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The symmetry of membrane protein polyhedral nanoparticles

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The symmetry of membrane protein polyhedral nanoparticles

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Content The Symmetry of Membrane Protein Polyhedral Nanoparticles by Mingyuan Ma A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Physics) August 2021 Copyright 2021 Mingyuan Ma Acknowledgements I would like to express deep and sincere gratitude to my supervisor, Christoph A. Haselwandter. The past few years of my PhD program would not have worked out so smoothly without his guidance. He has always been patient, helpful and insightful for questions, which drove me to make significant progress. I have been impressed particularly by his strong attention to every detail, which has inspired me to aim for perfection. I am indeed lucky and previlaged to have the chance to collaborate with him. I thank Prof. Aiichiro Nakano, Prof. James Boedicker, Prof. Moh El-Naggar and Prof. Clifford Johnson for serving in my committee on both my qualifying exam and thesis defense. Thank you all for your time and enlightening remarks. Thanks to O. Kahraman, D. Li, and M. H. B. Stowell for the illuminating discussions. The work in this thesis is supported by NSF award number DMR-1554716 and the USC Center for High-Performance Computing. I would want to thank my mom the most, who raised me and funded my education on her own. Any achievement I have accomplished would not be possible without her unconditional support. I also want to thank my partner Yu and best friend Yifei for their support and advice on the momentous decisions of my life. I wholeheartedly appreciate every little kindness that has ever made my life brighter and more colorful. Love you all. Mingyuan Ma April 2021 ii Table of Contents Acknowledgements ii List of Tables v List of Figures vi Abstract xiii Chapter 1: Introduction 1 1.1 Membrane protein polyhedral nanoparticles (MPPNs) . . . . . . . . . . . . . . . . 1 1.2 Extending of the mean-field model of MPPN self-assembly . . . . . . . . . . . . . 2 1.2.1 MPPNs formed from membrane proteins with large domains outside the membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 MPPNs formed from Piezo ion channels . . . . . . . . . . . . . . . . . . . 3 1.3 Minimal molecular model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2: Effect of protein steric constraints on the symmetry of membrane protein polyhedra 6 2.1 Calculation of MPPN self-assembly diagrams . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Thermal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 MPPN bilayer midplane deformations . . . . . . . . . . . . . . . . . . . . 9 2.1.3 MPPN bilayer thickness deformations . . . . . . . . . . . . . . . . . . . . 12 2.1.4 MPPN defect energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.5 Steric constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 MPPN self-assembly diagrams under protein steric constraints . . . . . . . . . . . 18 2.2.1 Modifying MPPN self-assembly diagrams throughr s andh s . . . . . . . . 18 2.2.2 Quantifying the effect of protein steric constraints on MPPN self-assembly diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 3: Self-assembly of polyhedral bilayer vesicles from Piezo ion channels 31 3.1 MPPN self-assembly diagrams for arbitrarily large membrane curvatures . . . . . . 31 3.1.1 Nonlinear MPPN shape equations . . . . . . . . . . . . . . . . . . . . . . 33 3.1.2 Small-gradient approximation for MPPNs . . . . . . . . . . . . . . . . . . 36 3.1.3 Topological defects in protein packing . . . . . . . . . . . . . . . . . . . . 37 iii 3.1.4 Constructing MPPN self-assembly diagrams . . . . . . . . . . . . . . . . 38 3.2 MPPN self-assembly from MscS . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 MPPN self-assembly from Piezo . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 4: Symmetry of membrane protein polyhedra with heterogeneous protein size 44 4.1 Modeling MPPN symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.1 Lipid-protein (LP) model . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.2 Composite particle (CP) model . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.3 Simulated annealing MC simulations . . . . . . . . . . . . . . . . . . . . 48 4.1.4 Quantifying MPPN symmetry . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Minimum-energy MPPN configurations . . . . . . . . . . . . . . . . . . . . . . . 57 4.2.1 Comparison of LP and CP models . . . . . . . . . . . . . . . . . . . . . . 58 4.2.2 Protein arrangement in MPPNs with heterogeneous protein size . . . . . . 60 4.2.3 Quantifying the symmetry of MPPNs with heterogeneous protein size . . . 65 Chapter 5: Conclusion 70 Appendices 76 A Physical model of the MPPN defect energy . . . . . . . . . . . . . . . . . . . . . 76 B Effect of changes in the lipid bilayer bending rigidity on MPPN self-assembly diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 C Vanishing protein height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 References 81 iv List of Tables 4.1 Symmetries and associated fit errorsE in Eq. (4.6) of the best two polyhedral fits to the minimum-energy MPPN configurations implied by the LP and CP models of MPPN symmetry for N = 12, 24, and 32 with n o = 0. All results were ob- tained through simulated annealing MC simulations (see Sec. 4.1.3). We denote [1, 2] the icosahedron by I, the snub cube by sC, the pentakis dodecahedron by kD, the metabiaugmented dodecahedron by J60, the truncated octahedron by tO, the pentagonal hexecontahedron by gD, the rhombic triacontahedron by jD, and the disdyakis triacontahedron by mD. We proceeded as described in Sec. 4.1.4 when searching for optimal polyhedral fits. The polyhedral chiralities result from the random numbers and initial conditions used and are not model predictions. . . . . . 58 4.2 Symmetries and associated root-mean-square fit errorsb s in Eq. (4.10) of the best two polyhedral fits to the minimum-energy MPPN configurations implied by the CP model of MPPN symmetry forN= 24 and the indicated values ofn o . All results were obtained through simulated annealing MC simulations (see Sec. 4.1.3). We use the same notation for polyhedral symmetries as in Table 4.1 [1, 2] with, in particular, gD corresponding to the pentagonal hexecontahedron, and denote the rhombicuboctahedron by eC and the gyroelongated square bicupola by J45. We proceeded as described in Sec. 4.1.4 when searching for optimal polyhedral fits. The polyhedral chiralities result from the random numbers and initial conditions used and are not model predictions. . . . . . . . . . . . . . . . . . . . . . . . . . 67 v List of Figures 2.1 Schematics of (a) protein-induced lipid bilayer deformations in MPPNs and (b) the geometry of MPPNs. We take each MPPN membrane patch to have a protein at its center at r = 0 and to be symmetric about the rotation axis h(0). As indi- cated in panel (a), we denote the protein radius in the bilayer midplane by r i , the MPPN membrane patch size by r o , the height of the bilayer midplane by h(r), and one-half the bilayer hydrophobic thickness by u(r)+m, where m is one-half the unperturbed bilayer thickness. As indicated in panel (b), we account for steric constraints on MPPN patch size due to protein domains outside the membrane through the in-plane protein radius outside the membraner s and the effective pro- tein height h s , which we define with respect to the protein hydrophobic midplane (dashed horizontal line). The protein height with respect to the MPPN center is given byh s +Re, whereR is the MPPN radius ande is an offset due to the finite protein size. The effective steric constraint on the membrane patch size in the bi- layer midplane,r s o , depends on r s and h s . The schematic in panel (b) corresponds to the smallest R allowed by steric constraints due to protein domains outside the membrane,R=R s , for which we haveR s =r s o =sin ¯ b (see Sec. 2.1.5). . . . . . . . 9 2.2 MPPN self-assembly diagrams as a function of the absolute value of the bilayer- protein hydrophobic mismatch,jUj, and the bilayer-protein contact angle, a, for the indicated values of r s and h s . The color maps show the maximum values of f(n) associated with the dominant MPPN n-states, obtained from Eq. (2.6) for 10n 80. Selected dominant MPPN n-states are indicated in the MPPN self- assembly diagrams. Orange shading shows regions of the MPPN self-assembly diagrams for which n= 12 is strongly penalized by large values of K s resulting from r min o !r i in Eq. (2.27) [3, 4]. To obtain these regions of the MPPN self- assembly diagrams we recalculated the MPPN self-assembly diagrams withK s = 0 forn= 12, and shaded in orange regions of the MPPN self-assembly diagrams for which these modified calculations yield a different dominant MPPN n-state. As noted in the main text, the continuum model of MPPN bilayer mechanics used here may not give reliable results in the orange-shaded regions of the MPPN self- assembly diagrams. Gray shading indicates regions of the MPPN self-assembly diagrams wheren= 80 gives the dominant MPPNn-state, which may be a spurious result of our constraint 10n 80. The dashed vertical line in the bottom-right panel corresponds toa = 0:62 rad. . . . . . . . . . . . . . . . . . . . . . . . . . 19 vi 2.3 MPPN energy per protein,E avg (n)=E min (n)=n, as a function of the absolute value of the bilayer-protein hydrophobic mismatch,jUj, fora = 0:62 rad, r s = 8:0 nm, and h s = 6:0 nm. The MPPN states n= 12 and n= 24 considered here are the dominant MPPNn-states along the dashed vertical line in the MPPN self-assembly diagram in the bottom-right panel of Fig. 2.2. The vertical solid line corresponds tojUj= 0:048 nm, for which the dominant MPPN n-state transitions from n= 24 ton= 12 in the MPPN self-assembly diagram in the bottom-right panel of Fig. 2.2. 21 2.4 MPPN self-assembly diagrams as in Fig. 2.2 but as a function of the bilayer-protein contact angle,a, and the in-plane protein radius outside the membrane,r s , atU= 0 with (a)h s = 3:0 nm, (b)h s = 5:0 nm, and (c)h s = 3:0 nm for an extendedr s -range, respectively. We use the same notation as in Fig. 2.2. The r s -coordinates of the horizontal red lines are given by r s 0 (n) in Eq. (2.34). Each red line is drawn from the left to the righta-boundary of the MPPN region dominated by a given MPPN n-state. We use the same color scheme for all three panels. The blue dashed vertical line in panel (b) corresponds toa = 0:55 rad. . . . . . . . . . . . . . . . . . . . . 23 2.5 Estimates of the smallest r s for which protein domains outside the membrane can affect MPPN self-assembly diagrams, r s 0 , in Eq. (2.34) versus n at h s = 3:0 nm (upper panel) and h s = 5:0 nm (lower panel). The highly symmetric MPPN n- states withn= 12, 20, and 24 are marked with star symbols. . . . . . . . . . . . . 24 2.6 Smallest r s estimated to affect the MPPN patch size at U = 0 in Eq. (2.36), r s h , versusn ath s = 3:0 nm (upper panel) andh s = 5:0 nm (lower panel). As in Fig. 2.5, the highly symmetric MPPNn-states withn= 12, 20, and 24 are marked with star symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 MPPN radiiR (upper panel) and corresponding MPPN energy per proteinE avg (n)= E min (n)=n (lower panel) versus in-plane protein radius outside the membrane, r s . As in Fig. 2.4(b), we seth s = 5:0 nm andU = 0. We use herea = 0:55 rad, which corresponds to the dashed vertical line in Fig. 2.4(b), and consider the first four dominant MPPNn-states (starting from smallr s ) along this line in Fig. 2.4(b). The dashed vertical lines indicate transitions from n= 20 to n= 24, from n= 24 to n= 30, and fromn= 30 ton= 32 (left to right) in the dominant MPPNn-state in the MPPN self-assembly diagram in Fig. 2.4(b). . . . . . . . . . . . . . . . . . . 26 2.8 MPPN self-assembly diagram as in Fig. 2.4(b) for h s = 5:0 nm but withjUj= 0:2 nm. The blue dashed vertical line corresponds toa = 0:55 rad, for which the dominant MPPN n-states are n= 20, 18, 17, and 12 (from bottom to top). We use the same notation as in Fig. 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 vii 2.9 MPPN radii R (upper panel) and corresponding MPPN energy per protein E avg = E(n)=n (lower panel) versus in-plane protein radius outside the membrane, r s . As in Fig. 2.8, we set h s = 5:0 nm andjUj= 0:2 nm. We use here a = 0:55 rad, which corresponds to the dashed vertical line in Fig. 2.8, and consider the first four dominant MPPN n-states (starting from small r s ) along this line in Fig. 2.8. The dashed vertical lines indicate transitions from n= 20 to n= 18, from n= 18 to n= 17, and fromn= 17 ton= 12 (left to right) in the dominant MPPNn-state in the MPPN self-assembly diagram in Fig. 2.8. . . . . . . . . . . . . . . . . . . . . 29 3.1 Schematic of MPPNs formed from Piezo ion channels. The thick gray curve shows the Piezo dome with radius of curvatureR D and cap anglea. The blue curve shows the membrane footprint of the Piezo dome with arclengths s= 0 and s=s b at the inner and outer membrane footprint boundaries, respectively. At s=s b , the Piezo membrane footprint is assumed to connect smoothly to the membrane footprints associated with the neighboring Piezo proteins on the MPPN surface, with contact angleb. We denote the inner and outer membrane footprint boundaries along the r-axis by r 0 and r b , respectively. The MPPN radius is given by R=r b =sinb. For each s b , Piezo’s membrane footprint is completely determined by a given set of values ofa, R D , r 0 =R D sina, andb, which are indicated in gray. For simplicity, we only show here one of then proteins on the MPPN surface. . . . . . . . . . . . 32 3.2 MPPN self-assembly diagrams for MPPNs formed from MscS obtained using (a) the arclength parameterization of Eq. (3.1) and (b) the Monge parameterization of Eq. (3.1). The horizontal axes show the bilayer-protein contact angle, a, and the vertical axes show the protein number fraction in solution,c. The dominantn-states of MPPNs are indicated by integers. The white dashed curves show transitions in the dominant MPPN n-states, with the colors indicating the maximum f(n) among all MPPNn-states considered. We use the same color bar in panels (a) and (b). The dashed horizontal lines indicate the parameter values c 7:8 10 8 and a 0:46–0:54 rad corresponding to experiments on MPPNs formed from MscS [5, 6], in which the snub cube withn= 24 MscS proteins was found to provide the dominant MPPN symmetry (models of the snub cube belown= 24). . . . . . . . 39 3.3 Fractional abundance of MPPN n-states, f(n), versus bilayer-protein contact an- gle, a, for selected (dominant) MPPN n-states in Fig. 3.2(a) (indicated by inte- gers) obtained with the arclength parameterization of Eq. (3.1) and the protein number fraction c 7:8 10 8 used for the MPPN self-assembly experiments in Refs. [5, 6]. As for Fig. 3.2(a), we calculated all curves through interpolation of numerical results at a resolutionDa= 0:01 rad [dots in panel (a)], using third-order splines. Panel (a) compares these interpolations with the corresponding results ob- tained at a higher resolution Da = 0:002 rad (crosses). Panel (b) compares the curves in panel (a) obtained using A= A D in Eq. (3.21) (solid curves) with the corresponding results obtained usingA=A S . . . . . . . . . . . . . . . . . . . . . 40 viii 3.4 MPPN self-assembly diagram for MPPNs formed from Piezo ion channels, ob- tained from the arclength parameterization of Eq. (3.1), as a function of the radius of curvature of the Piezo dome, R D , and the protein number fraction in solution, c. Colors indicate the maximum f(n) among all MPPN n-states considered. The color bar employed here is identical to the color bar employed in Fig. 3.2. As in Fig. 3.2, the dominant n-states of MPPNs are indicated by integers, and the white dashed curves show transitions in the dominant MPPN n-states. The hor- izontal dashed line indicates the protein number fraction c 7:8 10 8 used in experiments on MPPNs formed from MscS [5, 6], while the vertical dashed line shows the Piezo dome radius of curvature R D 10:2 nm observed for a closed state of Piezo [7, 8, 9, 10]. The polyhedra models show dominant MPPN sym- metries predicted by the MPPN self-assembly diagram, with n= 6 (octahedron), n= 12 (icosahedron), and n= 24 (snub cube), respectively. Gray shading indi- cates regions of the MPPN self-assembly diagram dominated by MPPN n-states withn= 80, which may be a spurious result of our constraint 3n 80. . . . . . 42 3.5 Fractional abundance of MPPN n-states obtained from the arclength parameter- ization of Eq. (3.1), f(n), versus radius of curvature of the Piezo dome, R D , for selected (dominant) MPPN n-states in Fig. 3.4 at the protein number frac- tion c 7:8 10 8 used in experiments on MPPNs formed from MscS [5, 6]. As in Fig. 3.4, the vertical dashed line shows the Piezo dome radius of curvature R D 10:2 nm observed for a closed state of Piezo [7, 8, 9, 10]. All curves were obtained through interpolation of numerical results at a resolution DR= 0:2 nm (dots), using third-order splines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Minimum-energy MPPN configurations obtained from simulated annealing MC simulations of the LP model in Sec. 4.1.1 with (a) 24 closed-state MscS proteins and (b) one MscS in the open state and 23 closed-state MscS. The small and large disks represent the lipids in a diC14:0 lipid bilayer [11] and MscS proteins [12, 13], respectively, with closed-state MscS corresponding to the large gray disks and open-state MscS to the large yellow disk. The closed-state MscS, open-state MscS, and lipid disk sizes are given by r c , r o , and r l , respectively (see Sec. 4.1.1). (c,d) Minimum-energy MPPN configurations obtained as in panels (a) and (b), respec- tively, but using the coarse-grained CP model in Sec. 4.1.2. Particles corresponding to closed-state and open-state MscS are illustrated by blue and red disks, respec- tively. For ease of visualization, the radii of these disks were decreased by some fixed scale factor relative to the disk radii implied byr 0 o andr 0 c (see Sec. 4.1.2). The green lines in panels (a–d) are obtained by connecting the centers of neighboring proteins. (e) 3D and (f) net representations of a snub cube. . . . . . . . . . . . . . 46 ix 4.2 Illustrative MC trajectories of the CP model for five independent MC simulations showing the total MPPN energy calculated from Eq. (4.3) versus number of MC steps. We set (N;n o )=(24;2), start from random initial conditions, and use our standard MC simulation procedure for the CP model including swapping moves (see Sec. 4.1.3). The inset shows the MPPN energy associated with the random initial conditions used for the five independent MC simulations. In the main panel, the curves leading to the first data point shown overlap for the five independent MC trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Square of the optimal MPPN radius, (R ) 2 , versus number of open-state MscS proteins, n 0 , for N = 24 obtained from simulated annealing MC simulations of the CP model and the estimate in Eq. (4.5). For the MC simulations we used our standard MC simulation procedure including swapping moves (see Sec. 4.1.3). . . 52 4.4 Total MPPN energy versus angular separation of open-state MscS in the final MPPN configuration,a=a final (see inset), in the CP model with(N;n o )=(24;2) obtained through simulated annealing MC simulations with no swapping moves (see Sec. 4.1.3). The dashed vertical lines show the values of a associated with perfect snub cube symmetry. Each cross symbol represents the result of one simu- lated annealing MC simulation. The insets show enlarged versions of the indicated regions in the plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.5 CP model with(N;n o )=(24;2). The results in panel (a), which correspond to the data in Fig. 4.4, were obtained through simulated annealing MC simulations with no swapping moves (see Sec. 4.1.3). (b)a final versusa init as in panel (a) and using the same initial conditions as in panel (a) but allowing for swapping moves. The dashed vertical and horizontal lines show the values of a associated with perfect snub cube symmetry. The solid lines in panels (a) and (b) indicate a final =a init . Each cross symbol represents the result of one simulated annealing MC simulation. The insets show enlarged versions of the indicated regions in the plots. . . . . . . 54 4.6 3D representations of (a) the icosahedron (I) and (b) the pentakis dodecahedron (kD) in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.7 Minimum-energy protein arrangements in MPPNs implied by the CP model of MPPN symmetry for N = 24. The numbers labeling the vertices denote the po- sitions of the n th o open-state MscS protein, with subscripts denoting the degree of degeneracy in placing then th o open-state MscS protein. We omit this subscript if the position of the n th o open-state MscS protein is uniquely determined by the protein arrangement in MPPNs with(n o 1) open-state MscS proteins. The degeneracy in placing open-state MscS proteins follows from the symmetry of the snub cube. The faces are colored according to their symmetry properties with red, yellow, and blue colors indicating two-fold, three-fold, and four-fold symmetry axes, respectively (see also Fig. 4.8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 x 4.8 3D illustrations of selected minimum-energy protein arrangements in Fig. 4.7 for (a) n o = 0 or n o = 24 (undeformed snub cube), (b) n o = 3, (c) n o = 12, and (d) n o = 16. Following the labeling scheme in Fig. 4.7, the faces of the snub cube are colored according to their symmetry properties. In panel (a), the locations of some of the two-, three-, and four-fold symmetry axes of the snub cube are indi- cated by arrow, triangle, and square symbols, respectively, with the symmetry axes perpendicularly intersecting these symbols at their geometric centers. In panels (b), (c), and (d), vertices of the snub cube occupied by open-state MscS proteins are indicated by disks. In panel (b), the two geometrically equivalent choices for placing the 3 rd open-state MscS protein are labeled as 3 and 3 0 , respectively. In panels (c) and (d) we highlight the positions of the open-state MscS proteins occu- pying the “front” and ”back” square faces of the snub cube through increased disk sizes with no labels (see main text), and label the positions of selected open-state MscS proteins with n o > 8 by n o . In panel (c), the closed zig-zag loop formed by the first eight open-state MscS proteins is shown in orange, while we indicate in green the nearest-neighbor bonds formed by the 9 th to 12 th open-state MscS pro- teins with the open-state MscS proteins occupying the front and back square faces of the snub cube. In panel (d), the white lines show the closed zig-zag loop of closed-state MscS proteins formed at n o = 16. Portions of the loops in panels (c) and (d) located at the back of the snub cube are indicated by dashed curves. . . . . 