimized membrane footprint energy in Eq. (5.1), and the effect of the caveola neck on the stability cup-like caveola shapes. We explore here different scenarios for the neck spontaneous curvatures C (neck) 0 , the boundary conditions at the outer caveola neck boundary at s= s b , and the caveola neck size. By combing the caveola (cap) energy from chapter 4 with the membrane footprint energy, we study how the interplay between the molecular properties of caveolae and their surrounding neck regions affects the minimum-energy caveola shapes. We first consider the interplay between the minimun-energy shape of caveola necks, the molec- ular composition of caveola necks, and the caveola shape, for large caveola neck regions with no constraints on the outer boundary of caveola necks (see Fig. 5.1). We find that, depending on the value and sign of the neck spontaneous curvature C (neck) 0 , the caveola neck can adopt drastically different shapes as the caveola shape parameterb is varied. The results in Fig. 5.1 suggest that the 63 Figure 5.2: Caveola neck profiles for cup-like caveola states withb = 0:25, 0:5 and 0:75 for zero, negative, and positive C (neck) 0 and for free and fixed boundary condition ony(s) at s= s b given by Eq. (5.21) and Eq. (5.22), respectively. We set C (neck) 0 =0:02 nm 1 for C (neck) 0 7 0. Caveolae are indicated in red and caveola necks are indicated in green. For all panels, we set L= 50 nm, g = 0:01 k B T=nm 2 , andk = 20 k B T . energy of the caveola neck depends crucially on C (neck) 0 , which is borne out by Fig. 5.2. We find in Fig. 5.2(a) that, for C (neck) 0 = 0, the caveola neck increases the energy of cup-like caveola states with b 0:5 compared to nearly flat or budded caveola states with b 0 and b 1. Increasing C (neck) 0 from zero can further destabilize cup-like caveola states compared to nearly flat or budded caveola states. Interestingly, we find in Fig. 5.2(a) that a negative C (neck) 0 < 0, which is the scenario thought to be most relevant for caveolae in cell membranes [92], can drastically shift the energy landscape of the caveola neck. Most notably, C (neck) 0 can lower the energy of cup-like caveola shapes compared to caveola states withb < 0:5, and even destabilize flat caveola states. 64 Figure 5.3: Caveola neck energy. (a) Minimum caveola neck energy G neck as a function of the caveola shape parameterb for the indicated values of C (neck) 0 , using natural (free) boundary condi- tions fory(s) at s= s b given by Eq. (5.21) with s b = 2L. (b) Minimum caveola neck energy G neck as a function ofb for free and fixed boundary condition ony(s b ) given by Eq. (5.21) and Eq (5.22), respectively, with the indicated values of s b at zero and non-zero neck spontaneous curvatures, us- ing C neck 0 =0:02 nm 1 for C neck 0 6= 0. For all panels, we set L= 50 nm,g = 0:01 k B T=nm 2 , and k = 20 k B T . In Fig. 5.2(a) we considered the particularly straightforward scenario of natural boundary con- ditions at outer caveola neck boundary with a large caveola neck region. As illustrated in Fig. 5.1 and shown in Fig. 5.2(b), effects arising from the finite size of membrane patches containing cave- olae and fixed boundary conditions at the outer caveola neck boundaries can produce an intricate spectrum of caveola neck shapes and further shift the energy landscape of the caveola neck. Such finite size effects are expected to arise in cell membranes containing caveolae through, for in- stance, interactions between the cell membrane and the cytoskeleton [95–98]. For example, with 65 Figure 5.4: Minimum-energy caveola shapes from the interplay of C (cap) 0 and C (neck) 0 . Total min- imum caveola energy in Eq. (5.23) as a function of b for different values of C neck 0 and C cap 0 as indicated. We use G cap for the caveola energy in Eq. (4.2) with zero line tension s = 0. We use natural (free) boundary condition fory(s) at s= s b given by Eq. (5.21) with s b = 2L, and we set L= 50 nm,g = 0:01 k B T=nm 2 , andk = 20 k B T . C (neck) 0 = 0 and a large enough membrane patch size (large enough s b ), we obtain similar caveola neck energies G neck with freey(s b ) and fixedy(s b )= 0, while the G neck associated with freey(s b ) and fixed y(s b )= 0 deviate for small enough s b , with the fixed boundary scenario further desta- bilizing cup-shaped caveolae. Moreover, for C (neck) 0 = 0, and small enough caveola neck regions, the fixed constraint y(s b )= 0 can further lower the energy of cup-like caveola shapes compared to caveola states withb < 0:5, and further destabilize flat caveola states. Finally, we consider the total caveola energy to examine the role of the caveola neck in stabi- lizing different caveola shapes. The total caveola energy G is given by G= G cap [L;C (cap) 0 ;g;b]+ G neck [L;C (neck) 0 ;g;b]; (5.23) where G cap is the caveola (cap) energy given by Eq. (4.2) with C 0 = C (cap) 0 and G neck is the min- imized caveola neck energy in Eq. (5.1). We note that in this chapter we refer by G to the total caveola energy, whereas in the previous chapter we neglected any contributions to G due to the caveola neck region. Firstly, we consider in Fig. 5.3 natural boundary conditions at the outer boundary of the membrane patch of the membrane neck region, and study G as a function of b 66 Figure 5.5: Effect of caveola neck on the minimum-energy caveola shape. Total minimum caveola energy in Eq. (5.23) as a function ofb for (a) free boundary condition ony(s b ) given by Eq. (5.21) and (b) fixed boundary condition on y(s b ) given by Eq (5.21), with the indicated values of s b at zero and negative neck spontaneous curvatures, using C neck 0 =0:02 nm 1 for C neck 0 < 0, for the indicated values of C cap 0 as indicated. The arrows on theb-axis indicates theb values minimizing the total energy of the caveola-neck system. We use G cap for the caveola energy in Eq. (4.2) with zero line tensions = 0. For all panels, we set L= 50 nm,g = 0:01 k B T=nm 2 , andk = 20 k B T . for different values of C (cap) 0 and C (neck) 0 . Figure 5.3 shows that the interplay of C (cap) 0 and C (neck) 0 can yield a range of minimum-energy caveola shapes, including a variety of cup-like shapes. For instance, with C (cap) 0 = 0:06 nm 1 and C (neck) 0 =0:02 nm 1 , we find that G is minimized at b 0:45. Conversely, other combinations of C (cap) 0 and C (neck) 0 can serve to stabilize flattened or budded caveola states. Figure 5.5 shows that change in membrane footprint size produces distinct shifts in caveola shape for distinct C (cap) 0 and boundary constraints. In particular, for negative neck spontaneous 67 curvature, C (neck) 0 < 0, the strength of finite size effects – that is the amount by which the minimum- energyb shifts due to a change in caveola neck size – depends strongly on the outer neck boundary conditions and on the value of C (cap) 0 . For instance, at C (cap) 0 = 0:07 nm 1 under free boundary conditions at s= s b , the minimum-energy caveola shape shifts fromb 0:50 tob 0:63, while under the fixed boundary conditions of y = 0 at s= s b we find stronger finite size effects with a larger shift in the minimum-energyb fromb 0:54 tob 0:81. Lowering the cap spontaneous curvature from C (cap) 0 = 0:07 nm 1 to C (cap) 0 = 0:05 nm 1 under the fixed boundary constraint y(s b )= 0, weakens the effects due to the finite caveola neck size, yielding a smaller shift in the minimum-energyb fromb 0:29 tob 0:40. These results suggest that C (cap) 0 and C (neck) 0 , which cells can control through the local lipid and protein composition of membranes, play an important role in the regulation of caveola shape, and may allow cells to prime caveola shape to produce controlled, and rapid, responses of caveolae to changes in membrane tension. 