Fluid lipid membranes: From differential geometry to curvature stresses

Transient pores in hemifusion diaphragms

Exchange of material across two membranes, as in the case of synaptic neurotransmitter release from a vesicle, involves the formation and poration of a hemifusion diaphragm (HD). The nontrivial geometry of the HD leads to environment-dependent control, regarding the stability and dynamics of the pores required for this kind of exocytosis. This work combines particle simulations, field-based calculations, and phenomenological modeling to explore the factors influencing the stability, dynamics, and possible control mechanisms of pores in HDs. We find that pores preferentially form at the HD rim, and that their stability is sensitive to a number of factors, including the three line tensions, membrane tension, HD size, and the ability of lipids to "flip-flop" across leaflets. Along with a detailed analysis of these factors, we discuss ways that vesicles or cells may use them to open and close pores and thereby quickly and efficiently transport material.

Biomembranes balance many types of leaflet asymmetries

Many biological membranes host different lipid species in their two leaflets. Since their spontaneous curvatures are typically not the same, this compositional asymmetry generally entails bending torques, which can be counteracted by differential stress-the difference between the two leaflet tensions. This stress, in turn, can affect elastic parameters or phase behavior of the membrane or each individual leaflet, or push easily flippable species, especially cholesterol, from the compressed leaflet into the tense leaflet. In short, breaking the symmetry of a single observable (to wit: composition), essentially breaks all other symmetries as well, with many potentially interesting consequences. This brief report examines the elastic aspects of this interplay, focusing on some elementary conditions of mechanical and thermodynamic equilibrium, but also shows how this poses novel questions that we are only beginning to appreciate.

The diffuse interface description of fluid lipid membranes captures key features of the hemifusion pathway and lateral stress profile

Topological transitions of lipid membranes are ubiquitous in key biological processes for cell life, like neurotransmission, fertilization, morphogenesis, and viral infections. Despite this, they are not well understood due to their multiscale nature, which limits the use of molecular models and calls for a mesoscopic approach such as the celebrated Canham-Helfrich one. Unfortunately, such a model cannot handle topological transitions, hiding the crucial involved forces and the appearance of the experimentally observed hemifused intermediates. In this work, we describe the membrane as a diffuse interface preserving the Canham-Helfrich elasticity. We show that pivotal features of the hemifusion pathway are captured by this mesoscopic approach, e.g. a (meta)stable hemifusion state and the fusogenic behavior of negative monolayer spontaneous curvatures. The membrane lateral stress profile is calculated as a function of the elastic rigidities, yielding a coarse-grained version of molecular models findings. Insights into the fusogenic mechanism are reported and discussed.

Shape equilibria of vesicles with rigid planar inclusions

Motivated by recent studies of two-phase lipid vesicles possessing 2D solid domains integrated within a fluid bilayer phase, we study the shape equilibria of closed vesicles possessing a single planar, circular inclusion. While 2D solid elasticity tends to expel Gaussian curvature, topology requires closed vesicles to maintain an average, non-zero Gaussian curvature leading to an elementary mechanism of shape frustration that increases with inclusion size. We study elastic ground states of the Helfrich model of the fluid-planar composite vesicles, analytically and computationally, as a function of planar fraction and reduced volume. Notably, we show that incorporation of a planar inclusion of only a few percent dramatically shifts the ground state shapes of vesicles from predominantly prolate to oblate, and moreover, shifts the optimal surface-to-volume ratio far from spherical shapes. We show that for sufficiently small planar inclusions, the elastic ground states break symmetry via a complex variety of asymmetric oblate, prolate, and triaxial shapes, while inclusion sizes above about 8% drive composite vesicles to adopt axisymmetric oblate shapes. These predictions cast useful light on the emergent shape and mechanical responses of fluid-solid composite vesicles.

Neural-network-based solver for vesicle shapes predicted by the Helfrich model

That a three-dimensional vesicle morphology can be modeled by an artificial neural network is proposed and demonstrated. In the phase-field representation, the Helfrich bending energy of a membrane is equivalently cast into field-based energy, which enables a more direct representation of a deformable, three-dimensional membrane surface. The core of our method is incorporating recent machine-learning techniques to perform the required energy minimization. The versatile ability of the method, to compute axisymmetric and nonsymmetric shapes, is discussed.

Single-cell topographical profiling of the immune synapse reveals a biomechanical signature of cytotoxicity

Immune cells have intensely physical lifestyles characterized by structural plasticity and force exertion. To investigate whether specific immune functions require stereotyped mechanical outputs, we used super-resolution traction force microscopy to compare the immune synapses formed by cytotoxic T cells with contacts formed by other T cell subsets and by macrophages. T cell synapses were globally compressive, which was fundamentally different from the pulling and pinching associated with macrophage phagocytosis. Spectral decomposition of force exertion patterns from each cell type linked cytotoxicity to compressive strength, local protrusiveness, and the induction of complex, asymmetric topography. These features were validated as cytotoxic drivers by genetic disruption of cytoskeletal regulators, live imaging of synaptic secretion, and in silico analysis of interfacial distortion. Synapse architecture and force exertion were sensitive to target stiffness and size, suggesting that the mechanical potentiation of killing is biophysically adaptive. We conclude that cellular cytotoxicity and, by implication, other effector responses are supported by specialized patterns of efferent force.