62 4.9 3D representations of the minimum-energy MPPN configurations implied by the CP model of MPPN symmetry for N = 24 and (a) n o = 2, (b) n o = 4, (c) n o = 8, (d)n o = 12, and (e)n o = 16 (see also Fig. 4.7). As in Fig. 4.1(d), closed-state and open-state MscS proteins are represented by blue and red disks, respectively, with the radii of these disks being decreased by some fixed scale factor relative to the disk radii implied byr 0 o andr 0 c . For clarity, polyhedral ridges enclosing (distorted) square faces of the snub cube are shown in bright green, while polyhedral ridges associated only with triangular faces of the snub cube are shown in dark green. . . 63 4.10 Root-mean-square fit errors b s in Eq. (4.10) of the best polyhedral fit (snub cube symmetry) to the minimum-energy MPPN configurations implied by the CP model of MPPN symmetry for N = 24 versus number of open-state MscS proteins, n o (blue data points). For each n o , we also show the range in b s associated with the ten lowest-energy MPPN configurations obtained in our simulated annealing MC simulations, which all correspond to snub cube symmetry. This range inb s is indicated by bars, with the red data points showing the averageb s for the ten lowest- energy MPPN configurations obtained in our simulated annealing MC simulations. See also Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 xi 4.11 Relative difference in BOO parameters between a snub cube and the MPPN con- figurations implied by the CP model, c Q l , versus number of open-state MscS, n o , at (a) order l = 4 and (b) order l = 6 in Eq. (4.9). For each n o , we show the range in c Q l associated with the ten lowest-energy MPPN configurations obtained in our simulated annealing MC simulations, which all correspond to snub cube symmetry. This range in c Q l is indicated by bars, with the red data points showing the average c Q l for the ten lowest-energy MPPN configurations and the blue data points showing the c Q l associated with the minimum-energy MPPN configurations. The plots in panels (a) and (b) are obtained from the data for low-energy MPPN configurations also used in Fig. 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . 69 B.1 MPPN self-assembly diagrams as a function of the absolute value of the bilayer- protein hydrophobic mismatch,jUj, and the bilayer-protein contact angle, a, for r s = 5:0 nm and h s = 6:0 nm calculated as in Fig. 2.2 with (a) K b = 7 k B T , (b) K b = 14k B T , and (c)K b = 28k B T . The MPPN self-assembly diagram in panel (b) is identical to that in the lower-left panel of Fig. 2.2 and reproduced here for ease of comparison. We use the same notation as in Fig. 2.2. . . . . . . . . . . . . . . 79 C.1 MPPN self-assembly diagram as in Fig. 2.4 as a function of the bilayer-protein contact angle,a, and the in-plane protein radius outside the membrane,r s , atU= 0 in the (nonphysical) caseh s = 0. The dotted curves delineate regions of the MPPN self-assembly diagram dominated by distinct MPPNn-states, while the solid lines show the corresponding estimates obtained from Eq. (C.2). The red horizontal lines are obtained as in Fig. 2.4. The small-a boundaries of the red horizontal lines are used to fixC in Eq. (C.2) for each n considered. We use the same notation as in Fig. 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 xii Abstract Experiments have shown that, in an aqueous environment, lipids and membrane proteins can self-assemble into membrane protein polyhedral nanoparticles (MPPNs). MPPNs are closed, spherical vesicles composed of a lipid bilayer membrane and membrane proteins, with a poly- hedral arrangement of membrane proteins. Because of these features, MPPNs have been proposed as a potential strategy for the high-resolution structrual study of membrane proteins in the pres- ence of physiologically relevant transmembrane gradients and targeted drug delivery. In this PhD thesis, we develop new methodologies for the prediction of the polyhedral symmetry of MPPNs for arbitrary lipid and protein compositions. In Chapter 2, we consider membrane proteins that have large domains outside the membrane, and discuss the effect of steric constraints arising from such domains on the symmetry and size of MPPNs. In Chapter 3, we use the arclength parameterization of surfaces to develop a mean-field model of MPPN self-assembly that allows for arbitrarily large (nonlinear) membrane curvatures. On this basis, we study the symmetry of MPPNs formed from Piezo ion channels, which have been found to underlie many forms of mechanosensation in vertebrates and bend the membrane into strongly curved dome shapes. In Chapter 4, we use kinetic Monte Carlo (MC) simulations to explore the symmetry of MPPNs composed of membrane proteins with heterogeneous structure, such as the closed and open states of ion channels. The work described here suggests that MPPNs can show a variety of distinct symmetries and provides a range of approaches for the control of MPPN symmetry, which may facilitate the further development of MPPNs for membrane protein structural analysis and targeted drug delivery. xiii The materials in this PhD thesis are also dicussed in the following publications: I. M. Ma and C. A. Haselwandter. Effect of protein steric constraints on the symmetry of membrane protein polyhedra. Phys. Rev. E, 102:042411, 2020. II. M. Ma and C. A. Haselwandter. Self-assembly of polyhedral bilayer vesicles from Piezo ion channels (in preparation). III. M. Ma, D. Li, O. Kahraman, and C. A. Haselwandter. Symmetry of membrane protein polyhedra with heterogeneous protein size. Phys. Rev. E, 101(2):022417, 2020. xiv Chapter 1 Introduction 1.1 Membrane protein polyhedral nanoparticles (MPPNs) Cell membranes are one of the fundamental hallmarks of life [14, 15]. Cell membranes typi- cally consist of a lipid bilayer, which self-assembles from lipids suspended in an aqueous environ- ment, and membrane proteins, which reside in the lipid bilayer and regulate many physiological processes [16, 17, 18]. Experiments have demonstrated that, if membrane proteins and suitably chosen lipids are mixed at a defined ratio in an aqueous environment, membrane proteins and lipids can self-assemble into membrane protein polyhedral nanoparticles (MPPNs) [5, 6]. MPPNs are closed, spherical vesicles composed of a lipid bilayer membrane and membrane proteins, with a polyhedral arrangement of membrane proteins. In particular, it was observed [5, 6] that mechanosensitive ion channels of small conductance (MscS) and diC14:0 lipids predominantly self-assemble into MPPNs composed of 24 MscS proteins, with each MscS protein being located at the vertex of a snub cube and a MPPN diameter of approximately 20 nm at the bilayer midplane. The closed surfaces of MPPNs can support transmembrane gradients mimicking the gradients in pH, voltage, or chemical composition typically found across cell membranes [6], while the regular polyhedral arrangement of membrane proteins in MPPNs could be utilized for structural studies of membrane proteins [19, 20, 21]. MPPNs have thus been proposed [6] as a strategy for the 1 structural analysis of membrane proteins under physiologically relevant transmembrane gradients. Furthermore, MPPNs have been proposed as a platform for targeted drug delivery with precisely controlled release mechanisms [6]. 1.2 Extending of the mean-field model of MPPN self-assembly A mean-field model of MPPN self-assembly [3, 4] successfully predicts the observed symme- try and size of MPPNs formed from MscS proteins [5, 6]. In this mean-field model, the dominant symmetry and size of MPPNs emerge from the interplay of protein-induced lipid bilayer defor- mations in MPPNs, topological defects in protein packing necessitated by the spherical shape of MPPNs, and thermal fluctuations in MPPN self-assembly. In this thesis, we develop, based on this mean-field model, new methodologies to study the symmetry of MPPNs that form from (i) mem- brane proteins that have large domains outside the mebrane and (ii) membrane proteins such as Piezo ion channels [22] that induce large curvatures in the surrounding membrane. 1.2.1 MPPNs formed from membrane proteins with large domains outside the membrane Structural biology has shown that membrane proteins often have large domains with well- defined shapes outside the membrane [14, 23, 24, 25]. Such protein domains outside the membrane impose restrictions on the arrangement of membrane proteins in MPPNs and, hence, are expected to affect MPPN shape. We have therefore extended the mean-field model of MPPN self-assembly [3, 4] to explore the effect of protein steric constraints on MPPN shape (see Chapter 2). Exam- ples of membrane proteins with large domains with well-defined shapes outside the membrane include some types of potassium channels, molecular machines such as ATP synthase, and, poten- tially, SARS-CoV-2 spike proteins [14, 23, 24, 25, 26, 27, 28, 29]. Effective steric constraints on membrane protein separation in MPPNs may also arise through the binding of other molecules to 2 membrane proteins or entropic protein repulsion due to large but flexible protein domains outside the membrane. 1.2.2 MPPNs formed from Piezo ion channels The ability to sense mechanical stimuli such as touch and changes in fluid pressure is funda- mental to life. Piezo ion channels [22] have recently been found to provide the molecular basis for a wide range of seemingly unrelated forms of mechanosensation in vertebrates [30, 31, 32, 33, 34, 35, 36, 37]. Structural studies [7, 8, 9, 10] have demonstrated that Piezo is an unusually large ion channel that locally bends the membrane into the shape of a spherical dome. Only closed-state structures of Piezo, obtained in the absence of transmembrane gradients, are currently available. The interaction of Piezo with the surrounding lipid membrane has been investigated through elec- tron microscopy experiments in which single Piezo proteins were embedded in unilamellar lipid bilayer vesicles, with a vesicle diameter 20 nm–60 nm [7, 38]. So far, these experiments have focused on vesicles containing a single Piezo protein. The highly curved shape of the (closed-state) Piezo dome induces pronounced membrane curvature deformations [7, 38, 39]. These curvature deformations may play an important role in Piezo gating [7, 8, 9, 10, 38, 39], with transitions from closed to open states of Piezo being accompanied by an increase in the Piezo dome radius of curvature. We have explored theoretically the self-assembly of MPPNs from Piezo ion channels (see Chapter 3). Our theoretical approach is based on a previous mean-field model [3, 4] (see also Sec.1.2.1). This previous model relied on the assumption that membrane proteins in MPPNs only weakly curve the membrane, which is a suitable assumption for MscS [12, 18, 40] but not Piezo [7, 8, 9, 10, 38, 39]. Allowing for arbitrarily large (nonlinear) membrane curvatures, we develop, motivated by experiments on Piezo vesicles [7, 38] and MPPNs formed from MscS [5, 6], a model of the self-assembly of MPPNs from Piezo ion channels. 3 1.3 Minimal molecular model Experiments on MPPNs have so far [5, 6] focused on MPPNs with homogeneous protein com- position. Intriguingly, the closed surface of MPPNs permits chemical or voltage gradients across the MPPN membrane, which could allow trapping of membrane proteins in distinct, physiologi- cally relevant conformational states [6]. Such transitions in protein conformational state are gen- erally accompanied by changes in protein size: for instance, mechanosensitive ion channels often have a larger size, when viewed perpendicularly to the cell membrane, in the open than the closed state [41]. Provided MPPNs with heterogeneous protein size show an ordered arrangement of proteins, MPPNs could be employed to elucidate membrane protein structures stabilized by trans- membrane gradients [6]. It is, however, unknown how heterogeneity in protein size affects MPPN symmetry. We have therefore developed a computational framework describing the symmetry of MPPNs with heterogeneous protein size (see Chapter 4). We thereby use as our benchmark previ- ous experiments on MPPNs containing 24 MscS proteins [5, 6], but allow for open-state as well as closed-state MscS [12, 13, 40] with a fixed total number of proteins in MPPNs. In MPPNs with heterogeneous protein size not all proteins are equivalent. We therefore do not use a mean-field approach to study MPPNs with heterogeneous protein size. Previous work in physical virology has shown that the symmetry of viral capsids can be captured through a minimal molecular model in which individual capsid subunits are represented by Lennard-Jones particles [42, 43, 44, 45]. For a given number of proteins per MPPN, a similar approach can also be used to successfully predict the symmetry of MPPNs with homogeneous protein size [3]. We have generalized this approach to model the symmetry of MPPNs with heterogeneous protein size. 1.4 Overview of this thesis The overall theme of this thesis is to explore, based on previous models of MPPNs [3, 4], the molecular mechanisms underlying the symmetry of MPPNs, and how these key molecular properties of MPPNs could be used to control MPPN symmetry and size. In Chapter 2, we consider 4 MPPNs formed from membrane proteins with large domains outside the membrane, and study the effect of protein steric constraints on MPPN symmetry. In particular, in Sec. 2.1 we describe in detail the mean-field model of MPPN self-assembly employed here. Section 2.2 surveys the effect of protein steric constraints on MPPN self-assembly diagrams. Subsequently, in Chapter 3, we consider MPPNs formed from membarne proteins that may induce large membrane curvatures, using Piezo ion channels [22] as a model system. We first develop a general methodology for predicting the symmetry and size of MPPNs composed of proteins that induce arbitrarily large membrane curvature deformations (see Sec. 3.1). We then validate this methodology for MPPNs formed from MscS (see Sec. 3.2). Finally, on this basis, we calculate the self-assembly diagram for MPPNs formed from Piezo ion channels, and explore the key parameters regulating the symmetry of MPPNs formed from Piezo (see Sec. 3.3). In Chapter 4 we go beyond the mean-field model of MPPN symmetry employed in Chapters 2 and 3, and study the symmetry of MPPNs with heterogenous protein size using simulated annealing MC simulations. In Sec. 4.1 we provide a detailed description of our modeling approach, the simu- lated annealing MC simulations we employ to obtain energetically favorable MPPN configurations, and the methods used here for quantifying MPPN symmetry. In Sec. 4.2 we survey, on this basis, the minimum-energy protein configurations in MPPNs with heterogeneous protein size. Based on previous experiments on MPPNs composed of MscS proteins [5, 6], we thereby focus on MPPNs with 24 proteins corresponding to closed-state or open-state MscS proteins [12, 13, 40]. We find that, as an increasing number of closed-state MscS transitions to the open state, the minimum- energy MscS arrangement in MPPNs follows a strikingly regular pattern, with the dominant MPPN symmetry always being provided by the snub cube. 5 Chapter 2 Effect of protein steric constraints on the symmetry of membrane protein polyhedra This chapter explores the effect of protein steric constraints on MPPN symmetry. In Sec. 2.1, we describe the contributions to the MPPN energy due to lipid bilayer midplane deformations in MPPNs, protein-induced lipid bilayer thickness deformations in MPPNs, and topological defects in protein packing in MPPNs. Importantly, we thereby allow for steric constraints due to large protein domains outside the membrane. In Sec. 2.2, we present the resulting self-assembly diagrams and summarize the effect of protein steric constraints on MPPN self-assembly. 2.1 Calculation of MPPN self-assembly diagrams Our calculations of MPPN self-assembly diagrams are based on the mean-field approach devel- oped in Refs. [3, 4]. The purpose of this section is to summarize this mean-field model of MPPN self-assembly, and to describe how this model can be extended to account for protein steric con- straints arising from protein domains outside the membrane. At the mean-field level, the observed MPPN symmetry and size [5, 6] can be understood [3, 4] by considering the interplay of thermal fluctuations in MPPN self-assembly, protein-induced lipid bilayer deformations in MPPNs, and 6 topological defects in protein packing arising from the spherical shape of MPPNs. Sections 2.1.1– 2.1.4 describe the resultant contributions to the MPPN energy. In Sec. 2.1.5 we discuss how steric constraints enter the calculation of MPPN self-assembly diagrams. We note that the mean-field approach employed here is, in general, not expected to apply to situations in which MPPNs self- assemble from distinct membrane proteins with, for instance, heterogeneous size, in which case a more detailed molecular model of MPPN symmetry is required [46] (see Chapter 4). 2.1.1 Thermal effects To ascertain the role of thermal effects in MPPN self-assembly, we note [3, 4] that MPPNs were obtained experimentally [5, 6] in dilute, aqueous solutions with a protein number fraction c= å n N n N w 7:8 10 8 1; (2.1) where N n denotes the total number of proteins bound in MPPNs with n proteins each and N w denotes the total number of solvent molecules in the system. We take N w to be dominated by contributions due to water. Furthermore, we take the temperature of the system, T , to be fixed. We assume that the system is in a thermodynamic equilibrium state minimizing the Helmholtz free energy, F =UTS, where U and S are the internal energy and the entropy of the system, respectively. In the dilute limitc 1 with no interactions between MPPNs,U is given by [3, 4] U =N wå n F(n)E min (n); (2.2) where the MPPN number fraction F(n)= N n =nN w and E min (n) is the minimum MPPN energy, andS is given by the mixing entropy [47, 48] S=N w k Bå n F(n)[lnF(n) 1]; (2.3) 7 where k B is Boltzmann’s constant. Minimization of F with respect toF(n) thus yields [3, 4, 47, 48, 49] F(n)=e [mnE min (n)]=k B T ; (2.4) where the protein chemical potentialm is determined by the constraint å n nF(n)=c (2.5) imposing a fixed protein number fraction in the system. We restrict n to the range 10n 80. For all the calculations described here we employ the protein number fraction c in Eq. (2.1) used in experiments on MPPNs formed from MscS proteins [5, 6]. For given E min (n), Eq. (2.5) thus fixes the protein chemical potentialm. We represent MPPN self-assembly diagrams in terms of the MPPN equilibrium distribution f(n)= F(n) å 80 n=10 F(n) : (2.6) For given E min (n), f(n) is determined via Eqs. (2.4)–(2.6) with Eq. (2.1). We obtain E min (n) by minimizing the MPPN energy E(n;R), at each n, with respect to the MPPN radius at the bilayer midplane, R, subject to steric constraints arising from the finite size of lipids and proteins (see Secs. 2.1.2–2.1.5). We writeE(n;R) as E(n;R)=E h (n;R)+E u (n;R)+E d (n;R); (2.7) where E h , E u , and E d denote contributions to E due to lipid bilayer midplane deformations in MPPNs, protein-induced lipid bilayer thickness deformations in MPPNs, and topological defects in protein packing in MPPNs, respectively. Sections 2.1.2–2.1.4 provide a discussion of these contributions to E. In Sec. 2.1.5 we discuss how steric constraints enter E min (n), and summarize the numerical procedure for the calculation ofE min (n) employed here. 8 Figure 2.1: Schematics of (a) protein-induced lipid bilayer deformations in MPPNs and (b) the geometry of MPPNs. We take each MPPN membrane patch to have a protein at its center atr = 0 and to be symmetric about the rotation axis h(0). As indicated in panel (a), we denote the protein radius in the bilayer midplane by r i , the MPPN membrane patch size by r o , the height of the bilayer midplane by h(r), and one-half the bilayer hydrophobic thickness by u(r)+m, where m is one-half the unperturbed bilayer thickness. As indicated in panel (b), we account for steric constraints on MPPN patch size due to protein domains outside the membrane through the in-plane protein radius outside the membrane r s and the effective protein height h s , which we define with respect to the protein hydrophobic midplane (dashed horizontal line). The protein height with respect to the MPPN center is given by h s +Re, where R is the MPPN radius ande is an offset due to the finite protein size. The effective steric constraint on the membrane patch size in the bilayer midplane,r s o , depends onr s andh s . The schematic in panel (b) corresponds to the smallest R allowed by steric constraints due to protein domains outside the membrane, R=R s , for which we haveR s =r s o =sin ¯ b (see Sec. 2.1.5). 