68 Chapter 6 Conclusion This chapter provides a summary and discussion of the results presented in this thesis and an outlook on the broader prospects of this work. Section 6.1 summarizes the motivation behind our work, highlights our key findings, and puts forth some concluding remarks. Finally, in section 6.2, we discuss the various future prospects of further expanding on this work and broadening the understanding of the physics of cell membranes. 6.1 Summary and discussion Cell membranes contain of a great variety of protein and lipid species with distinct unperturbed hydrophobic thicknesses, and show an intricate submicron organization of lipids and proteins into domains with defined composition [13, 15, 16, 18, 19, 41]. A wide range of experiments suggest that membrane hydrophobic thickness provides a key control parameter for cell membrane orga- nization [15, 16, 18, 23–25]. To achieve hydrophobic matching, the lipid bilayer tends to deform around membrane proteins so as to match the protein hydrophobic thickness at bilayer-protein in- terfaces [13, 39–41]. Such protein-induced distortions of the lipid bilayer hydrophobic thickness incur a substantial energy cost that depends critically on the hydrophobic mismatch between the membrane protein and the lipids localized around the membrane protein. We have combined here the membrane elasticity theory of protein-induced lipid bilayer thickness deformations with the 69 LG theory of lipid domain formation to systematically explore the mechanochemical coupling of lipid organization and protein function through membrane thickness deformations. Protein-induced lipid bilayer thickness deformations are localized over a scale of approxi- mately 4 nm about membrane proteins [2, 4, 39–41, 45, 47–50, 74], which corresponds to roughly one-half the typical protein-protein separation in cell membranes [13,19]. The resulting overlap in protein-induced bilayer thickness deformations gives rise to bilayer-thickness-mediated protein in- teractions that, at small enough protein separations, tend to be strongly favorable for proteins with identical hydrophobic thickness [4,40,47–50,55,57] and have been observed to yield protein aggre- gation in membranes [51–54, 56, 58]. Furthermore, protein-induced lipid bilayer thickness defor- mations induce a coupling of protein conformational state and lipid bilayer thickness, which pro- vides a general mechanism for protein regulation through lipid bilayer mechanics [2,13,39,41–46]. For membrane proteins at close enough separations, the combination of bilayer-thickness-mediated protein interactions and the coupling of protein conformational state to lipid bilayer thickness is expected to produce cooperativity in membrane protein function [13, 47–49, 54, 55, 57, 58]. We have extended here the classic theory of bilayer-thickness-mediated protein interactions and pro- tein cooperativity [4,40,47–50,55,57] to account for heterogeneous lipid bilayers composed of two lipid species with distinct unperturbed hydrophobic thicknesses. We find that lipid heterogeneity can yield colocalization of lipids and membrane proteins according to their preferred hydrophobic thickness, and can have intricate effects on membrane protein regulation, protein clustering, and protein cooperativity driven by bilayer thickness deformations. We considered here a purely mechanical coupling of lipid and protein composition through the energetics of protein-induced lipid bilayer thickness deformations as well as a chemical coupling driven by preferential interactions between particular lipid and protein species [99–106]. In gen- eral, both types of lipid-protein coupling are expected to occur in cell membranes. A chemical affinity between particular lipid and protein species could also be engineered synthetically. Our results show how the local lipid composition around membrane proteins depends on the protein hydrophobic thickness, the lipid-protein chemical affinity, and the local lipid chemical potential. 70 Employing MscL as a model system [34, 44, 45, 74], we find that the resultant lipid-protein or- ganization can induce transitions in the protein conformational state. Our calculations show that lipid heterogeneity can yield substantial modifications of bilayer-thickness-mediated protein in- teractions. Notably, we find that lipid heterogeneity can expand the range of attractive protein interactions and amplify membrane protein cooperativity. In the case of MscL, for instance, this amplification of protein cooperativity manifests itself as pronounced shifts in the MscL gating ten- sion. Taken together, our results suggest that membrane thickness deformations provide a physical mechanism for the formation of membrane domains with controlled mechanical properties that, in turn, can affect the membrane protein conformational state. The coupling of protein-induced bilayer thickness deformations [2, 4, 39–41, 47–50] and lipid domain formation [62, 65–73] may provide a general mechanism underlying the observed supramolecular organization of cell mem- branes [15, 16, 18, 19]. In addition to local lipid-protein organization and regulation through bilayer thickness de- formation, we also examined, in the context of caveolae, the mechanics of curved membrane domains. While early EM experiments indicated that caveola membrane domains either occur in approximately flat or budded spherical shapes, more recent EM as well as SRM experiments have demonstrated that caveolae can take a variety of cup-like shapes resembling spherical caps [11,76–78,82–86]. Motivated by these experimental observations, we have explored here the roles of membrane bending, membrane tension, and the line tension of caveola domains [30, 87, 88] in stabilizing cup-like caveola shapes. We find that, for the range in membrane tension relevant for cell membranes [13, 90, 91], the competition between membrane tension and membrane bending yields caveolae