Phase transitions of the coupled membrane-cytoskeleton modify cellular shape

Effect of relative timescale on a system of particles sliding on a fluctuating energy landscape: Exact derivation of product measure condition

We consider a system of hardcore particles advected by a fluctuating potential energy landscape, whose dynamics is in turn affected by the particles. Earlier studies have shown that as a result of two-way coupling between the landscape and the particles, the system shows an interesting phase diagram as the coupling parameters are varied. The phase diagram consists of various different kinds of ordered phases and a disordered phase. We introduce a relative timescale ω between the particle and landscape dynamics, and study its effect on the steady state properties. We find there exists a critical value ω=ω_{c} when all configurations of the system are equally likely in the steady state. We prove this result exactly in a discrete lattice system and obtain an exact expression for ω_{c} in terms of the coupling parameters of the system. We show that ω_{c} is finite in the disordered phase, diverges at the boundary between the ordered and disordered phase, and is undefined in the ordered phase. We also derive ω_{c} from a coarse-grained level description of the system using linear hydrodynamics. We start with the assumption that there is a specific value ω^{*} of the relative timescale when correlations in the system vanish, and mean-field theory gives exact expressions for the current Jacobian matrix A and compressibility matrix K. Our exact calculations show that Onsager-type current symmetry relation AK=KA^{T} can be satisfied if and only if ω^{*}=ω_{c}. Our coarse-grained model calculations can be easily generalized to other coupled systems.

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.

Motility of Synthetic Cells from Engineered Lipids

Synthetic cells are artificial systems that resemble natural cells. Significant efforts have been made over the years to construct synthetic protocells that can mimic biological mechanisms and perform various complex processes. These include compartmentalization, metabolism, energy supply, communication, and gene reproduction. Cell motility is also of great importance, as nature uses elegant mechanisms for intracellular trafficking, immune response, and embryogenesis. In this review, we discuss the motility of synthetic cells made from lipid vesicles and relevant molecular mechanisms. Synthetic cell motion may be classified into surface-based or solution-based depending on whether it involves interactions with surfaces or movement in fluids. Collective migration behaviors have also been demonstrated. The swarm motion requires additional mechanisms for intercellular signaling and directional motility that enable communication and coordination among the synthetic vesicles. In addition, intracellular trafficking for molecular transport has been reconstituted in minimal cells with the help of DNA nanotechnology. These efforts demonstrate synthetic cells that can move, detect, respond, and interact. We envision that new developments in protocell motility will enhance our understanding of biological processes and be instrumental in bioengineering and therapeutic applications.

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.

Applications of Atomic Force Microscopy in HIV-1 Research

Obtaining an understanding of the mechanism underlying the interrelations between the structure and function of HIV-1 is of pivotal importance. In previous decades, this mechanism was addressed extensively in a variety of studies using conventional approaches. More recently, atomic force microscopy, which is a relatively new technique with unique capabilities, has been utilized to study HIV-1 biology. Atomic force microscopy can generate high-resolution images at the nanometer-scale and analyze the mechanical properties of individual HIV-1 virions, virus components (e.g., capsids), and infected live cells under near-physiological environments. This review describes the working principles and various imaging and analysis modes of atomic force microscopy, and elaborates on its distinctive contributions to HIV-1 research in areas such as mechanobiology and the physics of infection.

Universal properties of active membranes

We put forward a general field theory for nearly flat fluid membranes with embedded activators and analyze their critical properties using renormalization group techniques. Depending on the membrane-activator coupling, we find a crossover between acoustic and diffusive scaling regimes, with mean-field dynamical critical exponents z=1 and 2, respectively. We argue that the acoustic scaling, which is exact in all spatial dimensions, leads to an early-time behavior, which is representative of the spatiotemporal patterns observed at the leading edge of motile cells, such as oscillations superposed on the growth of the membrane width. In the case of mean-field diffusive scaling, one-loop corrections to the mean-field exponents reveal universal behavior distinct from the Kardar-Parisi-Zhang scaling of passive interfaces and signs of strong-coupling behavior.

Renormalization group study of the dynamics of active membranes: Universality classes and scaling laws

Motivated by experimental observations of patterning at the leading edge of motile eukaryotic cells, we introduce a general model for the dynamics of nearly-flat fluid membranes driven from within by an ensemble of activators. We include, in particular, a kinematic coupling between activator density and membrane slope which generically arises whenever the membrane has a nonvanishing normal speed. We unveil the phase diagram of the model by means of a perturbative field-theoretical renormalization group analysis. Due to the aforementioned kinematic coupling the natural early-time dynamical scaling is acoustic, that is the dynamical critical exponent is 1. However, as soon as the the normal velocity of the membrane is tuned to zero, the system crosses over to diffusive dynamic scaling in mean field. Distinct critical points can be reached depending on how the limit of vanishing velocity is realized: in each of them corrections to scaling due to nonlinear coupling terms must be taken into account. The detailed analysis of these critical points reveals novel scaling regimes which can be accessed with perturbative methods, together with signs of strong coupling behavior, which establishes a promising ground for further nonperturbative calculations. Our results unify several previous studies on the dynamics of active membrane, while also identifying nontrivial scaling regimes which cannot be captured by passive theories of fluctuating interfaces and are relevant for the physics of living membranes.

