Softening in Two-Component Lipid Mixtures by Spontaneous Curvature Variance

The bending modulus of a lipid bilayer quantifies its mechanical resistance to curvature. It is typically understood in terms of thickness; e.g., thicker bilayers are usually stiffer. Here, we describe an additional and powerful molecular determinant of stiffness─the variance in the distribution of curvature sensitivity of lipids and lipid conformations. Zwitterionic choline and ethanolamine headgroups of glycerophospholipids dynamically explore inter- and intraspecies interactions, leading to transient clustering. We demonstrate that these clusters couple strongly to negative curvature, exciting undulatory membrane modes and reducing the apparent bending modulus. Three force fields (Martini 2, Martini 3, and all-atom CHARMM C36) each show the effect to a different extent, with the coarse-grained Martini models showing the most clustering and thus the most softening. The theory is a guide to understanding the stiffness of biological membranes with their complex composition, as well as how choices of force field parameterization are translated into mechanical stiffness.

Computing the influence of lipids and lipid complexes on membrane mechanics

Methodology for extracting the spontaneous curvature, bending modulus, and neutral surface of a lipid bilayer is described. The "SPEX" method is a robust technique for computing the bilayer bending modulus while allowing for resolution of the spontaneous curvature of specific interacting lipids and complexes, and the dependence of spontaneous curvature on wavelength. The method is described referring to the publicly available MembraneAnalysis.jl software package.

Stability of a biomembrane tube covered with proteins

Membrane tubes are essential structural features in cells that facilitate biomaterial transport and inter- and intracellular signaling. The shape of these tubes can be regulated by the proteins that surround and adhere to them. We study the stability of a biomembrane tube coated with proteins by combining linear stability analysis, out-of-equilibrium hydrodynamic calculations, and numerical solutions of a Helfrich-like membrane model. Our analysis demonstrates that both long- and short-wavelength perturbations can destabilize the tubes. Numerical simulations confirm the derived linear stability criteria and yield the nonlinearly perturbed vesicle shapes. Our study highlights the interplay between membrane shape and protein density, where the shape instability concurs with a redistribution of proteins into a banded pattern.

Comparison of structural effects of cholesterol, lanosterol, and oxysterol on phospholipid (POPC) bilayers

Membrane sterols contribute to the function of biomembranes by regulating the physical properties of the lipid bilayers. Cholesterol, a typical mammalian sterol, is biosynthesized by oxidation of lanosterol. From a molecular evolutionary perspective, lanosterol is considered the ancestral molecule of cholesterol. Here, we studied whether cholesterol is superior to lanosterol in regulating the physical properties of the lipid bilayer in terms of the structural effect on model biomembranes composed of a phospholipid. For comparison, oxysterol, which is formed by oxidation of cholesterol, was also studied. The phospholipid used was 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), which is abundantly found in mammalian biomembranes, and 7β-hydroxycholesterol, which is highly cytotoxic, was used as the oxysterol. The apparent molecular volume was calculated from the mass density determined by the flotation method using H2O and D2O, and the bilayer thickness was determined by reconstructing the electron density distribution from X-ray diffraction data of the POPC/sterol mixtures at a sterol concentration of 30 mol%. The apparent occupied area at the bilayer surface was calculated from the above two structural data. The cholesterol system had the thickest bilayer thickness and the smallest occupied area of the three sterols studied here. This indicates that the POPC/cholesterol bilayer has a better barrier property than the other two systems. Compared to cholesterol, the effects of lanosterol and 7β-hydroxycholesterol on lipid bilayer properties can be interpreted as suboptimal for the function of mammalian biomembranes.

Softening in two-component lipid mixtures by spontaneous curvature variance

The bending modulus of a lipid bilayer quantifies its mechanical resistance to curvature. It is typically understood in terms of thickness, e.g., thicker bilayers are stiffer. Here, we describe an additional and powerful molecular determinant of stiffness - the variance in the distribution of curvature sensitivity of lipids and lipid conformations. Zwitterionic choline and ethanolamine head-groups of glycero-phospholipids dynamically explore inter- and intra-species interactions, leading to transient clustering. We demonstrate that these clusters couple strongly to negative curvature, exciting undulatory membrane modes and reducing the apparent bending modulus. Three forcefields (Martini 2, Martini 3, and all-atom CHARMM C36) each show the effect to a different extent, with the coarse-grained Martini models showing the most clustering and thus the most softening. The theory is a guide to understanding the stiffness of biological membranes with their complex composition, as well as how choices of forcefield parameterization are translated into mechanical stiffness.

Forceful patterning: theoretical principles of mechanochemical pattern formation

Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.

Origin of the nonlinear structural and mechanical properties in oppositely curved lipid mixtures

Structural and mechanical properties of membranes such as thickness, tail order, bending modulus and curvature energetics play crucial role in controlling various cellular functions that depend on the local lipid organization and membrane reshaping. While behavior of these biophysical properties are well understood in single component membranes, very little is known about how do they change in the mixed lipid membranes. Often various properties of the mixed lipid bilayers are assumed to change linearly with the mole fractions of the constituent lipids which, however, is true for "ideal" mixing only. In this study, using molecular dynamics simulations, we show that structural and mechanical properties of binary lipid mixture change nonlinearly with the lipid mole fractions, and the strength of the nonlinearity depends on two factors - spontaneous curvature difference and locally inhomogeneous interactions between the lipid components.

The Membrane-Mediated Interaction of Liquid-Ordered Lipid Domains in the Presence of Amphipathic Peptides

The lipid membranes of living cells are composed of a large number of lipid types and can undergo phase separation with the formation of nanometer-scale liquid-ordered lipid domains, also called rafts. Raft coalescence, i.e., the fusion of lipid domains, is involved in important cell processes, such as signaling and trafficking. In this work, within the framework of the theory of elasticity of lipid membranes, we explore how amphipathic peptides adsorbed on lipid membranes may affect the domain-domain fusion processes. We show that the elastic deformations of lipid membranes drive amphipathic peptides to the boundary of lipid domains, which leads to an increase in the average energy barrier of the domain-domain fusion, even if the surface concentration of amphipathic peptides is low and the domain boundaries are only partially occupied by the peptides. This inhibition of the fusion of lipid domains may lead to negative side effects of using amphipathic peptides as antimicrobial agents.

