We introduce a simulation strategy to consistently couple continuum biomembrane dynamics to the motion of discrete biological macromolecules residing within or on the membrane. The methodology is used to study the diffusion of integral membrane proteins that impart a curvature on the bilayer surrounding them. Such proteins exhibit a substantial reduction in diffusion coefficient relative to "flat" proteins; this effect is explained by elementary hydrodynamic considerations.
Nonequilibrium dynamics of biomembranes with active inclusions is considered. The inclusions represent protein molecules which perform cyclic internal conformational motions driven by the energy brought with ATP ligands. As protein conformations cyclically change, this induces hydrodynamical flows and also directly affects the local curvature of a membrane. On the other hand, variations in the local curvature of the membrane modify the transitions rates between conformational states in a protein, leading to a feedback in the considered system. Moreover, active inclusions can move diffusively t…
We study the non-equilibrium dynamics of lipid membranes with proteins that actively pump ions across the membrane. We find that the activity leads to a fluctuating force distribution due to electrostatic interactions arising from variation in dielectric constant across the membrane. By applying a multipole expansion we find effects on both the tension and bending rigidity dominated parts of the membranes fluctuation spectrum. We discuss how our model compares with previous studies of force-multipole models.
We study a physical model for the interaction between general inclusions bound to fluid membranes that possess finite tension, as well as the usual bending rigidity. We are motivated by an interest in proteins bound to cell membranes that apply forces to these membranes, due to either entropic or direct chemical interactions. We find an exact analytic solution for the repulsive interaction between two similar circularly symmetric inclusions. This repulsion extends over length scales of order tens of nanometers, and contrasts with the membrane-mediated contact attraction for similar inclusions …
A very common strategy to penetrate the cell membrane and access the internal compartment, consists of using sharp tips or nano needles. However recent experiments of cell penetration by atomic force microscopy tips show, contrary to expectations, a weak dependence of penetration force on the curvature of the tip. Using molecular dynamics simulations and analytical arguments, here we show that membrane disruption can be driven either by elastic or entropic forces depending on the membrane size. Our findings have potentially relevant implications in tissue engineering and drug delivery, as they…
We investigate the effect on biomembrane mechanical properties due to the presence an external potential for a non-conductive non-compressible membrane surrounded by different electrolytes. By solving the Debye-Huckel and Laplace equations for the electrostatic potential and using the relevant stress-tensor we find: in (1.) the small screening length limit, where the Debye screening length is smaller than the distance between the electrodes, the screening certifies that all electrostatic interactions are short-range and the major effect of the applied potential is to decrease the membrane tens…
We discuss the hydrodynamic collective effects due to active protein molecules that are immersed in lipid bilayer membranes and modeled as stochastic force dipoles. We specifically take into account the presence of the bulk solvent which surrounds the two-dimensional fluid membrane. Two membrane geometries are considered: the free membrane case and the confined membrane case. Using the generalized membrane mobility tensors, we estimate the active diffusion coefficient and the drift velocity as a function of the size of a diffusing object. The hydrodynamic screening lengths distinguish the two …
We review the theoretical analyses and simulations of the interactions between curvature-inducing proteins and biomembranes. Laterally isotropic proteins induce spherical budding, whereas anisotropic proteins, such as Bin/Amphiphysin/Rvs (BAR) superfamily proteins, induce tabulation. Both types of proteins can sense the membrane curvature. We describe the theoretical analyses of various transitions of protein binding accompanied by a change in various properties, such as the number of buds, the radius of membrane tubes, and the nematic order of anisotropic proteins. Moreover, we explain the me…
This chapter summarizes several approaches combining theory, simulation and experiment that aim for a better understanding of phenomena in lipid bilayers and membrane protein systems, covering topics such as lipid rafts, membrane mediated interactions, attraction between transmembrane proteins, and aggregation in biomembranes leading to large superstructures such as the light harvesting complex of green plants. After a general overview of theoretical considerations and continuum theory of lipid membranes we introduce different options for simulations of biomembrane systems, addressing question…
How long a fluid membrane vesicle stressed with a steady ramp of micropipette last before rupture? Or conversely, how high the surface tension should be to rupture a membrane? To answer these challenging questions we have developed a theoretical framework that allows description and reproduction of Dynamic Tension Spectroscopy (DTS) observations. The kinetics of the membrane rupture under ramps of surface tension is described as a combination of initial pore formation followed by Brownian process of the pore radius crossing the time-dependent energy barrier. We present the formalism and derive…