Probing wrapping dynamics of spherical nanoparticles by 3D vesicles using force-based simulations

Nanoparticles present in various environments can interact with living organisms, potentially leading to deleterious effects. Understanding how these nanoparticles interact with cell membranes is crucial for rational assessment of their impact on diverse biological processes. While previous research has explored particle-membrane interactions, the dynamic processes of particle wrapping by fluid vesicles remain incompletely understood. In this study, we introduce a force-based, continuum-scale model utilizing triangulated mesh representation and discrete differential geometry to investigate particle-vesicle interaction dynamics. Our model captures the transformation of vesicle shape and nanoparticle wrapping by calculating the forces arising from membrane bending energy and particle adhesion energy. Inspired by cell phagocytosis of large particles, we focus on establishing a quantitative understanding of large-scale vesicle deformation induced by the interaction with particles of comparable sizes. We first examine the interactions between spherical vesicles and individual nanospheres, both externally and internally, and quantify energy landscapes across different wrapping fractions of the nanoparticles. Furthermore, we explore multiple particle interactions with biologically relevant fluid vesicles with nonspherical shapes. Our study reveals that initial particle positions and interaction sequences are critical in determining the final equilibrium shapes of the vesicle-particle complexes in these interactions. These findings emphasize the importance of nanoparticle positioning and wrapping fractions in the dynamics of particle-vesicle interactions, providing crucial insights for future research in the field.

Lipid osmosis, membrane tension, and other mechanochemical driving forces of lipid flow

Nonvesicular lipid transport among different membranes or membrane domains plays crucial roles in lipid homeostasis and organelle biogenesis. However, the forces that drive such lipid transport are not well understood. We propose that lipids tend to flow towards the membrane area with a higher membrane protein density in a process termed lipid osmosis. This process lowers the membrane tension in the area, resulting in a membrane tension difference called osmotic membrane tension. We examine the thermodynamic basis and experimental evidence of lipid osmosis and osmotic membrane tension. We predict that lipid osmosis can drive bulk lipid flows between different membrane regions through lipid transfer proteins, scramblases, or similar barriers that selectively pass lipids but not membrane proteins. We also speculate on the biological functions of lipid osmosis. Finally, we explore other driving forces for lipid transfer and describe potential methods and systems to further test our theory.

Methyl-β-cyclodextrin asymmetrically extracts phospholipid from bilayers, granting tunable control over differential stress in lipid vesicles

Lipid asymmetry - that is, a nonuniform lipid distribution between the leaflets of a bilayer - is a ubiquitous feature of biomembranes and is implicated in several cellular phenomena. Differential tension - that is, unequal lateral monolayer tensions comparing the leaflets of a bilayer- is closely associated with lipid asymmetry underlying these varied roles. Because differential tension is not directly measurable in combination with the fact that common methods to adjust this quantity grant only semi-quantitative control over it, a detailed understanding of lipid asymmetry and differential tension are impeded. To overcome these challenges, we leveraged reversible complexation of phospholipid by methyl-β-cyclodextrin (mbCD) to tune the direction and magnitude of lipid asymmetry in synthetic vesicles. Lipid asymmetry generated in our study induced (i) vesicle shape changes and (ii) gel-liquid phase coexistence in 1-component vesicles. By applying mass-action considerations to interpret our findings, we discuss how this approach provides access to phospholipid thermodynamic potentials in bilayers containing lipid asymmetry (which are coupled to the differential tension of a bilayer). Because lipid asymmetry yielded by our approach is (i) tunable and (ii) maintained over minute to hour timescales, we anticipate that this approach will be a valuable addition to the experimental toolbox for systematic investigation into the biophysical role(s) of lipid asymmetry (and differential tension).

Patterning of multicomponent elastic shells by gaussian curvature

Recent findings suggest that shell protein distribution and the morphology of bacterial microcompartments regulate the chemical fluxes facilitating reactions which dictate their biological function. We explore how the morphology and component patterning are coupled through the competition of mean and gaussian bending energies in multicomponent elastic shells that form three-component irregular polyhedra. We observe two softer components with lower bending rigidities allocated on the edges and vertices while the harder component occupies the faces. When subjected to a nonzero interfacial line tension, the two softer components further separate and pattern into subdomains that are mediated by the gaussian curvature. We find that this degree of fractionation is maximized when there is a weaker line tension and when the ratio of bending rigidities between the two softer domains ≈2. Our results reveal a patterning mechanism in multicomponent shells that can capture the observed morphologies of bacterial microcompartments, and moreover, can be realized in synthetic vesicles.

Lipid osmosis, membrane tension, and other mechanochemical driving forces of lipid flow

Nonvesicular lipid transport among different membranes or membrane domains plays crucial roles in lipid homeostasis and organelle biogenesis. However, the forces that drive such lipid transport are not well understood. We propose that lipids tend to flow towards the membrane area with a higher membrane protein density in a process termed lipid osmosis. This process lowers the membrane tension in the area, resulting in a membrane tension difference called osmotic membrane tension. We examine the thermodynamic basis and experimental evidence of lipid osmosis and osmotic membrane tension. We predict that lipid osmosis can drive bulk lipid flows between different membrane regions through lipid transfer proteins, scramblases, or other similar barriers that selectively pass lipids but not membrane proteins. We also speculate on the biological functions of lipid osmosis. Finally, we explore other driving forces for lipid transfer and describe potential methods and systems to further test our theory.

Dynamic framework for large-scale modeling of membranes and peripheral proteins

In this chapter, we present a novel computational framework to study the dynamic behavior of extensive membrane systems, potentially in interaction with peripheral proteins, as an alternative to conventional simulation methods. The framework effectively describes the complex dynamics in protein-membrane systems in a mesoscopic particle-based setup. Furthermore, leveraging the hydrodynamic coupling between the membrane and its surrounding solvent, the coarse-grained model grounds its dynamics in macroscopic kinetic properties such as viscosity and diffusion coefficients, marrying the advantages of continuum- and particle-based approaches. We introduce the theoretical background and the parameter-space optimization method in a step-by-step fashion, present the hydrodynamic coupling method in detail, and demonstrate the application of the model at each stage through illuminating examples. We believe this modeling framework to hold great potential for simulating membrane and protein systems at biological spatiotemporal scales, and offer substantial flexibility for further development and parametrization.