Model of membrane deformations driven by a surface pH gradient

Many cellular organelles are membrane-bound structures with complex membrane composition and shape. Their shapes have been observed to depend on the metabolic state of the organelle and the mechanisms that couple biochemical pathways and membrane shape are still actively investigated. Here, we study a model coupling inhomogeneities in the lipid composition and membrane geometry via a generalized Helfrich free energy. We derive the resulting stress tensor, the Green's function for a tubular membrane, and compute the phase diagram of the induced deformations. We then apply this model to study the deformation of mitochondria cristae described as membrane tubes supporting a pH gradient at its surface. This gradient in turn controls the lipid composition of the membrane via the protonation or deprotonation of cardiolipins, which are acid-based lipids known to be crucial for mitochondria shape and functioning. Our model predicts the appearance of tube deformations resembling the observed shape changes of cristea when submitted to a proton gradient.

Simulated dynamic cholesterol redistribution favors membrane fusion pore constriction

Endo- and exocytosis proceed through a highly strained membrane fusion pore topology regardless of the aiding protein machinery. The membrane's lipid components bias fusion pores toward expansion or closure, modifying the necessary work done by proteins. Cholesterol, a key component of plasma membranes, promotes both inverted lipid phases with concave leaflets (i.e., negative total curvature, which thins the leaflet) and flat bilayer phases with thick, ordered hydrophobic interiors. We demonstrate by theory and simulation that both leaflets of nascent catenoidal fusion pores have negative total curvature. Furthermore, the hydrophobic core of bilayers with strong negative Gaussian curvature is thinned. Therefore, it is an open question whether cholesterol will be enriched in these regions because of the negative total curvature or depleted because of the membrane thinning. Here, we compare all-atom molecular dynamics simulations (built using a procedure to create specific fusion pore geometries) and theory to understand the underlying reasons for lipid redistribution on fusion pores. Our all-atom molecular dynamics simulations resolve this question by showing that cholesterol is strongly excluded from the thinned neck of fusion and fission pores, revealing that thickness (and/or lipid order) influences cholesterol distributions more than curvature. The results imply that cholesterol exclusion can drive fusion pore closure by creating a small, cholesterol-depleted zone in the neck. This model agrees with literature evidence that membrane reshaping is connected to cholesterol-dependent lateral phase separation.

Demixing of homogeneous binary lipid membranes induced by protein inclusions

We study a model of a lipid bilayer membrane described by two order parameters: the chemical composition described using the Gaussian model and the spatial configuration described with the elastic deformation model of a membrane with a finite thickness or, equivalently, for an adherent membrane. We assume and explain on physical grounds the linear coupling between the two order parameters. Using the exact solution, we calculate the correlation functions and order parameter profiles. We also study the domains that form around inclusions on the membrane. We propose and compare six distinct ways to quantify the size of such domains. Despite its simplicity, the model has many interesting features like the Fisher-Widom line and two distinct critical regions.

Kinetic relaxation of giant vesicles validates diffusional softening in a binary lipid mixture

The stiffness of biological membranes determines the work required by cellular machinery to form and dismantle vesicles and other lipidic shapes. Model membrane stiffness can be determined from the equilibrium distribution of giant unilamellar vesicle surface undulations observable by phase contrast microscopy. With two or more components, lateral fluctuations of composition will couple to surface undulations depending on the curvature sensitivity of the constituent lipids. The result is a broader distribution of undulations whose complete relaxation is partially determined by lipid diffusion. In this work, kinetic analysis of the undulations of giant unilamellar vesicles made of phosphatidylcholine-phosphatidylethanolamine mixtures validates the molecular mechanism by which the membrane is made 25% softer than a single-component one. The mechanism is relevant to biological membranes, which have diverse and curvature-sensitive lipids.

On the deformability of additivated phosphatidylcholine liposomes: Molecular dynamic regimes and membrane elasticity

Liposomes with enhanced elasticity have been proven to increase the efficiency of drug transport across the skin. The understanding of the background physicochemical processes driving the liposome viscoelastic properties is an essential feature for the design of effective formulations involving different lipids and additive molecules. In this work we use field-cycled nuclear magnetic resonance relaxometry to analyze both the mechanical properties of liposome membranes, and their relationship with the involved molecular dynamics. Different liposomal formulations were considered. We show a correlation between the molecular dynamical regime and mesoscopic physical parameters that define the expected deformability of the vesicles. Results strongly suggest that the purity of the used lipids may influence the elastic properties of the membranes in an appreciable way. Common features in the behaviour of the involved dynamic variables were identified by comparing formulations with surfactants of similar molecular weight.

Systematic analysis of curvature-dependent lipid dynamics in a stochastic 3D membrane model

To minimize the free energy of the system, lipid membranes display curvature-dependent rearrangements at the local and global scale. The optimal membrane shape is generally approximated by averaging the curvature preference of individual lipids across the whole surface. Potential stress due to imperfections in lipid packing caused by local lipid inhomogeneities, however, is frequently neglected. Here, we developed a stochastic 3D membrane model to investigate the relevance of this parameter for shape-dependent lipid and membrane dynamics. A systematic analysis of the discretized Helfrich type Hamiltonian indicates that stress-energy arising from imperfections in packing is analogous to van der Waals interactions, jointly determining membrane shape and localization of curvature-sensitive lipids based on their relative strengths. Insights from this work can be used to characterize natural and design synthetic agents for membrane-shape changes.

Determination of elastic parameters of lipid membranes from simulation under varied external pressure

Many cellular processes such as endocytosis, exocytosis, and vesicle trafficking involve membrane deformations, which can be analyzed in the framework of the elastic theories of lipid membranes. These models operate with phenomenological elastic parameters. A connection between these parameters and the internal structure of lipid membranes can be provided by three-dimensional (3D) elastic theories. Considering a membrane as a 3D layer, Campelo et al. [F. Campelo et al., Adv. Colloid Interface Sci. 208, 25 (2014)10.1016/j.cis.2014.01.018] developed a theoretical basis for the calculation of elastic parameters. In this work we generalize and improve this approach by considering a more general condition of global incompressibility instead of local incompressibility. Crucially, we find an important correction to the theory of Campelo et al., which if not taken into account leads to a significant miscalculation of elastic parameters. With the total volume conservation taken into account, we derive an expression for the local Poisson's ratio, which determines how the local volume changes upon stretching and permits a more precise determination of elastic parameters. Also, we substantially simplify the procedure by calculating the derivatives of the moments of the local tension with respect to stretching instead of calculating the local stretching modulus. We obtain a relation between the Gaussian curvature modulus as a function of stretching and the bending modulus, showing that these two elastic parameters are not independent, as was previously assumed. The proposed algorithm is applied to membranes composed of pure dipalmitoylphosphatidylcholine (DPPC), dioleoylphosphatidylcholine (DOPC), and their mixture. The following elastic parameters of these systems are obtained: the monolayer bending and stretching moduli, spontaneous curvature, neutral surface position, and local Poisson's ratio. It is shown that the bending modulus of the DPPC/DOPC mixture follows a more complex trend than predicted by the classical Reuss averaging, which is often employed in theoretical frameworks.