Pattern formation, localized and running pulsation on active spherical membranes

Active force generation by an actin-myosin cortex coupled to a cell membrane allows the cell to deform, respond to the environment, and mediate cell motility and division. Several membrane-bound activator proteins move along it and couple to the membrane curvature. Besides, they can act as nucleating sites for the growth of filamentous actin. Actin polymerization can generate a local outward push on the membrane. Inward pull from the contractile actomyosin cortex can propagate along the membrane via actin filaments. We use coupled evolution of fields to perform linear stability analysis and numerical calculations. As activity overcomes the stabilizing factors such as surface tension and bending rigidity, the spherical membrane shows instability towards pattern formation, localized pulsation, and running pulsation between poles. We present our results in terms of phase diagrams and evolutions of the coupled fields. They have relevance for living cells and can be verified in experiments on artificial cell-like constructs.

A Monte Carlo study of giant vesicle morphologies in nonequilibrium environments

It is known that giant vesicles undergo dynamic morphological changes when exposed to a detergent. The solubilization process may take multiple pathways. In this work, we identify lipid vesicle shape dynamics before the solubilization of 1,2-dioleoyl-sn-glycero-3-phosphocholine giant vesicles with Triton X-100 (TR) detergent. The violent lipid vesicle dynamics was observed with laser confocal scanning microscopy and was qualitatively explained via a numerical simulation. A three-dimensional Monte Carlo scheme was constructed that emulated the nonequilibrium conditions at the beginning stages of solubilization, accounting for a gradual addition of TR detergent molecules into the lipid bilayers. We suggest that the main driving factor for morphology change in lipid vesicles is the associative tendency of the TR molecules, which induces spontaneous curvature of the detergent inclusions, an intrinsic consequence of their molecular shape. The majority of the observed lipid vesicle shapes in the experiments were found to correspond very well to the numerically calculated shapes in the phase space of possible solutions. The results give an insight into the early stages of lipid vesicle solubilization by amphiphilic molecules, which is nonequilibrium in nature and very difficult to study.

Curvature-driven feedback on aggregation-diffusion of proteins in lipid bilayers

Membrane bending is an extensively studied problem from both modeling and experimental perspectives because of the wide implications of curvature generation in cell biology. Many of the curvature generating aspects in membranes can be attributed to interactions between proteins and membranes. These interactions include protein diffusion and formation of aggregates due to protein-protein interactions in the plane of the membrane. Recently, we developed a model that couples the in-plane flow of lipids and diffusion of proteins with the out-of-plane bending of the membrane. Building on this work, here, we focus on the role of explicit aggregation of proteins on the surface of the membrane in the presence of membrane bending and diffusion. We develop a comprehensive framework that includes lipid flow, membrane bending, the entropy of protein distribution, along with an explicit aggregation potential and derive the governing equations for the coupled system. We compare this framework to the Cahn-Hillard formalism to predict the regimes in which the proteins form patterns on the membrane. We demonstrate the utility of this model using numerical simulations to predict how aggregation and diffusion, when coupled with curvature generation, can alter the landscape of membrane-protein interactions.

Mechanical and Electrical Interaction of Biological Membranes with Nanoparticles and Nanostructured Surfaces

In this review paper, we theoretically explain the origin of electrostatic interactions between lipid bilayers and charged solid surfaces using a statistical mechanics approach, where the orientational degree of freedom of lipid head groups and the orientational ordering of the water dipoles are considered. Within the modified Langevin Poisson–Boltzmann model of an electric double layer, we derived an analytical expression for the osmotic pressure between the planar zwitterionic lipid bilayer and charged solid planar surface. We also show that the electrostatic interaction between the zwitterionic lipid head groups of the proximal leaflet and the negatively charged solid surface is accompanied with a more perpendicular average orientation of the lipid head-groups. We further highlight the important role of the surfaces’ nanostructured topography in their interactions with biological material. As an example of nanostructured surfaces, we describe the synthesis of TiO₂ nanotubular and octahedral surfaces by using the electrochemical anodization method and hydrothermal method, respectively. The physical and chemical properties of these nanostructured surfaces are described in order to elucidate the influence of the surface topography and other physical properties on the behavior of human cells adhered to TiO₂ nanostructured surfaces. In the last part of the paper, we theoretically explain the interplay of elastic and adhesive contributions to the adsorption of lipid vesicles on the solid surfaces. We show the numerically predicted shapes of adhered lipid vesicles corresponding to the minimum of the membrane free energy to describe the influence of the vesicle size, bending modulus, and adhesion strength on the adhesion of lipid vesicles on solid charged surfaces.

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.

Membrane shape remodeling by protein crowding

The steric repulsion between proteins on biological membranes is one of the most generic mechanisms that cause membrane shape changes. We present a minimal model in which a spontaneous curvature is induced by asymmetric protein crowding. Our results show that the interplay between the induced spontaneous curvature and the membrane tension determines the energy-minimizing shapes, which describes the wide range of experimentally observed membrane shapes, i.e., flat membranes, spherical vesicles, elongated tubular protrusions, and pearling structures. Moreover, the model gives precise predictions on how membrane shape changes by protein crowding can be tuned by controlling the protein size, the density of proteins, and the size of the crowded domain.

Value of models for membrane budding

The budding of membranes and curvature generation is common to many forms of trafficking in cells. Clathrin-mediated endocytosis, as a prototypical example of trafficking, has been studied in great detail using a variety of experimental systems and methods. Recently, advances in experimental methods have led to great strides in insights on the molecular mechanisms and the spatiotemporal dynamics of the protein machinery associated with membrane curvature generation. These advances have been ably supported by computational models, which have given us insights into the underlying mechanical principles of clathrin-mediated endocytosis. On the other hand, targeted experimental perturbation of membranes has lagged behind that of proteins in cells. In this area, modeling is especially critical to interpret experimental measurements in a mechanistic context. Here, we discuss the contributions made by these models to our understanding of endocytosis and identify opportunities to strengthen the connections between models and experiments.

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.

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.