The fluctuation pressure of a lipid-bilayer membrane is important for the stability of lamellar phases and the adhesion of membranes to surfaces. In contrast to many theoretical studies, which predict a decrease of the pressure with the cubed inverse distance between the membranes, Freund suggested very recently a linear inverse distance dependence [Proc. Natl. Acad. Sci. U.S.A. 110, 2047 (2013)]. We address this discrepancy by performing Monte Carlo simulations for a membrane model discretized on a square lattice and employ the wall theorem to evaluate the pressure for a single membrane betwe…
We discuss a realistic scenario, accounting for the existence of sub-micrometric protein domains in cell membranes. At the biological level, such membrane domains have been shown to be specialized, in order to perform a determined biological task, in the sense that they gather one or a few protein species out of the hundreds of different ones that a cell membrane may contain. By analyzing the balance between mixing entropy and protein affinities, we propose that such protein sorting in distinct domains can be explained without appealing to pre-existing lipidic micro-phase separations, as in th…
Biomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging of nutrients, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modeling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model symmetry br…
Collective behavior of proteins on biomembranes is usually studied within the spontaneous curvature model. Here we consider an alternative phenomenological approach, which accounts consistently for partial ordering of proteins as well as the anchoring forces exerted on a membrane by layer of proteins. We show analytically that such anisotropic interactions can drive membrane bending, resulting in non-trivial equilibrium morphologies. The predicted instabilities can advance our conceptual understanding of physical mechanisms behind collective phenomena in biological systems, in particular those…
Ever since the raft model for biomembranes has been proposed, the traditional view of biomembranes based on the fluid-mosaic model has been altered. In the raft model, dynamical heterogeneities in multi-component lipid bilayers play an essential role. Focusing on the lateral phase separation of biomembranes and vesicles, we review some of the most relevant research conducted over the last decade. We mainly refer to those experimental works that are based on physical chemistry approach, and to theoretical explanations given in terms of soft matter physics. In the first part, we describe the pha…
Cell membranes are out of thermodynamic equilibrium notably because of membrane recycling, i.e. active exchange of material with the cytosol. We propose an analytically tractable model of biomembrane predicting the effects of recycling on the size of protein nanodomains. It includes a short-range attraction between proteins and a weaker long-range repulsion which ensures the existence of so-called cluster phases at equilibrium, where monomeric proteins coexist with finite-size domains. Our main finding is that when taking recycling into account, the typical cluster size increases logarithmical…
Cell plasma membranes display a dramatically rich structural complexity characterized by functional sub-wavelength domains with specific lipid and protein composition. Under favorable experimental conditions, patterned morphologies can also be observed in vitro on model systems such as supported membranes or lipid vesicles. Lipid mixtures separating in liquid-ordered and liquid-disordered phases below a demixing temperature play a pivotal role in this context. Protein-protein and protein-lipid interactions also contribute to membrane shaping by promoting small domains or clusters. Such phase s…
We present theoretical analyses and numerical simulations for the adhesion-induced phase separation of multi-component membranes with two types of ligand-receptor complexes (junctions). We show that after integrating all possible distributions of the junctions, the system can be regarded as a membrane under an effective external potential. Mean field theory and Gaussian approximation are used to analyze the effective membrane potential and we find (i) The height difference of the junctions is the main factor that drives phase separation at sufficiently large junction height difference. (ii) In…
Advances in nanotechnology lead to an increasing interest in how nanoparticles interact with biomembranes. Nanoparticles are wrapped spontaneously by biomembranes if the adhesive interactions between the particles and membranes compensate for the cost of membrane bending. In the last years, the cooperative wrapping of spherical nanoparticles in membrane tubules has been observed in experiments and simulations. For spherical nanoparticles, the stability of the particle-filled membrane tubules strongly depends on the range of the adhesive particle-membrane interactions. In this article, we show …
The stochastic dynamics of inclusions in a randomly fluctuating biomembrane is simulated. These inclusions can represent the embedded proteins and the external particles arriving at a cell membrane. The energetics of the biomembrane is modelled via the Canham-Helfrich Hamiltonian. The contributions of both the bending elastic-curvature energy and the surface tension of the biomembrane are taken into account. The biomembrane is treated as a two-dimensional sheet whose height variations from a reference frame is treated as a stochastic Wiener process. The lateral diffusion parameter associated w…