Close, but not too close: a mesoscopic description of (a)symmetry and membrane shaping mechanisms

Biomembranes are fundamental to our understanding of the cell, the basic building block of all life. An intriguing aspect of membranes is their ability to assume a variety of shapes, which is crucial for cell function. Here, we review various membrane shaping mechanisms with special focus on the current understanding of how local curvature and local rigidity induced by membrane proteins leads to emerging forces and consequently large-scale membrane deformations. We also argue that describing the interaction of rigid proteins with membranes purely in terms of local membrane curvature is incomplete and that changes in the membrane rigidity moduli must also be considered.

Modulation of self-organizing circuits at deforming membranes by intracellular and extracellular factors

Mechanical forces exerted to the plasma membrane induce cell shape changes. These transient shape changes trigger, among others, enrichment of curvature-sensitive molecules at deforming membrane sites. Strikingly, some curvature-sensing molecules not only detect membrane deformation but can also alter the amplitude of forces that caused to shape changes in the first place. This dual ability of sensing and inducing membrane deformation leads to the formation of curvature-dependent self-organizing signaling circuits. How these cell-autonomous circuits are affected by auxiliary parameters from inside and outside of the cell has remained largely elusive. Here, we explore how such factors modulate self-organization at the micro-scale and its emerging properties at the macroscale.

Determination of Elastic Parameters of Lipid Membranes with Molecular Dynamics: A Review of Approaches and Theoretical Aspects

Lipid membranes are abundant in living organisms, where they constitute a surrounding shell for cells and their organelles. There are many circumstances in which the deformations of lipid membranes are involved in living cells: fusion and fission, membrane-mediated interaction between membrane inclusions, lipid-protein interaction, formation of pores, etc. In all of these cases, elastic parameters of lipid membranes are important for the description of membrane deformations, as these parameters determine energy barriers and characteristic times of membrane-involved phenomena. Since the development of molecular dynamics (MD), a variety of in silico methods have been proposed for the determination of elastic parameters of simulated lipid membranes. These MD methods allow for the consideration of details unattainable in experimental techniques and represent a distinct scientific field, which is rapidly developing. This work provides a review of these MD approaches with a focus on theoretical aspects. Two main challenges are identified: (i) the ambiguity in the transition from the continuum description of elastic theories to the discrete representation of MD simulations, and (ii) the determination of intrinsic elastic parameters of lipid mixtures, which is complicated due to the composition-curvature coupling effect.

Stiffening or softening of elastic media: Anomalous elasticity near phase transitions

We present the general theory of Ising transitions in isotropic elastic media with vanishing thermal expansion. By constructing a minimal model with appropriate spin-lattice couplings, we show that in two dimensions near a continuous transition the elasticity is anomalous in unusual ways: the system either significantly stiffens with a hitherto unknown unique, positional order logarithmically stronger than quasi-long-range order, or, as the inversion asymmetry of the order parameter in its coupling with strain increases, it destabilizes when system size L exceeds a finite threshold. In three dimensions, stronger inversion-asymmetric couplings induce instability to the long-range positional order for all L. Sufficiently strong order parameter-displacement couplings can also turn the phase transition first order at all dimensions, concomitant with finite jumps in the elastic modulii across the transition. Our theory establishes a one-to-one correspondence between the order of the phase transitions and anomalous elasticity near the transitions.

Statistical mechanics of phase transitions in elastic media with vanishing thermal expansion

We consider a minimal spin model for Ising transitions in an isotropic elastic medium in the zero thermal expansion (ZTE) limit. We set up the elastic theory for this system. We use this theory to identify and study the nature of the fluctuations in the system near the second order phase transitions at T_{c} in the ZTE limit given by dT_{c}/dV=0, where V is the system volume, and explore anomalous elasticity. Allowing for the local strain to couple asymmetrically or selectively with the states of the order parameter, we uncover the dramatic effects of these couplings on the fluctuations of the local displacements near T_{c}, and also on the nature of the phase transition itself. Near second-order phase transitions and with weak asymmetry in the order parameter-strain couplings, the variance of the displacement fluctuations in two dimensions scale with the system size L in a universal fashion as [ln(L/a_{0})]^{2/3}; a_{0} is a small-scale cutoff. Likewise, the correlation functions of the difference of the local displacements at two different points separated by r scale as [ln(r/a_{0})]^{2/3} for large r. For stronger selectivity above a finite threshold, this variance diverge as L exceeds beyond a (nonuniversal) size, determined by the model parameters, signaling a transition to a phase with only short-range order or the loss of the positional order of the elastic medium. At dimensions higher than two, for sufficiently weak selectivity, the variance of the displacement fluctuations is L-independent corresponding to long-range order. However, if the selectivity parameters rise beyond a dimension-dependent threshold value, then again the positional order is lost with a concomitant transition to a phase with short-range order. Large values of the order parameter-strain couplings can turn the phase transition into a first order as well. Our theory establishes a one-to-one correspondence between the order of phase transitions and anomalous elasticity near the transitions. Our theory should be a useful guide to possible synthesis of appropriate ZTE materials.

Molecular mechanisms of spontaneous curvature and softening in complex lipid bilayer mixtures

Membrane reshaping is an essential biological process. The chemical composition of lipid membranes determines their mechanical properties and thus the energetics of their shape. Hundreds of distinct lipid species make up native bilayers, and this diversity complicates efforts to uncover what compositional factors drive membrane stability in cells. Simplifying assumptions, therefore, are used to generate quantitative predictions of bilayer dynamics based on lipid composition. One assumption commonly used is that "per lipid" mechanical properties are both additive and constant-that they are an intrinsic property of lipids independent of the surrounding composition. Related to this is the assumption that lipid bulkiness, or "shape," determines its curvature preference, independently of context. In this study, all-atom molecular dynamics simulations on three separate multilipid systems were used to explicitly test these assumptions, applying methodology recently developed to isolate properties of single lipids or nanometer-scale patches of lipids. The curvature preference experienced by populations of lipid conformations were inferred from their redistribution on a dynamically fluctuating bilayer. Representative populations were extracted by both structural similarity and semi-automated hidden Markov model analysis. The curvature preferences of lipid dimers were then determined and compared with an additive model that combines the monomer curvature preference of both the individual lipids. In all three systems, we identified conformational subpopulations of lipid dimers that showed non-additive curvature preference, in each case mediated by a special chemical interaction (e.g., hydrogen bonding). Our study highlights the importance of specific chemical interactions between lipids in multicomponent bilayers and the impact of interactions on bilayer stiffness. We identify two mechanisms of bilayer softening: diffusional softening, driven by the dynamic coupling between lipid distributions and membrane undulations, and conformational softening, driven by the inter-conversion between distinct dimeric conformations.

