Dependence of protein-induced lipid bilayer deformations on protein shape.

Membrane proteins typically deform the surrounding lipid bilayer membrane, which can play an important role in the function, regulation, and organization of membrane proteins. Membrane elasticity theory provides a beautiful description of protein-induced lipid bilayer deformations, in which all physical parameters can be directly determined from experiments. While analytic solutions of protein-induced elastic bilayer deformations are most easily developed for proteins with approximately circular cross sections, structural biology has shown that membrane proteins come in a variety of distinct shapes, with often considerable deviations from a circular cross section. We develop here a boundary value method (BVM) that permits the construction of analytic solutions of protein-induced elastic bilayer deformations for protein shapes with arbitrarily large deviations from a circular cross section, for constant as well as variable boundary conditions along the bilayer-protein interface. We apply this BVM to protein-induced lipid bilayer thickness deformations. Our BVM reproduces available analytic solutions for proteins with circular cross section and yields, for proteins with noncircular cross section, excellent agreement with numerical, finite element solutions. On this basis, we formulate a simple analytic approximation of the bilayer thickness deformation energy associated with general protein shapes and show that, for modest deviations from rotational symmetry, this analytic approximation is in good agreement with BVM solutions. Using the BVM, we survey the dependence of protein-induced elastic bilayer thickness deformations on protein shape, and thus explore how the coupling of protein shape and bilayer thickness deformations affects protein oligomerization and transitions in protein conformational state.

Stochastic effects in bacterial communication mediated by extracellular vesicles.

Quorum sensing (QS) allows bacterial cells to sense changes in local cell density and, hence, to regulate multicellular processes, including biofilm formation, regulation of virulence, and horizontal gene transfer. While, traditionally, QS was thought to involve the exchange of extracellular signal molecules free in solution, recent experiments have shown that for some bacterial systems a substantial fraction of signal molecules are packaged and delivered in extracellular vesicles. How the packaging of signal molecules in extracellular vesicles influences the ability of cells to communicate and coordinate multicellular behaviors remains largely unknown. We present here a stochastic reaction-diffusion model of QS that accounts for the exchange of both freely diffusing and vesicle-associated signal molecules. We find that the delivery of signal molecules via extracellular vesicles amplifies local fluctuations in the signal concentration, which can strongly affect the dynamics and spatial range of bacterial communication. For systems with multiple bacterial colonies, extracellular vesicles provide an alternate pathway for signal transport between colonies, and may be crucial for long-distance signal exchange in environments with strong degradation of free signal molecules.

Piezo1 as a force-through-membrane sensor in red blood cells.

Piezo1 is the stretch activated Ca2+ channel in red blood cells that mediates homeostatic volume control. Here, we study the organization of Piezo1 in red blood cells using a combination of super-resolution microscopy techniques and electron microscopy. Piezo1 adopts a non-uniform distribution on the red blood cell surface, with a bias toward the biconcave 'dimple'. Trajectories of diffusing Piezo1 molecules, which exhibit confined Brownian diffusion on short timescales and hopping on long timescales, also reflect a bias toward the dimple. This bias can be explained by 'curvature coupling' between the intrinsic curvature of the Piezo dome and the curvature of the red blood cell membrane. Piezo1 does not form clusters with itself, nor does it colocalize with F-actin, Spectrin, or the Gardos channel. Thus, Piezo1 exhibits the properties of a force-through-membrane sensor of curvature and lateral tension in the red blood cell.

Quantitative prediction and measurement of Piezo's membrane footprint.