Drops and fibers - how biomolecular condensates and cytoskeletal filaments influence each other

The cellular cytoskeleton self-organizes by specific monomer-monomer interactions resulting in the polymerization of filaments. While we have long thought about the role of polymerization in cytoskeleton formation, we have only begun to consider the role of condensation in cytoskeletal organization. In this review, we highlight how the interplay between polymerization and condensation leads to the formation of the cytoskeleton.

Active suppression of Ostwald ripening: Beyond mean-field theory

Active processes play a major role in the formation of membraneless cellular structures (biological condensates). Classical coarsening theory predicts that only a single droplet remains following Ostwald ripening. However, in both the cell nucleus and cytoplasm there coexist several membraneless organelles of the same basic composition, suggesting that there is some mechanism for suppressing Ostwald ripening. One potential candidate is the active regulation of liquid-liquid phase separation by enzymatic reactions that switch proteins between different conformational states (e.g., different levels of phosphorylation). Recent theoretical studies have used mean-field methods to analyze the suppression of Ostwald ripening in three-dimensional (3D) systems consisting of a solute that switches between two different conformational states, an S state that does not phase separate and a P state that does. However, mean-field theory breaks down in the case of 2D systems, since the concentration around a droplet varies as lnR rather than R^{-1}, where R is the distance from the center of the droplet. It also fails to capture finite-size effects. In this paper we show how to go beyond mean-field theory by using the theory of diffusion in domains with small holes or exclusions (strongly localized perturbations). In particular, we use asymptotic methods to study the suppression of Ostwald ripening in a 2D or 3D solution undergoing active liquid-liquid phase separation. We proceed by partitioning the region outside the droplets into a set of inner regions around each droplet together with an outer region where mean-field interactions occur. Asymptotically matching the inner and outer solutions, we derive leading-order conditions for the existence and stability of a multidroplet steady state. We also show how finite-size effects can be incorporated into the theory by including higher-order terms in the asymptotic expansion, which depend on the positions of the droplets and the boundary of the 2D or 3D domain. The theoretical framework developed in this paper provides a general method for analyzing active phase separation for dilute droplets in bounded domains such as those found in living cells.

Cell-Substrate Patterns Driven by Curvature-Sensitive Actin Polymerization: Waves and Podosomes

Cells adhered to an external solid substrate are observed to exhibit rich dynamics of actin structures on the basal membrane, which are distinct from those observed on the dorsal (free) membrane. Here we explore the dynamics of curved membrane proteins, or protein complexes, that recruit actin polymerization when the membrane is confined by the solid substrate. Such curved proteins can induce the spontaneous formation of membrane protrusions on the dorsal side of cells. However, on the basal side of the cells, such protrusions can only extend as far as the solid substrate and this constraint can convert such protrusions into propagating wave-like structures. We also demonstrate that adhesion molecules can stabilize localized protrusions that resemble some features of podosomes. This coupling of curvature and actin forces may underlie the differences in the observed actin-membrane dynamics between the basal and dorsal sides of adhered cells.

Modeling membrane nanotube morphology: the role of heterogeneity in composition and material properties

Membrane nanotubes are dynamic structures that may connect cells over long distances. Nanotubes are typically thin cylindrical tubes, but they may occasionally have a beaded architecture along the tube. In this paper, we study the role of membrane mechanics in governing the architecture of these tubes and show that the formation of bead-like structures along the nanotubes can result from local heterogeneities in the membrane either due to protein aggregation or due to membrane composition. We present numerical results that predict how membrane properties, protein density, and local tension compete to create a phase space that governs the morphology of a nanotube. We also find that there exists a discontinuity in the energy that impedes two beads from fusing. These results suggest that the membrane-protein interaction, membrane composition, and membrane tension closely govern the tube radius, number of beads, and the bead morphology.

Interface growth driven by a single active particle

We study pattern formation, fluctuations, and scaling induced by a growth-promoting active walker on an otherwise static interface. Active particles on an interface define a simple model for energy-consuming proteins embedded in the plasma membrane, responsible for membrane deformation and cell movement. In our model, the active particle overturns local valleys of the interface into hills, simulating growth, while itself sliding and seeking new valleys. In one dimension, this "overturn-slide-search" dynamics of the active particle causes it to move superdiffusively in the transverse direction while pulling the immobile interface upward. Using Monte Carlo simulations, we find an emerging tentlike mean profile developing with time, despite large fluctuations. The roughness of the interface follows scaling with the growth, dynamic, and roughness exponents, derived using simple arguments as β=2/3, z=3/2, and α=1/2, respectively, implying a breakdown of the usual scaling law β=α/z, due to very local growth of the interface. The transverse displacement of the puller on the interface scales as ∼t^{2/3} and the probability distribution of its displacement is bimodal, with an unusual linear cusp at the origin. Both the mean interface pattern and probability distribution display scaling. A puller on a static two-dimensional interface also displays aspects of scaling in the mean profile and probability distribution. We also show that a pusher on a fluctuating interface moves subdiffusively leading to a separation of timescale in pusher motion and interface response.

Interplay between surface and bending energy helps membrane protrusion formation

We consider a one-dimensional elastic membrane, which is pushed by growing filaments. The filaments tend to grow by creating local protrusions in the membrane and this process has surface energy and bending energy costs. Although it is expected that with increasing surface tension and bending rigidity, it should become more difficult to create a protrusion, we find that for a fixed bending rigidity, as the surface tension increases, protrusions are more easily formed. This effect also gives rise to nontrivial dependence of membrane velocity on the surface tension, characterized by a dip and a peak. We explain this unusual phenomenon by studying in detail the interplay of the surface and the bending energy and show that this interplay is responsible for a qualitative shape change of the membrane, which gives rise to the above effect.