2.1.2 MPPN bilayer midplane deformations Membrane proteins often produce elastic shape deformations of the surrounding lipid bilayer membrane [18, 50, 51, 52]. These bilayer shape deformations, and their associated elastic energy, depend on the elastic properties of the lipid bilayer under consideration, key properties of the protein structure such as the bilayer-protein contact anglea and the protein hydrophobic thickness w [see Fig. 2.1(a)], and the supramolecular protein arrangement. Based on the molecular structure of MscS [3, 12, 40] we consider here MPPNs formed from (approximately) rotationally symmetric membrane proteins, for which protein-induced elastic bilayer deformations are expected to favor hexagonal protein arrangements [53, 54, 55, 56, 57, 58, 59]. The elastic energy of lipid bilayer 9 deformations in MPPNs can then be estimated using a mean-field approach [3, 4, 53, 54, 55] in which the boundary of the hexagonal unit cell of the protein lattice is approximated by a circle. We thus divide the surface of MPPNs containing n proteins into n identical, circular membrane patches, each with a protein at its center. We use here the Monge parameterization of surfaces, and denote the in-plane radial coordinate in the membrane patch byr [Fig. 2.1(a)]. We haver i rr o , wherer i is the protein radius in the bilayer midplane and the patch size r o (n;R)= Rsinb, where b is the patch boundary angle mandated by the spherical shape of MPPNs [Fig. 2.1(a)]. We use here the value r i = 3:2 nm corresponding to closed-state MscS proteins [3, 12, 40]. We set b = arccos[(n 2)=n] [54] so that, for a spherical MPPN shape, the total area covered by membrane patches, 2npR 2 (1 cosb), is equal to the total MPPN area, 4pR 2 , at closest (hexagonal) protein packing. The spherical topology of MPPNs necessitates defects in this energetically preferred protein packing, which we return to in Sec. 2.1.4. One key contribution to the MPPN energy in Eq. (2.7) arises due to bending deformations of the lipid bilayer midplane field h(r) [Fig. 2.1(a)]. The energy cost associated with these lipid bilayer deformations is captured by the bilayer midplane deformation energy [17, 60, 61, 62] G h = K b 2 Z dA Ñ 2 h 2 ; (2.8) where K b is the lipid bilayer bending rigidity, the area element dA= 2prdr in polar coordinates, andÑ 2 is the Laplace operator in polar coordinates. For the diC14:0 lipids used for MPPNs formed from MscS proteins [5, 6] we have K b 14 k B T [63]. We set K b = 14 k B T throughout this thesis. We assume in Eq. (2.8), and throughout this thesis, that contributions to the MPPN energy due to lateral membrane tension can be neglected. Our analysis could be extended to allow for a non- zero membrane tension [4]. Furthermore, we assume throughout this thesis that the lipid bilayer membrane does not show a spontaneous curvature, and take the dominant bilayer shape to be the bilayer shape with lowest elastic energy. 10 The Euler-Lagrange equation associated with Eq. (2.8) is given by Ñ 4 h= 0, which has the general solution h(r)=A h r 2 +B h +C h lnr+D h r 2 lnr (2.9) for circular membrane patches with rotational symmetry about the protein center, whereA h ,B h ,C h , and D h are constants to be determined by the boundary conditions at the bilayer-protein interface and at the outer boundary of the membrane patch. As indicated in Fig. 2.1(a), we take the slope at the bilayer-protein interface to be fixed by the bilayer-protein contact anglea, h 0 (r i )=tanaa: (2.10) For reference, we note that for MscS we havea 0:46–0:54 rad [3, 12, 40]. The spherical geom- etry of MPPNs mandates that h 0 (r o )=tanbb: (2.11) We allowh(r o )h(r i ) to be adjusted freely as part of the minimization procedure, which amounts to the natural boundary condition [54, 64] d dr Ñ 2 h(r)= 0 (2.12) atr =r i or, equivalently,r =r o . Noting that the choice of the reference pointh(r)= 0 does not affect Eq. (2.8), substitution of the solution of the Euler-Lagrange equation for Eq. (2.8) [Eq. (2.9)] back into Eq. (2.8) [54] thus yields the MPPN midplane deformation energy E h (n;R)= 2npK b (br o ar i ) 2 r 2 o r 2 i (2.13) forn identical MPPN membrane patches. 11 2.1.3 MPPN bilayer thickness deformations In addition to the bilayer midplane deformations considered in Sec. 2.1.2, membrane proteins typically deform the hydrophobic thickness of lipid bilayer membranes [18, 50, 51, 52]. The corresponding elastic energy of membrane shape deformations in MPPNs [4] can be estimated from the bilayer thickness deformation energy [51, 65, 66] G u = 1 2 Z dA K b Ñ 2 u 2 +K t u m 2 ; (2.14) where u(r) is the lipid bilayer thickness deformation field, m is one-half the unperturbed bilayer thickness [Fig. 2.1(a)], andK t is the bilayer thickness deformation modulus. For the diC14:0 lipids used for MPPNs formed from MscS proteins [5, 6] we havem 1:76 nm andK t 56:5k B T=nm 2 [63, 67]. We setm= 1:76 nm andK t = 56:5k B T=nm 2 throughout this thesis. The Euler-Lagrange equation associated with Eq. (2.14) is given by Ñ 2 n + Ñ 2 n u= 0; (2.15) where n = r K t K b m 2 : (2.16) For circular membrane patches with rotational symmetry about the protein center, the general solution of Eq. (2.15) is given by [4] u(r) = A + u K 0 ( p n + r)+A u K 0 ( p n r) +B + u I 0 ( p n + r)+B u I 0 ( p n r); (2.17) where I j (x) and K j (x) are the j th -order modified Bessel functions of the first and second kind, and A u and B u are constants to be determined by the boundary conditions at the bilayer-protein interface and at the outer boundary of the membrane patch. 12 Under the assumption of hydrophobic matching, the value ofu(r) at the bilayer-protein bound- ary is fixed bym together with the protein hydrophobic thicknessw [Fig. 2.1(a)]: u(r i )=wmU; (2.18) whereU denotes the bilayer-protein hydrophobic mismatch. For reference, we note that for MscS we havew 1:8 nm [4, 12, 40]. Following previous work [65, 66, 67] we set here u 0 (r i )= 0; (2.19) but note that other choices foru 0 (r i ) are possible [68]. We allowu(r o ) to be adjusted freely as part of the minimization procedure, yielding the natural boundary condition [64] d dr Ñ 2 u(r)= 0 (2.20) atr =r o . Finally, by symmetry we have u 0 (r o )= 0: (2.21) With the boundary conditions in Eqs. (2.18)–(2.21), the solution of Eq. (2.15) is given by Eq. (2.17) with the coefficients A u = I 1o DQ C U; B u = K 1o DQ C U; (2.22) in whichI jh =I j p n r h ,K ih =K i p n r h , DQ =Q oi Q io ; Q hq = p n I 1h K 1q ; (2.23) and C=DQ + I 1o K 0i +I 0i K 1o DQ I + 1o K + 0i +I + 0i K + 1o ; (2.24) 13 with j = 1, 2, h = i; o, and q = i; o. Substitution of Eq. (2.17) with Eqs. (2.22)–(2.24) into Eq. (2.14) yields the MPPN thickness deformation energy [4] E u (n;R)= npK b U 2 r i C DQ + DQ (n + n ) (2.25) forn identical membrane patches. 2.1.4 MPPN defect energy In Secs. 2.1.2 and 2.1.3 we assumed a uniform hexagonal arrangement of membrane proteins in MPPNs and, on this basis, obtained [54] the mean-field MPPN midplane and thickness defor- mation energies in Eqs. (2.13) and (2.25), respectively. However, the spherical shape of MPPNs necessitates topological defects in the hexagonal packing of membrane proteins, which incur ann- dependent energy penalty. At the mean-field level, deviations from hexagonal protein packing due to the spherical shape of MPPNs can be quantified [3, 4, 49], for a givenn, through the fraction of the surface of a sphere enclosed byn identical non-overlapping circles at closest packing [69, 70], p(n). Approximating the spring network associated with the energetically preferred hexagonal protein arrangement by a uniform elastic sheet, the leading-order contribution to the MPPN defect energy is thus given by [3, 4, 49] E d (n;R)= K s 2 As 2 ; (2.26) where K s is the stretching modulus of the elastic sheet, the MPPN area A= 4pR 2 , and the areal strain s=[p max p(n)]=p max with p max =p=2 p 3 for uniform hexagonal protein arrangements (see Appendix A). The stretching modulus in Eq. (2.26) is given by [3, 4] K s = p 3 24n ¶ 2 E 0 ¶r 2 o r o =r min o ; (2.27) where E 0 = E h +E u [see Eqs. (2.13) and (2.25)] and r min o is the value of r o minimizing E 0 for r o r i at a givenn. In other words, we estimate then-dependent energy cost of topological defects 14 in protein packing in MPPNs [Eq. (2.26) with Eq. (2.27)] by expanding the MPPN energy about an ideally closed-packed (hexagonal) MPPN state with energyE 0 andr o =r min o . As mentioned in Sec. 2.1.1, we take here 10n 80. We use the values of p(n) compiled in Refs. [69, 70]. The expression for K s in Eq. (2.27) [3, 4] can be understood intuitively by dividing the ideally packed, hexagonal protein lattice into triangular area elements with one membrane protein at each corner, and taking the membrane proteins to be coupled by Hookean springs with force constantK 0 [3, 71]. In our mean-field model of lipid bilayer deformations in MPPNs the (in-plane) membrane protein separation is equal to 2r o (see Secs. 2.1.2 and 2.1.3). The side length of each triangular area element is therefore given byl= 2r o , withr o =r min o in the state of the spring network minimizing E 0 . Ifl is deformed byDl, the leading-order change in the elastic energy per area element is given by DE element = 3K 0 4 (Dl) 2 ; (2.28) where we have noted that each one of the three springs in each area element contributes equally to the energy of two area elements. Since each area element is associated with one-sixth of three hexagonal unit cells, we also haveE element =E 0 =2n. From Eq. (2.28) we thus find K 0 = 1 12n ¶ 2 E 0 ¶r 2 o r o =r min o ; (2.29) from which Eq. (2.27) follows in the continuum limit via K s = p 3K 0 =2 [3, 71]. As shown in Appendix A, the MPPN defect energy in Eq. (2.26) can be obtained following similar physical reasoning [3, 49, 71]. 2.1.5 Steric constraints As discussed in Secs. 2.1.1–2.1.4, the MPPN energy E min (n) in Eq. (2.6) is obtained, at each n, by minimizing E(n;R) in Eq. (2.7) with respect to R. The value of R and, hence, r o must thereby be large enough so as to satisfy steric constraints arising from the finite size of lipids and proteins. We assume that these steric constraints result in hardcore steric repulsion between 15 membrane proteins in MPPNs. We denote the smallest value of r o allowed by steric constraints byr s o [see Fig. 2.1(b)]. To determine the relation betweenr s o and the smallest value of R allowed by steric constraints, R s , we need to take into account topological defects in protein packing [3]. In particular, within the mean-field approach used here, the MPPN area occupied by the n non- overlapping circles associated with then proteins on the MPPN surface must be equal to 4pR 2 p(n) [69, 70] for spherical MPPN shapes. We thus have cos ¯ b = 1 2 p(n) n ; (2.30) where ¯ b E avg (24). This can be understood by noting that the Boltzmann weight for each MPPN n-state is given by exp n mE avg =k B T [see Eq. (2.4)]. In the dilute limit c 1, Eq. (2.5) impliesmE avg (n)< 0. As a result, thermal effects penalize MPPNn-states with larger n, and can make an MPPN n-state with decreased n (such as n= 12) dominant even if this MPPN n-state does not correspond to the state with minimalE avg . Thus, thermal effects can combine with protein steric constraints to bias MPPN self-assembly towards smaller, more symmetric MPPN states. 22 Figure 2.4: MPPN self-assembly diagrams as in Fig. 2.2 but as a function of the bilayer-protein contact angle, a, and the in-plane protein radius outside the membrane, r s , at U = 0 with (a) h s = 3:0 nm, (b) h s = 5:0 nm, and (c) h s = 3:0 nm for an extended r s -range, respectively. We use the same notation as in Fig. 2.2. The r s -coordinates of the horizontal red lines are given by r s 0 (n) in Eq. (2.34). Each red line is drawn from the left to the right a-boundary of the MPPN region dominated by a given MPPNn-state. We use the same color scheme for all three panels. The blue dashed vertical line in panel (b) corresponds toa = 0:55 rad. 2.2.2 Quantifying the effect of protein steric constraints on MPPN self-assembly diagrams Figure 2.4 shows MPPN self-assembly diagrams, plotted as a function of the bilayer-protein contact angle a and the in-plane protein radius outside the membrane r s , for h s = 3:0 nm and h s = 5:0 nm withU = 0 (see also Appendix C). Note from Fig. 2.4 that, in the small-r s regime, the dominant MPPN symmetry is independent ofr s . Conversely, Fig. 2.4 shows that, for large enough r s , the dominant MPPN symmetry has a strong dependence on r s . We find that, for large enough r s , steric constraints due to protein domains outside the membrane can substantially expand the regions of MPPN self-assembly diagrams dominated by highly symmetric MPPNn-states, such as MPPNs withn= 48 andn= 72 in Fig. 2.4. In Fig. 2.4(b), the ranges of MPPN radii associated with dominant MPPN n-states in the MPPN self-assembly diagram are given by 7:0 nm/R/ 8:2 nm (n= 12), 9:8 nm/R/ 17 nm (n= 24), 11 nm/R/ 20 nm (n= 32), 16 nm/R/ 26 nm (n= 48), and 25 nm/R/ 34 nm (n= 72). In Fig. 2.4(c), which shows the same data as Fig. 2.4(a) but with an extended r s -range, the corresponding ranges of MPPN radii are given by 7:0 nm/ R/ 23 Figure 2.5: Estimates of the smallest r s for which protein domains outside the membrane can affect MPPN self-assembly diagrams, r s 0 , in Eq. (2.34) versus n at h s = 3:0 nm (upper panel) and h s = 5:0 nm (lower panel). The highly symmetric MPPN n-states with n= 12, 20, and 24 are marked with star symbols. 8:1 nm (n= 12), 9:8 nm/R/ 18 nm (n= 24), 11 nm/R/ 24 nm (n= 32), 14 nm/R/ 54 nm (n= 48), and 21 nm/R/ 68 nm (n= 72). Following Sec. 2.2.1 it is instructive to consider the smallest r s for which protein domains outside the membrane can affect MPPN self-assembly diagrams,r s 0 . From Eq. (2.33) we find r s 0 (n)= r i +r l cos ¯ b +h s tan ¯ b: (2.34) At smallr s , the MPPN self-assembly diagrams in Fig. 2.4 are divided into bands along thea-axis within which distinct MPPNn-states are dominant. We indicate in Fig. 2.4 the correspondingr s 0 ob- tained from Eq. (2.34) (horizontal lines in Fig. 2.4). In general, Eq. (2.34) provides good estimates of the smallest values of r s in Fig. 2.4 for which steric constraints due to protein domains outside the membrane start to affect MPPN self-assembly diagrams. However, some notable discrepancies 24 Figure 2.6: Smallestr s estimated to affect the MPPN patch size atU = 0 in Eq. (2.36),r s h , versusn at h s = 3:0 nm (upper panel) and h s = 5:0 nm (lower panel). As in Fig. 2.5, the highly symmetric MPPNn-states withn= 12, 20, and 24 are marked with star symbols. are obtained for highly symmetric MPPNn-states, such asn= 12, 20, and 24 in Fig. 2.4. With in- creasingr s , the regions of the MPPN self-assembly diagrams in Fig. 2.4 dominated by such highly symmetric MPPN n-states are seen to “expand,” even at r s < r s 0 , into neighboring regions of the MPPN self-assembly diagrams dominated, asr s ! 0, by less symmetric MPPNn-states. This can be understood from Fig. 2.5, which showsr s 0 (n) in Eq. (2.34) for 10n 30. We find that, while r s 0 generally tends to decrease with increasing n, r s 0 (n) exhibits local maxima at n= 12, 20, and 24. Equation (2.30) implies that ¯ b monotonically increases with p(n) for 10n 80, with p(n) showing local maxima atn= 12, 20, and 24 [69, 70]. As a result, one finds from Eq. (2.34) that, for such highly symmetric MPPNn-states, largerr s are required to affect the minimum MPPN energy E min (n). Regions in the MPPN self-assembly diagrams dominated by highly symmetric MPPN 25 Figure 2.7: MPPN radii R (upper panel) and corresponding MPPN energy per protein E avg (n)= E min (n)=n (lower panel) versus in-plane protein radius outside the membrane,r s . As in Fig. 2.4(b), we seth s = 5:0 nm andU= 0. We use herea= 0:55 rad, which corresponds to the dashed vertical line in Fig. 2.4(b), and consider the first four dominant MPPN n-states (starting from small r s ) along this line in Fig. 2.4(b). The dashed vertical lines indicate transitions from n= 20 to n= 24, from n= 24 to n= 30, and from n= 30 to n= 32 (left to right) in the dominant MPPN n-state in the MPPN self-assembly diagram in Fig. 2.4(b). n-states are thus effectively less constrained by r s than neighboring regions in the MPPN self- assembly diagrams dominated by less symmetric MPPNn-states, allowing expansion of regions of MPPN self-assembly diagrams dominated by highly symmetric MPPNn-states with increasingr s . To understand some of the key features of Fig. 2.4 in the regime r s >r s 0 , we note that protein steric constraints only provide lower bounds on R. As explained in Sec. 2.1, the value of R and, hence, r o is determined at each n through minimization of E(n;R) in Eq. (2.7) with respect to R. Assuminga>b and neglecting, for simplicity, all contributions toE apart fromE h , MPPNs attain their minimum-energy state at the membrane patch radius r o = a b r i (2.35) 26 with E h = 0, which follows directly from Eq. (2.13). We define r s h as the value of r s for which the smallest value ofr o allowed by protein steric constraints is equal tor o in Eq. (2.35). For r s >r s h , protein steric constraints thus prohibit MPPNs from attaining the value ofr o in Eq. (2.35). From Eq. (2.33) with Eq. (2.35) we find r s h (n)= a bcos ¯ b r i +h s tan ¯ b: (2.36) Note that Eq. (2.35) does not include any contributions due to the defect energyE d . Even forU= 0, the simple estimate ofr s h in Eq. (2.36) therefore only holds approximately (see also Appendix C). Figure 2.6 showsr s h (n) in Eq. (2.36) for 10n 30. In contrast tor s 0 (n) in Eq. (2.34) (Fig. 2.5), we find that r s h generally tends to increase with increasing n. Thus, larger n require a larger value of r s to affect the estimate of r o in Eq. (2.35). This suggests that, in the regime r s >r s 0 , protein steric constraints bias MPPN self-assembly towards MPPNn-states with largern, which is indeed borne out by the results in Fig. 2.4. Note that, since we assumed E u = 0 in Eq. (2.36), Eq. (2.36) is not expected to apply to the large-jUj regime in Fig. 2.2. Similarly as r s 0 in Fig. 2.5, r s h in Fig. 2.6 exhibits local maxima at highly symmetric MPPN n-states. As for Eq. (2.34), this can be understood based on Eq. (2.36) by noting from Eq. (2.30) that ¯ b monotonically increases with p(n) for 10n 80, with p(n) showing local maxima at n= 12, 20, and 24 [69, 70]. Thus, not only forr s <r s 0 but also forr s >r s 0 highly symmetric MPPNn-states are less stringently restricted by protein steric constraints than less symmetric MPPNn-states with similarn, effectively biasing MPPN self-assembly towards highly symmetric MPPNn-states forr s >r s 0 as well as forr s <r s 0 in Fig. 2.4. Figure 2.7 shows the MPPN radius R (upper panel) and the MPPN energy per protein E avg (lower panel) as a function of r s along the linea = 0:55 rad in Fig. 2.4(b) for selected, dominant MPPN n-states. We find that, consistent with r s h in Eq. (2.36), smaller MPPN n-states start to increase in size at smaller r s . However, Fig. 2.7 also shows that transitions between different dominant MPPN n-states do not coincide exactly with the onset of r s -induced increases in MPPN size. This illustrates that, as anticipated above, Eq. (2.36) only provides an approximate measure 27 Figure 2.8: MPPN self-assembly diagram as in Fig. 2.4(b) forh s = 5:0 nm but withjUj= 0:2 nm. The blue dashed vertical line corresponds toa = 0:55 rad, for which the dominant MPPNn-states aren= 20, 18, 17, and 12 (from bottom to top). We use the same notation as in Fig. 2.2. of transitions between dominant MPPN n-states. The lower panel of Fig. 2.7 shows that, as r s is increased, MPPNs with smaller n can be dominant even if they have a larger energy per protein than competing MPPN n-states with larger n. As in Fig. 2.3, this effect results from entropic contributions to the free energy of the system. Finally, we consider the effect of a finiteU on the MPPN self-assembly diagrams in Fig. 2.4. In particular, we setjUj= 0:2 nm with, as in Fig. 2.4(b), h s = 5:0 nm (see Fig. 2.8). Comparing Fig. 2.4(b) and Fig. 2.8 we find that, in the smallr s -regime, a finiteU can strongly bias MPPN self- assembly towards highly symmetric MPPN n-states [4]. Furthermore, Fig. 2.8 shows that large r s can prevent the dominance of highly symmetric MPPN n-states at finitejUj. This is in marked contrast to the results obtained in Fig. 2.4 atU = 0. But we also note from Fig. 2.2 that, for large enough a, the interplay of large r s and largejUj can strongly amplify the dominance of highly 28 Figure 2.9: MPPN radii R (upper panel) and corresponding MPPN energy per protein E avg = E(n)=n (lower panel) versus in-plane protein radius outside the membrane, r s . As in Fig. 2.8, we set h s = 5:0 nm andjUj= 0:2 nm. We use here a = 0:55 rad, which corresponds to the dashed vertical line in Fig. 2.8, and consider the first four dominant MPPN n-states (starting from small r s ) along this line in Fig. 2.8. The dashed vertical lines indicate transitions fromn= 20 ton= 18, from n= 18 to n= 17, and from n= 17 to n= 12 (left to right) in the dominant MPPN n-state in the MPPN self-assembly diagram in Fig. 2.8. symmetric MPPN n-states, such as n= 12. The ranges of MPPN radii associated with dominant MPPNn-states in the MPPN self-assembly diagram in Fig. 2.8 are given by 7:0 nm/R/ 8:4 nm (n= 12), 9:8 nm/R/ 12 nm (n= 24), 11 nm/R/ 13 nm (n= 32), and 14 nm/R/ 15 nm (n= 48). Similarly as in Fig. 2.7, we plot in Fig. 2.9 the MPPN radius R (upper panel) and the MPPN energy per protein E avg (lower panel) as a function of r s along the line a = 0:55 rad in Fig. 2.8 for selected, dominant MPPN n-states. The upper panel in Fig. 2.9 shows that, contrary to the results obtained atU = 0 in Fig. 2.7, MPPNn-states with smallern start to expand in size at larger r s than MPPN n-states with larger n along the line a = 0:55 rad in Fig. 2.8. As expected from Fig. 2.7 the lower panel of Fig. 2.9 shows that, as one increasesr s , MPPNs with largern can cease 29 to be dominant even if they have a smaller energy per protein than competing MPPNn-states with smallern. Again, this effect results from entropic contributions to the free energy of the system. 30 Chapter 3 Self-assembly of polyhedral bilayer vesicles from Piezo ion channels In this chapter, we adapt the formalism described in Refs. [3, 4] as well as in Chapter 2 to explore the symmetry and size of MPPNs formed from Piezo ion channels [22]. We first de- velop a methodology for predicting the symmetry and size of MPPNs composed of proteins that induce arbitrarily large membrane curvature deformations (see Secs. 3.1). Next, we validate this methodology for MPPNs formed from MscS (see Sec. 3.2). Finally, we calculate the self-assembly diagram for MPPNs formed from Piezo (see Sec. 3.3). 3.1 MPPN self-assembly diagrams for arbitrarily large mem- brane curvatures The dominant symmetry and size of MPPNs observed in experiments on MPPNs formed from MscS are successfully predicted by only considering contributions toE due to membrane bending deformations,E b , and topological defects in protein packing in MPPNs,E d , such thatE=E b +E d [3, 4, 5, 6]. For simplicity, we therefore focus on these two contributions to the MPPN energy, and contrast MPPN self-assembly diagrams obtained using two distinct mathematical representations 31 Figure 3.1: Schematic of MPPNs formed from Piezo ion channels. The thick gray curve shows the Piezo dome with radius of curvatureR D and cap anglea. The blue curve shows the membrane footprint of the Piezo dome with arclengths s= 0 and s= s b at the inner and outer membrane footprint boundaries, respectively. At s=s b , the Piezo membrane footprint is assumed to connect smoothly to the membrane footprints associated with the neighboring Piezo proteins on the MPPN surface, with contact angle b. We denote the inner and outer membrane footprint boundaries along the r-axis by r 0 and r b , respectively. The MPPN radius is given by R = r b =sinb. For each s b , Piezo’s membrane footprint is completely determined by a given set of values of a, R D , r 0 =R D sina, andb, which are indicated in gray. For simplicity, we only show here one of the n proteins on the MPPN surface. of the MPPN energy [73]: On the one hand, we consider the Monge parameterization of the MPPN energy described in Chapter 2. This paramaterization is only expected to give accurate results in the limit of small membrane gradients. We therefore also consider, on the other hand, the arclength parameterization of surfaces, which allows for arbitrarly large membrane deformations. 