This article summarizes a variety of physical mechanisms proposed in the literature, which can generate micro- and nanodomains in multicomponent lipid bilayers and biomembranes. It mainly focusses on lipid-driven mechanisms that do not involve direct protein-protein interactions. Specifically, it considers (i) equilibrium mechanisms based on lipid-lipid phase separation such as critical cluster formation close to critical points, and multiple domain formation in curved geometries, (ii) equilibrium mechanisms that stabilize two-dimensional microemulsions, such as the effect of linactants and th…
Respiration in bacteria involves a sequence of energetically-coupled electron and proton transfers creating an electrochemical gradient of protons (a proton-motive force) across the inner bacterial membrane. With a simple kinetic model we analyze a redox loop mechanism of proton-motive force generation mediated by a molecular shuttle diffusing inside the membrane. This model, which includes six electron-binding and two proton-binding sites, reflects the main features of nitrate respiration in E. coli bacteria. We describe the time evolution of the proton translocation process. We find that the…
The statistical physics and dynamics of double supported bilayers are studied theoretically. The main goal in designing double supported lipid bilayers is to obtain model systems of biomembranes: the upper bilayer is meant to be almost freely floating, the substrate being screened by the lower bilayer. The fluctuation-induced repulsion between membranes and between the lower membrane and the wall are explicitly taken into account using a Gaussian variational approach. It is shown that the variational parameters, the "effective" adsorption strength and the average distance to the substrate, dep…
Proteins often form chiral assembly structures on a biomembrane. However, the role of the chirality in the interaction with an achiral membrane is poorly understood. Here, the differential behavior between chiral and achiral crescent-shaped protein rods was investigated using meshless membrane simulations. The achiral rods deformed the membrane tube into an elliptical shape by stabilizing the edges of the ellipse. In contrast, the chiral rods formed a helical assembly that generated a cylindrical membrane tube with a constant radius in addition to the elliptical tube. This helical assembly cou…
Colloidal membranes, self assembled monolayers of aligned rod like molecules, offer a template for designing membranes with definite shapes and curvature, and possibly new functionalities in the future. Often the constituent rods, due to their molecular chirality, are tilted with respect to the membrane normal. Spatial patterns of this tilt on curved membranes result from a competition among depletion forces, nematic interaction, molecular chirality and boundary effects. We present a covariant theory for the tilt pattern on minimal surfaces, like helicoids and catenoids, which have been genera…
The bioactivity of nanoparticles crucially depends on their ability to cross biomembranes. Recent simulations indicate the cooperative wrapping and internalization of spherical nanoparticles in tubular membrane structures. In this article, we systematically investigate the energy gain of this cooperative wrapping by minimizing the energies of the rotationally symmetric shapes of the membrane tubes and of membrane segments wrapping single particles. We find that the energy gain for the cooperative wrapping of nanoparticles in membrane tubes relative to their individual wrapping as single partic…
The entropic effects of anchored polymers on biomembranes are studied using simulations of a meshless membrane model combined with anchored linear polymer chains. The bending rigidity and spontaneous curvature are investigated for anchored ideal and excluded-volume polymer chains. Our results are in good agreement with the previous theoretical predictions. It is found that the polymer reduces the line tension of membrane edges, as well as the interfacial line tension between membrane domains, leading to microdomain formation. Instead of the mixing of two phases as observed in typical binary fl…
Motivated by applications to seed germination, we consider the transverse deflection that results from the axisymmetric indentation of an elastic membrane by a rigid body. The elastic membrane is fixed around its boundary, with or without an initial pre-stretch, and may be initially curved prior to indentation. General indenter shapes are considered, and the load-indentation curves that result for a range of spheroidal tips are obtained for both flat and curved membranes. Wrinkling may occur when the membrane is initially curved, and a relaxed strain-energy function is used to calculate the de…
We investigate how an externally imposed curvature influences lipid segregation on two-phase-coexistent membranes. We show that the bending-modulus contrast of the two phases and the curvature act together to yield a reduced effective line tension. On largely curved membranes, a state of multiple domains (or rafts) forms due to a mechanism analogous to that causing magnetic-vortex formation in type-II superconductors. We determine the criterion for such multi-domain state to occur; we then calculate respectively the size of the domains formed on cylindrically and spherically curved membranes.