Piezo proteins are mechanosensitive ion channels that can locally curve the membrane into a dome shape [Y. R. Guo, R. MacKinnon, eLife 6, e33660 (2017)]. The curved shape of the Piezo dome is expected to deform the surrounding lipid bilayer membrane into a membrane footprint, which may serve to amplify Piezo's sensitivity to applied forces [C. A. Haselwandter, R. MacKinnon, eLife 7, e41968 (2018)]. If Piezo proteins are embedded in lipid bilayer vesicles, the membrane shape deformations induced by the Piezo dome depend on the vesicle size. We employ here membrane elasticity theory to predict, with no free parameters, the shape of such Piezo vesicles outside the Piezo dome, and show that the predicted vesicle shapes agree quantitatively with the corresponding measured vesicle shapes obtained through cryoelectron tomography, for a range of vesicle sizes [W. Helfrich, Z. Naturforsch. C 28, 693-703 (1973)]. On this basis, we explore the coupling between Piezo and membrane shape and demonstrate that the features of the Piezo dome affecting Piezo's membrane footprint approximately follow a spherical cap geometry. Our work puts into place the foundation for deducing key elastic properties of the Piezo dome from membrane shape measurements and provides a general framework for quantifying how proteins deform bilayer membranes.

Elastic properties and shape of the Piezo dome underlying its mechanosensory function.

We show in the companion paper that the free membrane shape of lipid bilayer vesicles containing the mechanosensitive ion channel Piezo can be predicted, with no free parameters, from membrane elasticity theory together with measurements of the protein geometry and vesicle size [C. A. Haselwandter, Y. R. Guo, Z. Fu, R. MacKinnon, Proc. Natl. Acad. Sci. U.S.A., 10.1073/pnas.2208027119 (2022)]. Here we use these results to determine the force that the Piezo dome exerts on the free membrane and hence, that the free membrane exerts on the Piezo dome, for a range of vesicle sizes. From vesicle shape measurements alone, we thus obtain a force-distortion relationship for the Piezo dome, from which we deduce the Piezo dome's intrinsic radius of curvature, [Formula: see text] nm, and bending stiffness, [Formula: see text], in freestanding lipid bilayer membranes mimicking cell membranes. Applying these estimates to a spherical cap model of Piezo embedded in a lipid bilayer, we suggest that Piezo's intrinsic curvature, surrounding membrane footprint, small stiffness, and large area are the key properties of Piezo that give rise to low-threshold, high-sensitivity mechanical gating.

Mechanochemical coupling of lipid organization and protein function through membrane thickness deformations.

Cell membranes are composed of a great variety of protein and lipid species with distinct unperturbed hydrophobic thicknesses. To achieve hydrophobic matching, the lipid bilayer tends to deform around membrane proteins so as to match the protein hydrophobic thickness at bilayer-protein interfaces. Such protein-induced distortions of the lipid bilayer hydrophobic thickness incur a substantial energy cost that depends critically on the bilayer-protein hydrophobic mismatch, while distinct conformational states of membrane proteins often show distinct hydrophobic thicknesses. As a result, hydrophobic interactions between membrane proteins and lipids can yield a rich interplay of lipid-protein organization and transitions in protein conformational state. We combine here the membrane elasticity theory of protein-induced lipid bilayer thickness deformations with the Landau-Ginzburg theory of lipid domain formation to systematically explore the coupling between local lipid organization, lipid and protein hydrophobic thickness, and protein-induced lipid bilayer thickness deformations in membranes with heterogeneous lipid composition. We allow for a purely mechanical coupling of lipid and protein composition through the energetics of protein-induced lipid bilayer thickness deformations as well as a chemical coupling driven by preferential interactions between particular lipid and protein species. We find that the resulting lipid-protein organization can endow membrane proteins with diverse and controlled mechanical environments that, via protein-induced lipid bilayer thickness deformations, can strongly influence protein function. The theoretical approach employed here provides a general framework for the quantitative prediction of how membrane thickness deformations influence the joint organization and function of lipids and proteins in cell membranes.

Self-assembly of polyhedral bilayer vesicles from Piezo ion channels.