Lipid redistribution in the highly curved footprint of Piezo1

We investigate the effects on the distribution of lipids in the plasma membrane that are caused by the insertion of a protein, Piezo1, that significantly distorts the membrane toward the cytosol. From coarse-grained molecular dynamics simulations, we find that the major effects occur in the outer, extracellular, leaflet. The mol fraction of cholesterol increases significantly in the curved region of the membrane close to Piezo1, while those of phosphatidylcholine and of sphingomyelin decrease. In the inner leaflet, mol fractions of cholesterol and of phosphatidylethanolamine decrease slightly as the protein is approached, while that of phosphatidylserine increases slightly. The mol fraction of phosphatidylcholine decreases markedly as the protein is approached. Most of these results are understood in the context of a theoretical model that utilizes two elements; (i) a coupling between the leaflets' actual curvatures and their compositionally-dependent spontaneous curvatures and, (ii) the dependence of the spontaneous curvatures not only on the mol fractions of the phospholipids, but also on the effect that cholesterol has on the spontaneous curvatures of the phospholipids.

Molecular Shape Solution for Mesoscopic Remodeling of Cellular Membranes

Cellular membranes self-assemble from and interact with various molecular species. Each molecule locally shapes the lipid bilayer, the soft elastic core of cellular membranes. The dynamic architecture of intracellular membrane systems is based on elastic transformations and lateral redistribution of these elementary shapes, driven by chemical and curvature stress gradients. The minimization of the total elastic stress by such redistribution composes the most basic, primordial mechanism of membrane curvature-composition coupling (CCC). Although CCC is generally considered in the context of dynamic compositional heterogeneity of cellular membrane systems, in this article we discuss a broader involvement of CCC in controlling membrane deformations. We focus specifically on the mesoscale membrane transformations in open, reservoir-governed systems, such as membrane budding, tubulation, and the emergence of highly curved sites of membrane fusion and fission. We reveal that the reshuffling of molecular shapes constitutes an independent deformation mode with complex rheological properties.This mode controls effective elasticity of local deformations as well as stationary elastic stress, thus emerging as a major regulator of intracellular membrane remodeling.

A Theoretical Basis for Nanodomains

We review the current theories of nanodomain, or "raft," formation. We emphasize that the idea that they are co-exisiting Lo and Ld phases is fraught with difficulties, as is the closely related idea that they are due to critical fluctuations. We then review an alternate theory that the plasma membrane is a two-dimensional microemulsion, and that the mechanism that drives to zero the line tension between Lo and Ld phases is the coupling of height and composition fluctuations. The theory yields rafts of SM and cholesterol in the outer leaf and POPS and POPC in the inner leaf. The "sea" between rafts consists of POPC in the outer leaf and POPE and cholesterol in the inner leaf. The characteristic size of the domain structures is tens of nanometers.

Sterols lower energetic barriers of membrane bending and fission necessary for efficient clathrin-mediated endocytosis

Clathrin-mediated endocytosis (CME) is critical for cellular signal transduction, receptor recycling, and membrane homeostasis in mammalian cells. Acute depletion of cholesterol disrupts CME, motivating analysis of CME dynamics in the context of human disorders of cholesterol metabolism. We report that inhibition of post-squalene cholesterol biosynthesis impairs CME. Imaging of membrane bending dynamics and the CME pit ultrastructure reveals prolonged clathrin pit lifetimes and shallow clathrin-coated structures, suggesting progressive impairment of curvature generation correlates with diminishing sterol abundance. Sterol structural requirements for efficient CME include 3′ polar head group and B-ring conformation, resembling the sterol structural prerequisites for tight lipid packing and polarity. Furthermore, Smith-Lemli-Opitz fibroblasts with low cholesterol abundance exhibit deficits in CME-mediated transferrin internalization. We conclude that sterols lower the energetic costs of membrane bending during pit formation and vesicular scission during CME and suggest that reduced CME activity may contribute to cellular phenotypes observed within disorders of cholesterol metabolism.

Anderson et al. demonstrate that sterol abundance and identity play a dominant role in facilitating clathrin-mediated endocytosis. Detailed analyses of clathrin-coated pits under sterol depletion support a requirement for sterol-mediated membrane bending during multiple stages of endocytosis, implicating endocytic dysfunction within the pathogenesis of disorders of cholesterol metabolism.

FisB relies on homo-oligomerization and lipid binding to catalyze membrane fission in bacteria

Little is known about mechanisms of membrane fission in bacteria despite their requirement for cytokinesis. The only known dedicated membrane fission machinery in bacteria, fission protein B (FisB), is expressed during sporulation in Bacillus subtilis and is required to release the developing spore into the mother cell cytoplasm. Here, we characterized the requirements for FisB-mediated membrane fission. FisB forms mobile clusters of approximately 12 molecules that give way to an immobile cluster at the engulfment pole containing approximately 40 proteins at the time of membrane fission. Analysis of FisB mutants revealed that binding to acidic lipids and homo-oligomerization are both critical for targeting FisB to the engulfment pole and membrane fission. Experiments using artificial membranes and filamentous cells suggest that FisB does not have an intrinsic ability to sense or induce membrane curvature but can bridge membranes. Finally, modeling suggests that homo-oligomerization and trans-interactions with membranes are sufficient to explain FisB accumulation at the membrane neck that connects the engulfment membrane to the rest of the mother cell membrane during late stages of engulfment. Together, our results show that FisB is a robust and unusual membrane fission protein that relies on homo-oligomerization, lipid binding, and the unique membrane topology generated during engulfment for localization and membrane scission, but surprisingly, not on lipid microdomains, negative-curvature lipids, or curvature sensing.

The Mechanics and Thermodynamics of Tubule Formation in Biological Membranes

Membrane tubulation is a ubiquitous process that occurs both at the plasma membrane and on the membranes of intracellular organelles. These tubulation events are known to be mediated by forces applied on the membrane either due to motor proteins, by polymerization of the cytoskeleton, or due to the interactions between membrane proteins binding onto the membrane. The numerous experimental observations of tube formation have been amply supported by mathematical modeling of the associated membrane mechanics and have provided insights into the force-displacement relationships of membrane tubes. Recent advances in quantitative biophysical measurements of membrane-protein interactions and tubule formation have necessitated the need for advances in modeling that will account for the interplay of multiple aspects of physics that occur simultaneously. Here, we present a comprehensive review of experimental observations of tubule formation and provide context from the framework of continuum modeling. Finally, we explore the scope for future research in this area with an emphasis on iterative modeling and experimental measurements that will enable us to expand our mechanistic understanding of tubulation processes in cells.