Modeling the mechanochemical feedback for membrane-protein interactions using a continuum mesh model

The Helfrich free energy is widely used to model the generation of membrane curvature due to different physical and chemical components. The governing equations resulting from the energy minimization procedure are a system of coupled higher order partial differential equations. Simulations of membrane deformation for obtaining quantitative comparisons against experimental observations require computational schemes that will allow us to solve these equations without restrictions to axisymmetric coordinates. Here, we describe one such tool that we developed in our group based on discrete differential geometry to solve these equations along with examples.

Binding equations for the lipid composition dependence of peripheral membrane-binding proteins

The specific recognition of peripheral membrane-binding proteins for their target membranes is mediated by a complex constellation of various lipid contacts. Despite the inherent complexities of the heterogeneous protein-membrane interface, the binding dependence of such proteins is, surprisingly, often reliably described by simple models such as the Langmuir Adsorption Isotherm or the Hill equation. However, these models were not developed to describe associations with two-dimensional, highly concentrated heterogeneous ligands such as lipid membranes. In particular, these models fail to capture the dependence on the lipid composition, a significant determinant of binding that distinguishes target from non-target membranes. In this work, we present a model that describes the dependence of peripheral proteins on lipid composition through an analytic expression for their association. The resulting membrane-binding equation retains the features of these simple models but completely describes the binding dependence on multiple relevant variables in addition to the lipid composition, such as protein and vesicle concentration. Implicit in this lipid composition dependence is a new form of membrane-based cooperativity that significantly differs from traditional solution-based cooperativity. We introduce the Membrane-Hill number as a measure of this cooperativity and describe its unique properties. We illustrate the utility and interpretational power of our model by analyzing previously published data on two peripheral proteins that associate with phosphatidylserine-containing membranes: The transmembrane immunoglobulin and mucin domain-containing protein 3 (TIM3) that employs calcium in its association, and milk fat globulin epidermal growth factor VIII (MFG-E8) which is completely insensitive to calcium. We also provide binding equations for systems that exhibit more complexity in their membrane-binding.

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.

Membrane fission via transmembrane contact

Division of intracellular organelles often correlates with additional membrane wrapping, e.g., by the endoplasmic reticulum or the outer mitochondrial membrane. Such wrapping plays a vital role in proteome and lipidome organization. However, how an extra membrane impacts the mechanics of the division has not been investigated. Here we combine fluorescence and cryo-electron microscopy experiments with self-consistent field theory to explore the stress-induced instabilities imposed by membrane wrapping in a simple double-membrane tubular system. We find that, at physiologically relevant conditions, the outer membrane facilitates an alternative pathway for the inner-tube fission through the formation of a transient contact (hemi-fusion) between both membranes. A detailed molecular theory of the fission pathways in the double membrane system reveals the topological complexity of the process, resulting both in leaky and leakless intermediates, with energies and topologies predicting physiological events.

Biophysical Modeling of Actin-Mediated Structural Plasticity Reveals Mechanical Adaptation in Dendritic Spines

Synaptic plasticity is important for learning and memory formation; it describes the strengthening or weakening of connections between synapses. The postsynaptic part of excitatory synapses resides in dendritic spines, which are small protrusions on the dendrites. One of the key features of synaptic plasticity is its correlation with the size of these spines. A long-lasting synaptic strength increase [long-term potentiation (LTP)] is only possible through the reconfiguration of the actin spine cytoskeleton. Here, we develop an experimentally informed three-dimensional computational model in a moving boundary framework to investigate this reconfiguration. Our model describes the reactions between actin and actin-binding proteins leading to the cytoskeleton remodeling and their effect on the spine membrane shape to examine the spine enlargement upon LTP. Moreover, we find that the incorporation of perisynaptic elements enhances spine enlargement upon LTP, exhibiting the importance of accounting for these elements when studying structural LTP. Our model shows adaptation to repeated stimuli resulting from the interactions between spine proteins and mechanical forces.

Synthesis, insertion, and characterization of SARS-CoV-2 membrane protein within lipid bilayers

Throughout history, coronaviruses have posed challenges to both public health and the global economy; nevertheless, methods to combat them remain rudimentary, primarily due to the absence of experiments to understand the function of various viral components. Among these, membrane (M) proteins are one of the most elusive because of their small size and challenges with expression. Here, we report the development of an expression system to produce tens to hundreds of milligrams of M protein per liter of Escherichia coli culture. These large yields render many previously inaccessible structural and biophysical experiments feasible. Using cryo-electron microscopy and atomic force microscopy, we image and characterize individual membrane-incorporated M protein dimers and discover membrane thinning in the vicinity, which we validated with molecular dynamics simulations. Our results suggest that the resulting line tension, along with predicted induction of local membrane curvature, could ultimately drive viral assembly and budding.

The local variation of the Gaussian modulus enables different pathways for fluid lipid vesicle fusion

Viral infections, fertilization, neurotransmission, and many other fundamental biological processes rely on membrane fusion. Straightforward calculations based on the celebrated Canham-Helfrich elastic model predict a large topological energy barrier that prevents the fusion process from being thermally activated. While such high energy is in accordance with the physical barrier function of lipid membranes, it is difficult to reconcile with the biological mechanisms involved in fusion processes. In this work, we use a Ginzburg-Landau type of free energy that recovers the Canham-Helfrich model in the limit of small width-to-vesicle-extension ratio, with the additional ability to handle topological transitions. We show that a local modification of the Gaussian modulus in the merging region both dramatically lowers the elastic energy barrier and substantially changes the minimal energy pathway for fusion, in accordance with experimental evidence. Therefore, we discuss biological examples in which such a modification might play a crucial role.

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.