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.

Inception Mechanisms of Tunneling Nanotubes

Tunneling nanotubes (TNTs) are thin membranous tubes that interconnect cells, representing a novel route of cell-to-cell communication and spreading of pathogens. TNTs form between many cell types, yet their inception mechanisms remain elusive. We review in this study general concepts related to the formation and stability of membranous tubular structures with a focus on a deviatoric elasticity model of membrane nanodomains. We review experimental evidence that tubular structures initiate from local membrane bending facilitated by laterally distributed proteins or anisotropic membrane nanodomains. We further discuss the numerical results of several theoretical and simulation models of nanodomain segregation suggesting the mechanisms of TNT inception and stability. We discuss the coupling of nanodomain segregation with the action of protruding cytoskeletal forces, which are mostly provided in eukaryotic cells by the polymerization of f-actin, and review recent inception mechanisms of TNTs in relation to motor proteins.

Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces

Eukaryote cells have flexible membranes that allow them to have a variety of dynamical shapes. The shapes of the cells serve important biological functions, both for cells within an intact tissue, and during embryogenesis and cellular motility. How cells control their shapes and the structures that they form on their surface has been a subject of intensive biological research, exposing the building blocks that cells use to deform their membranes. These processes have also drawn the interest of theoretical physicists, aiming to develop models based on physics, chemistry and nonlinear dynamics. Such models explore quantitatively different possible mechanisms that the cells can employ to initiate the spontaneous formation of shapes and patterns on their membranes. We review here theoretical work where one such class of mechanisms was investigated: the coupling between curved membrane proteins, and the cytoskeletal forces that they recruit. Theory indicates that this coupling gives rise to a rich variety of membrane shapes and dynamics, while experiments indicate that this mechanism appears to drive many cellular shape changes.This article is part of the theme issue 'Self-organization in cell biology'.

Modeling Membrane Curvature Generation due to Membrane⁻Protein Interactions

To alter and adjust the shape of the plasma membrane, cells harness various mechanisms of curvature generation. Many of these curvature generation mechanisms rely on the interactions between peripheral membrane proteins, integral membrane proteins, and lipids in the bilayer membrane. Mathematical and computational modeling of membrane curvature generation has provided great insights into the physics underlying these processes. However, one of the challenges in modeling these processes is identifying the suitable constitutive relationships that describe the membrane free energy including protein distribution and curvature generation capability. Here, we review some of the commonly used continuum elastic membrane models that have been developed for this purpose and discuss their applications. Finally, we address some fundamental challenges that future theoretical methods need to overcome to push the boundaries of current model applications.

Mechanics of Cell Mechanosensing in Protrusion and Retraction of Lamellipodium

Lamellipodia (LP), a subcellular structure at cell front, plays a key role in cell spreading and migration. And its mechanosensing function is of crucial importance for cell activities. But the mechanism of the mechanosensing function remains poorly understood. Here we developed a multiscale model to consider its protrusion and retraction processes, and analyzed the forces acted on the key structural components of the LP and the effect of these forces on LP movement. Our results show that raising substrate rigidity increases the force acting on the focal adhesion (FA) and decreases the force on LP actin, thus promoting the maturation of FA while suppressing the detachment of LP actin from the cell membrane. The membrane tension also influences the LP movement, but its effect is opposite to that of the substrate rigidity. It turns out that the substrate rigidity and membrane tension together regulate the dynamics of FAs and the detachment of LP actin, which in turn determine the LP movement. Interestingly, we found that the effect of substrate rigidity and membrane tension on the LP movement both exhibit a biphasic manner. We show that our predictions agree, in general, with the experiments on cell mechanosensing behaviors at both subcellular and cellular levels.

Ordered phases in coupled nonequilibrium systems: Static properties

We study a coupled driven system in which two species of particles are advected by a fluctuating potential energy landscape. While the particles follow the potential gradient, each species affects the local shape of the landscape in different ways. As a result of this two-way coupling between the landscape and the particles, the system shows interesting new phases, characterized by different sorts of long-ranged order in the particles and in the landscape. In all these ordered phases, the two particle species phase separate completely from each other, but the underlying landscape may either show complete ordering, with macroscopic regions with distinct average slopes, or may show coexistence of ordered and disordered regions, depending on the differential nature of effect produced by the particle species on the landscape. We discuss several aspects of static properties of these phases in this paper, and we discuss the dynamics of these phases in the sequel.

On the role of external force of actin filaments in the formation of tubular protrusions of closed membrane shapes with anisotropic membrane components

Biological membranes are composed of different components and there is no a priori reason to assume that all components are isotropic. It was previously shown that the anisotropic properties of membrane components may explain the stability of membrane tubular protrusions even without the application of external force. Our theoretical study focuses on the role of anisotropic membrane components in the stability of membrane tubular structures generated or stabilized by actin filaments. We show that the growth of the actin cytoskeleton inside the vesicle can induce the partial lateral segregation of different membrane components. The entropy of mixing of membrane components hinders the total lateral segregation of the anisotropic and isotropic membrane components. Self-assembled aggregates formed by anisotropic membrane components facilitate the growth of long membrane tubular protrusions. Protrusive force generated by actin filaments favors strong segregation of membrane components by diminishing the opposing effect of mixing entropy.