Piezo ion channels underlie many forms of mechanosensation in vertebrates and have been found to bend the membrane into strongly curved dome shapes. We develop a methodology describing the self-assembly of lipids and Piezo proteins into polyhedral bilayer vesicles. We validate this methodology for bilayer vesicles formed from bacterial mechanosensitive channels of small conductance, for which experiments found a polyhedral arrangement of proteins with snub cube symmetry and a well-defined characteristic vesicle size. On this basis, we calculate the self-assembly diagram for polyhedral bilayer vesicles formed from Piezo proteins. We find that the radius of curvature of the Piezo dome provides a critical control parameter for the self-assembly of Piezo vesicles, with high abundances of Piezo vesicles with octahedral, icosahedral, and snub cube symmetry with increasing Piezo dome radius of curvature.

Mechanics of cup-shaped caveolae.

Caveolae are cell membrane invaginations of defined lipid and protein composition that flatten with increasing membrane tension. Super-resolution light microscopy and electron microscopy have revealed that caveolae can take a variety of cuplike shapes. We show here that, for the range in membrane tension relevant for cell membranes, the competition between membrane tension and membrane bending yields caveolae with cuplike shapes similar to those observed experimentally. We find that the caveola shape and its sensitivity to changes in membrane tension can depend strongly on the caveola spontaneous curvature and on the size of caveola domains. Our results suggest that heterogeneity in caveola shape produces a staggered response of caveolae to mechanical perturbations of the cell membrane, which may facilitate regulation of membrane tension over the wide range of scales thought to be relevant for cell membranes.

Stochastic self-assembly of reaction-diffusion patterns in synaptic membranes.

Synaptic receptor and scaffold molecules self-assemble into membrane protein domains, which play an important role in signal transmission across chemical synapses. Experiment and theory have shown that the formation of receptor-scaffold domains of the characteristic size observed in nerve cells can be understood from the receptor and scaffold reaction and diffusion processes suggested by experiments. We employ here kinetic Monte Carlo (KMC) simulations to explore the self-assembly of synaptic receptor-scaffold domains in a stochastic lattice model of receptor and scaffold reaction-diffusion dynamics. For reaction and diffusion rates within the ranges of values suggested by experiments we find, in agreement with previous mean-field calculations, self-assembly of receptor-scaffold domains of a size similar to that observed in experiments. Comparisons between the results of our KMC simulations and mean-field solutions suggest that the intrinsic noise associated with receptor and scaffold reaction and diffusion processes accelerates the self-assembly of receptor-scaffold domains, and confers increased robustness to domain formation. In agreement with experimental observations, our KMC simulations yield a prevalence of scaffolds over receptors in receptor-scaffold domains. Our KMC simulations show that receptor and scaffold reaction-diffusion dynamics can inherently give rise to plasticity in the overall properties of receptor-scaffold domains, which may be utilized by nerve cells to regulate the receptor number at chemical synapses.

Nanodissected elastically loaded clathrin lattices relax to increased curvature.

Clathrin-mediated endocytosis (CME) is the major endocytosis pathway for the specific internalization of large compounds, growth factors, and receptors. Formation of internalized vesicles from the flat plasma membrane is accompanied by maturation of cytoplasmic clathrin coats. How clathrin coats mature and the mechanistic role of clathrin coats are still largely unknown. Maturation models proposed clathrin coats to mature at constant radius or constant area, driven by molecular actions or elastic energy. Here, combining high-speed atomic force microscopy (HS-AFM) imaging, HS-AFM nanodissection, and elasticity theory, we show that clathrin lattices deviating from the intrinsic curvature of clathrin form elastically loaded assemblies. Upon nanodissection of the clathrin network, the stored elastic energy in these lattices drives lattice relaxation to accommodate an ideal area-curvature ratio toward the formation of closed clathrin-coated vesicles. Our work supports that the release of elastic energy stored in curvature-frustrated clathrin lattices could play a major role in CME.

Effect of protein steric constraints on the symmetry of membrane protein polyhedra.