32 3.1.1 Nonlinear MPPN shape equations We describe here the Piezo dome as a spherical cap with area S cap 390 nm 2 and radius of curvatureR D [7, 8, 9, 10] (see Fig. 3.1). Assuming that MPPNs are under negligible membrane ten- sion [4], the shape and energy of Piezo’s membrane footprint can be estimated [39] by minimizing the bending energy of the lipid membrane [60, 61, 62], G b = K b 2 Z dA(c 1 +c 2 ) 2 ; (3.1) where K b is the lipid bilayer bending rigidity, c 1 and c 2 are the local principal curvatures of the mid-bilayer surface, and the integral runs over Piezo’s membrane footprint. We use the same K b 14k B T for diC14:0 lipids as in Chapter 2 [5, 6, 63]. Membrane-mediated interactions between Piezo proteins are expected to favor approximately hexagonal protein arrangements [53, 54, 55, 57, 58] (see also Sec. 2.1.2). Following Chapter 2, the bending energy of MPPNs can thus be estimated from a mean-field approach [3, 4, 53, 54, 55, 74] in which the boundary of the hexagonal unit cell of the protein lattice is approximated by a circle. In particular, we divide the surface of MPPNs containing n Piezo proteins into n identical, circular membrane patches, each with a Piezo dome at its center (Fig. 3.1). Using the arclength parameterization of surfaces, Eq. (3.1) can be rewritten as [73, 75, 76, 77] G b = Z s b 0 ds pK b r ˙ y+ siny r 2 +l r (s)(˙ r cosy)+l h (s)( ˙ h siny) (3.2) for each membrane patch, wheres is the arclength along the profile of Piezo’s membrane footprint, s = 0 at the inner boundary of Piezo’s membrane footprint (the boundary of the Piezo dome) and s=s b at the outer boundary of Piezo’s membrane footprint away from the Piezo dome, h(s) denotes the height of Piezo’s membrane footprint along its symmetry axis h, r(s) denotes the radial coordinate of Piezo’s membrane footprint perpendicular to theh-axis,y(s) denotes the angle 33 between the tangent to the profile of Piezo’s membrane footprint and the r-axis, and the Lagrange parameter functions l r (s) and l h (s) enforce the geometric constraints ˙ r = cosy and ˙ h= siny inherent in the arclength parameterization of surfaces (Fig. 3.1). The boundary conditions on the Piezo membrane footprint at the Piezo dome boundary follow from the assumption that the membrane surface is smooth ats= 0 [7, 39] (Fig. 3.1): r(0) r 0 =R D sina; (3.3) h(0) = R D cosa; (3.4) y(0) = a; (3.5) with the membrane-Piezo dome contact angle [78] a = cos 1 1 S cap 2pR 2 D : (3.6) We use the same value of boundary angle b as in Sec. 2.1.2, which in this chapter denotes the contact angle at the outer boundary of Piezo’s membrane footprint. Following Chapter 2, we note that the solid angle associated with each unit cell on the MPPN surfaceW is related tob through W= 2p(1 cosb). Since each unit cell on the MPPN surface contains one protein, we also have W= 4p=n, which implies the same b = cos 1 (1 2 n ). Thus, the resulting boundary condition is given by [3, 4, 54, 74] y(s b )= cos 1 1 2 n (3.7) at s=s b (Fig. 3.1). Equations (3.3)–(3.7) encapsulate the effect of a particular Piezo dome shape and protein number per MPPN onE b . With the (arbitrary) origin of ther-h coordinate system fixed via Eqs. (3.3) and (3.4) (Fig. 3.1), we assume that the values of r(s b )r b and h(s b ) can be freely 34 varied when finding the extremal functions of Eq. (3.2) [64, 79], resulting in the natural boundary conditions p r (s b ) = l r (s b )= 0; (3.8) p h (s b ) = l h (s b )= 0 (3.9) at s=s b , where p r (s)¶L=¶ ˙ r and p h (s)¶L=¶ ˙ h are the generalized momenta associated with the generalized displacements r(s) and h(s), in which the Lagrangian function L is given by the integrand in Eq. (3.2). To determine the stationary shapes of MPPNs we solve the Hamilton equations associated with the membrane bending energy in Eq. (3.2) [39, 80, 81, 82, 83, 84, 85, 86] subject to the boundary conditions on Piezo’s membrane footprint in Eqs. (3.3)–(3.9). Compared to the corresponding Euler-Lagrange equations associated with Eq. (3.2) [64, 75, 76, 77, 79, 80, 87, 88], these Hamilton equations are of first rather than second order in derivatives. The Hamilton equations for Eq. (3.2) are given by ˙ y = p y 2r siny r ; (3.10) ˙ r = cosy; (3.11) ˙ h = siny; (3.12) ˙ p y = p y r p h cosy+p r siny; (3.13) ˙ p r = p y r p y 4r siny r ; (3.14) ˙ p h = 0; (3.15) where p y (s)¶L=¶ ˙ y is the generalized momentum associated with the generalized displacement y(s). The boundary condition in Eq. (3.9) and the Hamilton equation in Eq. (3.15) imply that p h (s)= 0 for 0ss b . The solution of the remaining five Hamilton equations in Eqs. (3.10)– (3.14) is specified by the five boundary conditions in Eqs. (3.3)–(3.8). A numerical difficulty arises 35 here in that some of these boundary conditions are specified at s= 0, while others are specified at s=s b . We thus solve Eqs. (3.10)–(3.14) using a shooting method [39, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90], for which we introduce the boundary conditions p y (0) = p y;0 ; (3.16) p r (0) = p r;0 ; (3.17) where p y;0 and p r;0 must be adjusted so as to satisfy the boundary conditions in Eqs. (3.7) and (3.8). The values of p y;0 and p r;0 can be conveniently determined through the FindRoot command in Mathematica [91]. We obtain E b by substituting the solutions of Eqs. (3.10)–(3.14) into Eq. (3.2) and (numerically) integrating with respect tos. In the above calculation ofE b , the size of Piezo’s membrane footprint enters through the value of s b . We note that the MPPN radius R is related to the in-plane membrane patch radius r b via R=r b =sinb (Fig. 3.1). In general, different values ofs b yield different values ofr b and, hence,R for the stationary membrane footprints. We therefore minimize, at each n, E b with respect to the MPPN size by adjusting s b so that E b is minimal, and then determine the values of r b and, hence, R associated with this value of s b . For the scenarios considered here, we considered the range 0:01 nms b 20 nm. Noting that E b may vary rapidly at 0:01 nms b 2 nm, we set a finer resolution ins b ofDs b = 0:05 nm within this range andDs b = 0:2 nm for the rest. We interpolated E b between these values of s b using third-order splines. For each Piezo dome shape and protein number per MPPN considered, the value of s b minimizing E b thus specifies the MPPN size with minimal bending energy. 3.1.2 Small-gradient approximation for MPPNs As described in Chapter 2,h is regarded as a single-valued function ofr,h(r) [73] in the Monge parameterization of Eq. (3.1). In the small-gradient limit of the Monge parameterization,jÑhj 1, the stationary membrane shapes implied by Eq. (3.1) can be solved for analytically, yielding an 36 exact expression for the membrane bending energy in MPPNs, which is stated in Sec. 2.1.2. Noting that the variablesr i andr o in Chapter 2 are equivalent tor 0 andr b =Rsinb in Fig. 3.1, we have ¯ E b (n;R)= 2npK b (r 0 tanaRsinb tanb) 2 R 2 sin 2 br 2 0 : (3.18) For the purposes of this chapter, the analytic solution in Eq. (3.18) presents a useful counterpoint to the fully nonlinear, numerical solutions obtained from Eq. (3.2). 3.1.3 Topological defects in protein packing Following the same reasoning as in Sec. 2.1.4 we have the topological defect energy E d (n;R)= K s 2 A p max p(n) p max 2 ; (3.19) where K s is the stretching modulus of the elastic sheet, A is the MPPN surface area, and p max = p=2 p 3 denotes the optimal packing fraction associated with a uniform hexagonal protein arrange- ment. Similar as in Sec. 2.1.4, the stretching modulus is given by K s = p 3 24n ¶ 2 E b ¶r 2 b r b =r min b ; (3.20) wherer min b is the value ofr b that yields, for a givenn, the minimalE b . Most straightforwardly, the MPPN surface areaA in Eq. (3.19) can be approximated viaA=A S [3, 4, 74], where A S = 4pR 2 is the surface area associated with a spherical MPPN shape as in Sec. 2.1.4. However, the approximation A=A S breaks down for large enough deviations from a spherical shape, which is expected to be the case for MPPNs formed from Piezo. As an alternative to A= A S , we therefore consider here the choice A= A D with the area of the deformed MPPN surface,A D , being given by A D =nS cap + 2pn Z s b 0 rds; (3.21) 37 where, as noted above, S cap = 390 nm 2 for the Piezo dome. In the case of MPPNs formed from MscS, we approximate the MPPN surface area associated with MscS proteins by modeling MscS as a spherical cap with areaS cap = 2pR 2 1 cos[sin 1 (r 0 =R)] for the MPPN sphere radiusR and the MscS base radiusr 0 3:2 nm [18]. Unless stated otherwise, we use hereA=A D in Eq. (3.19). 3.1.4 Constructing MPPN self-assembly diagrams As in Chapter 2, we construct MPPN self-assembly diagrams from the fraction of MPPNs con- taining n proteins each,f(n) in Eq. (2.6). To this end, we obtain the minimal MPPN energy, E min in Eq. (2.2), at each n by minimizing the sum of E b , calculated from the stationary G b implied by the arclength or Monge parameterization of Eq. (3.1), and the corresponding E d in Eq. (3.19) with respect to the MPPN radiusR. For MPPNs formed from Piezo ion channels, we perform this minimization subject to steric constraints due to the finite size of lipids and proteins only, which corresponds to the second term in Eq. (2.33). The parameters ¯ b and r l are as in Chapter 2, and r i is replaced by r 0 in Eq. (3.3). Once E min (n) is calculated, the f(n) are conveniently obtained via Eq. (2.6) for arbitrary protein number fractions c. But, in the arclength parameterization of Eq. (3.1), changes in R D or a necessitate repeated numerical solution of the Hamilton equations in Eqs. (3.10)–(3.14), which can be computationally intensive. For increased computational effi- ciency, we determine the dominant f(n) for a discrete (relatively sparse) set of values of R D or a, and interpolate the dominantf(n) between these values, using third-order splines, to construct MPPN self-assembly diagrams (see below). 3.2 MPPN self-assembly from MscS Since MscS proteins only weakly curve the membrane [12, 18, 40], the Monge parameteriza- tion of Eq. (3.1) is expected to yield a good approximation of the self-assembly diagram of MPPNs formed from MscS. Figures 3.2(a) and 3.2(b) show the self-assembly diagrams for MPPNs formed from MscS obtained from the arclength and Monge parameterizations of Eq. (3.1), respectively. 38 Figure 3.2: MPPN self-assembly diagrams for MPPNs formed from MscS obtained using (a) the arclength parameterization of Eq. (3.1) and (b) the Monge parameterization of Eq. (3.1). The horizontal axes show the bilayer-protein contact angle, a, and the vertical axes show the protein number fraction in solution, c. The dominant n-states of MPPNs are indicated by integers. The white dashed curves show transitions in the dominant MPPN n-states, with the colors indicating the maximumf(n) among all MPPN n-states considered. We use the same color bar in panels (a) and (b). The dashed horizontal lines indicate the parameter values c 7:8 10 8 anda 0:46– 0:54 rad corresponding to experiments on MPPNs formed from MscS [5, 6], in which the snub cube with n= 24 MscS proteins was found to provide the dominant MPPN symmetry (models of the snub cube belown= 24). Both parameterizations of Eq. (3.1) predict, with no free parameters, that the snub cube (n= 24) provides the dominant MPPN symmetry for thea-rangea 0:46–0:54 rad associated with MscS [3, 12, 18, 40] and the protein number fraction c 7:8 10 8 used in experiments on MPPNs formed from MscS [5, 6] (dashed horizontal lines in Fig. 3.2). Furthermore, the arclength and 39 Figure 3.3: Fractional abundance of MPPN n-states, f(n), versus bilayer-protein contact angle, a, for selected (dominant) MPPN n-states in Fig. 3.2(a) (indicated by integers) obtained with the arclength parameterization of Eq. (3.1) and the protein number fraction c 7:8 10 8 used for the MPPN self-assembly experiments in Refs. [5, 6]. As for Fig. 3.2(a), we calculated all curves through interpolation of numerical results at a resolutionDa = 0:01 rad [dots in panel (a)], using third-order splines. Panel (a) compares these interpolations with the corresponding results obtained at a higher resolutionDa= 0:002 rad (crosses). Panel (b) compares the curves in panel (a) obtained usingA=A D in Eq. (3.21) (solid curves) with the corresponding results obtained usingA=A S . Monge parameterizations of Eq. (3.1) predict that the dominant MPPNs with snub cube symme- try have a characteristic bilayer midplane radius 9:8 nm/R/ 10 nm and 9:8 nm/R/ 11 nm, respectively. These predictions of Eq. (2.6) are in quantitative agreement with experiments on MPPNs formed from MscS [5, 6]. A notable discrepancy between the MPPN self-assembly dia- grams predicted by the arclength and Monge parameterizations of Eq. (3.1) is that, for the Monge parameterization, a given dominant MPPN n-state tends to appear at slightly smaller values ofa. For thea-range relevant for MscS this means that, for instance, subdominant MPPNs tend to have 40 largern in Fig. 3.2(a) than in Fig. 3.2(b). Since the Monge parameterization becomes less accurate asa is increased, these shifts tend to become more pronounced asa is increased. As described above, the self-assembly diagram in Fig. 3.2(a) was obtained by interpolating the dominant f(n) between a discrete set of values of a. Figure 3.3(a) compares the interpolated f at the resolutionDa = 0:01 rad used for Fig. 3.2(a) for selected, dominant MPPN n-states at the protein number fractionc 7:810 8 [5, 6] with the corresponding results of calculations done at a finer resolutionDa = 0:002 rad. Figure 3.3(a) indicates that the interpolation scheme employed here provides accurate estimates off(n) for continuousa. Finally, Fig. 3.3(b) compares the results forf(n) in Fig. 3.2(a), obtained withA=A D in Eq. (3.21), with the corresponding results obtained with A = A S . As expected, A = A S provides a good approximation of the MPPN area for the weak membrane curvatures induced by MscS. But this approach for estimating A is, in general, not expected to give accurate results for membrane proteins such as Piezo that strongly curve the membrane. In the following we therefore focus onA=A D in Eq. (3.21). 3.3 MPPN self-assembly from Piezo Figure 3.4 shows the self-assembly diagram of MPPNs formed from Piezo ion channels as a function of the radius of curvature of the Piezo dome and the protein number fraction in solution. We obtained the results in Fig. 3.4 using the arclength parameterization of Eq. (3.1) with A=A D in Eq. (3.21). The MPPN self-assembly diagram in Fig. 3.4 includes highly curved MPPN states with, for instance, a contact anglea 1:3 rad atR D 9:0 nm. The vertical dashed line in Fig. 3.4 indicates the Piezo dome radius of curvature R D 10:2 nm observed for a closed state of Piezo [7, 8, 9, 10], while the horizontal dashed line indicates the protein number fractionc 7:8 10 8 used in experiments on MPPNs formed from MscS [5, 6]. Figure 3.5 shows, for this protein number fraction, the f(n) associated with dominant MPPN n-states in Fig. 3.4 as a function R D . The results in Figs. 3.4 and 3.5 were calculated using a resolution DR D = 0:2 nm in R D , with a 41 Figure 3.4: MPPN self-assembly diagram for MPPNs formed from Piezo ion channels, obtained from the arclength parameterization of Eq. (3.1), as a function of the radius of curvature of the Piezo dome,R D , and the protein number fraction in solution,c. Colors indicate the maximumf(n) among all MPPN n-states considered. The color bar employed here is identical to the color bar employed in Fig. 3.2. As in Fig. 3.2, the dominantn-states of MPPNs are indicated by integers, and the white dashed curves show transitions in the dominant MPPN n-states. The horizontal dashed line indicates the protein number fraction c 7:8 10 8 used in experiments on MPPNs formed from MscS [5, 6], while the vertical dashed line shows the Piezo dome radius of curvature R D 10:2 nm observed for a closed state of Piezo [7, 8, 9, 10]. The polyhedra models show dominant MPPN symmetries predicted by the MPPN self-assembly diagram, withn= 6 (octahedron),n= 12 (icosahedron), and n= 24 (snub cube), respectively. Gray shading indicates regions of the MPPN self-assembly diagram dominated by MPPN n-states with n= 80, which may be a spurious result of our constraint 3n 80. third-order spline interpolation of f(n) between these data points. Similar results are obtained with a finer resolution inR D . We find in Fig. 3.4 that the dominant MPPNn-state depends only weakly on the protein number fraction in solution, but strongly on the Piezo dome radius of curvature. As R D is increased, Piezo’s membrane footprint becomes less curved [39], yielding larger and less curved MPPNs that incorporate more Piezo proteins. Most notably, the MPPN self-assembly diagram in Fig. 3.4 is dominated by highly symmetric MPPN n-states with octahedral (O h ) (n= 6), icosahedral (I h ) (n= 12), and snub cube (O) (n= 12) symmetry. Forc 7:810 8 , MPPN octahedra are dominant in Fig. 3.4 for 8:0 nm/R D / 13 nm, MPPN icosahedra are dominant for 14 nm/R D / 18 nm, and 42 Figure 3.5: Fractional abundance of MPPN n-states obtained from the arclength parameterization of Eq. (3.1),f(n), versus radius of curvature of the Piezo dome,R D , for selected (dominant) MPPN n-states in Fig. 3.4 at the protein number fraction c 7:8 10 8 used in experiments on MPPNs formed from MscS [5, 6]. As in Fig. 3.4, the vertical dashed line shows the Piezo dome radius of curvatureR D 10:2 nm observed for a closed state of Piezo [7, 8, 9, 10]. All curves were obtained through interpolation of numerical results at a resolution DR= 0:2 nm (dots), using third-order splines. MPPN snub cubes are dominant for 22 nm/R D / 25 nm. The dominant MPPN states in Fig. 3.4 withn= 6, 12, and 24 have the characteristic MPPN radii 14 nm/R/ 15 nm, 21 nm/R/ 26 nm, and 30 nm/R/ 34 nm with the top of the Piezo dome being located approximately(9:2–1:1) nm, (4:5–0:52) nm, and(1:5–0:61) nm above the spherical surface defined byR (Fig. 3.1), respectively. Figures 3.4 and 3.5 show that, in addition to n= 6, 12, and 24, MPPNs with n= 20, 32, and 48 can also be abundant at c 7:8 10 8 . These MPPN n-states have D 3h (n= 20), face-capped icosahedron (D 3 ) (n= 32), and snub cube (O) (n= 48) symmetry [69, 70] with 20 nm/ R D / 22 nm, 27 nm/ R D / 29 nm, and 33 nm/ R D / 36 nm and the top of the Piezo dome being located approximately(1:35–0:73) nm,(0:69–0:53) nm, and(0:58–0:37) nm above the spherical surface defined by R, respectively. Overall, Figs. 3.4 and 3.5 thus predict that self-assembly of MPPNs from Piezo can yield highly symmetric and highly curved MPPN states, with the radius of curvature of the Piezo dome providing the critical control parameter for MPPN symmetry and size. 43 Chapter 4 Symmetry of membrane protein polyhedra with heterogeneous protein size In this chapter we employ simulated annealing Monte Carlo simulations to describe the sym- metry of MPPNs with heterogeneous protein size. We first provide a detailed discussion of the modeling approach used here, and how the symmetry of MPPNs with heterogeneous protein com- position can be quantified (see Sec. 4.1). We then apply this modeling methodology to MPPNs formed from 24 MscS proteins [5, 6], allowing for MscS to be in open as well as closed confor- mational states (see Sec. 4.2). On this basis, we predict the symmetry of partially gated MPPNs composed of MscS proteins. 4.1 Modeling MPPN symmetry Proceeding in analogy to previous work on viral capsid symmetry [42, 43], MPPN symmetry can be described through a simple particle-based model [3] in which lipids and MscS proteins are represented by differently-sized disks on the surface of a sphere. In this section, we first review this previous model of MPPN symmetry [3], which we term the lipid-protein (LP) model (see Sec. 4.1.1). In Sec. 4.1.2 we formulate the composite particle (CP) model of MPPN symmetry, 44 which is the primary focus of this chapter and which provides a simplified, coarse-grained rep- resentation of the LP model. In the CP model, each membrane protein and its surrounding lipid environment are represented by a single disk on the surface of a sphere. We use simulated an- nealing MC simulations [92, 93] to find the minimum-energy states implied by the LP and CP models. We summarize the pertinent computational methods in Sec. 4.1.3. Finally, in Sec. 4.1.4 we describe the mathematical approaches used here to quantify MPPN symmetry. Throughout this chapter, we denote the total number of MscS proteins per MPPN by N and the number of open-state MscS proteins per MPPN byn o , such that the number of closed-state MscS proteins per MPPN is given byNn o . 