Curvature-Induced Spatial Ordering of Composition in Lipid Membranes

Phase segregation of membranal components, such as proteins, lipids, and cholesterols, leads to the formation of aggregates or domains that are rich in specific constituents. This process is important in the interaction of the cell with its surroundings and in determining the cell's behavior and fate. Motivated by published experiments on curvature-modulated phase separation in lipid membranes, we formulate a mathematical model aiming at studying the spatial ordering of composition in a two-component biomembrane that is subjected to a prescribed (imposed) geometry. Based on this model, we identified key nondimensional quantities that govern the biomembrane response and performed numerical simulations to quantitatively explore their influence. We reproduce published experimental observations and extend them to surfaces with geometric features (imposed geometry) and lipid phases beyond those used in the experiments. In addition, we demonstrate the possibility for curvature-modulated phase separation above the critical temperature and propose a systematic procedure to determine which mechanism, the difference in bending stiffness or difference in spontaneous curvatures of the two phases, dominates the coupling between shape and composition.

Actin Cytoskeleton-Mediated Constriction of Membrane Organelles via Endoplasmic Reticulum Scaffolding

Intracellular organelles constantly undergo fission to facilitate turnover, transport, and functional changes. The cytoskeleton has long been understood to play a role in these events, and recent work strongly suggests that several conserved molecular players cooperate with the cytoskeleton to mediate the fission process. Membrane curvature-inducing, membrane scission proteins, and force-inducing cytoskeletal proteins all cooperate to drive the fission process. Recent work suggests that the endoplasmic reticulum serves as the linchpin that orchestrates and spatially organizes fission via these curvature-inducing, scission, and force-producing molecules. This all leads us to postulate a "universal theory" of organelle fission with distinct biophysical and biochemical features mediated by a finite number of physical and molecular constraints. This new physical paradigm deserves special attention from those who wish to model these processes, since previous theoretical and experimental attempts to elucidate these fission mechanisms have not included the organizing factor of the endoplasmic reticulum. Here we review the basic concepts of this new model for organelle fission, and explore the implications thereof. Previous studies that didn't include this component can now be interpreted in light of these new data and serve as a useful guide for understanding how this process happens in vivo. Thus, this review provides direction for future modeling and experimental efforts to better understand how these complex systems and processes are regulated in both healthy and diseased biological systems.

Toward the reconstitution of synthetic cell motility

Cellular motility is a fundamental process essential for embryonic development, wound healing, immune responses, and tissues development. Cells are mostly moving by crawling on external, or inside, substrates which can differ in their surface composition, geometry, and dimensionality. Cells can adopt different migration phenotypes, e.g., bleb-based and protrusion-based, depending on myosin contractility, surface adhesion, and cell confinement. In the few past decades, research on cell motility has focused on uncovering the major molecular players and their order of events. Despite major progresses, our ability to infer on the collective behavior from the molecular properties remains a major challenge, especially because cell migration integrates numerous chemical and mechanical processes that are coupled via feedbacks that span over large range of time and length scales. For this reason, reconstituted model systems were developed. These systems allow for full control of the molecular constituents and various system parameters, thereby providing insight into their individual roles and functions. In this review we describe the various reconstituted model systems that were developed in the past decades. Because of the multiple steps involved in cell motility and the complexity of the overall process, most of the model systems focus on very specific aspects of the individual steps of cell motility. Here we describe the main advancement in cell motility reconstitution and discuss the main challenges toward the realization of a synthetic motile cell.

Closed membrane shapes with attached BAR domains subject to external force of actin filaments

Membrane deformations induced by attached BAR superfamily domains could trigger or facilitate the growth of plasma membrane protrusions. The BAR domain family consists of BAR, F-BAR and I-BAR domains, each enforcing a different local curvature when attached to the membrane surface. Our theoretical study mainly focuses on the role of I-BAR in the membrane tubular deformations generated or stabilised by actin filaments. The influence of the area density of membrane attached BAR domains and their intrinsic curvature on the closed membrane shapes (vesicles) was investigated numerically. We derived an analytical approximative expression for the critical relative area density of BARs at which the membrane tubular protrusions on vesicles are most prominent. We have shown that the BARs with a higher intrinsic curvature induce thinner and longer cylindrical protrusions. The average orientation of the membrane attached BARs is altered when the vesicle shape is subjected to external force of growing actin rod-like structure inside a vesicle. The average orientation angle of membrane attached BARs may indicate whether the actin filaments are just stabilising the protrusion or generating it by stretching the vesicle.

Time-resolved neurite mechanics by thermal fluctuation assessments

In the absence of simple noninvasive measurements, the knowledge of temporal and spatial variations of axons mechanics remains scarce. By extending thermal fluctuation spectroscopy (TFS) to long protrusions, we determine the transverse amplitude thermal fluctuation spectra that allow direct and simultaneous access to three key mechanics parameters: axial tension, bending flexural rigidity and plasma membrane tension. To test our model, we use PC12 cell protrusions-a well-know biophysical model of axons-in order to simplify the biological system under scope. For instance, axial and plasma membrane tension are found in the range of nano Newton and tens of pico Newtons per micron respectively. Furthermore, our results shows that the TFS technique is capable to distinguish quasi-identical protrusions. Another advantage of our approach is the time resolved nature of the measurements. Indeed, in the case of long term experiments on PC12 protrusions, TFS has revealed large temporal, correlated variations of the protrusion mechanics, displaying extraordinary feedback control over the axial tension in order to maintain a constant tension value.

Physical model for the geometry of actin-based cellular protrusions

Actin-based cellular protrusions are a ubiquitous feature of cell morphology, e.g., filopodia and microvilli, serving a huge variety of functions. Despite this, there is still no comprehensive model for the mechanisms that determine the geometry of these protrusions. We present here a detailed computational model that addresses a combination of multiple biochemical and physical processes involved in the dynamic regulation of the shape of these protrusions. We specifically explore the role of actin polymerization in determining both the height and width of the protrusions. Furthermore, we show that our generalized model can explain multiple morphological features of these systems, and account for the effects of specific proteins and mutations.