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. The observed symmetry and size of MPPNs can be understood from the interplay of protein-induced lipid bilayer deformations in MPPNs, topological defects in protein packing necessitated by the spherical shape of MPPNs, and thermal fluctuations in MPPN self-assembly. We explore here the effect of protein steric constraints on MPPN shape. The protein steric constraints considered here may arise 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. Calculating MPPN self-assembly diagrams under protein steric constraints we find that protein steric constraints can strongly affect MPPN self-assembly. Depending on the specific scenario considered, protein steric constraints can leave large portions of the MPPN self-assembly diagrams with no clearly defined MPPN symmetry or substantially expand the regions of MPPN self-assembly diagrams dominated by highly symmetric MPPN states, such as MPPNs with icosahedral or snub cube symmetry. Our results suggest that modification of protein steric constraints may allow the directed self-assembly of MPPNs with specified symmetry, size, and protein composition and may thus facilitate the further utilization of MPPNs for membrane protein structural analysis or targeted drug delivery.

Symmetry of membrane protein polyhedra with heterogeneous protein size.

In experiments on membrane protein polyhedral nanoparticles (MPPNs) [Basta et al., Proc. Natl. Acad. Sci. USA 111, 670 (2014)PNASA60027-842410.1073/pnas.1321936111], it has been observed that membrane proteins and lipids can self-assemble into closed lipid bilayer vesicles with a polyhedral arrangement of membrane proteins. In particular, MPPNs formed from the mechanosensitive channel of small conductance (MscS) were found to have the symmetry of the snub cube-a chiral, Archimedean solid-with one MscS protein located at each one of the 24 vertices of the snub cube. It is currently unknown whether MPPNs with heterogeneous protein composition maintain a high degree of symmetry. Inspired by previous work on viral capsid symmetry, we employ here computational modeling to study the symmetry of MPPNs with heterogeneous protein size. We focus on MPPNs formed from MscS proteins, which can exist in closed or open conformational states with distinct sizes. We find that, as an increasing number of closed-state MscS proteins transitions to the open conformational state of MscS, the minimum-energy MscS arrangement in MPPNs follows a strikingly regular pattern, with the dominant MPPN symmetry always being provided by the snub cube. Our results suggest that MPPNs with heterogeneous protein size can be highly symmetric, with a well-defined polyhedral ordering of membrane proteins of different sizes.

Regulation of membrane proteins through local heterogeneity in lipid bilayer thickness.

Cell membranes show an intricate organization of lipids and membrane proteins into domains with distinct composition and hydrophobic thickness. Using mechanosensitive ion channels as a model system, we employ the membrane elasticity theory of lipid-protein interactions together with the Landau-Ginzburg theory of lipid domain formation to quantify protein-induced lipid bilayer thickness deformations in lipid bilayers with heterogeneous hydrophobic thickness. We show that protein-induced lipid bilayer thickness deformations yield, without any assumptions about preferential interactions between particular lipid and protein species, organization of lipids and membrane proteins according to their preferred hydrophobic thickness, and couple the conformational states of membrane proteins to the local membrane composition. Our calculations suggest that protein-induced lipid bilayer thickness deformations endow proteins in cell membranes with diverse and controlled mechanical environments that, in turn, allow targeted regulation of membrane proteins.

Supramolecular organization of membrane proteins with anisotropic hydrophobic thickness.

Experiments have revealed that membrane proteins often self-assemble into locally ordered clusters. Such membrane protein lattices can play key roles in the functional organization of cell membranes. Membrane protein organization can be driven, at least in part, by bilayer-mediated elastic interactions between membrane proteins. For membrane proteins with anisotropic hydrophobic thickness, bilayer-mediated protein interactions are inherently directional. Here we establish general relations between anisotropy in membrane protein hydrophobic thickness and supramolecular membrane protein organization. We show that protein symmetry is distinctively reflected in the energy landscape of bilayer-mediated protein interactions, favoring characteristic lattice architectures of membrane protein clusters. We find that, in the presence of thermal fluctuations, anisotropy in protein hydrophobic thickness can induce membrane proteins to form mesh-like structures dividing the membrane into compartments. Our results help to elucidate the physical principles and mechanisms underlying the functional organization of cell membranes.