Spatial extent of a single lipid's influence on bilayer mechanics

To what spatial extent does a single lipid affect the mechanical properties of the membrane that surrounds it? The lipid composition of a membrane determines its mechanical properties. The shapes available to the membrane depend on its compositional material properties, and therefore, the lipid environment. Because each individual lipid species' chemistry is different, it is important to know its range of influence on membrane mechanical properties. This is defined herein as the lipid's mechanical extent. Here, a lipid's mechanical extent is determined by quantifying lipid redistribution and the average curvature that lipid species experience on fluctuating membrane surfaces. A surprising finding is that, unlike unsaturated lipids, saturated lipids have a complicated, nonlocal effect on the surrounding surface, with the interaction strength maximal at a finite length-scale. The methodology provides the means to substantially enrich curvature-energy models of membrane structures, quantifying what was previously only conjecture.

Flexible pivoting of dynamin pleckstrin homology domain catalyzes fission: insights into molecular degrees of freedom

The neuronal dynamin1 functions in the release of synaptic vesicles by orchestrating the process of GTPase-dependent membrane fission. Dynamin1 associates with the plasma membrane–localized phosphatidylinositol-4,5-bisphosphate (PIP₂) through the centrally located pleckstrin homology domain (PHD). The PHD is dispensable as fission (in model membranes) can be managed, even when the PHD-PIP₂ interaction is replaced by a generic polyhistidine- or polylysine-lipid interaction. However, the absence of the PHD renders a dramatic dampening of the rate of fission. These observations suggest that the PHD-PIP₂–containing membrane interaction could have evolved to expedite fission to fulfill the requirement of rapid kinetics of synaptic vesicle recycling. Here, we use a suite of multiscale modeling approaches to explore PHD–membrane interactions. Our results reveal that 1) the binding of PHD to PIP₂-containing membranes modulates the lipids toward fission-favoring conformations and softens the membrane, and 2) PHD associates with membrane in multiple orientations using variable loops as pivots. We identify a new loop (VL4), which acts as an auxiliary pivot and modulates the orientation flexibility of PHD on the membrane—a mechanism that we believe may be important for high-fidelity dynamin collar assembly. Together, these insights provide a molecular-level understanding of the catalytic role of PHD in dynamin-mediated membrane fission.

Comparing physical mechanisms for membrane curvature-driven sorting of BAR-domain proteins

Protein enrichment at specific membrane locations in cells is crucial for many cellular functions. It is well-recognized that the ability of some proteins to sense membrane curvature contributes partly to their enrichment in highly curved cellular membranes. In the past, different theoretical models have been developed to reveal the physical mechanisms underlying curvature-driven protein sorting. This review aims to provide a detailed discussion of the two continuous models that are based on the Helfrich elasticity energy, (1) the spontaneous curvature model and (2) the curvature mismatch model. These two models are commonly applied to describe experimental observations of protein sorting. We discuss how they can be used to explain the curvature-induced sorting data of two BAR proteins, amphiphysin and centaurin. We further discuss how membrane rigidity, and consequently the membrane curvature generated by BAR proteins, could influence protein organization on the curved membranes. Finally, we address future directions in extending these models to describe some cellular phenomena involving protein sorting.

Optimality in Self-Organized Molecular Sorting

We introduce a simple physical picture to explain the process of molecular sorting, whereby specific proteins are concentrated and distilled into submicrometric lipid vesicles in eukaryotic cells. To this purpose, we formulate a model based on the coupling of spontaneous molecular aggregation with vesicle nucleation. Its implications are studied by means of a phenomenological theory describing the diffusion of molecules toward multiple sorting centers that grow due to molecule absorption and are extracted when they reach a sufficiently large size. The predictions of the theory are compared with numerical simulations of a lattice-gas realization of the model and with experimental observations. The efficiency of the distillation process is found to be optimal for intermediate aggregation rates, where the density of sorted molecules is minimal and the process obeys simple scaling laws. Quantitative measures of endocytic sorting performed in primary endothelial cells are compatible with the hypothesis that these optimal conditions are realized in living cells.

On the Role of Curved Membrane Nanodomains, and Passive and Active Skeleton Forces in the Determination of Cell Shape and Membrane Budding

Biological membranes are composed of isotropic and anisotropic curved nanodomains. Anisotropic membrane components, such as Bin/Amphiphysin/Rvs (BAR) superfamily protein domains, could trigger/facilitate the growth of membrane tubular protrusions, while isotropic curved nanodomains may induce undulated (necklace-like) membrane protrusions. We review the role of isotropic and anisotropic membrane nanodomains in stability of tubular and undulated membrane structures generated or stabilized by cyto- or membrane-skeleton. We also describe the theory of spontaneous self-assembly of isotropic curved membrane nanodomains and derive the critical concentration above which the spontaneous necklace-like membrane protrusion growth is favorable. We show that the actin cytoskeleton growth inside the vesicle or cell can change its equilibrium shape, induce higher degree of segregation of membrane nanodomains or even alter the average orientation angle of anisotropic nanodomains such as BAR domains. These effects may indicate whether the actin cytoskeleton role is only to stabilize membrane protrusions or to generate them by stretching the vesicle membrane. Furthermore, we demonstrate that by taking into account the in-plane orientational ordering of anisotropic membrane nanodomains, direct interactions between them and the extrinsic (deviatoric) curvature elasticity, it is possible to explain the experimentally observed stability of oblate (discocyte) shapes of red blood cells in a broad interval of cell reduced volume. Finally, we present results of numerical calculations and Monte-Carlo simulations which indicate that the active forces of membrane skeleton and cytoskeleton applied to plasma membrane may considerably influence cell shape and membrane budding.

Domain formation in bicomponent vesicles induced by composition-curvature coupling

Lipid vesicles composed of a mixture of two types of lipids are studied by intensive Monte Carlo numerical simulations. The coupling between the local composition and the membrane shape is induced by two different spontaneous curvatures of the components. We explore the various morphologies of these biphasic vesicles coupled to the observed patterns such as nano-domains or labyrinthine mesophases. The effect of the difference in curvatures, the surface tension, and the interaction parameter between components is thoroughly explored. Our numerical results quantitatively agree with the previous analytical results obtained by Gueguen et al. [Eur. Phys. J. E 37, 76 (2014)] in the disordered (high temperature) phase. Numerical simulations allow us to explore the full parameter space, especially close to and below the critical temperature, where analytical results are not accessible. Phase diagrams are constructed and domain morphologies are quantitatively studied by computing the structure factor and the domain size distribution. This mechanism likely explains the existence of nano-domains in cell membranes as observed by super-resolution fluorescence microscopy.