Membrane-wrapping contributions to malaria parasite invasion of the human erythrocyte

The blood stage malaria parasite, the merozoite, has a small window of opportunity during which it must successfully target and invade a human erythrocyte. The process of invasion is nonetheless remarkably rapid. To date, mechanistic models of invasion have focused predominantly on the parasite actomyosin motor contribution to the energetics of entry. Here, we have conducted a numerical analysis using dimensions for an archetypal merozoite to predict the respective contributions of the host-parasite interactions to invasion, in particular the role of membrane wrapping. Our theoretical modeling demonstrates that erythrocyte membrane wrapping alone, as a function of merozoite adhesive and shape properties, is sufficient to entirely account for the first key step of the invasion process, that of merozoite reorientation to its apex and tight adhesive linkage between the two cells. Next, parasite-induced reorganization of the erythrocyte cytoskeleton and release of parasite-derived membrane can also account for a considerable energetic portion of actual invasion itself, through membrane wrapping. Thus, contrary to the prevailing dogma, wrapping by the erythrocyte combined with parasite-derived membrane release can markedly reduce the expected contributions of the merozoite actomyosin motor to invasion. We therefore propose that invasion is a balance between parasite and host cell contributions, evolved toward maximal efficient use of biophysical forces between the two cells.

Membrane sorting via the extracellular matrix

We consider the coupling between a membrane and the extracellular matrix. Computer simulations demonstrate that the latter coupling is able to sort lipids. It is assumed that membranes are elastic manifolds, and that this manifold is disrupted by the extracellular matrix. For a solid-supported membrane with an actin network on top, regions of positive curvature are induced below the actin fibers. A similar mechanism is conceivable by assuming that the proteins which connect the cytoskeleton to the membrane induce local membrane curvature. The regions of non-zero curvature exist irrespective of any phase transition the lipids themselves may undergo. For lipids that prefer certain curvature, the extracellular matrix thus provides a spatial template for the resulting lateral domain structure of the membrane.

Dynamics and instabilities of lipid bilayer membrane shapes

Biological membranes undergo constant shape remodeling involving the formation of highly curved structures. The lipid bilayer represents the fundamental architecture of the cellular membrane with its shapes determined by the Helfrich curvature bending energy. However, the dynamics of bilayer shape transitions, especially their modulation by membrane proteins, and the resulting shape instabilities, are still not well understood. Here, we review in a unifying manner several theories that describe the fluctuations (i.e. undulations) of bilayer shapes as well as their local coupling with lipid or protein density variation. The coupling between local membrane curvature and lipid density gives rise to a 'slipping mode' in addition to the conventional 'bending mode' for damping the membrane fluctuation. This leads to a number of interesting experimental phenomena regarding bilayer shape dynamics. More importantly, curvature-inducing proteins can couple with membrane shape and eventually render the membrane unstable. A criterion for membrane shape instability is derived from a linear stability analysis. The instability criterion reemphasizes the importance of membrane tension in regulating the stability and dynamics of membrane geometry. Recent progresses in understanding the role of membrane tension in regulating dynamical cellular processes are also reviewed. Protein density is emphasized as a key factor in regulating membrane shape transitions: a threshold density of curvature coupling proteins is required for inducing membrane morphology transitions.

Exploring the elasticity and adhesion behavior of cardiac fibroblasts by atomic force microscopy indentation

AFM was used to collect the whole force-deformation cell curves. They provide both the elasticity and adhesion behavior of mouse primary cardiac fibroblasts. To confirm the hypothesis that a link exists between the membrane receptors and the cytoskeletal filaments causing therefore changing in both elasticity and adhesion behavior, actin-destabilizing Cytochalsin D was administrated to the fibroblasts. From immunofluorescence observation and AFM loading/unloading curves, cytoskeletal reorganization as well as a change in the elasticity and adhesion was indeed observed. Elasticity of control fibroblasts is three times higher than that for fibroblasts treated with 0.5 μM Cytochalasin. Moreover, AFM loading-unloading curves clearly show the different mechanical behavior of the two different cells analyzed: (i) for control cells the AFM cantilever rises during the dwell time while cells with Cytochalasin fail to show such an active resistance; (ii) the maximum force to deform control cells is quite higher and as far as adhesion is concern (iii) the maximum separation force, detachment area and the detachment process time are much larger for control compared to the Cytochalasin treated cells. Therefore, alterations in the cytoskeleton suggest that a link must exist between the membrane receptors and the cytoskeletal filaments beneath the cellular surface and inhibition of actin polymerization has effects on the whole cell mechanical behavior as well as adhesion.

A theoretical study of biological membrane response to temperature gradients at the single-cell level

Recent experimental studies provide evidence for the existence of a spatially non-uniform temperature field in living cells and in particular in their plasma membrane. These findings have led to the development of a new and exciting field: thermal biology at the single-cell level. Here, we examine theoretically a specific aspect of this field, i.e. how temperature gradients at the single-cell level affect the phase behaviour and geometry of heterogeneous membranes. We address this issue by using the Onsager reciprocal relations combined with a simple model for a binary lipid mixture. We demonstrate that even small temperature variations along the membrane may introduce intriguing phenomena, such as phase separation above the critical temperature and unusual shape response. These results also suggest that the shape of a membrane can be manipulated by dynamically controlling the temperature field in its vicinity. The effects of intramembranous temperature gradients have never been studied experimentally. Thus, the predictions of the current contribution are of a somewhat speculative nature. Experimental verification of these results could mark the beginning of a new line of research in the field of biological membranes. We report our findings with the hope of inspiring others to perform such experiments.