Piezo's membrane footprint and its contribution to mechanosensitivity.

Piezo1 is an ion channel that gates open when mechanical force is applied to a cell membrane, thus allowing cells to detect and respond to mechanical stimulation. Molecular structures of Piezo1 reveal a large ion channel with an unusually curved shape. This study analyzes how such a curved ion channel interacts energetically with the cell membrane. Through membrane mechanical calculations, we show that Piezo1 deforms the membrane shape outside the perimeter of the channel into a curved 'membrane footprint'. This membrane footprint amplifies the sensitivity of Piezo1 to changes in membrane tension, rendering it exquisitely responsive. We assert that the shape of the Piezo channel is an elegant example of molecular form evolved to optimize a specific function, in this case tension sensitivity. Furthermore, the predicted influence of the membrane footprint on Piezo gating is consistent with the demonstrated importance of membrane-cytoskeletal attachments to Piezo gating.

Directed Supramolecular Organization of N-BAR Proteins through Regulation of H0 Membrane Immersion Depth.

Many membrane remodeling events rely on the ability of curvature-generating N-BAR membrane proteins to organize into distinctive supramolecular configurations. Experiments have revealed a conformational switch in N-BAR proteins resulting in vesicular or tubular membrane shapes, with shallow membrane immersion of the H0 amphipathic helices of N-BAR proteins on vesicles but deep H0 immersion on tubes. We develop here a minimal elastic model of the local thinning of the lipid bilayer resulting from H0 immersion. Our model predicts that the observed conformational switch in N-BAR proteins produces a corresponding switch in the bilayer-mediated N-BAR interactions due to the H0 helices. In agreement with experiments, we find that bilayer-mediated H0 interactions oppose N-BAR multimerization for the shallow H0 membrane immersion depths measured on vesicles, but promote self-assembly of supramolecular N-BAR chains for the increased H0 membrane immersion depths measured on tubes. Finally, we consider the possibility that bilayer-mediated H0 interactions might contribute to the concerted structural reorganization of N-BAR proteins suggested by experiments. Our results indicate that the membrane immersion depth of amphipathic protein helices may provide a general molecular control parameter for membrane organization.

Stochastic lattice model of synaptic membrane protein domains.

Neurotransmitter receptor molecules, concentrated in synaptic membrane domains along with scaffolds and other kinds of proteins, are crucial for signal transmission across chemical synapses. In common with other membrane protein domains, synaptic domains are characterized by low protein copy numbers and protein crowding, with rapid stochastic turnover of individual molecules. We study here in detail a stochastic lattice model of the receptor-scaffold reaction-diffusion dynamics at synaptic domains that was found previously to capture, at the mean-field level, the self-assembly, stability, and characteristic size of synaptic domains observed in experiments. We show that our stochastic lattice model yields quantitative agreement with mean-field models of nonlinear diffusion in crowded membranes. Through a combination of analytic and numerical solutions of the master equation governing the reaction dynamics at synaptic domains, together with kinetic Monte Carlo simulations, we find substantial discrepancies between mean-field and stochastic models for the reaction dynamics at synaptic domains. Based on the reaction and diffusion properties of synaptic receptors and scaffolds suggested by previous experiments and mean-field calculations, we show that the stochastic reaction-diffusion dynamics of synaptic receptors and scaffolds provide a simple physical mechanism for collective fluctuations in synaptic domains, the molecular turnover observed at synaptic domains, key features of the observed single-molecule trajectories, and spatial heterogeneity in the effective rates at which receptors and scaffolds are recycled at the cell membrane. Our work sheds light on the physical mechanisms and principles linking the collective properties of membrane protein domains to the stochastic dynamics that rule their molecular components.