with cup-like shapes similar to those observed experimentally [11, 76–78, 82–86], and that cup-like caveola shapes tend to be ruled out as the line tension of caveola domains comes to dominate the energy budget of caveolae. Our results suggest that the size and the spontaneous curvature of caveola domains are key control parameters for the stability of cup-shaped caveolae, and for the sensitivity of cup-shaped caveolae to changes in membrane tension. Heterogeneity in the size of caveola domains and heterogeneity in caveola spontaneous curvature due to, for 71 instance, variations in the concentration of caveolin-1 in caveola domains, as well as spatial het- erogeneity in membrane tension within cell membranes [91, 141] or variations in the caveola line tension, may thus produce heterogeneity in caveola shape. Cup-shaped caveolae are expected to deform the surrounding membrane [92], leaving a mem- brane footprint. Additionally, curvature-sensing or curvature-generating lipids or proteins may be enriched in the curved neck region of caveolae, and thus modify stable caveola shapes [93,94]. We studied how the curved caveola neck, through the mechanics of caveola-induced membrane shape deformations, affects the stable shapes of caveolae. We find that the contribution of the caveola neck to the overall energy budget of the caveola-neck system depends crucially on the membrane spontaneous curvature and on the local membrane organization around caveolae neck. Moreover, we considered the effect of boundary constraints on the caveola neck provided by, for instance, membrane-cytoskeletal interactions [95–98], on the shape and energetics of caveolae and the cave- ola neck. Our work suggests that the interplay of caveola and neck spontaneous curvature, which cells can control through the local lipid and protein composition of membranes, plays an important role in the regulation of caveola shape, and may allow cells to prime caveola shape to produce controlled, and rapid, responses to changes in membrane tension. Taken together, our calculations predict that variations in caveola shape yield heterogeneity in the response of caveolae to mechan- ical perturbations of the cell membrane, which may facilitate regulation of membrane tension over the wide range of scales thought to be relevant for cell membranes [13, 90, 91]. Overall, we have studied in this thesis emergent membrane phenomena that are independent of most molecular details. The physical mechanisms and principles for membrane organization and function explored here may thus be broadly applicable to cell membranes. Taken together, the work described here provides a theoretical framework bridging the physical, chemical and biological aspects of cell membranes. 72 6.2 Outlook on future prospects The role of local lipid-protein interactions in membrane protein regulation can be further ex- panded, beyond approximately cylindrical and axisymmetric membrane proteins. It would be interesting to consider membrane proteins or protein conformational states that have structures which induce bilayer mid-plane deformations in the membrane surrounding the protein or break rotational symmetry about the protein center. Our model in chapter 2 can be extended to account for the contributions arising form the different possible shapes and symmetries of membrane pro- teins. Membrane proteins such as mechanosensitive ion channels of small conductance (MscS), for instance, have been modeled as conical membrane inclusions which induce bilayer mid-plane deformations in the surrounding membrane [2, 13, 64]. In the Monge representation of the bi- layer energy, the variables that characterize the bilayer thickness deformation h and the bilayer mid-plane deformation u are decoupled to leading order [4, 5, 50]. However, an indirect coupling can arise between h and u a dependence of the lipid bilayer bending rigidity on lipid composi- tion, K b (c) [1] (see also Table 2.1 and Appendix C). Furthermore, an additional coupling term cÑ 2 u can also be included in the bilayer energy due the interaction between bilayer mid-plane and lipid composition as invoked in previous works [64, 142, 143]. Beyond axisymmetric protein shapes, considering a bilayer-protein boundary with a hydrophobic thickness and lipid composi- tion that varies azimuthally can induce additional effects. For instance, a cloverleaf or a crown model of membrane proteins embarks an anisotropic bilayer thickness deformation pattern around the protein. Such protein-induced asymmetries has been shown, in bilayers of homogeneous lipid composition, to modify the bilayer energy, to influence bilayer-mediated protein interactions, and to produce varying gating curves for mechanosensitive ion channels [49, 50, 59]. Incorporation of such contributions into our existing framework can broaden the scope of our predictions on the ef- fects of lipid-protein heterogeneity on membrane protein regulation by accommodating for a wide range of membrane protein structures. 73 Previous theoretical work has shown that stable caveola shapes can arise through a coupling between membrane bending and nematic ordering on the caveola surface [126]. Intriguingly, ex- periments [84] suggest that coat proteins form spiral patterns on caveolae that are, at least at a qual- itative level, consistent with theoretical predictions. To quantify the extent to which the nematic model of caveola energy developed in previous works [126] can capture the observed cup-like shapes and plasticity of caveolae domains, we can account for contributions to the caveolae due to tilt and chirality of caveola molecules through the Frank energy, and for the coupling between bending and nematic degrees of freedom through the Helfrich-Prost energy [126]. Following previ- ous works [126,144,145], such additional contributions to the caveola energy can be parameterized and, for a given set of parameter values, the energetically optimal caveola shapes can thus be cal- culated. Moreover, for many mammalian cell types, caveolae not only exist as individual entities but also form superstructures or clusters of caveolae [76, 78, 92, 146]. The shapes of these caveola clusters or rosettes have also been shown to respond to changes in membrane tension [92, 146]. Expanding upon our work on caveolae in this context, it would be interesting to examine caveola- caveola interactions through caveola membrane footprint. In this way, the caveola model developed in chapter 4 and 5 can be further extended to explore the role of compositional heterogeneity of caveola domains as well as caveola clustering in the response of caveolae shapes to mechanical perturbations. Biological processes are often dynamic. A straightforward way to account for the basic dy- namics of the membrane systems described here, is by considering our variables or