Annexin A4 trimers are recruited by high membrane curvatures in giant plasma membrane vesicles

The plasma membrane (PM) of eukaryotic cells consists of a crowded environment comprised of a high diversity of proteins in a complex lipid matrix. The lateral organization of membrane proteins in the PM is closely correlated with biological functions such as endocytosis, membrane budding and other processes which involve protein mediated shaping of the membrane into highly curved structures. Annexin A4 (ANXA4) is a prominent player in a number of biological functions including PM repair. Its binding to membranes is activated by Ca2+ influx and it is therefore rapidly recruited to the cell surface near rupture sites where Ca2+ influx takes place. However, the free edges near rupture sites can easily bend into complex curvatures and hence may accelerate recruitment of curvature sensing proteins to facilitate rapid membrane repair. To analyze the curvature sensing behavior of curvature inducing proteins in crowded membranes, we quantifify the affinity of ANXA4 monomers and trimers for high membrane curvatures by extracting membrane nanotubes from giant PM vesicles (GPMVs). ANXA4 is found to be a sensor of negative membrane curvatures. Multiscale simulations, in which we extract molecular information from atomistic scale simulations as input to our macroscopic scale simulations, furthermore predicted that ANXA4 trimers generate membrane curvature upon binding and have an affinity for highly curved membrane regions only within a well defined membrane curvature window. Our results indicate that curvature sensing and mobility of ANXA4 depend on the trimer structure of ANXA4 which could provide new biophysical insight into the role of ANXA4 in membrane repair and other biological processes.

Diffuso-kinetic membrane budding dynamics

A wide range of proteins are known to create shape transformations of biological membranes, where the remodelling is a coupling between the energetic costs from deforming the membrane, the recruitment of proteins that induce a local spontaneous curvature C₀ and the diffusion of proteins along the membrane. We propose a minimal mathematical model that accounts for these processes to describe the diffuso-kinetic dynamics of membrane budding processes. By deploying numerical simulations we map out the membrane shapes, the time for vesicle formation and the vesicle size as a function of the dimensionless kinetic recruitment parameter K₁ and the proteins sensitivity to mean curvature. We derive a time for scission that follows a power law ∼K₁^-2/3, a consequence of the interplay between the spreading of proteins by diffusion and the kinetic-limited increase of the protein density on the membrane. We also find a scaling law for the vesicle size ∼1/([small sigma, Greek, macron]_avC₀), with [small sigma, Greek, macron]_av the average protein density in the vesicle, which is confirmed in the numerical simulations. Rescaling all the membrane profiles at the time of vesicle formation highlights that the membrane adopts a self-similar shape.

Diffusion on Membrane Domes, Tubes, and Pearling Structures

Diffusion is a fundamental mechanism for protein distribution in cell membranes. These membranes often exhibit complex shapes, which range from shallow domes to elongated tubular or pearl-like structures. Shape complexity of the membrane influences the diffusive spreading of proteins and molecules. Despite the importance membrane geometry plays in these diffusive processes, it is challenging to establish the dependence between diffusion and membrane morphology. We solve the diffusion equation numerically on various static curved shapes representative for experimentally observed membrane shapes. Our results show that membrane necks become diffusion barriers. We determine the diffusive half-time, i.e., the time that is required to reduce the amount of protein in the budded region by one half, and find a quadratic relation between the diffusive half-time and the averaged mean curvature of the membrane shape, which we rationalize by a scaling law. Our findings thus help estimate the characteristic diffusive timescale based on the simple measure of membrane mean curvature.

Additional contributions to elastic energy of lipid membranes: Tilt-curvature coupling and curvature gradient

Lipid bilayer membranes under biologically relevant conditions are flexible thin laterally fluid films consisting of two unimolecular layers (monolayers) each about 2 nm thick. On spatial scales much larger than the bilayer thickness, the membrane elasticity is well determined by its shape. The classical Helfrich theory considers the membrane as an elastic two-dimensional (2D) film, which has no particular internal structure. However, various local membrane heterogeneities can result in a lipids tilt relative to the membrane surface normal. On the basis of the classical elasticity theory of 3D bodies, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)10.1007/s101890070003] derived the most general energy functional, taking into account the tilt and lipid monolayer curvature. Recently, Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)10.1063/1.4990404] showed that Hamm and Kozlov's derivation was incomplete because the tilt-curvature coupling term had been missed. However, the energy functional derived by Terzi and Deserno appeared to be unstable, thereby being invalid for applications that require minimizations of the overall energy of deformations. Here, we derive a stable elastic energy functional, showing that the squared gradient of the curvature was missed in both of these works. This change in the energy functional arises from a more accurate consideration of the transverse shear deformation terms and their influence on the membrane stability. We also consider the influence of the prestress terms on the stability of the energy functional, and we show that it should be considered small and the effective Gaussian curvature should be neglected because of the stability requirements. We further generalize the theory, including the stretching-compressing deformation modes, and we provide the geometrical interpretation of the terms that were previously missed by Hamm and Kozlov. The physical consequences of the new terms are analyzed in the case of a membrane-mediated interaction of two amphipathic peptides located in the same monolayer. We also provide the expression for director fluctuations, comparing it with that obtained by Terzi and Deserno.

Effect of gold nanoparticle incorporation into oil-swollen surfactant lamellar membranes

An oil-swollen surfactant membrane is employed to measure the effects of incorporated hydrophobically functionalized gold nanoparticles (AuNPs) on the structure and dynamics of the membranes. While maintaining an average AuNP diameter of approximately 5 nm, the membrane thickness was varied from 5 nm to 7.5 nm by changing the amount of oil in the membrane. The membranes become softer as the proportion of oil is increased, while the thickness fluctuations become slower. We attribute this to an increased fluctuation wavelength. Incorporation of AuNPs in the membrane induces membrane thinning and softening. Oil molecules surround the nanoparticles in the membrane and help their relatively homogeneous distribution. AuNPs significantly alter the membrane's structure and dynamics through thinning of the membrane, increased compressibility, and possible diffusion of AuNPs inside the membrane.