Effective cell membrane tension protects red blood cells against malaria invasion

A critical step in how malaria parasites invade red blood cells (RBCs) is the wrapping of the membrane around the egg-shaped merozoites. Recent experiments have revealed that RBCs can be protected from malaria invasion by high membrane tension. While cellular and biochemical aspects of parasite actomyosin motor forces during the malaria invasion have been well studied, the important role of the biophysical forces induced by the RBC membrane-cytoskeleton composite has not yet been fully understood. In this study, we use a theoretical model for lipid bilayer mechanics, cytoskeleton deformation, and membrane-merozoite interactions to systematically investigate the influence of effective RBC membrane tension, which includes contributions from the lipid bilayer tension, spontaneous tension, interfacial tension, and the resistance of cytoskeleton against shear deformation on the progression of membrane wrapping during the process of malaria invasion. Our model reveals that this effective membrane tension creates a wrapping energy barrier for a complete merozoite entry. We calculate the tension threshold required to impede the malaria invasion. We find that the tension threshold is a nonmonotonic function of spontaneous tension and undergoes a sharp transition from large to small values as the magnitude of interfacial tension increases. We also predict that the physical properties of the RBC cytoskeleton layer-particularly the resting length of the cytoskeleton-play key roles in specifying the degree of the membrane wrapping. We also found that the shear energy of cytoskeleton deformation diverges at the full wrapping state, suggesting the local disassembly of the cytoskeleton is required to complete the merozoite entry. Additionally, using our theoretical framework, we predict the landscape of myosin-mediated forces and the physical properties of the RBC membrane in regulating successful malaria invasion. Our findings on the crucial role of RBC membrane tension in inhibiting malaria invasion can have implications for developing novel antimalarial therapeutic or vaccine-based strategies.

Geometry-structure models for liquid crystal interfaces, drops and membranes: wrinkling, shape selection and dissipative shape evolution

We review our recent contributions to anisotropic soft matter models for liquid crystal interfaces, drops and membranes, emphasizing validations with experimental and biological data, and with related theory and simulation literature. The presentation aims to illustrate and characterize the rich output and future opportunities of using a methodology based on the liquid crystal-membrane shape equation applied to static and dynamic pattern formation phenomena. The geometry of static and kinetic shapes is usually described with dimensional curvatures that co-mingle shape and curvedness. In this review, we systematically show how the application of a novel decoupled shape-curvedness framework to practical and ubiquitous soft matter phenomena, such as the shape of drops and tactoids and bending of evolving membranes, leads to deeper quantitative insights than when using traditional dimensional mean and Gaussian curvatures. The review focuses only on (1) statics of wrinkling and shape selection in liquid crystal interfaces and membranes; (2) kinetics and dissipative dynamics of shape evolution in membranes; and (3) computational methods for shape selection and shape evolution; due to various limitations other important topics are excluded. Finally, the outlook follows a similar structure. The main results include: (1) single and multiple wavelength corrugations in liquid crystal interfaces appear naturally in the presence of surface splay and bend orientation distortions with scaling laws governed by ratios of anchoring-to-isotropic tension energy; adding membrane elasticity to liquid crystal anchoring generates multiple scales wrinkling as in tulips; drops of liquid crystals encapsulates in membranes can adopt, according to the ratios of anchoring/tension/bending, families of shapes as multilobal, tactoidal, and serrated as observed in biological cells. (2) Mapping the liquid crystal director to a membrane unit normal. The dissipative shape evolution model with irreversible thermodynamics for flows dominated by bending rates, yields new insights. The model explains the kinetic stability of cylinders, while spheres and saddles are attractors. The model also adds to the evolving understanding of outer hair cells in the inner ear. (3) Computational soft matter geometry includes solving shape equations, trajectories on energy and orientation landscapes, and shape-curvedness evolutions on entropy production landscape with efficient numerical methods and adaptive approaches.

Structural and Mechanical Response of Two-Component Photoswitchable Lipid Bilayer Vesicles

Optical control of phospholipids is an attractive option for the rapid, reversible, and tunable manipulation of membrane structure and dynamics. Azo-PC, a lipid with an azobenzene group within one acyl chain, undergoes a light-induced trans-to-cis isomerization and thus arises as a powerful tool for manipulating lipid order and dynamics. Here, we report on vesicle-scale micropipette measurements and atomistic simulations to probe the elastic stretching modulus, water permeability, toughness, thickness, and membrane area upon isomerization. We investigated both dynamics and steady-state properties. In pure azo-PC membranes, we found that the molecular area in trans was 16% smaller than that in cis, the membrane's stretching modulus kA was 2.5 ± 0.3 times greater, and the water permeability PW was 3.5 ± 0.5 times smaller. We also studied mixtures of azo-PC with the miscible, unsaturated lipid DOPC. Atomistic molecular dynamics simulations show how the membrane thickness, chain order, and correlations across membrane leaflets explain the experimental data. Together, these data show how one rotating bond changes the molecular- and membrane-scale properties. These results will be useful for photopharmacology and for developing new materials whose permeability, elasticity, and toughness may be switched on demand.

Effects of lipid saturation on bicelle to vesicle transition of a binary phospholipid mixture: a molecular dynamics simulation study

Controlling the transition from lipid bicelles to vesicles is essential for producing engineered vesicles. We perform coarse-grained molecular dynamics (CGMD) simulations of unsaturated/saturated lipid mixtures to clarify the effects of lipid unsaturation on vesiculation at the molecular scale. The results demonstrate that vesiculation depends on the concentration of unsaturated lipids and the degree of unsaturation. The probability of vesiculation increases linearly with the apparent unsaturated lipid concentration at a low degree of unsaturation. Higher degrees of unsaturation lead to phase segregation within the binary bicelles, reducing the probability of vesiculation. A comparison between CGMD simulations and the conventional theory of vesiculation shows that the theoretical predictions of binary lipid systems must explicitly include phase segregation effects. Furthermore, simulations with biased lipid distributions reveal that vesiculation is facilitated by the preconcentration of unsaturated lipids in the core region of the bicelle but is then temporally limited as the unsaturated lipids move to the bicelle edges. These findings advance theoretical and experimental studies on binary lipid systems and promote the development of tailor-made vesicles.