parameters to be time-dependent functions that satisfy classical time evolution equations. The observed coars- ening dynamics of lipid domain formation [21, 68, 69] can be modeled as a binary fluid or alloy described by the time-dependent LG equation for non-conserved lipid composition or the Cahn- Hillard equation for conserved lipid composition [70, 71, 107]. This time-dependent composition c(r;t) can be coupled to a time-dependent bilayer thickness h(r;t) that evolves through relaxational kinetics [60, 61]. Thus, in this manner, one can study the dynamics of lipid domain formation 74 around membrane proteins. Furthermore, in a similar fashion, the dynamics of the caveola re- sponse to changes in membrane tension can also be studied by allowing for a time-dependent caveola shape. Such a caveola shape evolution can be described using classical kinetics [30, 88]. Capturing such general dynamical features of our systems would allow a more extensive range of predictions than considered here. Many emerging properties of cell membranes rely on electrostatic interactions. In particu- lar, charged membranes in ionic solutions forms a surrounding layer of oppositely charged ions. The mechanism underlying the formation of such layers of counterions can be described through the Poisson-Boltzmann theory of the electric double layer, where the competition between the electrostatic Coulomb interactions and the Boltzmann entropy of the ions in the solution deter- mines the charge distribution around the membrane [147, 148]. This electric double layer sur- rounding charged membranes in ionic solutions can interact with the mechanical properties of the membrane [149–151]. In particular, an electromechanical coupling can occur in cell membranes, for instance, through the coupling between the electrostatic potential and the membrane curva- ture [152–154]. It is intriguing to consider, for a charged membrane in ionic solution, the elec- trostatic contributions of curved membrane domains. 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Handbook of Heuristics. Springer, Cham, 2018. 86 Appendices A Derivation of bilayer energy and natural boundary condi- tions This appendix provides a derivation of the free energy and the natural boundary conditions for the lipid-protein domains described in chapter 2. In particular, we derive the bilayer thickness deformation energy from the elasticity theory of lipid bilayers, the Landau-Ginzburg (LG) energy of lipid domain formation from the lattice gas model of binary mixtures, and the natural (free) boundary conditions from the calculus of variations. A.1 Free energy of elastic thickness deformations In classical elasticity theory, thickness deformations of an elastic membrane are associated with a specific energy cost. In the Monge parameterization of surfaces, the elastic bending energy can be written as the energy functional [39, 74], G ben = Z dxdy K b 4 (Ñ 2 h + ) 2 +(Ñ 2 h ) 2 ; (A.1) where h + (x;y) and h (x;y) are scalar fields that characterize the thickness of the upper and lower leaflets, respectively, with respect to the mid-plane of the lipid bilayer. K b is the bending rigidity and we have ignored any spontaneous curvature. To account for the hydrophobic mismatch at the 87 boundary, we need to add the energy functional due to the stretching and compressing of the lipid bilayer of unperturbed thickness a, which is given by [39, 74] G str = Z dxdy K t 2 h + h 2a 2a 2 ; (A.2) where K t is the thickness deformation modulus. Finally, the energy cost associated with a non-zero membrane tension is given by [39, 74] G ten = Z dxdy t 4 ((Ñh + ) 2 +(Ñh ) 2 )+t h + h 2a 2a ; (A.3) wheret is the membrane tension. The total elastic energy is thus given by G el = G ben + G str + G ten : (A.4) Equation (A.4) can be simplified by making the substitutions u(x;y)= 1 2 (h + (x;y)+ h (x;y)) and h(x;y)= 1 2 (h + (x;y) h (x;y)), yielding G el = Z dxdy " K b 2 ((Ñ 2 u) 2 +(Ñ 2 h) 2 )+ K t 2 h a a 2 + t 2 (Ñu) 2 +(Ñh) 2 + 2 h a a # ; (A.5) where u(x;y) characterizes the mid-plane deformations and h(x;y) corresponds to the thickness de- formations of the bilayer. Here we assume up-down symmetry of the leaflets, h + (x;y)=h (x;y) h(x;y). In the following we focus on bilayer thickness deformations, and therefore set u(x;y)= 0 in Eq. (A.5). We thus have G h = Z dxdy " K b 2 (Ñ 2 h) 2 + K t 2 h a a 2 + t 2 (Ñh) 2 + 2 h a a + t 2 2K t # ; (A.6) where we have added a constant term G 0 = R dxdy(t 2 =2K t ) such that G h 0 [50]. 88 A.2 Free energy of lipid domain formation To describe lipid domain formation, we employ a LG model for a two-component fluid system [70]. The LG energy for a total number of lipids N l in an area A of one bilayer leaflet is based on the lattice gas model for binary mixtures with Hamiltonian [67] H = 1 2 å i j J i j s i (1 s j ); (A.7) where J i j is the interaction coefficient and s i is the local concentration variable at the i-th site such that s i = 0 and s i = 1 refers to the two different lipid species, respectively. Now consider a non-interacting Hamiltonian given by [67] H 0 = k B T å i q i s i (A.8) for some lipid dependent parameter q i . Then the partition function forH 0 becomes Z 0 = N l Õ k=1 å s k =0;1 e q i s i = N l Õ k=1 1 1 c i ; (A.9) where c i =< s i >= 1 1+e q i the average area fraction of a lipid species. From the partition function we obtain the free energy via F 0 =k B T ln(Z 0 ): (A.10) Using a variational approximation [67], the free energy for our system described byH in Eq. (A.7) is hence given by F = F 0 +<HH 0 >= k B T å i c i ln(c i )+(1 c i )ln(1 c i )+ 1 2 å i å j J i j c i (1 c j ) = å i h k B T(c i ln(c i )+(1 c i )ln(1 c i )) + 1 4 å j J i j ((c i c j ) 2 c 2 i c 2 j + 2c i ) i : (A.11) 89 In the continuum limit, c i ! c(x;y) and (c i c j )! lÑc, where l is the lipid-lipid separation. So, we define J =å j J i j such that Jc is the mean interaction of a lipid with the mixture of other lipids [67]. In this limit F! G gl and thus we obtain the LG energy, G gl = Z dxdy h e 2 (Ñc) 2 +f(c) i ; (A.12) where c(x;y) is a scalar field that indicates the lipid concentration at position(x;y). The parameter e =x 2 n 0 k B T , wherex l is our molecular length scale andf(c) is our mean field potential given by f(c) n 0 k B T = cln(c)+(1 c)ln(1 c)+cc(1 c); (A.13) where n 0 = N l =A andc= J=2k B T is the lipid-lipid interaction parameter such that phase separation is favored ifc> 0 [60,61,67]. To simplify the model, it is useful to expand Eq. (A.13) to a fourth- order polynomial in c [70, 107] centered at the critical point c 0 : f(c) b 0 b 1 2 (c c 0 ) 2 + b 2 4 (c c 0 ) 4 ; (A.14) where b 1 =(4 2c)n 0 k B T and b 2 =(16=3)n 0 k B T , and b 0 =(b 1 =8 b 2 =64) is a constant that shifts the mean field potential such that it is non-negative and is equal to zero at its minima at c= 0 and c= 