Influence of nanoparticle-membrane electrostatic interactions on membrane fluidity and bending elasticity

The aim of this work is to investigate the effect of electrostatic interactions between the nanoparticles and the membrane lipids on altering the physical properties of the liposomal membrane such as fluidity and bending elasticity. For this purpose, we have used nanoparticles and lipids with different surface charges. Positively charged iron oxide (γ-Fe2O3) nanoparticles, neutral and negatively charged cobalt ferrite (CoFe2O4) nanoparticles were encapsulated in neutral lipid 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine and negatively charged 1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-L-serine lipid mixture. Membrane fluidity was assessed through the anisotropy measurements using the fluorescent probe 1,6-diphenyl-1,3,5-hexatriene. Though the interaction of both the types of nanoparticles reduced the membrane fluidity, the results were more pronounced in the negatively charged liposomes encapsulated with positively charged iron oxide nanoparticles due to strong electrostatic attractions. X-ray photoelectron spectroscopy results also confirmed the presence of significant quantity of positively charged iron oxide nanoparticles in negatively charged liposomes. Through thermally induced shape fluctuation measurements of the giant liposomes, a considerable reduction in the bending elasticity modulus was observed for cobalt ferrite nanoparticles. The experimental results were supported by the simulation studies using modified Langevin-Poisson-Boltzmann model.

Linking actin networks and cell membrane via a reaction-diffusion-elastic description of nonlinear filopodia initiation

Reaction-diffusion models have been used to describe pattern formation on the cellular scale, and traditionally do not include feedback between cellular shape changes and biochemical reactions. We introduce here a distinct reaction-diffusion-elasticity approach: The reaction-diffusion part describes bistability between two actin orientations, coupled to the elastic energy of the cell membrane deformations. This coupling supports spatially localized patterns, even when such solutions do not exist in the uncoupled self-inhibited reaction-diffusion system. We apply this concept to describe the nonlinear (threshold driven) initiation mechanism of actin-based cellular protrusions and provide support by several experimental observations.

Interaction between dipolar lipid headgroups and charged nanoparticles mediated by water dipoles and ions

In this work, a theoretical model describing the interaction between a positively or negatively charged nanoparticle and neutral zwitterionic lipid bilayers is presented. It is shown that in the close vicinity of the positively charged nanoparticle, the zwitterionic lipid headgroups are less extended in the direction perpendicular to the membrane surface, while in the vicinity of the negatively charged nanoparticle, the headgroups are more extended. This result coincides with the calculated increase in the osmotic pressure between the zwitterionic lipid surface and positively charged nanoparticle and the decrease of osmotic pressure between the zwitterionic lipid surface and the negatively charged nanoparticle. Our theoretical predictions agree well with the experimentally determined fluidity of a lipid bilayer membrane in contact with positively or negatively charged nanoparticles. The prospective significance of the present work is mainly to contribute to better understanding of the interactions of charged nanoparticles with a zwitterionic lipid bilayer, which may be important in the efficient design of the lipid/nanoparticle nanostructures (like liposomes with encapsulated nanoparticles), which have diverse biomedical applications, including targeted therapy (drug delivery) and imaging of cancer cells.

Signaling networks and cell motility: a computational approach using a phase field description

The processes of protrusion and retraction during cell movement are driven by the turnover and reorganization of the actin cytoskeleton. Within a reaction-diffusion model which combines processes along the cell membrane with processes within the cytoplasm a Turing type instability is used to form the necessary polarity to distinguish between cell front and rear and to initiate the formation of different organizational arrays within the cytoplasm leading to protrusion and retraction. A simplified biochemical network model for the activation of GTPase which accounts for the different dimensionality of the cell membrane and the cytoplasm is used for this purpose and combined with a classical Helfrich type model to account for bending and stiffness effects of the cell membrane. In addition streaming within the cytoplasm and the extracellular matrix is taken into account. Combining these phenomena allows to simulate the dynamics of cells and to reproduce the primary phenomenology of cell motility. The coupled model is formulated within a phase field approach and solved using adaptive finite elements.

Cell migration with multiple pseudopodia: temporal and spatial sensing models

Cell migration triggered by pseudopodia (or "false feet") is the most used method of locomotion. A 3D finite element model of a cell migrating over a 2D substrate is proposed, with a particular focus on the mechanical aspects of the biological phenomenon. The decomposition of the deformation gradient is used to reproduce the cyclic phases of protrusion and contraction of the cell, which are tightly synchronized with the adhesion forces at the back and at the front of the cell, respectively. First, a steady active deformation is considered to show the ability of the cell to simultaneously initiate multiple pseudopodia. Here, randomness is considered as a key aspect, which controls both the direction and the amplitude of the false feet. Second, the migration process is described through two different strategies: the temporal and the spatial sensing models. In the temporal model, the cell "sniffs" the surroundings by extending several pseudopodia and only the one that receives a positive input will become the new leading edge, while the others retract. In the spatial model instead, the cell senses the external sources at different spots of the membrane and only protrudes one pseudopod in the direction of the most attractive one.

Mesoscale simulations of curvature-inducing protein partitioning on lipid bilayer membranes in the presence of mean curvature fields

The membrane-surface migration of curvature-inducing proteins in response to membrane curvature gradients has been investigated using Monte Carlo simulations of a curvilinear membrane model based on the Helfrich Hamiltonian. Consistent with theoretical and experimental data, we find the proteins that generate curvature can also sense the background membrane curvature, wherein they preferentially partition to the high curvature regions. The partitioning strength depends linearly on local membrane curvature and the slope (or the coupling constant) of the partitioning probability versus mean curvature depends on the membrane bending rigidity and instantaneous curvature field caused by different proteins. Our simulation study allows us to quantitatively characterize and identify the important factors affecting the coupling constant (slope), which may be difficult to determine in experiments. Furthermore, the membrane model is used to study budding of vesicles where it is found that in order to stabilize a mature vesicle with a stable 'neck-region' (or stable membrane overhangs), the area (extent) of the intrinsic curvature region needs to exceed a threshold-critical value. The migration and partitioning of curvature-inducing proteins in a budding vesicle with a stable neck (with a characteristic negative value of the Gaussian curvature) is investigated.