Three membrane fusion pore families determine the pathway to pore dilation

During exocytosis secretory vesicles fuse with a target membrane and release neurotransmitters, hormones, or other bioactive molecules through a membrane fusion pore. The initially small pore may subsequently dilate for full contents release, as commonly observed in amperometric traces. The size, shape, and evolution of the pore is critical to the course of contents release, but exact fusion pore solutions accounting for membrane tension and bending energy constraints have not been available. Here, we obtained exact solutions for fusion pores between two membranes. We find three families: a narrow pore, a wide pore, and an intermediate tether-like pore. For high tensions these are close to the catenoidal and tether solutions recently reported for freely hinged membrane boundaries. We suggest membrane fusion initially generates a stable narrow pore, and the dilation pathway is a transition to the stable wide pore family. The unstable intermediate pore is the transition state that sets the energy barrier for this dilation pathway. Pore dilation is mechanosensitive, as the energy barrier is lowered by increased membrane tension. Finally, we study fusion pores in nanodiscs, powerful systems for the study of individual pores. We show that nanodiscs stabilize fusion pores by locking them into the narrow pore family.

Statics and diffusive dynamics of surfaces driven by p-atic topological defects

Inspired by epithelial morphogenesis, we consider a minimal model for the shaping of a surface driven by p-atic topological defects. We show that a positive (negative) defect can dynamically generate a (hyperbolic) cone whose shape evolves diffusively, and predict that a defect of charge + 1/p leads to a final semi-cone angle β which satisfies the inequality . By exploiting the fact that for axisymmetric surfaces, the extrinsic geometry is tightly coupled to the intrinsic geometry, we further show that the resulting stationary shape of a membrane with negligible bending modulus and embedded polar order is a deformed lemon with two defects at antipodal points. Finally, we close by pointing out that our results may be relevant beyond epithelial morphogenesis in such contexts as shape transitions in macroscopic closed spheroidal surfaces such as pollen grains.

Clustering of lipids driven by integrin

Integrin is an important transmembrane receptor protein which remodels the actin network and anchors the cell membrane towards the extracellular matrix via mechanochemical pathways. The clustering of specific lipids and lipid-anchored proteins, which is essential for a certain type of endocytosis process, is facilitated at integrin-mediated active regions. To study this, we propose a minimal exactly solvable model which includes the interplay of stochastic shuttling between integrin on and off states with the intrinsic dynamics of the membrane. We propose a two-step mechanism in which the integrin induces an aster-like arrangement in the actin network, followed by clustering of lipids in that region. We obtain an analytic expression for the deformation and local membrane velocity, and thereby the evolution of clustering mediated by a single integrin. The deformation evolves nonmonotonically and its dependence on the stochastic shuttling timescales and membrane properties is elucidated. Our estimates of the area of the deformed region and the number of lipids in it indicate strong clustering.

Conformal order and Poincaré-Klein mapping underlying electrostatics-driven inhomogeneity in tethered membranes

Understanding the organization of matter under the long-range electrostatic force is a fundamental problem in multiple fields. In this work, based on the electrically charged tethered membrane model, we reveal regular structures underlying the lowest-energy states of inhomogeneously stretched planar lattices by a combination of numerical simulation and analytical geometric analysis. Specifically we show the conformal order characterized by the preserved bond angle in the lattice deformation and reveal the Poincaré-Klein mapping underlying the electrostatics-driven inhomogeneity. The discovery of the Poincaré-Klein mapping, which connects the Poincaré disk and the Klein disk for the hyperbolic plane, implies the connection of long-range electrostatic force and hyperbolic geometry. We also discuss lattices with patterned charges of opposite signs for modulating in-plane inhomogeneity and even creating 3D shapes, which may have a connection to metamaterials design. This work suggests the geometric analysis as a promising approach for elucidating the organization of matter under the long-range force.

A minimal cell model for lamellipodia-based cellular dynamics and migration

One ubiquitous cellular structure for performing various tasks, such as spreading and migration over external surfaces, is the sheet-like protrusion called a lamellipodium, which propels the leading edge of the cell. Despite the detailed knowledge about the many components of this cellular structure, it is not yet fully understood how these components self-organize spatiotemporally to form lamellipodia. We review here recent theoretical works where we have demonstrated that membrane-bound protein complexes that have intrinsic curvature and recruit the protrusive forces of the cytoskeleton result in a simple, yet highly robust, organizing feedback mechanism that organizes the cytoskeleton and the membrane. This self-organization mechanism accounts for the formation of flat lamellipodia at the leading edge of cells spreading over adhesive substrates, allowing for the emergence of a polarized, motile 'minimal cell' model. The same mechanism describes how lamellipodia organize to drive robust engulfment of particles during phagocytosis and explains in simple physical terms the spreading and migration of cells over fibers and other curved surfaces. This Review highlights that despite the complexity of cellular composition, there might be simple general physical principles that are utilized by the cell to drive cellular shape dynamics.

Geometry-induced patterns through mechanochemical coupling

Intracellular protein patterns regulate a variety of vital cellular processes such as cell division and motility, which often involve dynamic cell-shape changes. These changes in cell shape may in turn affect the dynamics of pattern-forming proteins, hence leading to an intricate feedback loop between cell shape and chemical dynamics. While several computational studies have examined the rich resulting dynamics, the underlying mechanisms are not yet fully understood. To elucidate some of these mechanisms, we explore a conceptual model for cell polarity on a dynamic one-dimensional manifold. Using concepts from differential geometry, we derive the equations governing mass-conserving reaction-diffusion systems on time-evolving manifolds. Analyzing these equations mathematically, we show that dynamic shape changes of the membrane can induce pattern-forming instabilities in parts of the membrane, which we refer to as regional instabilities. Deformations of the local membrane geometry can also (regionally) suppress pattern formation and spatially shift already existing patterns. We explain our findings by applying and generalizing the local equilibria theory of mass-conserving reaction-diffusion systems. This allows us to determine a simple onset criterion for geometry-induced pattern-forming instabilities, which is linked to the phase-space structure of the reaction-diffusion system. The feedback loop between membrane shape deformations and reaction-diffusion dynamics then leads to a surprisingly rich phenomenology of patterns, including oscillations, traveling waves, and standing waves, even if these patterns do not occur in systems with a fixed membrane shape. Our paper reveals that the local conformation of the membrane geometry acts as an important dynamical control parameter for pattern formation in mass-conserving reaction-diffusion systems.