1. The values c= 0 and c= 1 correspond here to the two species of lipid in the membrane. To incorporated a non-zero chemical potential, we add the term m R cdxdy to our LG energy and obtain, G c = Z dxdy h e 2 (Ñc) 2 +f(c)+mc+m 0 i ; (A.15) wherem 0 is a constant such that G c 0. A.3 Natural boundary conditions We need to minimize Eq. (2.4) subject to suitable boundary conditions at r= R and r= R+ L with L > 0. The form of these boundary conditions, as well as the Euler-Lagrange equations 90 associated with Eq. (2.4), is derived from the calculus of variations [110]. We start by writing Eq. (2.4) as G= Z R+L R dr f(h;h 0 ;h 00 ;c;c 0 ;r); (A.16) where f is the functional inside the integral in Eq. (2.4). We then consider the variations ˜ h(r)= h(r)+e 1 h(r) and ˜ c(r)= c(r)+e 2 z(r) for arbitrary functions h(r), z(r) and small constants e 1 , e 2 such that G([ ˜ h; ˜ c])= G([h;c])+dG(e 1 ;e 2 )+O(e 2 1 ;e 2 2 ) where dG= Z R+L R dr " e 1 h ¶ f ¶h +e 1 h 0 ¶ f ¶h 0 +e 1 h 00 ¶ f ¶h 00 +e 2 z ¶ f ¶c +e 2 z 0 ¶ f ¶c 0 # =e 1 h ¶ f ¶h 0 d dr ¶ f ¶h 00 R+L R +e 1 h 0 ¶ f ¶h 00 R+L R +e 2 z ¶ f ¶c 0 R+L R +e 1 Z R+L R drh " ¶ f ¶h d dr ¶ f ¶h 0 + d 2 dr 2 ¶ f ¶h 00 # +e 2 Z R+L R drz " ¶ f ¶h d dr ¶ f ¶h 00 # : (A.17) SettingdG= 0, we thus find the coupled Euler-Lagrange equations ¶ f ¶h d dr ¶ f ¶h 0 + d 2 dr 2 ¶ f ¶h 00 = 0; (A.18) ¶ f ¶h d dr ¶ f ¶c 0 = 0: (A.19) According to Eq. (A.17), the natural boundary conditions associated with the functional in Eq. (A.16) correspond to ¶ f ¶h 00 r=R = ¶ f ¶h 00 r=R+L = 0; (A.20a) ¶ f ¶h 0 d dr ¶ f ¶h 00 r=R = ¶ f ¶h 0 d dr ¶ f ¶h 00 r=R+L = 0; (A.20b) ¶ f ¶c 0 r=R = ¶ f ¶c 0 r=R+L = 0; (A.20c) 91 yielding the natural boundary conditions, 1 r d dr r dh dr r=r b = 0; (A.21a) d dr th K b 1 r d dr r dh dr r=r b = 0; (A.21b) dc dr r=r b = 0: (A.21c) B One-dimensional model of lipid-protein domains Previous work [4, 40, 47–50] has shown that generic properties of bilayer-thickness-mediated protein interactions in homogeneous lipid bilayers—such as the order of magnitude, sign, and approximate range of bilayer-mediated protein interactions—are already captured by a highly sim- plified model in which the membrane is described as an effectively one-dimensional (1D) system. Such a 1D approach also successfully captures basic properties of the tension-dependent gating of MscL [45, 74]. In this appendix we generalize this approach to protein-induced bilayer thickness deformations in heterogeneous lipid bilayers composed of two lipid species with distinct unper- turbed hydrophobic thicknesses, and thereby complement the axisymmetric, two-dimensional (2D) model developed in Sec. 2.1. In analogy to the 2D system in Sec. 2.1, we take our 1D system to have a lateral extent L, as measured from the edge of the protein to the system boundary. We approximate the total bilayer energy by the integral over the 1D energy density g (1D) (x) multiplied by the circumference of the protein at the center of the membrane patch, 2pR: G (1D) = 2pR Z L 0 g (1D) (x)dx; (B.1) 92 Figure B.1: Comparing 1D and 2D model results. (a) Total 1D bilayer energy G (1D) in Eq. (B.1) and total 2D bilayer energy G in Eq. (2.4) as a function of H 0 with R= R c for a heterogeneous bilayer containing lipid species A and B (AB) att = 0 andt = 1 k B T=nm 2 , and for homogeneous bilayers composed solely of lipid species A or B at t = 0. The dashed vertical lines indicate H 0 = H 0 , where the dominant lipid composition in the bilayer patch changes from lipid species A to lipid species B, for t = 0 and t = 1 k B T=nm 2 (right to left). The 1D and 2D values of H 0 are identical within numerical accuracy. We set L= 20 nm. (b) Bilayer-mediated protein interactions G int as a function of L att= 0 for proteins with identical hydrophobic thickness H 0 = H L = 2:0 nm (left panel) and distinct hydrophobic thicknesses H 0 = 2:2 nm and H L = H o (right panel) obtained from the 1D and 2D models for heterogeneous and homogeneous lipid bilayers. To calculate G int (L), we subtracted from the total 1D bilayer energy G (1D) in Eq. (B.1) and the total 2D bilayer energy G in Eq. (2.4) the respective values of G (1D) and G obtained in the large-L, non-interacting regimes (see also Appendix E). (c) MscL gating probability P o in Eq. (2.9) as a function oft with natural boundary conditions on h at the outer membrane patch boundary for L= 20 nm (left panel), and with the fixed-value boundary condition H L = H o at the outer membrane patch boundary for L= L f = 12 nm and L= L c = 3 nm (right panel). We set m = 0 throughout and, unless indicated otherwise, used natural boundary conditions for h and c. 93 where the 1D energy density is given by g (1D) (x)= K b 2 d 2 h dx 2 2 + K t 2 h a a 2 + t 2 dh dx 2 +t h a a + t 2 2K t + e 2 dc dx 2 + b 0 b 1 2 c 1 2 2 + b 2 4 c 1 2 4 mc+m 0 ; (B.2) with the 1D scalar functions h(x) and c(x). As in Eq. (2.4) in Sec. 2.1, we assume that the unper- turbed bilayer leaflet thickness, a(c), is a linear function of the lipid composition [see Eq. (2.5)], while K b and K t are constants. Similarly as in Sec. 2.1, we minimize the energy functional in Eq. (B.1) subject to natural (free) or fixed-value boundary conditions on h(x), dh=dx, and c(x) [110, 111], using the L-BFGS-B solver [108, 109]. In particular, the 1D natural boundary conditions are given by d 2 h dx 2 x=x b = 0; (B.3a) d dx th K b d 2 h dx 2 x=x b = 0; (B.3b) dc dx x=x b = 0 (B.3c) with x b = 0 or x b = L. The corresponding 1D fixed-value boundary conditions are given by h(x b )= H b ; (B.4a) dh dx x=x b = s b ; (B.4b) c(x b )= c b (B.4c) 94 with x b = 0 or x b = L, where H b , s b , and c b take given, fixed values (see Sec. 2.1). As for the 2D model developed in Sec. 2.1, we use here Eq. (B.4b) with s b = 0 throughout. The 1D model in Eqs. (B.1) and (B.2) is not expected to yield precise estimates of the nu- merical values of the bilayer energy G. For instance, the magnitude of G as a function of protein hydrophobic thickness tends to be larger for the 2D model than for the 1D model [see Fig. B.1(a)]. Similarly, we also find shifts in the predicted bilayer-mediated protein interactions [see Fig. B.1(b)] as well as in the predicted MscL gating tensions in the non-interacting and interacting regimes [see Fig. B.1(c)]. But, consistent with previous work on protein-induced bilayer thickness deformations in homogeneous bilayers [4, 40, 45, 74], we find that Eqs. (B.1) and (B.2) do capture important generic features of lipid-protein interactions in heterogeneous bilayers. In particular, we find that most of the conclusions we arrived at in chapters 2 and 3 based on the 2D model in Eqs. (2.1) and (2.2), which do not rely on the precise numerical values of G, remain unchanged if Eqs. (2.1) and (2.2) are replaced by Eqs. (B.1) and (B.2). Broadly speaking, the main difference between our 1D