A combined molecular/continuum-modeling approach to predict the small-angle neutron scattering of curved membranes

This paper develops a framework to compute the small-angle neutron scattering (SANS) from highly curved, dynamically fluctuating, and potentially inhomogeneous membranes. This method is needed to compute the scattering from nanometer-scale membrane domains that couple to curvature, as predicted by molecular modeling. The detailed neutron scattering length density of a small planar bilayer patch is readily available via molecular dynamics simulation. A mathematical, mechanical transformation of the planar scattering length density is developed to predict the scattering from curved bilayers. By simulating a fluctuating, curved, surface-continuum model, long time- and length-scales can be reached while, with the aid of the planar-to-curved transformation, the molecular features of the scattering length density can be retained. A test case for the method is developed by constructing a coarse-grained lipid vesicle following a protocol designed to relieve both the osmotic stress inside the vesicle and the lipid-number stress between the leaflets. A question was whether the hybrid model would be able to replicate the scattering from the highly deformed inner and outer leaflets of the small vesicle. Matching the scattering of the full (molecular vesicle) and hybrid (continuum vesicle) models indicated that the inner and outer leaflets of the full vesicle were expanded laterally, consistent with previous simulations of the Martini forcefield that showed thinning in small vesicles. The vesicle structure is inconsistent with a zero-tension leaflet deformed by a single set of elastic parameters, and the results show that this is evident in the scattering. The method can be applied to translate observations of any molecular model's neutron scattering length densities from small patches to large length and timescales.

Recent Experiments Support a Microemulsion Origin of Plasma Membrane Domains: Dependence of Domain Size on Physical Parameters

It is widely, but not universally, believed that the lipids of the plasma membrane are not uniformly distributed, but that “rafts” of sphingolipids and cholesterol float in a “sea” of unsaturated lipids. The physical origin of such heterogeneities is often attributed to a phase coexistence between the two different domains. We argue that this explanation is untenable for several reasons. Further, we note that the results of recent experiments are inconsistent with this picture. However, they are quite consistent with an alternate explanation, namely, that the plasma membrane is a microemulsion of the two kinds of regions. To show this, we briefly review a simplified version of this theory and its phase diagram. We also explicate the dependence of the predicted domain size on four physical parameters. They are the energy cost of gradients in the composition, the spontaneous curvature of the membrane, its bending modulus and its surface tension. Taking values of the latter two from experiment, we obtain domain sizes for several different cell types that vary from 58 to 88 nm.

Non-uniform distribution of myosin-mediated forces governs red blood cell membrane curvature through tension modulation

The biconcave disk shape of the mammalian red blood cell (RBC) is unique to the RBC and is vital for its circulatory function. Due to the absence of a transcellular cytoskeleton, RBC shape is determined by the membrane skeleton, a network of actin filaments cross-linked by spectrin and attached to membrane proteins. While the physical properties of a uniformly distributed actin network interacting with the lipid bilayer membrane have been assumed to control RBC shape, recent experiments reveal that RBC biconcave shape also depends on the contractile activity of nonmuscle myosin IIA (NMIIA) motor proteins. Here, we use the classical Helfrich-Canham model for the RBC membrane to test the role of heterogeneous force distributions along the membrane and mimic the contractile activity of sparsely distributed NMIIA filaments. By incorporating this additional contribution to the Helfrich-Canham energy, we find that the RBC biconcave shape depends on the ratio of forces per unit volume in the dimple and rim regions of the RBC. Experimental measurements of NMIIA densities at the dimple and rim validate our prediction that (a) membrane forces must be non-uniform along the RBC membrane and (b) the force density must be larger in the dimple than the rim to produce the observed membrane curvatures. Furthermore, we predict that RBC membrane tension and the orientation of the applied forces play important roles in regulating this force-shape landscape. Our findings of heterogeneous force distributions on the plasma membrane for RBC shape maintenance may also have implications for shape maintenance in different cell types.

The spectrin-actin network of the membrane skeleton plays an important role in controlling specialized cell membrane morphology. In the paradigmatic red blood cell (RBC), where actin filaments are present exclusively in the membrane skeleton, recent experiments reveal that nonmuscle myosin IIA (NMIIA) motor contractility maintains the RBC biconcave disk shape. In this study, we have identified criteria for micron-scale distributions of NMIIA forces at the membrane required to maintain the biconcave disk shape of an RBC in the resting condition. Supported by experimental measurements of RBC NMIIA distribution, we showed that a heterogeneous force distribution with a larger force density at the dimple is able to capture the experimentally observed biconcave morphology of an RBC with better accuracy compared to previous models that did not consider the heterogeneity in the force distribution. Furthermore, we showed that the biconcave geometry of the RBC is closely regulated by the effective membrane tension and the direction of applied forces on the membrane. These findings can be generalized to any force-mediated membrane shape, providing insight into the role of actomyosin forces in prescribing and maintaining the morphology of different cell types.

Model Plasma Membrane Exhibits a Microemulsion in Both Leaves Providing a Foundation for "Rafts"

We consider a model lipid plasma membrane, one that describes the outer leaf as consisting of sphingomyelin, phosphatidylcholine, and cholesterol and the inner leaf of phosphatidylethanolamine, phosphatidylserine, phosphatidylcholine, and cholesterol. Their relative compositions are taken from experiment; the cholesterol freely interchanges between leaves. Fluctuations in local composition are coupled to fluctuations in the local membrane curvature, as in the Leibler-Andelman mechanism. Structure factors of components in both leaves display a peak at nonzero wavevector. This indicates that the disordered fluid membrane is characterized by structure of the corresponding wavelength. The scale is given by membrane properties: its bending modulus and its surface tension, which arises from the membrane's connections to the cytoskeleton. From measurements on the plasma membrane, this scale is on the order of 100 nm. We find that the membrane can be divided into two different kinds of domains that differ not only in their composition but also in their curvature. The first domain in the outer, exoplasmic leaf is rich in cholesterol and sphingomyelin, whereas the inner, cytoplasmic leaf is rich in phosphatidylserine and phosphatidylcholine. The second kind of domain is rich in phosphatidylcholine in the outer leaf and in cholesterol and phosphatidylethanolamine in the inner leaf. The theory provides a tenable basis for the origin of structure in the plasma membrane and an illuminating picture of the organization of lipids therein.

Saffman-Delbrück and beyond: A pointlike approach

We show that a very good analytical approximation of Saffman-Delbrück's (SD) law (mobility of a bio-membrane inclusion) can be obtained easily from the velocity field produced by a pointlike force in a 2D fluid embedded in a solvent, by using a small wavelength cutoff of the order of the particle's radius a . With this method, we obtain analytical generalizations of the SD law that take into account the bilayer nature of the membrane and the intermonolayer friction b . We also derive, in a calculation that consistently couples the quasi-planar two-dimensional (2D) membrane flow with the 3D solvent flow, the correction to the SD law arising when the inclusion creates a local spontaneous curvature. For an inclusion spanning a flat bilayer, the SD law is found to hold simply upon replacing the 2D viscosity [Formula: see text] of the membrane by the sum of the monolayer viscosities, without influence of b as long as b is above a threshold in practice well below known experimental values. For an inclusion located in only one of the two monolayers (or adhering to one monolayer), the SD law is influenced by b when b < [Formula: see text]/(4a²) . In this case, the mobility can be increased by up to a factor of two, as the opposite monolayer is not fully dragged by the inclusion. For an inclusion creating a local spontaneous curvature, we show that the total friction is the sum of the SD friction and that due to the pull-back created by the membrane deformation, a point that was assumed without demonstration in the literature.