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.

High curvature promotes fusion of lipid membranes: Predictions from continuum elastic theory

The fusion of lipid membranes progresses through a series of hemifusion intermediates with two significant energy barriers related to the formation of stalk and fusion pore, respectively. These energy barriers determine the speed and success rate of many critical biological processes, including the fusion of highly curved membranes, for example synaptic vesicles and enveloped viruses. Here we use continuum elastic theory of lipid monolayers to determine the relationship between membrane shape and energy barriers to fusion. We find that the stalk formation energy decreases with curvature by up to 31 kBT in a 20-nm-radius vesicle compared with planar membranes and by up to 8 kBT in the fusion of highly curved, long, tubular membranes. In contrast, the fusion pore formation energy barrier shows a more complicated behavior. Immediately after stalk expansion to the hemifusion diaphragm, the fusion pore formation energy barrier is low (15-25 kBT) due to lipid stretching in the distal monolayers and increased tension in highly curved vesicles. Therefore, the opening of the fusion pore is faster. However, these stresses relax over time due to lipid flip-flop from the proximal monolayer, resulting in a larger hemifusion diaphragm and a higher fusion pore formation energy barrier, up to 35 kBT. Therefore, if the fusion pore fails to open before significant lipid flip-flop takes place, the reaction proceeds to an extended hemifusion diaphragm state, which is a dead-end configuration in the fusion process and can be used to prevent viral infections. In contrast, in the fusion of long tubular compartments, the surface tension does not accumulate due to the formation of the diaphragm, and the energy barrier for pore expansion increases with curvature by up to 11 kBT. This suggests that inhibition of polymorphic virus infection could particularly target this feature of the second barrier.

A Membrane Biophysics Perspective on the Mechanism of Alcohol Toxicity

Motivations for understanding the underlying mechanisms of alcohol toxicity range from economical to toxicological and clinical. On the one hand, acute alcohol toxicity limits biofuel yields, and on the other hand, acute alcohol toxicity provides a vital defense mechanism to prevent the spread of disease. Herein the role that stored curvature elastic energy (SCE) in biological membranes might play in alcohol toxicity is discussed, for both short and long-chain alcohols. Structure-toxicity relationships for alcohols ranging from methanol to hexadecanol are collated, and estimates of alcohol toxicity per alcohol molecule in the cell membrane are made. The latter reveal a minimum toxicity value per molecule around butanol before alcohol toxicity per molecule increases to a maximum around decanol and subsequently decreases again. The impact of alcohol molecules on the lamellar to inverse hexagonal phase transition temperature (T_H) is then presented and used as a metric to assess the impact of alcohol molecules on SCE. This approach suggests the nonmonotonic relationship between alcohol toxicity and chain length is consistent with SCE being a target of alcohol toxicity. Finally, in vivo evidence for SCE-driven adaptations to alcohol toxicity in the literature are discussed.

A Review of Continuum Mechanics for Mechanical Deformation of Lipid Membranes

Mechanical deformation of lipid membranes plays important roles in various cellular tasks. Curvature deformation and lateral stretching are two major energy contributions to the mechanical deformation of lipid membranes. In this paper, continuum theories for these two major membrane deformation events were reviewed. Theories based on curvature elasticity and lateral surface tension were introduced. Numerical methods as well as biological applications of the theories were discussed.

Enzymatic Janus Liposome Micromotors

A liposome-based micromotor system that utilizes regional enzymatic conversion and gas generation to achieve directional motion in water is presented. Constituted mainly of a low-melting lipid and a high-melting lipid together with cholesterol, these liposomes maintain stable Janus configuration at room temperature as a result of lipid liquid-liquid phase separation. Local placement of enzymes such as horseradish peroxidase is realized via affinity binding between avidin and biotin, the latter as a lipid conjugate sorted specifically into one domain of these Janus liposomes as a minor component. In the presence of the substrate, hydrogen peroxide, these enzyme-decorated Janus liposomes undergo directional motion, yielding velocities exceeding thermal diffusion by three folds in some cases. Experimental details on liposome size control, motor assembly, and substrate distribution are presented; effects of key experimental factors on liposome motion, such as substrate concentration and liposome Janus ratio, are also examined. This work thus provides a viable approach to building asymmetrical lipid-assembled, enzyme-attached colloids and, in addition, stresses the importance of asymmetry in achieving particle directional motion.

A Note on Vestigial Osmotic Pressure

Recent experiments have indicated that at least a part of the osmotic pressure across the giant unilamellar vesicle (GUV) membrane was balanced by the rapid formation of the monodisperse daughter vesicles inside the GUVs through an endocytosis-like process. Therefore, we investigated a possible osmotic role played by these daughter vesicles for the maintenance of osmotic regulation in the GUVs and, by extension, in living cells. We highlighted a mechanism whereby the daughter vesicles acted as osmotically active solutes (osmoticants), contributing an extra vestigial osmotic pressure component across the membrane of the parent vesicle, and we showed that the consequences were consistent with experimental observations. Our results highlight the significance of osmotic regulation in cellular processes, such as fission/fusion, endocytosis, and exocytosis.