and 2D results is that the predicted effects of lipid bilayer heterogeneity on lipid-protein orga- nization and regulation through membrane thickness deformations tend to be more pronounced in the 2D model than in the (less accurate) 1D model. C Dependence of bilayer bending rigidity and thickness defor- mation modulus on lipid composition The purpose of this appendix is to examine the relative importance of the dependence of K b , K t , and a in Eq. (2.1) on lipid composition [1, 2, 13, 41] for lipid-protein organization and regulation through membrane thickness deformations. In analogy to a(c) in Eq. (2.5), we assume that K b (c) and K t (c) can be approximated by linear functions of c, K b (c)b 1 c+b 0 ; (C.1) 95 Figure C.1: Total bilayer energy with c-dependent K b or K t . (a) Bilayer energy G in Eq. (2.4) for a single membrane protein as a function of protein hydrophobic thickness H 0 with the expressions for K b (c) and K t (c) in Eqs. (C.1) and (C.2), as well as the constant K b = 20 k B T and K t = 60 k B T=nm 2 [1,13] considered in chapters 2 and 3. The dashed vertical lines show H 0 = H 0 , where the dominant lipid composition in the membrane patch changes from lipid species A to lipid species B. For constant K b and K t we have H 0 1:950 nm (black dashed line), and with the c-dependent K b and K t in Eqs. (C.1) and (C.2) we find H 0 1:954 nm (red dashed line). We set L= 20 nm. (b,c) Percentage difference between the total bilayer energy G in Eq. (2.4) obtained with constant K b and K t as described in Sec. 2.1, and the corresponding total bilayer energy obtained with the expressions for K b (c) or K t (c) in Eqs. (C.1) and (C.2), d in Eq. (C.3), for non-interacting [panel (b)] and interacting [panel (c)] membrane proteins in heterogeneous bilayers. In panel (b) we used m = 10 2 k B T=nm 2 for m > 0 and set L= 20 nm. In panel (c) we used H 0 = H L = 2:0 nm for H 0 = H L , and H 0 = 2:2 nm and H L = 1:3 nm for H 0 6= H L . In all panels we set, unless specified otherwise,m= 0 and used natural boundary conditions on h,Ñh, and c. We sett= 0 and employed a(c) in Eq. (2.5). 96 whereb 1 = 15 k B T andb 0 = 14 k B T such that K b (0) 14 k B T and K b (1) 29 k B T [1, 2], and K t (c)g 1 c+g 0 ; (C.2) whereg 1 = 5 k B T=nm 2 andg 0 = 58 k B T=nm 2 such that K t (0) 58 k B T=nm 2 and K t (1) 63 k B T=nm 2 [1, 2] (see also Table 2.1). In Fig. C.1(a) we illustrate the effect of the dependence of K b and K t on c in Eqs. (C.1) and (C.2) on the total bilayer energy in Eq. (2.4) for a single membrane protein. Figure C.1(a) suggests that variations in K b (c) or K t (c) only produce minor shifts in the energy landscape of bilayer-protein interactions in heterogeneous bilayers. To further quantify the ramifications of a dependence of K b or K t on c we consider the percentage difference between the total bilayer energy obtained with constant K b and K t as described in Sec. 2.1, G, and the corresponding total bilayer energy obtained with the expressions for K b (c) or K t (c) in Eqs. (C.1) and (C.2), ¯ G, d = 100 G ¯ G G : (C.3) Figures C.1(b) and C.1(c) show Eq. (C.3) for selected scenarios corresponding to non-interacting and interacting membrane proteins in heterogeneous bilayers. We find d < 20% in Figs. C.1(b) and C.1(c), withd < 10% for many of the scenarios considered here. D Numerical minimization of bilayer energy We employ [3] the L-BFGS-B solver [109]—a low memory (L) version of the Broyden- Fletcher-Goldfarb-Shanno (BFGS) algorithm [108] with bounded (B) constraints—to directly min- imize the functional G in Eq. (2.4) and thus to numerically find h(r) and c(r). Numerical mini- mization methods that implement this solver were also used in previous work on bilayer-thickness- mediated protein interactions [50]. The L-BFGS-B solver aims to find a local minimum of a given functionalG[v] with respect to a function v(t). We discretize the energy functional in Eq. (2.6) with 97 coordinates r! r R using finite differences [157]. The discretized function v(t i )! v i thereby becomes the ith entry of the input vector~ v. Since we have two functions h(r) and c(r) in the model in Sec. 2.1, we consider a vector~ v of length N partitioned into two domains of length n= N=2, (v 0 ;:::;v n1 ;v n ;:::;v N1 )=(h 0 ;::;h n1 ;c 0 ;:::;c n1 ). We discretize the integral in Eq. (2.6) as a sum over grid points with lattice spacingDr= L=(n 1). Starting from a given initial value of the input vector~ v 0 , the L-BFGS-B solver iteratively finds better estimates of~ v so as to minimizeG . To this end, the L-BFGS-B solver employs a steepest-decent method based on the discretized energy functional ~ ÑG with entries given byÑG i =¶G=¶v i [108, 109]. Unless mandated by fixed-value boundary conditions, we do not restrict the h i -variables in~ v. We always restrict the c i -variables in ~ v to the range 0 c i 1. A complication arises here in that the derivatives in Eq. (2.4) can yield exterior “ghost” points lying outside the grid used for the numerical minimization procedure. We determine the values of v i at these exterior points from the boundary conditions on the h- and c-fields. For instance, employing the forward and central discretizations [157] for the first and second derivatives of h with respect to r, we obtain the following expression for the discretized Laplacian of h: Ñ 2 h i = (1+ w i )h i+1 (2+ w i )h i + h i1 (Dr) 2 ; (D.1) where w i =Dr=(R+ iDr). Evaluating Eq. (D.1) at the boundaries with i= 0 and i= n 1, gives two undefined exterior values h 1 and h n . If, for instance, natural boundary conditions are imposed on h at the system boundaries, h 1 and h n are specified by Eq. (2.7a), which gives (1+ w 0 )h 1 (2+ w 0 )h 0 + h 1 = 0; (D.2a) (1+ w n1 )h n (2+ w n1 )h n1 + h n2 = 0: (D.2b) 98 Figure D.1: Convergence tests for the total bilayer energy G in Eq. (2.4) for membrane proteins in heterogeneous bilayers. We consider here (a) single proteins with H 0 = 2:0 nm att= 0 (left panel) andt= 1 k B T=nm 2 (right panel), and (b) interacting proteins with identical hydrophobic thickness (H 0 = H L = 2:0 nm; left panel) and distinct hydrophobic thickness (H 0 = 2:2 nm and H L = 1:3 nm; right panel) with c 0 = 1 and c L = 0. Results obtained using the L-BFGS-B solver with the multistart method are indicated by red data points and plotted as a function of the logarithm (base 10) of the number of grid points used in the L-BFGS-B solver, log 10 (N). For comparison, we also show the corresponding results obtained by numerically solving the Euler-Lagrange equations in Eqs. (D.4) and (D.5) using the NDSolve-command in Mathematica [14] (dashed blue horizontal lines). We set here e = 100 k B T , m = 0, R= 3 nm, and L= 5 nm for all panels and used, unless specified otherwise, natural boundary conditions on h,Ñh, and c. Similarly, if fixed-value boundary conditions