4.1.1 Lipid-protein (LP) model As described previously [3], MPPN symmetry can be captured by a minimal molecular model— the LP model—in which proteins and lipids are represented by distinct particles moving on the surface of a sphere. We take this spherical surface to correspond to the outer membrane leaflet of MPPNs [see Figs. 4.1(a) and 4.1(b)] and denote its radius by R. In the LP model, lipids interact with other lipids as well as proteins through Lennard-Jones potentials, V (LP) i;j (r)=e k " ¯ r i;j r 12 2 ¯ r i;j r 6 # ; (4.1) where we use the notation (i; j)=(l;l), (l;c), and (l;o) to denote interactions between lipids, lipids and closed-state MscS, and lipids and open-state MscS, respectively, the index k = 1, 2 denotes lipid-lipid and lipid-protein interactions, r is the Euclidean particle separation in three- dimensional (3D) space, and the ¯ r i;j denote the energetically most favorable particle separations implied by the Lennard-Jones potentials in Eq. (4.1). MscS proteins are not expected to aggregate in the absence of lipids, and are therefore assumed here [3] to interact with other MscS proteins only through hardcore steric constraints so that all MscS-MscS separations r > r m;n , where the 45 Figure 4.1: Minimum-energy MPPN configurations obtained from simulated annealing MC simu- lations of the LP model in Sec. 4.1.1 with (a) 24 closed-state MscS proteins and (b) one MscS in the open state and 23 closed-state MscS. The small and large disks represent the lipids in a diC14:0 lipid bilayer [11] and MscS proteins [12, 13], respectively, with closed-state MscS corresponding to the large gray disks and open-state MscS to the large yellow disk. The closed-state MscS, open- state MscS, and lipid disk sizes are given by r c , r o , and r l , respectively (see Sec. 4.1.1). (c,d) Minimum-energy MPPN configurations obtained as in panels (a) and (b), respectively, but using the coarse-grained CP model in Sec. 4.1.2. Particles corresponding to closed-state and open-state MscS are illustrated by blue and red disks, respectively. For ease of visualization, the radii of these disks were decreased by some fixed scale factor relative to the disk radii implied by r 0 o and r 0 c (see Sec. 4.1.2). The green lines in panels (a–d) are obtained by connecting the centers of neighboring proteins. (e) 3D and (f) net representations of a snub cube. indices (m;n)=(c;c), (c;o), and (o;o) denote pairs of closed-state, closed-state and open-state, and open-state MscS, respectively. The parameterse k in Eq. (4.1) set the energy scale of lipid-lipid and lipid-protein interactions, and can be viewed as the energy penalty for exposing lipids or membrane proteins to an aqueous environment. Experiments and previous calculations [17, 94, 95] suggest [3]e 1 10k B T ande 2 20k B T for the diC14:0 lipids and MscS proteins used in experiments on MPPNs [5, 6], wherek B is 46 Boltzmann’s constant andT is the room temperature. We employ these values ofe k throughout this chapter. It was found previously [3, 96] that the minimum-energy MPPN configurations implied by the LP model are robust with respect to the values of e k . Approximating the shapes of lipids and proteins by disks that are tangent to the spherical MPPN surface, the values of ¯ r i;j in Eq. (4.1) can be estimated from the lipid radiusr l , the closed-state MscS radiusr c , and the open-state MscS radius r o . Assuming that, in their energetically most favorable configuration, lipid and protein disks touch each other but do not overlap, we have ¯ r i;j = 2Rsin 1 2 arctan r i R + arctan r j R : (4.2) Experiments on the diC14:0 lipids used for MPPNs formed from MscS [5, 6] give r l 0:45 nm [11] while structural studies of MscS yield r c 4:0 nm and r o 4:5 nm [12, 13] for the outer membrane leaflet of MPPNs, respectively. In experiments on MPPNs formed from 24 MscS proteins, each MPPN was found to be com- posed of approximately 1700 lipids [6], which corresponds to approximately 1200 lipids in the outer membrane leaflet of MPPNs [96, 97]. To be consistent with these previous experiments on MPPNs [5, 6], we therefore use here a fixed lipid-protein ratio 1200 : 24. Note that lipids are much more abundant in MPPNs than proteins. As a result, while the protein configuration in MPPNs is of primary interest, MC simulations of the LP model devote considerable computational resources to updating the lipid configuration, which can make it computationally challenging to escape from local energy minima in the particle configuration. This issue becomes particularly significant for MPPNs with heterogeneous protein composition. In Sec. 4.1.2 we thus develop a simplified model of MPPN symmetry. 4.1.2 Composite particle (CP) model The LP model successfully predicts the dominant symmetry of MPPNs composed solely of closed-state MscS proteins [3]. However, as pointed out in Sec. 4.1.1, the explicit representation 47 of lipids in the LP model leads to difficulties when simulating MPPNs with heterogeneous protein size. Indeed, most of the degrees of freedom in the LP model correspond to lipid positions, which are not of primary interest. In both the LP model [3] and experiments [6], MscS proteins tend to be surrounded by an annulus of lipids. This motivates us to formulate the CP model, which provides a simplified, coarse-grained description of MPPN symmetry. In the CP model, we take each particle on the MPPN surface to be composed of one protein surrounded by an annulus of lipids [see Figs. 4.1(c) and 4.1(d)]. Similarly as in Sec. 4.1.1, we let these particles interact with each other via Lennard-Jones potentials, V (CP) i;j (r)=e 2 4 ¯ r 0 i;j r ! 12 2 ¯ r 0 i;j r ! 6 3 5 ; (4.3) where the parameter e sets the energy scale of particle interactions and, in analogy to Sec. 4.1.1, we use the notation (i; j)=(c;c), (c;o), and (o;o) to denote interactions between particles cor- responding to two closed-state MscS, one closed- and one open-state MscS, and two open-state MscS, respectively. We take here each MscS protein to be surrounded by one layer of lipids. The values of ¯ r 0 i;j in Eq. (4.3) are then fixed via an expression analogous to Eq. (4.2), but using the effective particle radii r 0 c =r c + 2r l 4:9 nm and r 0 o =r o + 2r l 5:4 nm instead of r l , r c , or r o in Eq. (4.2). Based on Sec. 4.1.1 we use here e = 20 k B T for the energy scale in Eq. (4.3), but the minimum-energy MPPN configurations implied by the CP model are independent of the value of e. From a practical standpoint, the central advantage of the CP model over the LP model is that the CP model focuses on the protein configuration, which is what defines the MPPN symmetry, and does not allow for any additional degrees of freedom. 4.1.3 Simulated annealing MC simulations For a given set of values of(N;n o ), we employ simulated annealing MC simulations [92, 98] with linear cooling to numerically determine the minimum-energy configurations associated with the LP and CP models described in Secs. 4.1.1 and 4.1.2 [3]. For the LP model with n o = 0 we 48 generate the initial conditions for our MC simulations from a random, uniform distribution of lipids and proteins on the MPPN surface with no overlap of proteins. For the CP model with n o = 0 we employ a uniform, random distribution of CPs with no constraints on the relative particle positions. In our simulated annealing MC simulations [92, 98] of the LP and CP models we randomly pick, in each MC step, one of the particles on the MPPN surface as the target particle for this MC step. In particular, in the LP model we first randomly decide whether to update the position of a lipid or protein particle (with probabilities 0:8 and 0:2, respectively) and then pick with equal probability a lipid or protein particle among all lipid or protein particles on the MPPN surface. In the CP model we choose the target particle with equal probability among all particles. Next, we generate a unit vector with its initial point at the MPPN center and a target point that is chosen randomly from a uniform angular distribution. We update the MPPN configuration by rotating the target particle about the axis defined by this unit vector through an angular step size dq = 0:005 rad. The MC move is accepted with probability p= min 1;e DG=k B T sys ; (4.4) whereT sys is the system temperature andDG is the difference in MPPN energy between the MPPN configurations after and before the attempted MC move. The maximum strength of the interaction potentials in Eqs. (4.1) and (4.3) is set bye 2 ande. As discussed in Secs. 4.1.1 and 4.1.2, we use here e 2 =e = 20 k B T . In our simulated annealing MC simulations of the LP and CP models we therefore evolve, for the first 10 5 MC steps, the MPPN configurations using a system temperature that is increased twenty-fold compared to the room temperature,T sys = 20T , so as to allow thermal fluctuations to compete with the Lennard-Jones interactions considered here. To access minimum- energy MPPN configurations we then linearly decrease T sys from T sys = 20T to T sys = 0 over 10 5 additional MC steps. Finally, we let the system evolve for 10 4 further MC steps with T sys = 0, accepting only MC steps that result in a lower-energy MPPN configuration. Figure 4.2 illustrates the MC cooling procedure outlined above for the CP model. In particu- lar, Fig. 4.2 shows the total MPPN energy as a function of MC steps for five representative MC 49 Figure 4.2: Illustrative MC trajectories of the CP model for five independent MC simulations showing the total MPPN energy calculated from Eq. (4.3) versus number of MC steps. We set (N;n o )=(24;2), start from random initial conditions, and use our standard MC simulation pro- cedure for the CP model including swapping moves (see Sec. 4.1.3). The inset shows the MPPN energy associated with the random initial conditions used for the five independent MC simulations. In the main panel, the curves leading to the first data point shown overlap for the five independent MC trajectories. simulations of the CP model at (N;n o )=(24;2). As illustrated in Fig. 4.2, our MC trajectories show a rapid transition from the random initial conditions to energetically more favorable protein configurations. Over the first 10 5 MC steps, we find large fluctuations in the MPPN energy, with substantial overlap in the energy fluctuations for the five representative MC trajectories shown in Fig. 4.2. This illustrates that T sys = 20T is large enough for the system to explore different en- ergetically favorable protein configurations irrespective of the initial conditions used. Once the linear cooling process is started (after 10 5 MC steps), the MC trajectories converge to the same minimum-energy MPPN configuration with, within numerical accuracy, the same MPPN energy. This convergence of different MC trajectories illustrates that our cooling process is slow enough to allow distinct MC trajectories to “find” the same energy minimum. 50 For both the LP and CP models, we repeat the above simulated annealing MC procedure for a range of MPPN radii R to find the optimal MPPN radius R minimizing the MPPN energy. The value of R depends on(N;n o ) as well as the model under consideration. For the LP model with N = 24 we find, in agreement with previous work [3, 96], R 12:3 nm for n o = 0 and R 12:4 nm forn o = 1 with a snub cube symmetry of protein centers [see Figs. 4.1(a) and 4.1(b)]. To obtainR in the CP model forN= 24 andn o 1 we carry out simulated annealing MC simulations first with the value ofR=R found forn o 1, and then increaseR to determineR with a resolution of 0:01 nm. For n o = 0 we start our search for R at R= 11 nm. We thus find the optimal MPPN radii 12:06 nm/R / 13:29 nm for 0n o 24. A simple way to rationalize this increase in R with n o is to regard the ratio of MPPN surface area to the area occupied by the disks representing closed-state and open-state MscS proteins as being approximately constant, which implies (Nn o + 1)pr 02 c +(n o 1)pr 02 o 4p[R (n o 1)] 2 (Nn o )pr 02 c +n o pr 02 o 4p[R (n o )] 2 (4.5) for R =R (n o 1) and R =R (n o ). Using the value R (0)= 12:06 nm found in our simulated annealing MC simulations, Eq. (4.5) allows us to recursively estimate R (n o ) for 1 n o 24. Note that, as shown in Fig. 4.3, Eq. (4.5) implies that (R ) 2 , which is proportional to the MPPN surface area, depends approximately linearly onn o . Equation (4.5) yields the optimal MPPN radii 12:11 nm/ R / 13:29 nm for 1n o 24, which is in approximate agreement with the range of R (n o ) obtained through our simulated annealing MC simulations (Fig. 4.3). Proceeding as for N = 24, we find R 7:84 nm and R 8:6 nm for the CP and LP models with (N;n o )= (12;0), and R 14:07 nm and R 15:0 nm for the CP and LP models with (N;n o )=(32;0), respectively (see Sec. 4.2). For both the LP and CP models, we estimated R on the basis of 50 independent simulated annealing MC simulations for each MPPN radius considered, for which we used different random seeds. To determine the lowest-energy MPPN configuration implied by the LP model, we selected from the resulting set of 50 MPPN configurations atR=R the MPPN 51 Figure 4.3: Square of the optimal MPPN radius,(R ) 2 , versus number of open-state MscS proteins, n 0 , forN= 24 obtained from simulated annealing MC simulations of the CP model and the estimate in Eq. (4.5). For the MC simulations we used our standard MC simulation procedure including swapping moves (see Sec. 4.1.3). configuration with the lowest energy. To determine the lowest-energy MPPN configuration implied by the CP model, we carried out 200 further simulated annealing MC simulations at R=R , and selected from the resulting set of 250 MPPN configurations at R= R the MPPN configuration with the lowest energy. For simulations of the LP and CP models with n o = 1 we use as initial conditions the lowest- energy MPPN configurations obtained atn o = 0 and randomly replace one closed-state MscS pro- tein by an open-state MscS protein. When using the CP model to simulate MPPNs with n o > 1 we take MscS proteins to gate sequentially, i.e., we use as the initial conditions for MPPNs with (N;n o ) and n o > 1 the minimum-energy configurations found for (N;n o 1) and randomly re- place a closed-state MscS protein with an open-state MscS protein. Our simulations suggest that similar results for the minimum-energy MPPN configurations are obtained if one does not make this assumption. Here a technical difficulty arises in that the simulated annealing MC procedure described above fails to robustly identify the minimum-energy arrangements of open-state MscS 52 Figure 4.4: Total MPPN energy versus angular separation of open-state MscS in the final MPPN configuration,a =a final (see inset), in the CP model with(N;n o )=(24;2) obtained through sim- ulated annealing MC simulations with no swapping moves (see Sec. 4.1.3). The dashed vertical lines show the values ofa associated with perfect snub cube symmetry. Each cross symbol repre- sents the result of one simulated annealing MC simulation. The insets show enlarged versions of the indicated regions in the plots. in MPPNs with n o > 1. To illustrate this issue we consider (N;n o )=(24;2), in which case the arrangement of open-state MscS can be characterized by the anglea between the vectors pointing from the MPPN center to the particles representing open-state MscS [see inset in Fig. 4.4]. The minimum-energy MPPN configuration corresponds to a snub cube arrangement of protein centers with the two open-state MscS being located across the diagonal of one of the square faces of the snub cube. This configuration corresponds toa 1:12 rad [see Figs. 4.1(e) and 4.4]. Plotting the values of a associated with the final MPPN configurations obtained in our simulated annealing MC simulations, a final , versus the values of a associated with the initial MPPN configurations, a init , we find that our simulated annealing MC simulations are, in general, unable to overcome the energy barriers for interchanging closed-state and open-state MscS and, hence, fail to yield the minimum-energy MPPN configuration [see Figs. 4.4 and 4.5(a)]. 53 Figure 4.5: CP model with (N;n o )=(24;2). The results in panel (a), which correspond to the data in Fig. 4.4, were obtained through simulated annealing MC simulations with no swapping moves (see Sec. 4.1.3). (b) a final versus a init as in panel (a) and using the same initial conditions as in panel (a) but allowing for swapping moves. The dashed vertical and horizontal lines show the values of a associated with perfect snub cube symmetry. The solid lines in panels (a) and (b) indicatea final =a init . Each cross symbol represents the result of one simulated annealing MC simulation. The insets show enlarged versions of the indicated regions in the plots. To address the computational issue described above and illustrated in Figs. 4.4 and 4.5(a), we augment our simulated annealing MC simulations to allow for “swapping moves,” in which the positions of closed-state and open-state MscS are randomly interchanged. In particular, we allow the “newly gated” MscS protein to randomly swap its position with a randomly selected closed- state MscS protein every 2 10 3 MC steps. The swapping move is accepted with a probability of the same form as in Eq. (4.4), but withDG being given by the difference in MPPN energy between the MPPN configurations after and before the attempted swapping move. If this MC move is not accepted, we let the system evolve as if it was accepted and, after 100 further MC steps, again apply Eq. (4.4), whereDG is now calculated with respect to this evolved MPPN configuration. If, after this second attempt, the swapping move is still rejected, we revert the MPPN configuration to its state immediately prior to the attempted swapping move, and attempt the next swapping move after 2 10 3 further MC steps. The motivation behind the “staggered” swapping procedure described above is that, since open- state MscS proteins have a larger size than closed-state MscS proteins, swapping of closed-state 54 and open-state MscS proteins tends to strongly increase the MPPN energy unless the MPPN con- figuration is allowed to relax following the swapping move. Spurious rejection of swapping moves is thus avoided, speeding up our MC procedure. Our choice to evolve the system over 100 MC steps following an attempted swapping move does not have any deeper significance and, indeed, no such relaxation steps are needed if the number of independent MC simulations is large enough. We find that our results concerning energetically optimal MPPN configurations in Sec. 4.2 are robust with respect to the number of MC relaxation steps used. Figure 4.5(b) shows that, upon imple- mentation of the swapping procedure, our simulated annealing MC simulations robustly identify the minimum-energy arrangement of open-state MscS proteins irrespective of the initial conditions used, and only in rare instances fail to produce the energetically preferred value ofa final . Our simulations suggest that if one allows any open-state MscS protein—rather than just the newly opened MscS protein—to engage in swapping moves one obtains similar results for the minimum-energy MPPN configurations as with the sequential gating procedure used here. We find, however, that if any open-state MscS protein is allowed to engage in swapping moves a given simulated annealing MC simulation is less likely to successfully identify the energetically preferred MPPN configuration, making the sequential gating procedure employed here more ef- ficient from a computational perspective. Furthermore, our simulations indicate that if one uses as the “initial” MPPN state (N;n o )=(24;24)—rather than (N;n o )=(24;0)—and sequentially closes open-state MscS proteins—rather than opens closed-state MscS proteins—one obtains sim- ilar minimum-energy MPPN configurations as those described here (see Sec. 4.2). 4.1.4 Quantifying MPPN symmetry The most straightforward approach for quantifying the symmetry of the MPPN configurations obtained through our simulated annealing MC simulations of the LP and CP models is to fit the protein centers to polyhedral vertices. In particular, we proceed as in experiments on MPPNs [6] and previous simulations of the LP model [3] and compare the protein arrangements in MPPNs with the symmetries implied by the 132 convex polyhedra with regular faces [72]: the Platonic 55 (P), Archimedean (A), Catalan (C), and Johnson (J) solids. We denote the Platonic, Archimedean, and Catalan solids using the Conway polyhedron notation [1], and the Johnson solids using the indexing scheme developed in Ref. [2]. For each simulated MPPN configuration we quantify the fits to the aforementioned 132 polyhedral symmetries by calculating the fit error E = N å i=1 (~ v i ~ v 0i ) 2 ; (4.6) where the vectors~ v i point from the MPPN center to the positions of the protein centers on the MPPN surface obtained in our simulated annealing MC simulations, and the vectors ~ v 0i denote the corresponding positions of the closest fitted polyhedron vertices. The latter are obtained [6, 99] by freely moving, rotating, and rescaling the polyhedron models untilE is minimized for each polyhedral symmetry considered. Note that this fitting procedure yields for each MscS particle a closest polyhedron vertex. If the number of polyhedron vertices is smaller thanN, each polyhedron vertex is associated with multiple MscS particles. In contrast, if the number of polyhedron vertices is greater than N, the fitting procedure used here determines the subset of polyhedron vertices yielding the best fit to the MscS positions. We also note that in our simulated annealing MC simulations the preference of the minimum-energy MPPN configuration for one chiral polyhedral symmetry over its mirror-symmetric configuration results from the random numbers and initial conditions used [3], and is therefore not a model prediction. In addition to Eq. (4.6), we characterize MPPN symmetry through bond-orientational order (BOO) parameters. BOO parameters have been employed to quantify local ordering in liquids and glasses [100, 101, 102] and, more recently, have been used to characterize the symmetry of protein shells [43]. BOO parameters are rotational invariants and, for MPPNs, can be constructed from the spherical harmonics of the protein positions on the MPPN surface. In particular, we employ here the BOO parametersQ l [100], Q l = 4p 2l+ 1 l å m=l jQ lm j 2 ! 1=2 ; (4.7) 56 where Q lm = 1 N N å i=1 Y lm (~ v i ); (4.8) in which, as in Eq. (4.6), the vectors~ v i point from the MPPN center to the centers of the proteins on the MPPN surface obtained in our simulated annealing MC simulations, and theY lm (~ v i ) denote the corresponding spherical harmonics. We take herel in Eqs. (4.7) and (4.8) to be even so thatQ l is independent of the direction of a particular bond [100]. In Sec. 4.2.3 we employ the BOO parametersQ l in Eq. (4.7) to quantify how closely the MPPN configurations implied by the CP model resemble a snub cube [Fig. 4.1(e)]. To this end, we first use Eq. (4.7) to calculate Q l for a perfect snub cube for even l starting from l= 0. We denote the values of Q l associated with a perfect snub cube by Q (sc) l . We note that these values of Q (sc) l are independent of the chirality of the snub cube. The relative difference between the Q l associated with a simulated MPPN configuration andQ (sc) l can then be expressed in the form c Q l = 1 Q l Q (sc) l : (4.9) We find that for even l> 0 the first non-zero Q (sc) l occurs at l= 4, and Q (sc) l = 0:0525 and 0:0412 for l= 4 and l= 6. In Sec. 4.2.3 we use c Q 4 and c Q 6 to characterize the symmetry of MPPNs with N= 24. 