Distribution of randomly diffusing particles in inhomogeneous media.

Diffusion can be conceptualized, at microscopic scales, as the random hopping of particles between neighboring lattice sites. In the case of diffusion in inhomogeneous media, distinct spatial domains in the system may yield distinct particle hopping rates. Starting from the master equations (MEs) governing diffusion in inhomogeneous media we derive here, for arbitrary spatial dimensions, the deterministic lattice equations (DLEs) specifying the average particle number at each lattice site for randomly diffusing particles in inhomogeneous media. We consider the case of free (Fickian) diffusion with no steric constraints on the maximum particle number per lattice site as well as the case of diffusion under steric constraints imposing a maximum particle concentration. We find, for both transient and asymptotic regimes, excellent agreement between the DLEs and kinetic Monte Carlo simulations of the MEs. The DLEs provide a computationally efficient method for predicting the (average) distribution of randomly diffusing particles in inhomogeneous media, with the number of DLEs associated with a given system being independent of the number of particles in the system. From the DLEs we obtain general analytic expressions for the steady-state particle distributions for free diffusion and, in special cases, diffusion under steric constraints in inhomogeneous media. We find that, in the steady state of the system, the average fraction of particles in a given domain is independent of most system properties, such as the arrangement and shape of domains, and only depends on the number of lattice sites in each domain, the particle hopping rates, the number of distinct particle species in the system, and the total number of particles of each particle species in the system. Our results provide general insights into the role of spatially inhomogeneous particle hopping rates in setting the particle distributions in inhomogeneous media.

Symmetry and Size of Membrane Protein Polyhedral Nanoparticles.

In recent experiments [T. Basta et al., Proc. Natl. Acad. Sci. U.S.A. 111, 670 (2014)] lipids and membrane proteins were observed to self-assemble into membrane protein polyhedral nanoparticles (MPPNs) with a well-defined polyhedral protein arrangement and characteristic size. We develop a model of MPPN self-assembly in which the preferred symmetry and size of MPPNs emerge from the interplay of protein-induced lipid bilayer deformations, topological defects in protein packing, and thermal effects. With all model parameters determined directly from experiments, our model correctly predicts the observed symmetry and size of MPPNs. Our model suggests how key lipid and protein properties can be modified to produce a range of MPPN symmetries and sizes in experiments.

Bilayer-thickness-mediated interactions between integral membrane proteins.

Hydrophobic thickness mismatch between integral membrane proteins and the surrounding lipid bilayer can produce lipid bilayer thickness deformations. Experiment and theory have shown that protein-induced lipid bilayer thickness deformations can yield energetically favorable bilayer-mediated interactions between integral membrane proteins, and large-scale organization of integral membrane proteins into protein clusters in cell membranes. Within the continuum elasticity theory of membranes, the energy cost of protein-induced bilayer thickness deformations can be captured by considering compression and expansion of the bilayer hydrophobic core, membrane tension, and bilayer bending, resulting in biharmonic equilibrium equations describing the shape of lipid bilayers for a given set of bilayer-protein boundary conditions. Here we develop a combined analytic and numerical methodology for the solution of the equilibrium elastic equations associated with protein-induced lipid bilayer deformations. Our methodology allows accurate prediction of thickness-mediated protein interactions for arbitrary protein symmetries at arbitrary protein separations and relative orientations. We provide exact analytic solutions for cylindrical integral membrane proteins with constant and varying hydrophobic thickness, and develop perturbative analytic solutions for noncylindrical protein shapes. We complement these analytic solutions, and assess their accuracy, by developing both finite element and finite difference numerical solution schemes. We provide error estimates of our numerical solution schemes and systematically assess their convergence properties. Taken together, the work presented here puts into place an analytic and numerical framework which allows calculation of bilayer-mediated elastic interactions between integral membrane proteins for the complicated protein shapes suggested by structural biology and at the small protein separations most relevant for the crowded membrane environments provided by living cells.