are imposed on h at the system boundaries, h 1 and h n are specified by Eq. (2.8b), which gives h 0 h 1 = 0; (D.3a) h n h n1 = 0: (D.3b) Analogous considerations apply to all combinations of boundary conditions considered here. 99 The L-BFGS-B solver employed here gives a local energy minimum near the initial values of the input vector~ v that, in principle, may not correspond to the global energy minimum of the system. We address this issue through the multistart method [158, 159], which provides a simple approach for determining a global minimum within a bounded range through a local-minimum solver. We thereby test different initial values for~ v corresponding to 0 c 1 using increments of 0:1, with h= a(c). For the boundary conditions and system sizes considered here, we generally find distinct local energy minima with the initial trial values c= 0 and c= 1. We also find that, for protein hydrophobic thicknesses H 0 1:95 nm, a third local energy minimum may appear for the initial trial value c= 0:5. From the sets of local energy minima determined through the L-BFGS-B solver with the multistart method, we take the solutions with the smallest energy to correspond to the global energy minima. It is instructive to compare the results obtained through direct numerical minimization of Eq. (2.4) with the corresponding solutions of the Euler-Lagrange equations associated with Eq. (2.4), which follow from the calculus of variations [110, 111]: K b Ñ 4 htÑ 2 h+ K t a(c) 2 [h a(c)]+ t a(c) = 0; (D.4) meÑ 2 c b 1 (c c 0 )+ b 2 (c c 0 ) 3 K t a(c) 3 h[h a(c)]a 0 (c) t a(c) 2 a 0 (c)h = 0: (D.5) For the parameter values used here, Eqs. (D.4) and (D.5) can be conveniently solved using the NDSolve-command in Mathematica [14] subject to the boundary conditions in Eqs. (2.7a)–(2.7c) or Eqs. (2.8a)–(2.8c), provided that L is small enough with L/ 5 nm and that e in Eq. (2.2) is increased from the valuee= 1 k B T employed in chapters 2 and 3 toe' 100 k B T . As illustrated in Fig. D.1, the results obtained through direct numerical minimization of Eq. (2.4) via the L-BFGS- B solver with the multistart method agree with the corresponding results obtained by numerically solving Eqs. (D.4) and (D.5), provided that the number of grid points N used for the L-BFGS-B solver is large enough. For all the calculations described in chapters 2 and 3 we used N 500, and 100 tested for convergence of the solutions obtained through the L-BFGS-B solver with the multistart method as N is increased. E Calculation of protein interaction potentials in heterogeneous bilayers In this appendix we elaborate on the method by which we obtain the protein interaction poten- tials G int (L) in Sec. 3.2 for heterogeneous lipid bilayers (Figs. 3.4 and 3.6). To calculate G int (L), we subtract from the total bilayer energy G in Eq. (2.4) the value of G obtained in the large-L, non-interacting regime. With the mean-field model in Fig. 3.1, a complication arises here in that G tends to increase linearly with L as L!¥ (see Fig. E.1), because in the far-field limit the bilayer energy associated with the bilayer-protein interactions at the outer membrane patch boundary is approximately proportional to the circumference of the outer membrane patch boundary. We thus set G int (L)= G(L) G f (L); (E.1) Figure E.1: Total bilayer energy G in Eq. (2.4), protein interaction potential in Eq. (E.1), and far- field bilayer energy G f in Eq. (E.2) as a function of protein separation L for membrane proteins with identical hydrophobic thickness H 0 = H L = 2:0 nm (left panel) and distinct hydrophobic thicknesses H 0 = 2:2 nm and H L = 1:3 nm (right panel) in heterogeneous bilayers. For both panels we used natural boundary conditions on c at all bilayer-protein interfaces, and setm= 0 andt= 0. 101 where we take the far-field bilayer energy G f to be of the form G f (L)= AL+ B; (E.2) where A and B are constants. Equation (E.1) ensures that G int (L)! 0 as L!¥. For each G int (L) considered in Figs. 3.4 and 3.6, we determine the values of A and B by fitting Eq. (E.2) to G(L) in the range 15 nm L 20 nm, sampling G(L) at increments of 0:5 nm. For instance, for m = 0 and t = 0, we have A 1:2 k B T=nm and B 6:0 k B T in Eq. (E.2) for membrane proteins with natural boundary conditions on c and identical hydrophobic thickness H 0 = H L = 2:0 nm (left panel in Fig. E.1), while for membrane proteins with natural boundary conditions on c and distinct hydrophobic thicknesses H 0 = 2:2 nm and H L = 1:3 nm we have A 15:8 k B T=nm and B 22:5 k B T in Eq. (E.2) (right panel in Fig. E.1). We do not subtract G f from G for any calculations in the main text of this thesis other than those connected to Figs. 3.4 and 3.6. In particular, we consider the total bilayer energy G in Eq. (2.4) for the cooperative gating curves in Fig. 3.8. F Discontinuous transitions in caveola shape at constant line tension We follow here chapter 4 and only consider contributions to the caveola energy due to the caveola membrane domain, and denote the caveola energy by G [see Eq. (4.2)]. In particular, we neglect any contributions to the caveola energy due to the caveola neck. For s > 0 there can be a discontinuous transition in the energetically most favorable caveola shape with increasing membrane tension. Such a discontinuous transition is a generic prediction of the classic model of caveola shape [30], in which only flat (b 0) and fully budded (b 1) caveola states are stable. Instead of such sharp transitions between these two extreme caveolae shapes, more recent exper- iments rather show that caveolae can exist as stable intermediate cup-like shapes. As we show in chapter 4, such cup-like caveola shapes can be explained through a non-zero spontaneous curvature 102 of caveolae. Depending on the parameter values considered, the model we have employed can also show discontinuous transitions between caveola states with different b, but between states with b = 1 andb > 0 (rather thanb = 0). However, such discontinuous transitions are not an essential feature of our model. To gain some more (mathematical) insight into these discontinuous jumps inb min , it is useful to systematically consider the following four distinct scenarios:(1)s = 0, C 0 = 0, (2)s > 0, C 0 = 0, (3) s = 0, C 0 > 0, and (4) s > 0, C 0 > 0. Scenario (1) corresponds to the trivial case in which the flat caveola state is always energetically favorable, that is,b min = 0 irrespective ofg. Scenario (2) corresponds to the classic model of caveola shape [30] yielding (local) energy minima atb = 0 andb = 1, with a discontinuous jump in the stable caveola shape betweenb = 1 andb = 0 with increasing membrane tension. Scenario (3) corresponds to the one of our key results discussed in the section 4.2.1 of chapter 4, as it is the simplest scenario in our model giving stable cup-like caveola