4.2 Minimum-energy MPPN configurations In this section we discuss the results of our simulated annealing MC simulations of the LP and CP models of MPPN symmetry. We first compare the MPPN symmetries predicted by the LP and CP models (see Sec. 4.2.1). We then use the CP model to survey the minimum-energy MscS configurations in MPPNs with N = 24 and 06 n o 6 24. We thereby first consider qualitative 57 features of the protein arrangement in MPPNs with heterogeneous protein size (see Sec. 4.2.2) and then quantify the symmetry of MPPNs with heterogeneous protein size (see Sec. 4.2.3). 4.2.1 Comparison of LP and CP models As discussed in Sec. 4.1, we consider here two models of MPPN symmetry: the LP model (see Sec. 4.1.1) and the CP model (see Sec. 4.1.2). The LP model separately accounts for the lipids and proteins in MPPNs. In contrast, the CP model focuses on the protein configurations in MPPNs and does not explicitly consider the arrangement of lipids in MPPNs. Table 4.1 shows the polyhedral symmetries obtained through simulated annealing MC simulations of the LP and CP models (see Sec. 4.1.3) for N = 12, 24, and 32 with n o = 0. We fitted the simulated MPPN configurations to polyhedral symmetries as described in Sec. 4.1.4. For both the LP and CP models we find the minimum-energy MPPNs to have the symmetry of the icosahedron forN= 12 [see Fig. 4.6(a)], the snub cube forN= 24 [see Fig. 4.1(e)], and the pentakis dodecahedron forN= 32 [see Fig. 4.6(b)]. For N = 12 and N = 32, the LP and CP models give different results for the 2 nd -best polyhedral fits to the minimum-energy MPPNs, but identical results forN= 24. Best polyhedral fit 2 nd -best polyhedral fit N Model Symmetry E [nm 2 ] Symmetry E [nm 2 ] 12 CP I 1:50 10 2 J60 6:07 10 12 LP I 4:18 tO 1:21 10 24 CP sC (dextro) 4:70 10 2 gD (laevo) 3:62 10 24 LP sC (dextro) 2:96 10 gD (laevo) 6:60 10 32 CP kD 1:06 10 jD 1:95 10 32 LP kD 7:27 10 mD 7:78 10 Table 4.1: Symmetries and associated fit errorsE in Eq. (4.6) of the best two polyhedral fits to the minimum-energy MPPN configurations implied by the LP and CP models of MPPN symmetry for N= 12, 24, and 32 withn o = 0. All results were obtained through simulated annealing MC simu- lations (see Sec. 4.1.3). We denote [1, 2] the icosahedron by I, the snub cube by sC, the pentakis dodecahedron by kD, the metabiaugmented dodecahedron by J60, the truncated octahedron by tO, the pentagonal hexecontahedron by gD, the rhombic triacontahedron by jD, and the disdyakis triacontahedron by mD. We proceeded as described in Sec. 4.1.4 when searching for optimal poly- hedral fits. The polyhedral chiralities result from the random numbers and initial conditions used and are not model predictions. 58 Figure 4.6: 3D representations of (a) the icosahedron (I) and (b) the pentakis dodecahedron (kD) in Table 4.1. The icosahedron is a Platonic solid with 12 vertices, while the pentakis dodecahedron is a lower-symmetric Catalan solid with 32 vertices. Experiments on MPPNs formed from MscS [5, 6] yielded MPPNs withN= 24 and snub cube symmetry as the dominant MPPN symmetry, which we also find through our simulated annealing MC simulations of the LP and CP models. We note that the snub cube is a chiral polyhedron. The right-handed (dextro) and left-handed (laevo) chiralities of the snub cube can be constructed by translating outward the faces of a cube, and rotating them clockwise (dextro) or counter-clockwise (laevo) as viewed from the polyhedron center until the polyhedral shell can be closed up with equilateral triangles. As noted in Sec. 4.1.4, the preference of the minimum-energy MPPN configuration for one chiral polyhedral symmetry over its mirror- symmetric configuration results from the random numbers and initial conditions used [3] for our simulated annealing MC simulations, and is therefore not a model prediction. The polyhedra in Table 4.1 that provide the best fits to the simulated MPPN configurations all have N vertices. Interestingly, the 2 nd -best polyhedral fits in Table 4.1 do not necessarily corre- spond to polyhedra with the same number of vertices as the number of proteins in MPPNs. For instance, the 2 nd -best polyhedral fits in Table 4.1 for(N;n o )=(12;0) are provided by the metabi- augmented dodecahedron (J60) and the truncated octahedron (tO), which have 22 and 24 vertices, 59 respectively, and not by the truncated tetrahedron or the cuboctahedron, which both have 12 ver- tices. While we considered in Table 4.1 MPPNs withN= 12, 24, and 32, we find similar agreement of the dominant symmetries predicted by the LP and CP models for other values of N. Further- more, we checked whether the LP and CP models yield identical results for the dominant MPPN symmetry for MPPNs with heterogeneous protein size. In particular, we find that the LP and CP models both yield the snub cube as the dominant symmetry of MPPNs with(N;n o )=(24;1) [Figs. 4.1(b) and 4.1(d)]. As discussed in Secs. 4.1.1 and 4.1.2, the CP model is conceptually sim- pler than the LP model and avoids some of the computational difficulties associated with finding the minimum-energy MPPN configurations in the LP model. In the remainder of this chapter we therefore focus on the CP model. 4.2.2 Protein arrangement in MPPNs with heterogeneous protein size To determine the minimum-energy arrangement of MscS proteins in MPPNs [5, 6] with het- erogeneous MscS size, we carried out simulated annealing MC simulations of the CP model for N = 24 and 06n o 6 24. Independent of the value of n o considered, we find that the minimum- energy MPPNs have the symmetry of a snub cube with one (closed-state or open-state) MscS protein being located at each one of the 24 vertices of the snub cube (see also Sec. 4.2.3). We find that, as n o is increased from n o = 0, the minimum-energy MscS arrangement in MPPNs follows a strikingly regular pattern (see Figs. 4.7 and 4.8). To specify, at eachn o , the minimum-energy MscS arrangement obtained in our simulated annealing MC simulations, we label in Figs. 4.7 and 4.8 the vertices of the snub cube byn o to denote the position of then th o open-state MscS protein. If there is more than one equivalent choice for the position of then th o open-state MscS protein we introduce a subscript specifying the degree of degeneracy. In Fig. 4.7 we show the pattern of open-state MscS proteins found in our simulated annealing MC simulations. In Fig. 4.8 we provide 3D illustra- tions of selected MPPN configurations in Fig. 4.7, and in Fig. 4.9 we show some of the MPPN configurations corresponding to Fig. 4.7. 60 Figure 4.7: Minimum-energy protein arrangements in MPPNs implied by the CP model of MPPN symmetry for N = 24. The numbers labeling the vertices denote the positions of the n th o open- state MscS protein, with subscripts denoting the degree of degeneracy in placing then th o open-state MscS protein. We omit this subscript if the position of then th o open-state MscS protein is uniquely determined by the protein arrangement in MPPNs with (n o 1) open-state MscS proteins. The degeneracy in placing open-state MscS proteins follows from the symmetry of the snub cube. The faces are colored according to their symmetry properties with red, yellow, and blue colors indicating two-fold, three-fold, and four-fold symmetry axes, respectively (see also Fig. 4.8). To understand the protein arrangement in MPPNs withN= 24 and heterogeneous protein size it is instructive to briefly recall the symmetry properties of the (undeformed) snub cube [103]. The snub cube is an Archimedean solid with 24 vertices and 38 faces corresponding to six squares, none of which share a vertex, and 32 equilateral triangles [Fig 4.1(f)]. All vertices in the snub cube are equivalent. The six square faces of the snub cube are associated with six four-fold rotational symmetry axes. Eight of the 32 triangular faces of the snub cube are associated with three-fold rotational symmetry axes, while 24 of the 32 triangular faces of the snub cube are associated with two-fold rotational symmetry axes. Figure 4.8(a) provides an illustration of the symmetry properties of the snub cube. 61 Figure 4.8: 3D illustrations of selected minimum-energy protein arrangements in Fig. 4.7 for (a) n o = 0 or n o = 24 (undeformed snub cube), (b) n o = 3, (c) n o = 12, and (d) n o = 16. Following the labeling scheme in Fig. 4.7, the faces of the snub cube are colored according to their symmetry properties. In panel (a), the locations of some of the two-, three-, and four-fold symmetry axes of the snub cube are indicated by arrow, triangle, and square symbols, respectively, with the symme- try axes perpendicularly intersecting these symbols at their geometric centers. In panels (b), (c), and (d), vertices of the snub cube occupied by open-state MscS proteins are indicated by disks. In panel (b), the two geometrically equivalent choices for placing the 3 rd open-state MscS protein are labeled as 3 and 3 0 , respectively. In panels (c) and (d) we highlight the positions of the open-state MscS proteins occupying the “front” and ”back” square faces of the snub cube through increased disk sizes with no labels (see main text), and label the positions of selected open-state MscS pro- teins with n o > 8 by n o . In panel (c), the closed zig-zag loop formed by the first eight open-state MscS proteins is shown in orange, while we indicate in green the nearest-neighbor bonds formed by the 9 th to 12 th open-state MscS proteins with the open-state MscS proteins occupying the front and back square faces of the snub cube. In panel (d), the white lines show the closed zig-zag loop of closed-state MscS proteins formed atn o = 16. Portions of the loops in panels (c) and (d) located at the back of the snub cube are indicated by dashed curves. For n o = 1, the single open-state MscS protein can equivalently occupy each one of the 24 vertices of the snub cube. We therefore have a 24-fold degeneracy in the position of the first open- state MscS protein (Fig. 4.7). In contrast, once the position of the first open-state MscS protein has been set, the energetically most favorable position of the 2 nd open-state MscS protein is fixed (Fig. 4.7): The 2 nd open-state MscS protein is arranged so that it is located diagonally across the square-shaped face of the snub cube from the first open-state MscS protein [see Fig. 4.9(a)]. In other words, the two open-state MscS proteins form a next-nearest-neighbor pair across a square face of the snub cube. As a result, the symmetry axis associated with this square face is changed from a four-fold symmetry axis (for a perfect snub cube) to an axis with approximate two-fold rotational symmetry. The 3 rd open-state MscS protein forms a nearest-neighbor pair across a trian- gular face of the snub cube with either one of the two other open-state MscS proteins so as to trace 62 Figure 4.9: 3D representations of the minimum-energy MPPN configurations implied by the CP model of MPPN symmetry for N = 24 and (a) n o = 2, (b) n o = 4, (c) n o = 8, (d) n o = 12, and (e) n o = 16 (see also Fig. 4.7). As in Fig. 4.1(d), closed-state and open-state MscS proteins are represented by blue and red disks, respectively, with the radii of these disks being decreased by some fixed scale factor relative to the disk radii implied byr 0 o andr 0 c . For clarity, polyhedral ridges enclosing (distorted) square faces of the snub cube are shown in bright green, while polyhedral ridges associated only with triangular faces of the snub cube are shown in dark green. out a “zig-zag” pattern (Fig. 4.7). The two equivalent nearest-neighbor “bonds” associated with the 3 rd open-state MscS protein intersect two-fold symmetry axes of the snub cube [see Fig. 4.8(b)]. The 4 th open-state MscS protein is again located diagonally across a square face of the snub cube from an open-state MscS protein [see Fig. 4.9(b)]. Hence, its position is uniquely determined by the position of the 3 rd open-state MscS protein (Fig. 4.7). The aforementioned zig-zag pattern of alternating next-nearest-neighbor bonds of open-state MscS proteins across the square faces of the snub cube and nearest-neighbor bonds intersecting two-fold symmetry axes of the snub cube continues for 0n o 6 8 (Fig. 4.7). Atn o = 8, the eight open-state MscS proteins thus connect up to form a closed loop [see Fig. 4.8(c)]. As a result, two square faces of the snub cube, located at opposite sides of the snub cube, are left devoid of any open-state MscS proteins [see the lower panel in Fig. 4.9(c)]. At n o = 9, one of the eight vertices associated with these two square faces is occupied by the 9 th open-state MscS protein, resulting in a degeneracy of eight for placing the 9 th open-state MscS protein (Fig. 4.7). Note that the 9 th open-state MscS protein forms a nearest-neighbor pair with some other open-state MscS protein 63 located across a polyhedral ridge separating triangular faces associated with two- and three-fold symmetry axes [Fig. 4.8(c)]. The 10 th open-state MscS protein forms a next-nearest-neighbor bond across a square face of the snub cube with the 9 th open-state MscS protein and, hence, has a position that is uniquely de- termined by the position of the 9 th open-state MscS protein (Fig. 4.7). The polyhedral geometry of the snub cube mandates that, similarly as the 9 th open-state MscS protein, the 10 th open-state MscS protein forms a nearest-neighbor bond with some other open-state MscS protein located across a polyhedral ridge separating triangular faces with two- and three-fold symmetry axes [Fig. 4.8(c)]. As a result, the square face of the snub cube containing the 9 th and 10 th open-state MscS proteins connects two square faces of the snub cube, each containing two open-state MscS proteins, that are part of the zig-zag loop formed atn o = 8. Following Fig. 4.8(c), we denote the latter two faces as the “front” and “back” faces of the zig-zag loop formed atn o = 8, respectively. Atn o = 11, the last remaining square face of the snub cube containing only closed-state MscS proteins starts to be occupied by open-state MscS proteins [Figs. 4.7 and 4.8(c)]. Two vertices on that square face are—via open-state MscS proteins that are part of the front and back faces of the zig-zag loop of open-state MscS proteins formed at n o = 8 in Fig. 4.8(c)—next-next-nearest neighbors of the 9 th and 10 th open-state MscS proteins. We find that either one of these two vertices is occupied by the 11 th open-state MscS protein. Hence, there is a two-fold degeneracy in the position of the 11 th open-state MscS protein. The 12 th open-state MscS protein is again located diagonally across a square face of the snub cube from an open-state MscS protein, and its position is therefore uniquely determined by the position of the 11 th open-state MscS protein [Figs. 4.7 and 4.8(c)]. Thus, the square face occupied by open-state MscS proteins at n o = 11 and n o = 12 connects, similarly as the square face containing the 9 th and 10 th open-state MscS proteins, the front and back faces of the zig-zag loop of open-state MscS proteins formed atn o = 8 in Fig. 4.8(c). Atn o = 12, all square faces of the snub cube contain two open-state MscS proteins, which form next-nearest-neighbor pairs across the diagonals of the square faces [see Figs. 4.8(c) and 4.9(d)]. 64 As n o is increased beyond n o = 12, open-state MscS proteins start to form nearest-neighbor bonds on the square faces of the snub cube. We find that first the front and back faces of the zig-zag loop of open-state MscS proteins formed at n o = 8 in Fig. 4.8(c) are fully occupied by open-state MscS proteins. In particular, atn o = 13, there are four geometrically equivalent choices for the location of the 13 th open-state MscS protein, resulting in a four-fold degeneracy for placing the 13 th open-state MscS protein (Fig. 4.7). The 14 th open-state MscS protein is located diagonally across a square face of the snub cube from an open-state MscS protein, and its position is therefore uniquely determined by the position of the 13 th open-state MscS protein (Fig. 4.7). Similarly, we have a two-fold degeneracy in the protein position at n o = 15, with the position of the 16 th open- state MscS protein being determined uniquely by the position of the 15 th open-state MscS protein (Fig. 4.7). Note that, at n o = 16, there are four geometrically equivalent square faces of the snub cube containing two closed-state MscS proteins each. Connecting the vertices associated with these closed-state MscS proteins, we obtain a closed zig-zag loop with similar geometric properties as the loop of open-state MscS proteins found at n o = 8 [see Fig. 4.8(d)]. As n o is increased beyond n o = 16, the vertices of this loop are occupied by open-state MscS proteins following a pattern that is analogous to that obtained for 0 n o 8 (Fig. 4.7). We find an eight-fold degeneracy in the protein position atn o = 17 and two-fold degeneracies in the protein position atn o = 19, 21, and 23, respectively, with the positions of open-state MscS proteins at even n o being determined uniquely by the positions of open-state MscS proteins at odd n o . At n o = 24, all vertices of the snub cube are occupied by open-state MscS proteins. 4.2.3 Quantifying the symmetry of MPPNs with heterogeneous protein size In this section we employ the mathematical approaches described in Sec. 4.1.4 to quantify the symmetry of MPPNs with heterogeneous protein size. To determine how closely the protein 65 configurations in MPPNs follow polyhedral symmetry, it is convenient to introduce, based on the fit errorE in Eq. (4.6), the dimensionless root-mean-square fit error b s = 1 l r E N (4.10) withl = 10 2 r 0 c as the characteristic length scale. In Table 4.2 we listb s in Eq. (4.10) for the best two polyhedral fits for MPPNs withN= 24 and 06n o 6 24. We calculated these values ofb s from the minimum-energy MPPN configurations obtained in our simulated annealing MC simulations of the CP model. As already noted in Sec. 4.2.2 we find that, independent of the value of n o considered, the best polyhedral fits in Table 4.2 always correspond to snub cube (sC) symmetry. Figure 4.10 shows b s in Table 4.2 versus n o for the best polyhedral fits. We also provide in Fig. 4.10 the range in b s associated with the ten lowest-energy MPPN configurations obtained, at eachn o , in our simulated annealing MC simulations, as well as the averageb s associated with these ten lowest-energy MPPN configurations. Each one of these ten lowest-energy MPPN configura- tions corresponds to one independent MC trajectory. At eachn o , the best polyhedral fits to the ten lowest-energy MPPN configurations in Fig. 4.10 all correspond to snub cube symmetry. Note from Fig. 4.10 that the root-mean-square fit error in Eq. (4.10) is smallest for MPPNs with homogeneous protein composition (n o = 0 and n o = 24). Similarly, comparison of fits of the simulated MPPN configurations to the snub cube with fits to competing polyhedral symmetries shows that the snub cube symmetry is most dominant for homogeneous or nearly homogeneous protein compositions (see Table 4.2). Starting from n o = 0, the increase in b s with n o in Fig. 4.10 and Table 4.2 can be understood by noting that for MPPNs with heterogeneous protein composition the polyhedral symmetry must deform so as to accommodate proteins of different size. For instance, the square faces of the snub cube in Fig. 4.9 containing a mixture of closed-state and open-state MscS proteins are seen to deviate from a perfect square. We find a maximum inb s in Fig. 4.10 at n o = 8. As already noted in Sec. 4.2.2, n o = 8 yields an MPPN configuration with four deformed square faces of the snub cube forming a closed loop, which “flattens” the polyhedron and increases the fit error [Figs. 4.7 66 Best polyhedral fit 2 nd -best polyhedral fit n o Symmetry b s Symmetry b s=10 0 sC (dextro) 1:13 gD (laevo) 3:06 1 sC (laevo) 3:16 eC 3:66 2 sC (dextro) 5:52 gD (laevo) 3:31 3 sC (laevo) 7:21 gD (dextro) 3:22 4 sC (dextro) 8:31 gD (dextro) 3:24 5 sC (laevo) 8:93 gD (dextro) 3:34 6 sC (laevo) 1:08 10 gD (dextro) 3:04 7 sC (dextro) 1:21 10 gD (laevo) 3:03 8 sC (dextro) 1:44 10 sC (laevo) 3:15 9 sC (laevo) 1:37 10 sC (dextro) 3:25 10 sC (laevo) 1:29 10 gD (laevo) 3:67 11 sC (laevo) 1:24 10 gD (laevo) 3:35 12 sC (laevo) 1:20 10 sC (dextro) 3:58 13 sC (dextro) 1:23 10 J45 (dextro) 3:56 14 sC (laevo) 1:30 10 gD (dextro) 3:4 15 sC (dextro) 1:30 10 gD (laevo) 3:14 16 sC (laevo) 1:29 10 J45 (laevo) 3:52 17 sC (dextro) 1:18 10 gD (laevo) 3:21 18 sC (laevo) 1:07 10 gD (dextro) 3:15 19 sC (dextro) 9:23 gD (laevo) 3:24 20 sC (laevo) 8:23 gD (laevo) 3:49 21 sC (dextro) 7:04 gD (dextro) 3:48 22 sC (laevo) 6:16 gD (dextro) 3:42 23 sC (laevo) 3:47 gD (dextro) 3:75 24 sC (dextro) 1:08 gD (dextro) 3:54 Table 4.2: Symmetries and associated root-mean-square fit errors b s in Eq. (4.10) of the best two polyhedral fits to the minimum-energy MPPN configurations implied by the CP model of MPPN symmetry for N = 24 and the indicated values of n o . All results were obtained through simulated annealing MC simulations (see Sec. 4.1.3). We use the same notation for polyhedral symmetries as in Table 4.1 [1, 2] with, in particular, gD corresponding to the pentagonal hexecontahedron, and de- note the rhombicuboctahedron by eC and the gyroelongated square bicupola by J45. We proceeded as described in Sec. 4.1.4 when searching for optimal polyhedral fits. The polyhedral chiralities result from the random numbers and initial conditions used and are not model predictions. and 4.9(c)]. An analogous protein configuration is obtained atn o = 16 [Figs. 4.7 and 4.9(e)], which may explain the large values of b s found in our simulations in the vicinity of n o = 16. A (weak) local minimum occurs in Fig. 4.10 atn o = 12. As noted in Sec. 4.2.2, atn o = 12 all square faces of the snub cube are composed of two closed-state and two open-state MscS proteins arranged in the same pattern [Figs. 4.7 and 4.9(d)], which may explain the relatively small value ofb s atn o = 12. 