Effect of monolayer spontaneous curvature on constant tension-induced pore formation in lipid bilayers

The methodology of constant tension-induced rupture of giant unilamellar vesicles (GUVs) has provided information on tension-induced pore formation. This method was used to investigate the effect of spontaneous curvature (H₀) for a lipid monolayer on the rate constant (k_r) for constant tension (σ)-induced rupture, which originates from pore formation in lipid bilayers. Lipids were incorporated with different H₀ values into GUV membranes to change the overall H₀ value for the GUV monolayer. The dioleoylphosphatidylglycerol (DOPG)/dioleoylphosphatidylethanolamine (DOPE) (4/6, molar ratio, here and elsewhere) monolayer has a negative H₀, whereas the DOPG/dioleoylphosphatidylcholine (DOPC) (4/6) monolayer has an essentially zero H₀. A higher tension was required to induce the rupture of DOPG/DOPE (4/6)-GUVs compared with DOPG/DOPC (4/6)-GUVs. The line tension (Γ) for a pre-pore in DOPG/DOPE (4/6)-GUVs, determined by the analysis of the tension dependence of k_r, was 1.5 times larger than that in DOPG/DOPC (4/6)-GUVs. The k_r values for GUVs comprising DOPG/DOPC/18:1 lysophosphatidylcholine (LPC) (40/55/10), which has a positive H₀, were larger than those for DOPG/DOPC (4/6)-GUVs under the same tension. The Γ value for DOPG/DOPC/LPC (40/55/10)-GUVs was almost half that for DOPG/DOPC (4/6)-GUVs. These results indicate that Γ decreases with increasing H₀, which results in an increase in k_r. Based on these results, the effect of H₀ on k_r and Γ is discussed.

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.

Geometric control of myosin II orientation during axis elongation

The actomyosin cytoskeleton is a crucial driver of morphogenesis. Yet how the behavior of large-scale cytoskeletal patterns in deforming tissues emerges from the interplay of geometry, genetics, and mechanics remains incompletely understood. Convergent extension in Drosophila melanogaster embryos provides the opportunity to establish a quantitative understanding of the dynamics of anisotropic non-muscle myosin II. Cell-scale analysis of protein localization in fixed embryos suggests that gene expression patterns govern myosin anisotropy via complex rules. However, technical limitations have impeded quantitative and dynamic studies of this process at the whole embryo level, leaving the role of geometry open. Here, we combine in toto live imaging with quantitative analysis of molecular dynamics to characterize the distribution of myosin anisotropy and the corresponding genetic patterning. We found pair rule gene expression continuously deformed, flowing with the tissue frame. In contrast, myosin anisotropy orientation remained approximately static and was only weakly deflected from the stationary dorsal-ventral axis of the embryo. We propose that myosin is recruited by a geometrically defined static source, potentially related to the embryo-scale epithelial tension, and account for transient deflections by cytoskeletal turnover and junction reorientation by flow. With only one parameter, this model quantitatively accounts for the time course of myosin anisotropy orientation in wild-type, twist, and even-skipped embryos, as well as embryos with perturbed egg geometry. Geometric patterning of the cytoskeleton suggests a simple physical strategy to ensure a robust flow and formation of shape.

Remodeling of the Plasma Membrane by Surface-Bound Protein Monomers and Oligomers: The Critical Role of Intrinsically Disordered Regions

The plasma membrane (PM) of cells is a dynamic structure whose morphology and composition is in constant flux. PM morphologic changes are particularly relevant for the assembly and disassembly of signaling platforms involving surface-bound signaling proteins, as well as for many other mechanochemical processes that occur at the PM surface. Surface-bound membrane proteins (SBMP) require efficient association with the PM for their function, which is often achieved by the coordinated interactions of intrinsically disordered regions (IDRs) and globular domains with membrane lipids. This review focuses on the role of IDR-containing SBMPs in remodeling the composition and curvature of the PM. The ability of IDR-bearing SBMPs to remodel the Gaussian and mean curvature energies of the PM is intimately linked to their ability to sort subsets of phospholipids into nanoclusters. We therefore discuss how IDRs of many SBMPs encode lipid-binding specificity or facilitate cluster formation, both of which increase their membrane remodeling capacity, and how SBMP oligomers alter membrane shape by monolayer surface area expansion and molecular crowding.

Membrane-mediated protein interactions drive membrane protein organization

The plasma membrane's main constituents, i.e., phospholipids and membrane proteins, are known to be organized in lipid-protein functional domains and supercomplexes. No active membrane-intrinsic process is known to establish membrane organization. Thus, the interplay of thermal fluctuations and the biophysical determinants of membrane-mediated protein interactions must be considered to understand membrane protein organization. Here, we used high-speed atomic force microscopy and kinetic and membrane elastic theory to investigate the behavior of a model membrane protein in oligomerization and assembly in controlled lipid environments. We find that membrane hydrophobic mismatch modulates oligomerization and assembly energetics, and 2D organization. Our experimental and theoretical frameworks reveal how membrane organization can emerge from Brownian diffusion and a minimal set of physical properties of the membrane constituents.

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.

Activation energy and force fields during topological transitions of fluid lipid vesicles

Topological transitions of fluid lipid membranes are fundamental processes for cell life. For example, they are required for endo- and exocytosis or to enable neurotransmitters to cross the neural synapses. Here, inspired by the idea that fusion and fission proteins could have evolved in Nature in order to carry out a minimal work expenditure, we evaluate the minimal free energy pathway for the transition between two spherical large unilamellar vesicles and a dumbbell-shaped one. To address the problem, we propose and successfully use a Ginzburg-Landau type of free energy, which allows us to uniquely describe without interruption the whole, full-scale topological change. We also compute the force fields needed to overcome the involved energy barriers. The obtained forces are in excellent agreement, in terms of intensity, scale, and spatial localization with experimental data on typical fission protein systems, whereas they suggest the presence of additional features in fusion proteins.