'Run-and-tumble' or 'look-and-run'? A mechanical model to explore the behavior of a migrating amoeboid cell

Single cell migration constitutes a fundamental phenomenon involved in many biological events. Amoeboid cells are single cell organisms that migrate in a cyclic manner like worms. In this paper, we propose a 3D finite element model of an amoeboid cell migrating over a 2D surface. In particular, we focus on the mechanical aspect of the problem. The cell is able to generate cyclic active deformations, such as protrusion and contraction, in any direction. The progression of the cell is governed by a tight synchronization between the adhesion forces, which are alternatively applied at the front and at the rear edges of the cell, and the protrusion-contraction phases of the cell body. Finally, two important aspects have been taken into account: (1) the external stimuli in response to which the cell migrates (e.g. need to feed, morphogenetic events, normal or abnormal environment cues), (2) the heterogeneity of the 2D substrate (e.g. obstacles, rugosity, slippy regions) for which two distinct approaches have been evaluated: the 'run-and-tumble' strategy and the 'look-and-run' strategy. Overall, the results show a good agreement with respect to the experimental observations and the data from the literature (e.g. velocity and strains). Therefore, the present model helps, on one hand, to better understand the intimate relationship between the deformation modes of a cell and the adhesion strength that is required by the cell to crawl over a substrate, and, on the other hand, to put in evidence the crucial role played by mechanics during the migration process.

Excitable actin dynamics in lamellipodial protrusion and retraction

Many animal cells initiate crawling by protruding lamellipodia, consisting of a dense network of actin filaments, at their leading edge. We imaged XTC cells that exhibit flat lamellipodia on poly-L-lysine-coated coverslips. Using active contours, we tracked the leading edge and measured the total amount of F-actin by summing the pixel intensities within a 5-μm band. We observed protrusion and retraction with period 130-200 s and local wavelike features. Positive (negative) velocities correlated with minimum (maximum) integrated actin concentration. Approximately constant retrograde flow indicated that protrusions and retractions were driven by fluctuations of the actin polymerization rate. We present a model of these actin dynamics as an excitable system in which a diffusive, autocatalytic activator causes actin polymerization; F-actin accumulation in turn inhibits further activator accumulation. Simulations of the model reproduced the pattern of actin polymerization seen in experiments. To explore the model's assumption of an autocatalytic activation mechanism, we imaged cells expressing markers for both F-actin and the p21 subunit of the Arp2/3 complex. We found that integrated Arp2/3-complex concentrations spike several seconds before spikes of F-actin concentration. This suggests that the Arp2/3 complex participates in an activation mechanism that includes additional diffuse components. Response of cells to stimulation by fetal calf serum could be reproduced by the model, further supporting the proposed dynamical picture.

Mechanical modulation of receptor-ligand interactions at cell-cell interfaces

Cell surface receptors have been extensively studied because they initiate and regulate signal transduction cascades leading to a variety of functional cellular outcomes. An important class of immune receptors (e.g., T-cell antigen receptors) whose ligands are anchored to the surfaces of other cells remain poorly understood. The mechanism by which ligand binding initiates receptor phosphorylation, a process termed "receptor triggering", remains controversial. Recently, direct measurements of the (two-dimensional) receptor-ligand complex lifetimes at cell-cell interface were found to be smaller than (three-dimensional) lifetimes in solution but the underlying mechanism is unknown. At the cell-cell interface, the receptor-ligand complex spans a short intermembrane distance (15 nm) compared to long surface molecules (LSMs) whose ectodomains span >40 nm and these LSMs include phosphatases (e.g., CD45) that dephosphorylate the receptor. It has been proposed that size-based segregation of LSMs from a receptor-ligand complex is a mechanism of receptor triggering but it is unclear whether the mechanochemistry supports such small-scale segregation. Here we present a nanometer-scale mathematical model that couples membrane elasticity with the compressional stiffness and lateral mobility of LSMs. We find robust supradiffusive segregation of LSMs from a single receptor-ligand complex. The model predicts that LSM redistribution will result in a time-dependent tension on the complex leading to a decreased two-dimensional lifetime. Interestingly, the model predicts a nonlinear relationship between the three- and two-dimensional lifetimes, which can enhance the ability of receptors to discriminate between similar ligands.

A review of models of fluctuating protrusion and retraction patterns at the leading edge of motile cells

A characteristic feature of motile cells as they undergo a change in motile behavior is the development of fluctuating exploratory motions of the leading edge, driven by actin polymerization. We review quantitative models of these protrusion and retraction phenomena. Theoretical studies have been motivated by advances in experimental and computational methods that allow controlled perturbations, single molecule imaging, and analysis of spatiotemporal correlations in microscopic images. To explain oscillations and waves of the leading edge, most theoretical models propose nonlinear interactions and feedback mechanisms among different components of the actin cytoskeleton system. These mechanisms include curvature-sensing membrane proteins, myosin contraction, and autocatalytic biochemical reaction kinetics. We discuss how the combination of experimental studies with modeling promises to quantify the relative importance of these biochemical and biophysical processes at the leading edge and to evaluate their generality across cell types and extracellular environments.