67 Figure 4.10: Root-mean-square fit errorsb s in Eq. (4.10) of the best polyhedral fit (snub cube sym- metry) to the minimum-energy MPPN configurations implied by the CP model of MPPN symmetry forN= 24 versus number of open-state MscS proteins,n o (blue data points). For eachn o , we also show the range in b s associated with the ten lowest-energy MPPN configurations obtained in our simulated annealing MC simulations, which all correspond to snub cube symmetry. This range in b s is indicated by bars, with the red data points showing the average b s for the ten lowest-energy MPPN configurations obtained in our simulated annealing MC simulations. See also Table 4.2. In addition to the dimensionless root-mean-square fit error in Eq. (4.10), the BOO parameters Q l in Eq. (4.7) provide a mathematical approach for quantifying MPPN symmetry. In the case of the snub cube, the first two non-zero values of the BOO parameters Q l occur at l = 4 and l = 6 (see Sec. 4.1.4). In Figs. 4.11(a) and 4.11(b) we plot c Q l in Eq. (4.9), which corresponds to the relative difference between the values of Q l associated with a (perfect) snub cube and the minimum-energy MPPN configurations implied by the CP model, for 06n o 6 24 and l = 4 and l = 6, respectively. The parameters d Q 4;6 do not depend on the chirality of the snub cube. For completeness, we also show in Fig. 4.11 the range in d Q 4;6 associated with the ten lowest-energy MPPN configurations obtained, at eachn o , in our simulated annealing MC simulations, as well as the corresponding average values of d Q 4;6 . With the definition of c Q l in Eq. (4.9), smaller values of c Q l indicate a closer resemblance of the protein arrangement in MPPNs to a snub cube. Consistent with the results in Fig. 4.10, Fig. 4.11 shows local peaks at n o = 8 and n o = 16, as well as a local minimum at n o = 12. Similarly as for Fig. 4.10, these features of Fig. 4.11 can be understood by 68 Figure 4.11: Relative difference in BOO parameters between a snub cube and the MPPN config- urations implied by the CP model, c Q l , versus number of open-state MscS, n o , at (a) order l = 4 and (b) order l = 6 in Eq. (4.9). For each n o , we show the range in c Q l associated with the ten lowest-energy MPPN configurations obtained in our simulated annealing MC simulations, which all correspond to snub cube symmetry. This range in c Q l is indicated by bars, with the red data points showing the average c Q l for the ten lowest-energy MPPN configurations and the blue data points showing the c Q l associated with the minimum-energy MPPN configurations. The plots in panels (a) and (b) are obtained from the data for low-energy MPPN configurations also used in Fig. 4.10. noting that the protein arrangements found forn o = 8 andn o = 16 correspond to deformed square faces of the snub cube forming closed loops [Figs. 4.7, 4.9(c), and 4.9(e)], while for n o = 12 all square faces of the snub cube are occupied by two closed-state and two open-state MscS proteins [Figs. 4.7 and 4.9(d)]. Note that the c Q 4 -versus-n o curve in Fig. 4.11(a) shows two peaks atn o = 8 and n o = 16 of approximately equal height. In contrast, the c Q 6 -versus-n o curve in Fig. 4.11(b) shows a more pronounced peak at n o = 8 than at n o = 16. At a qualitative level, the c Q 6 -versus-n o curve in Fig. 4.11(b) thus resembles theb s-versus-n o curve in Fig. 4.10. 69 Chapter 5 Conclusion MPPNs have been proposed [6] as a strategy for the structural analysis of membrane proteins under physiologically relevant transmembrane gradients, and as a method for targeted drug deliv- ery with precisely controlled release mechanisms through, for instance, the gating of membrane channels in MPPNs. To this end, one must arrive at a quantitative understanding of MPPN self- assembly for general membrane proteins, with potentially more than one distinct protein shape in MPPNs. We have generalized here the theory of MPPN self-assembly [3, 4] to aid these goals. In particular, in Chapter 2, we explored the effect of protein steric constraints on the symmetry of MPPNs [5, 6]. Steric constraints on the separation of membrane proteins in MPPNs arise, on the one hand, within the membrane due to the finite size of proteins and lipids [3, 4]. On the other hand, protein domains outside the membrane can produce steric constraints on the separation of membrane proteins in MPPNs. Such protein steric constraints may originate from a well-defined shape of protein domains outside the membrane, entropic repulsion between membrane proteins with flexible domains outside the membrane, or binding of other molecules to membrane proteins. We describe here protein steric constraints in MPPNs based on the in-plane protein radius outside the membrane, r s , and the effective protein height above the midplane of the protein hydrophobic region, h s . We assume that protein steric constraints induce hardcore steric repulsion between membrane proteins in MPPNs. 70 The mean-field approach in Refs. [3, 4] successfully predicts, with all model parameters de- termined directly by experiments, the observed symmetry and size of MPPNs formed from MscS proteins [5, 6]. Our calculations show that, for the molecular structure of MscS [12, 40], steric constraints due to protein regions outside the membrane do not affect MPPN self-assembly dia- grams. Our results on MPPNs formed from MscS [5, 6] therefore agree with those in Refs. [3, 4] and experiments on MPPNs formed from MscS [12, 40]. However, we also find that protein steric constraints can strongly affect MPPN self-assembly. On the one hand, our calculations show that protein steric constraints can effectively rule out dominant MPPN states, leaving large portions of the MPPN self-assembly diagrams with no clearly defined MPPN symmetry. On the other hand, we find that suitable values of r s and h s can substantially expand the regions of MPPN self-assembly diagrams dominated by highly symmetric MPPN states, such as MPPNs with icosahedral or snub cube symmetry. For small enough r s or large enough h s , MPPN self-assembly is independent of protein steric constraints. For scenarios with negligible bilayer-protein hydrophobic mismatch we find that, as r s is increased or h s decreased, protein steric constraints tend to imply more stringent constraints on membrane protein separation in MPPNs for larger, less symmetric MPPN states, thus biasing MPPN self-assembly towards smaller, more symmetric MPPN states. Our calcula- tions suggest that, upon further increasing r s or decreasing h s , the dominant MPPN states can be shifted towards larger, highly symmetric MPPNs. Finally, we find that a substantial bilayer-protein hydrophobic mismatch biases MPPN self-assembly towards highly symmetric MPPN states for weak protein steric constraints [4] and can, depending on the bilayer-protein contact angle, prevent or amplify the dominance of highly symmetric MPPN states for strong protein steric constraints. In Chapter 3, we developed a methodology for predicting the symmetry and size of MPPNs with arbitrarily large (nonlinear) membrane curvature deformations. For MPPNs formed from MscS [12, 40] this methodology predicts, with no adjustable parameters, the observed symmetry and size of MPPNs [5, 6]. Since MscS proteins only weakly curve the membrane [12, 18, 40], similar conclusions were reached previously using a small-gradient approximation [3, 4]. In con- trast, (closed-state) Piezo proteins have a highly curved structure [7, 8, 9, 10], and the resulting 71 membrane shape deformations are highly nonlinear [39]. We find here that the self-assembly di- agram for MPPNs formed from Piezo critically depends on the Piezo dome radius of curvature. In particular, for the Piezo dome radius of curvature R D 10:2 nm observed for a closed state of Piezo [7, 8, 9, 10], we generally find MPPNs with six Piezo proteins and octahedral symmetry to be dominant, with only a weak dependence of the dominant MPPN symmetry on the protein number fraction in solution. As the value of R D is increased, we find dominant MPPN states with icosahedral and snub cube symmetry, composed of 12 and 24 Piezo proteins, respectively. Such highly symmetric MPPN states may allow high-resolution structural studies of MPPNs formed from Piezo [19, 20, 21], potentially even in the presence of transmembrane gradients similar to those typically found in cellular environments [5, 6]. Intriguingly, if gating of Piezo is accompa- nied by an increase inR D [7, 8, 9, 10, 38, 39], the distinct symmetries we predict here for MPPNs formed from Piezo may be associated with distinct, biologically relevant conformational states of Piezo. In addition to calculating MPPN self-assembly diagrams, we also used computational model- ing to explore the symmetry of MPPNs with heterogeneous protein size. Our work on MPPNs with heterogeneous protein size is described in Chapter 4. Our computational modeling approach is closely related to previous models describing the symmetry of viral capsids [42, 43, 44, 45]. Although, from an experimental perspective, MPPNs and viral capsids are quite distinct—with proteins in MPPNs being embedded in a lipid bilayer environment but viral capsids being com- posed solely of proteins—our results suggest that MPPN symmetry and viral capsid symmetry are governed by similar physical principles [3]. Analogous modeling approaches may also be appli- cable to other kinds of systems forming polyhedral shells [43, 104, 105, 106, 107]. Motivated by previous experimental studies of MPPNs [5, 6], we have focused here on MPPNs composed of 24 closed-state or open-state MscS proteins [12, 13, 40]. However, our modeling approach is easily generalized to other types of MPPNs. An important distinction between previous studies of protein shells [42, 43, 44, 45, 104, 107] and the model of MPPN symmetry developed here is that, in the former case, a key question 72 concerns the symmetry of protein shells as a function of the number of protein subunits. In con- trast, a central question for MPPNs is how the protein arrangement in MPPNs changes after some proteins in MPPNs transition to a different conformational state following, for instance, osmotic shock, while leaving the membrane intact [5, 6]. We have therefore focused here on MPPNs con- taining a fixed number of proteins. In the spirit of previous work on the symmetry of closed protein shells [42, 43, 44, 45], we have employed highly idealized models of MPPN symmetry. Our mod- eling approach could be extended in various ways to allow more detailed predictions. For instance, we assumed here similar interactions between closed-state and open-state MscS proteins, with the only difference in their interaction potentials stemming from the distinct sizes of closed-state and open-state MscS proteins [12, 13, 40]. In general, different conformational states of a given mem- brane protein or distinct kinds of membrane proteins may show distinct interactions in MPPNs. Furthermore, we assumed here that the particles representing membrane proteins (and lipids) are confined to the surface of a sphere. While this assumption is justified for the observed MPPNs formed from MscS proteins [5, 6], it may not hold in general. For instance, for large enough protein numbers MPPNs may, in analogy to protein shells [108, 109], buckle into faceted shapes. A key outcome of our study is that MPPNs with heterogeneous protein size can be highly symmetric, with a well-defined polyhedral (snub cube) ordering of membrane proteins of different sizes. MPPNs have been proposed [6] as a means for the structural analysis of membrane proteins in the presence of physiologically relevant transmembrane gradients. Such transmembrane gra- dients are expected to result in heterogeneous protein size, with different proteins being trapped in different conformational states, while leaving the membrane intact. Our finding that MPPNs with heterogeneous protein size can be highly symmetric suggests that it may be feasible to utilize MPPNs for structural studies [6, 20] even if not all membrane proteins in MPPNs are trapped in the same conformational state. In particular, for MPPNs formed from 24 MscS proteins [5, 6] we predict that the first eight gated (open-state) MscS proteins form a closed zig-zag loop, resulting in two square faces of the snub cube that are devoid of any open-state MscS proteins (Figs. 4.7–4.9). For more than eight open-state MscS proteins, these two square faces are gradually filled with 73 open-state MscS proteins until, for twelve open-state MscS proteins, all square faces of the snub cube are occupied by two open-state MscS proteins located diagonally across the square faces of the snub cube. As the number of open-state MscS proteins is increased further, the square faces of the snub cube that connect—via bonds between open-state MscS proteins—the square faces of the snub cube that were not part of the original closed zig-zag loop of open-state MscS proteins are filled with open-state MscS proteins until, for more than sixteen open-state MscS proteins, the remaining vertices of the snub cube are populated by open-state MscS proteins. Besides their potential use for structural studies, MPPNs have also been proposed as a novel vehicle for targeted drug delivery with precisely controlled release mechanisms [6]. Such applica- tions of MPPNs require control over the number of proteins in MPPNs, and the MPPN symmetry and size. One avenue for controlling MPPN protein composition and shape is to tune the bilayer- protein hydrophobic mismatch in MPPNs [4] through changes in lipid composition [63, 110] or repositioning of amphipathic protein residues [111]. Our work on the effect of protein steric con- straints on MPPN symmetry suggests that protein steric constraints provide a complementary ap- proach for the control of MPPN protein composition and shape. For instance, our results imply that targeted binding of proteins of specified size, or of nanoparticles such as quantum dots [112], to membrane proteins could be used to bias MPPN self-assembly towards highly symmetric MPPN states with, depending on the protein or nanoparticle size, small or large MPPN sizes and pro- tein numbers. Modification of protein steric constraints may thus allow the directed self-assembly of MPPNs with specified symmetry, size, and protein composition, facilitating the utilization of MPPNs for high-resolution structural studies of membrane proteins or targeted drug delivery. The mean-field approach for the description of MPPN self-assembly employed here cannot be directly applied to MPPNs with heterogeneous protein composition, because in MPPNs with heterogeneous protein composition not all proteins are equivalent. However, the well-defined and regular protein arrangements in MPPNs with heterogeneous protein size found in Chapter 4 suggest that it may be practicable to generalize the theory of MPPNs [3, 4] to allow for heterogeneous 74 protein compositions. Such a generalized theory of MPPNs may allow prediction of how MPPN self-assembly must be directed to produce MPPNs with given release mechanisms [6]. Our mean-field model of MPPN self-assembly could be developed further in a number of dif- ferent ways. In particular, a more detailed molecular model of protein steric constraints in MPPNs would, for instance, allow for a competition between attractive and repulsive protein interactions outside the membrane. Moreover, our mean-field model of MPPN self-assembly allows for ther- mal effects during MPPN self-assembly [3, 4, 47, 48, 49] but assumes that, for each MPPNn-state, the dominant MPPN shape corresponds to that with the lowest (elastic) energy. Thermal fluctu- ations are expected to perturb the MPPN shape about the lowest-energy state. Furthermore, we note that topological defects in protein packing in MPPNs are captured in our mean-field model via the packing fraction p(n) [49, 69, 70], which has two key limitations. On the one hand, as al- ready noted above, our mean-field approach implicitly assumes a uniform protein composition. On the other hand, to make our calculations more tractable we assumed in our mean-field approach a circular unit cell. In principle, our approach could be extended to consider, for instance, the polyg- onal unit cells suggested by the polyhedral symmetry of MPPNs. In analogy to recent work on the physics of protein shells [107, 113], such an extension of our approach would allow investigation of the interaction of protein packing defects in MPPNs. 75 Appendices A Physical model of the MPPN defect energy Following Ref. [3], we develop here a physical model of the MPPN defect energy E d (n;R) in Eq. (2.26). We first note that, in the continuum limit, the elastic (compression or expansion) energy of a hexagonal network of harmonic springs is given by [71] H = K 2 Z dx 1 dx 2 (Ñr r r) 2 ; (A.1) whereK is the continuum force constant and(Ñr r r) 2 =(¶r r r=¶x 1 ) 2 +(¶r r r=¶x 2 ) 2 , in whichx 1 andx 2 are internal coordinates andr r r=r r r(x 1 ;x 2 ) denotes the external coordinate specifying the location of the surface in the (three-dimensional) embedding space. We relate the elastic energy of the spring network to its areal strain by considering a flat, rectangular patch of a uniform, elastic sheet. We denote the width and height of the rectangular patch by L 1 and L 2 , respectively. If L 1 and L 2 are slightly compressed or expanded byDL 1 andDL 2 , respectively, a point in the patch originally at r r r 0 (x 1 ;x 2 ) is translated to r r r(x 1 ;x 2 )=r r r 0 0 0 + DL 1 L 1 x 1 i i i+ DL 2 L 2 x 2 j j j; (A.2) wherei i i and j j j are orthogonal unit vectors. We thus have (Ñr r r) 2 = DL 1 L 1 2 + DL 2 L 2 2 : (A.3) 76 Furthermore, since the original (unperturbed) area of the rectangular patch is given by A=L 1 L 2 , the perturbationsDL 1 andDL 2 change, to leading order, the area of the rectangular patch by DA=L 1 DL 2 +L 2 DL 1 : (A.4) For uniform strain we haveDL 1 =L 1 =DL 2 =L 2 , in which case Eqs. (A.3) and (A.4) yield DA A 2 = 2(Ñr r r) 2 : (A.5) The continuum force constant K in Eq. (A.1) is related to the discrete force constant of the Hookean springs,K 0 in Eq. (2.29), viaK= p 3K 0 [71]. Thus, Eq. (A.5) implies that Eq. (A.1) can be rewritten as H = p 3K 0 4 Z dx 1 dx 2 DA A 2 : (A.6) For uniform areal strainDA=A is constant and, to leading order, Eq. (A.6) can be simplified [14] to H = K s 2 A DA A 2 (A.7) with, as in Sec. 2.1.4, the stretching modulus K s = p 3K 0 2 (A.8) of the elastic sheet. Similarly as in previous work on viral capsid self-assembly [49] we approx- imate, at the mean-field level, the areal strain due to topological defects in protein packing in MPPNs by DA A = p max p(n) p max : (A.9) In Eq. (A.9) it is assumed that the average energy penalty due to topological defects in protein packing in MPPNs can be captured, for each MPPNn-state, by a uniform areal strain corresponding to the percentage difference between the optimal protein packing fraction allowed by topological 77 constraints and the packing fraction associated with the hexagonal protein arrangement, p max , favored in the absence of topological constraints [49, 69, 70]. Substitution of Eq. (A.9) into the expression forH in Eq. (A.7) results in the MPPN defect energyE d in Eq. (2.26). B Effect of changes in the lipid bilayer bending rigidity on MPPN self-assembly diagrams Our focus in this thesis is on lipid bilayers with elastic properties corresponding to the diC14:0 lipids used for MPPNs formed from MscS proteins [5, 6, 63]. In particular, we use the lipid bilayer bending rigidityK b = 14k B T [63]. To explore to what extent our results are sensitive to changes in K b , we consider in Fig. B.1 MPPN self-assembly diagrams for the same scenario as in the lower-left panel of Fig. 2.2 (r s = 5:0 nm and h s = 6:0 nm) with K b = 7 k B T [see Fig. B.1(a)], K b = 14 k B T [see Fig. B.1(b)], and K b = 28 k B T [see Fig. B.1(c)]. The results in Fig. B.1(b) are identical to those in the lower-left panel of Fig. 2.2, and are reproduced here for ease of comparison. The range in K b considered in Fig. B.1 approximately corresponds to the range in K b measured for phospholipid bilayers with distinct lipid tail lengths [18, 63]. We find in Fig. B.1 that the broad features of the MPPN self-assembly diagrams obtained here do not depend on the precise value of K b used [4]. However, we also find that changes in K b can shift the boundaries separating regions of the MPPN self-assembly diagrams dominated by distinct MPPN n-states. In particular, as K b is increased in Fig. B.1, largerjUj are required to produce transitions in MPPN symmetry from large-n (lower symmetry) MPPN states to small-n (higher symmetry) MPPN states. Thus, the MPPN self-assembly diagrams in Fig. B.1 are, from left to right, effectively “stretched out” in the direction of increasingjUj. 78 Figure B.1: MPPN self-assembly diagrams as a function of the absolute value of the bilayer- protein hydrophobic mismatch,jUj, and the bilayer-protein contact angle,a, for r s = 5:0 nm and h s = 6:0 nm calculated as in Fig. 2.2 with (a) K b = 7 k B T , (b) K b = 14 k B T , and (c) K b = 28 k B T . The MPPN self-assembly diagram in panel (b) is identical to that in the lower-left panel of Fig. 2.2 and reproduced here for ease of comparison. We use the same notation as in Fig. 2.2. C Vanishing protein height In this appendix we consider the limiting case of a vanishing protein height, h s ! 0, in our model of MPPN self-assembly (Fig. 4.1). While this limiting case has no direct physical sig- nificance, it provides insight into the mathematical properties of the mean-field model of MPPN self-assembly considered here. We focus on the small-r s regime, where protein steric constraints due to r s only produce small perturbations of MPPN self-assembly diagrams. For simplicity, we also setU = 0 throughout this appendix. Forh s = 0, Eq. (2.36) simplifies to r s h (n)= a bcos ¯ b r i : (C.1) The boundaries delineating regions in thea-r s plane of MPPN self-assembly diagrams dominated by distinct MPPNn-states can thus be estimated from B(a;n)=r s h +C; (C.2) whereC depends on n but is constant with a. Figure C.1 shows Eq. (C.2) together with the cor- responding MPPN self-assembly diagram at h s = 0. We thereby fixC so that, at each n,B(a;n) 79 Figure C.1: MPPN self-assembly diagram as in Fig. 2.4 as a function of the bilayer-protein contact angle, a, and the in-plane protein radius outside the membrane, r s , atU = 0 in the (nonphysical) case h s = 0. The dotted curves delineate regions of the MPPN self-assembly diagram dominated by distinct MPPN n-states, while the solid lines show the corresponding estimates obtained from Eq. (C.2). The red horizontal lines are obtained as in Fig. 2.4. The small-a boundaries of the red horizontal lines are used to fixC in Eq. (C.2) for each n considered. 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Abstract (if available)