From LUVs to GUVs─How to Cover Micrometer-Sized Pores with Membranes

Pore-spanning membranes (PSMs) are a versatile tool to investigate membrane-confined processes in a bottom-up approach. Pore sizes in the micrometer range are most suited to visualize PSMs using fluorescence microscopy. However, the preparation of these PSMs relies on the spreading of giant unilamellar vesicles (GUVs). GUV production faces several limitations. Thus, alternative ways to generate PSMs starting from large or small unilamellar vesicles that are more reproducibly prepared are highly desirable. Here we describe a method to produce PSMs obtained from large unilamellar vesicles, making use of droplet-stabilized GUVs generated in a microfluidic device. We analyzed the lipid diffusion in the free-standing and supported parts of the PSMs using z-scan fluorescence correlation spectroscopy and fluorescence recovery after photobleaching experiments in combination with finite element simulations. Employing atomic force indentation experiments, we also investigated the mechanical properties of the PSMs. Both lipid diffusion constants and lateral membrane tension were compared to those obtained on PSMs derived from electroformed GUVs, which are known to be solvent- and detergent-free, under otherwise identical conditions. Our results demonstrate that the lipid diffusion, as well as the mechanical properties of the resulting PSMs, is almost unaffected by the GUV formation procedure but depends on the chosen substrate functionalization. With the new method in hand, we were able to reconstitute the syntaxin-1A transmembrane domain in microfluidic GUVs and PSMs, which was visualized by fluorescence microscopy.

Cholesterol Stiffening of Lipid Membranes

Biomembrane order, dynamics, and other essential physicochemical parameters are controlled by cholesterol, a major component of mammalian cell membranes. Although cholesterol is well known to exhibit a condensing effect on fluid lipid membranes, the extent of stiffening that occurs with different degrees of lipid acyl chain unsaturation remains an enigma. In this review, we show that cholesterol locally increases the bending rigidity of both unsaturated and saturated lipid membranes, suggesting there may be a length-scale dependence of the bending modulus. We review our published data that address the origin of the mechanical effects of cholesterol on unsaturated and polyunsaturated lipid membranes and their role in biomembrane functions. Through a combination of solid-state deuterium NMR spectroscopy and neutron spin-echo spectroscopy, we show that changes in molecular packing cause the universal effects of cholesterol on the membrane bending rigidity. Our findings have broad implications for the role of cholesterol in lipid–protein interactions as well as raft-like mixtures, drug delivery applications, and the effects of antimicrobial peptides on lipid membranes.

Multiscale Modeling of Hepatitis B Virus Capsid Assembly and Its Dimorphism

Hepatitis B virus (HBV) is an endemic, chronic virus that leads to 800000 deaths per year. Central to the HBV lifecycle, the viral core has a protein capsid assembled from many copies of a single protein. The capsid protein adopts different (quasi-equivalent) conformations to form icosahedral capsids containing 180 or 240 proteins: T = 3 or T = 4, respectively, in Caspar-Klug nomenclature. HBV capsid assembly has become an important target for recently developed antivirals; nonetheless, the assembly pathways and mechanisms that control HBV dimorphism remain unclear. We describe computer simulations of the HBV assembly, using a coarse-grained model that has parameters learned from all-atom molecular dynamics simulations of a complete HBV capsid and yet is computationally tractable. Dynamical simulations with the resulting model reproduce experimental observations of HBV assembly pathways and products. By constructing Markov state models and employing transition path theory, we identify pathways leading to T = 3, T = 4, and other experimentally observed capsid morphologies. The analysis shows that capsid polymorphism is promoted by the low HBV capsid bending modulus, where the key factors controlling polymorphism are the conformational energy landscape and protein-protein binding affinities.

Mem3DG: Modeling membrane mechanochemical dynamics in 3D using discrete differential geometry

Biomembranes adopt varying morphologies that are vital to cellular functions. Many studies use computational modeling to understand how various mechanochemical factors contribute to membrane shape transformations. Compared with approximation-based methods (e.g., finite element method [FEM]), the class of discrete mesh models offers greater flexibility to simulate complex physics and shapes in three dimensions; its formulation produces an efficient algorithm while maintaining coordinate-free geometric descriptions. However, ambiguities in geometric definitions in the discrete context have led to a lack of consensus on which discrete mesh model is theoretically and numerically optimal; a bijective relationship between the terms contributing to both the energy and forces from the discrete and smooth geometric theories remains to be established. We address this and present an extensible framework, Mem3DG, for modeling 3D mechanochemical dynamics of membranes based on discrete differential geometry (DDG) on triangulated meshes. The formalism of DDG resolves the inconsistency and provides a unifying perspective on how to relate the smooth and discrete energy and forces. To demonstrate, Mem3DG is used to model a sequence of examples with increasing mechanochemical complexity: recovering classical shape transformations such as 1) biconcave disk, dumbbell, and unduloid; and 2) spherical bud on spherical, flat-patch membrane; investigating how the coupling of membrane mechanics with protein mobility jointly affects phase and shape transformation. As high-resolution 3D imaging of membrane ultrastructure becomes more readily available, we envision Mem3DG to be applied as an end-to-end tool to simulate realistic cell geometry under user-specified mechanochemical conditions.

Membrane area gain and loss during cytokinesis

In cytokinesis of animal cells, the cell is symmetrically divided into two. Since the cell's volume is conserved, the projected area has to increase to allow for the change of shape. Here we aim to predict how membrane gain and loss adapt during cytokinesis. We work with a kinetic model in which membrane turnover depends on membrane tension and cell shape. We apply this model to a series of calculated vesicle shapes as a proxy for the shape of dividing cells. We find that the ratio of kinetic turnover parameters changes nonmonotonically with cell shape, determined by the dependence of exocytosis and endocytosis on membrane curvature. Our results imply that controlling membrane turnover will be crucial for the successful division of artificial cells.