Fluctuations and conformational stability of a membrane patch with curvature inducing inclusions

Membranes with curvature inducing inclusions display a range of cooperative phenomena, which can be linked to biomembrane function, e.g. membrane tubulation, vesiculation, softening and spontaneous tension. We investigate how these phenomena are related for a fluctuating, framed membrane through analysis of a descretized membrane model by Monte Carlo simulation techniques. The membrane model is based on a dynamically triangulated surface equipped with non-interacting, up-down symmetry breaking inclusions where only terms coupled linearly to mean-curvature are maintained. We show that the lateral configurational entropy plays a key role for the mechanical properties of the semi-flexible membrane, e.g. a pronounced softening at intermediate inclusion coverages of the membrane and generation of membrane tension. Tensionless framed membranes will remain quasi-flat up to some threshold coverage, where a shape instability occurs with formation of pearling or tubular membranes, which below full coverage is associated with segregation of inclusions between the curved and flat membrane geometries. For inclusions with preference for highly curved membranes the instability appears at dilute inclusion coverages and is accompanied by strong configurational fluctuations.

Thermodynamic equilibrium of binary mixtures on curved surfaces

We study the global influence of curvature on the free energy landscape of two-dimensional binary mixtures confined on closed surfaces. Starting from a generic effective free energy, constructed on the basis of symmetry considerations and conservation laws, we identify several model-independent phenomena, such as a curvature-dependent line tension and local shifts in the binodal concentrations. To shed light on the origin of the phenomenological parameters appearing in the effective free energy, we further construct a lattice-gas model of binary mixtures on nontrivial substrates, based on the curved-space generalization of the two-dimensional Ising model. This allows us to decompose the interaction between the local concentration of the mixture and the substrate curvature into four distinct contributions, as a result of which the phase diagram splits into critical subdiagrams. The resulting free energy landscape can admit, as stable equilibria, strongly inhomogeneous mixed phases, which we refer to as "antimixed" states below the critical temperature. We corroborate our semianalytical findings with phase-field numerical simulations on realistic curved lattices. Despite this work being primarily motivated by recent experimental observations of multicomponent lipid vesicles supported by colloidal scaffolds, our results are applicable to any binary mixture confined on closed surfaces of arbitrary geometry.

Theoretical study of vesicle shapes driven by coupling curved proteins and active cytoskeletal forces

Eukaryote cells have a flexible shape, which dynamically changes according to the function performed by the cell. One mechanism for deforming the cell membrane into the desired shape is through the expression of curved membrane proteins. Furthermore, these curved membrane proteins are often associated with the recruitment of the cytoskeleton, which then applies active forces that deform the membrane. This coupling between curvature and activity was previously explored theoretically in the linear limit of small deformations, and low dimensionality. Here we explore the unrestricted shapes of vesicles that contain active curved membrane proteins, in three-dimensions, using Monte-Carlo numerical simulations. The activity of the proteins is in the form of protrusive forces that push the membrane outwards, as may arise from the cytoskeleton of the cell due to actin or microtubule polymerization occurring near the membrane. For proteins that have an isotropic convex shape, the additional protrusive force enhances their tendency to aggregate and form membrane protrusions (buds). In addition, we find another transition from deformed spheres with necklace type aggregates, to flat pancake-shaped vesicles, where the curved proteins line the outer rim. This second transition is driven by the active forces, coupled to the spontaneous curvature, and the resulting configurations may shed light on the formation of sheet-like protrusions and lamellipodia of adhered and motile cells.

Statistical mechanics of asymmetric tethered membranes: Spiral and crumpled phases

We develop the elastic theory for inversion-asymmetric tethered membranes and use it to identify and study their possible phases. Asymmetry in a tethered membrane causes spontaneous curvature, which in general depends on the local in-plane dilation of the tethered network. This in turn leads to long-range interactions between the local mean and Gaussian curvatures, which are not present in symmetric tethered membranes. This interplay between asymmetry and Gaussian curvature leads to a double-spiral phase not found in symmetric tethered membranes. At temperature T=0, tethered membranes of arbitrarily large size are always rolled up tightly into a conjoined pair of Archimedes' spirals. At finite T this spiral structure swells up significantly into algebraic spirals characterized by universal exponents, which we calculate. These spirals have long-range orientational order, and are the asymmetric analogs of statistically flat symmetric tethered membranes. We also find that sufficiently strong asymmetry can trigger a structural instability leading to crumpling of these membranes as well. This provides a mechanism for crumpling of asymmetric tethered membranes which is not present for symmetric membranes. We calculate the maximum linear extent L_{c} beyond which the membrane crumples, and calculate the universal dependence of L_{c} on the membrane parameters. By tuning the asymmetry parameter, L_{c} can be continuously varied, implying a scale-dependent crumpling. Our theory can be tested in controlled experiments on lipids with artificial deposits of spectrin filaments, in in vitro experiments on red blood cell membrane extracts, and on graphene coated on one side.

Rolled Up or Crumpled: Phases of Asymmetric Tethered Membranes

We show that inversion-asymmetric tethered membranes exhibit a new double-spiral phase with long range orientational order not present in symmetric membranes. We calculate the universal algebraic spiral shapes of these membranes in this phase. Asymmetry can trigger the crumpling of these membranes as well. In vitro experiments on lipid membranes, red blood cell membrane extracts, and on graphene coated on one side, could test these predictions.

Scaling of lipid membrane rigidity with domain area fraction

Biological membranes are highly heterogeneous in composition which in turn leads to local variations in the physical properties. Here we quantify how heterogeneity in stiffness determines the effective bending modulus, κ_eff, of model phase-separated membranes with coexisting soft fluid and rigid gel domains. We find that the temperature- and composition- dependent trends in membrane rigidity collapse onto a single curve, such that κ_eff directly scales with the area fraction of the rigid gel domains. Using no adjustable parameters, the measurements are found to agree with theoretical predictions for inhomogenous membranes and indicate that κ_eff is sensitive to the lateral distribution of the rigid phase within the membrane. This key finding confirms that the properties of heterogeneous membranes can be quantitatively predicted if the area fraction and properties of the individual phases are known.