Abstract Experiments have shown that, in an aqueous environment, lipids and membrane proteins can self-assemble into membrane protein polyhedral nanoparticles (MPPNs). MPPNs are closed, spherical vesicles composed of a lipid bilayer membrane and membrane proteins, with a polyhedral arrangement of membrane proteins. Because of these features, MPPNs have been proposed as a potential strategy for the high-resolution structural study of membrane proteins in the presence of physiologically relevant transmembrane gradients and targeted drug delivery. In this PhD thesis, we develop new methodologies for the prediction of the polyhedral symmetry of MPPNs for arbitrary lipid and protein compositions. ? In Chapter 2, we consider membrane proteins that have large domains outside the membrane, and discuss the effect of steric constraints arising from such domains on the symmetry and size of MPPNs. In Chapter 3, we use the arclength parameterization of surfaces to develop a mean-field model of MPPN self-assembly that allows for arbitrarily large (nonlinear) membrane curvatures. On this basis, we study the symmetry of MPPNs formed from Piezo ion channels, which have been found to underlie many forms of mechanosensation in vertebrates and bend the membrane into strongly curved dome shapes. In Chapter 4, we use kinetic Monte Carlo (MC) simulations to explore the symmetry of MPPNs composed of membrane proteins with heterogeneous structure, such as the closed and open states of ion channels. The work described here suggests that MPPNs can show a variety of distinct symmetries and provides a range of approaches for the control of MPPN symmetry, which may facilitate the further development of MPPNs for membrane protein structural analysis and targeted drug delivery. ? The materials in this PhD thesis are also discussed in the following publications: ? I. M. Ma and C. A. Haselwandter. Effect of protein steric constraints on the symmetry of membrane protein polyhedra. Phys. Rev. E, 102:042411, 2020. ? II. M. Ma and C. A. Haselwandter. Self-assembly of polyhedral bilayer vesicles from Piezo ion channels (in preparation). ? III. M. Ma, D. Li, O. Kahraman, and C. A. Haselwandter. Symmetry of membrane protein polyhedra with heterogeneous protein size. Phys. Rev. E, 101(2):022417, 2020.

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Asset Metadata

Creator Ma, Mingyuan (author)

Core Title The symmetry of membrane protein polyhedral nanoparticles

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2021-08

Publication Date 07/19/2021

Defense Date 05/03/2021

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

Tag biomolecular self-assembly,mean-field and cluster methods,membrane structure,nanoparticles,OAI-PMH Harvest,proteins

Format application/pdf (imt)

Language English

Advisor Haselwandter, Christoph (committee chair), Boedicker, James (committee member), El-Naggar, Moh (committee member), Johnson, Clifford (committee member), Nakano, Aiichiro (committee member)

Creator Email mingyuam@usc.edu,victor49152@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC15612359

Unique identifier UC15612359

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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