shapes. Here, energy minimization yields a gradual, continuous shift ofb min fromb = 1 tob = 0 with increasingg. In this case, there is no critical membrane tension at which a discontinuous jump in caveola shape occurs. Instead, for L> 4=C 0 , we find that there is a finite threshold membrane tension ¯ g [given by Eq. (4.4)], which is the membrane tension for which caveolae start to flatten. While b min is a continuous function ofg, its derivative db min =dg (tension sensitivity) is discontinuous atg = ¯ g. In scenario (4), energy minimization for a constant s > 0 yields a quartic polynomial, from which one finds four critical pointsb i with i= 1;:::;4 (see Fig. F.1). For a givenb i to be physically relevant and to correspond to a (local or global) energy minimum, we need: (i) 0b i 1 and (ii) d 2 G db 2 b i > 0. The resulting b i , together with the boundary points b = 0 and b = 1, are the candidates for the global energy minimumb min (Fig. F.1). Asg is increased from zero, one finds a discontinuous transition in the minimum-energy caveola shape if the energy of the fully budded state is equal to the energy of one of the competing minima withb6= 1 at some membrane tension g 0. The value ofg and the magnitude of the discontinuous jump length 1b i depend on the values of L, C 0 ands. In particular, for a given L ands such thatg 0, a decrease in C 0 results 103 Figure F.1: The real solutionsb 1 ,b 2 ,b 3 , andb 4 of the quartic equation arising from the minimiza- tion of G in Eq. (2) together with the boundary statesb = 0;1 (upper panel) and the corresponding values of G (lower panel) as a function ofg for L= 100 nm, C 0 = 0:04 nm 1 , ands = 1 k B T=nm. The gray dashed vertical line drawn across the panels indicates the critical tension g 0:028 k B T=nm 2 for which the fully budded caveola state withb = 1 becomes unstable for the parameter values used here. We setk = 20 k B T . 104 in a smallerg and a greater discontinuous jump length 1b i . As C 0 ! 0 withs > 0, we recover the bistable system in scenario (2), with the only two stable caveola shapes corresponding tob 0 andb 1 at large and small membrane tension, respectively. In contrast, for small enoughs, the discontinuity in the dependence of caveola shape ong disappears, yielding a continuous spectrum of stable cup-like caveola shapes as in scenario (3). 105

Abstract (if available)

Abstract Cell membranes are composed of a great variety of lipid and protein species that organize into distinct domains which vary in composition and structure. This thesis explores the physical mechanisms and principles underlying the emergent properties of heterogeneous membrane domains, at scales of tens of nanometers in the context of lipid-protein organization through hydrophobic matching, and at scales of hundreds of nanometers in the context of caveola membrane domains.
To achieve hydrophobic matching, the lipid bilayer tends to deform around membrane proteins so as to match the protein hydrophobic thickness at bilayer-protein interfaces. Such protein-induced distortions of the lipid bilayer hydrophobic thickness incur a substantial energy cost that depends critically on the bilayer-protein hydrophobic mismatch, while distinct conformational states of membrane proteins often show distinct hydrophobic thicknesses. As a result, hydrophobic interactions between membrane proteins and lipids can yield a rich interplay of lipid-protein organization and transitions in protein conformational state. We combine here the membrane elasticity theory of protein-induced lipid bilayer thickness deformations with the Landau-Ginzburg theory of lipid domain formation to systematically explore the coupling between local lipid organization, lipid and protein hydrophobic thickness, and protein-induced lipid bilayer thickness deformations in membranes with heterogeneous lipid composition. We allow for a purely mechanical coupling of lipid and protein composition through the energetics of protein-induced lipid bilayer thickness deformations as well as a chemical coupling driven by preferential interactions between particular lipid and protein species. We find that the resulting lipid-protein organization can endow membrane proteins with diverse and controlled mechanical environments that, via protein-induced lipid bilayer thickness deformations, can strongly influence protein function. The theoretical approach employed here provides a general framework for the quantitative prediction of how membrane thickness deformations influence the joint organization and function of lipids and proteins in cell membranes.
A striking feature of cell membranes in many mammalian cell types is the abundance of small membrane invaginations of defined lipid and protein composition, called caveolae, which flatten with increasing membrane tension. Superresolution light microscopy and electron microscopy have revealed that caveolae can take a variety of cup-like shapes. We show here that, for the range in membrane tension relevant for cell membranes, the competition between membrane tension and membrane bending yields caveolae with cup-like shapes similar to those observed experimentally. We find that the caveola shape and its sensitivity to changes in membrane tension can depend strongly on the caveola spontaneous curvature and on the size of caveola domains. Cup-shaped caveolae are also expected to deform the surrounding membrane into curved caveolae neck region. We show that the contribution of the caveola neck to the overall energy cost of caveolae depends on the membrane intrinsic curvature and the local membrane organization around caveolae. Our results suggest that heterogeneity in caveola shape produces a staggered response of caveolae to mechanical perturbations of the cell membrane, which may facilitate regulation of membrane tension over the wide range of scales thought to be relevant for cell membranes.

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Asset Metadata

Creator Shrestha, Ahis (author)

Core Title The physics of emergent membrane phenomena

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2022-05

Publication Date 04/16/2022

Defense Date 03/07/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag biological physics,cell membranes,emergent phenomena,lipid bilayer,lipid-protein interactions,membrane mechanics,membrane protein,membranes,OAI-PMH Harvest,Physics,soft matter,statistical physics

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haselwandter, Christoph (committee chair), Boedicker, James (committee member), El-Naggar, Moh (committee member), Pinaud, Fabien (committee member), Saleur, Hubert (committee member)

Creator Email abshrest@usc.edu,ahis55@hotmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC110964849

Unique identifier UC110964849

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