A mesoscopic coarse-grain model for computationally-efficient simulations of biomembranes is presented. It combines molecular dynamics simulations for the lipids, modeled as elastic chains of beads, with multiparticle collision dynamics for the solvent. Self-assembly of a membrane from a uniform mixture of lipids is observed. Simulations at different temperatures demonstrate that it reproduces the gel and liquid phases of lipid bilayers. Investigations of lipid diffusion in different phases reveals a crossover from subdiffusion to normal diffusion at long times. Macroscopic membrane properties…
Recent in vitro experiments with FtsZ polymers show self-organization into different dynamic patterns, including structures reminiscent of the bacterial Z-ring. We model FtsZ polymers as active particles moving along chiral, circular paths by Brownian dynamics simulations and a Boltzmann approach. Our two conceptually different methods point to a generic phase behavior. At intermediate particle densities, we find self-organization into vortex structures including closed rings. Moreover, we show that the dynamics at the onset of pattern formation is described by a generalized complex Ginzburg-L…
In spite of the great success that all-atom molecular dynamics simulations have seen in revealing the nature of the lipid bilayer, the interplay between a membrane's curvature and dynamics remains elusive. This is largely due to the computational challenges involved in simulating a highly curved membrane, as the one found in a small vesicle. In the present work, thanks to the computing power of Anton2, we present the first all-atom molecular dynamics simulation of a full-scale, realistically composed (both heterogeneous and asymmetric) vesicle of a meaningful time scale (over 10 microseconds),…
Lipid bilayers forming biological membranes are known to behave as viscous 2D fluids on submicrometer scales; usually they contain a large number of active protein inclusions. Recently, it has been shown [Proc. Nat. Acad. Sci. USA 112, E3639 (2015)] that such active proteins should in- duce non-thermal fluctuating lipid flows leading to diffusion enhancement and chemotaxis-like drift for passive inclusions in biomembranes. Here, a detailed analytical and numerical investigation of such effects is performed. The attention is focused on the situations when proteins are concentrated within lipid …
The theory of irreversible thermodynamics for arbitrarily curved lipid membranes is presented here. The coupling between elastic bending and irreversible processes such as intra-membrane lipid flow, intra-membrane phase transitions, and protein binding and diffusion is studied. The forms of the entropy production for the irreversible processes are obtained, and the corresponding thermodynamic forces and fluxes are identified. Employing the linear irreversible thermodynamic framework, the governing equations of motion along with appropriate boundary conditions are provided.
Cell membranes interact with a myriad of curvature-active proteins that control membrane morphology and are responsible for mechanosensation and mechanotransduction. Some of these proteins, such as those containing BAR domains, are curved and elongated, and hence may adopt different states of orientational order, from isotropic to maximize entropy to nematic as a result of crowding or to adapt to the curvature of the underlying membrane. Here, extending the work of [Nascimento et. al, Phys. Rev. E, 2017, 96, 022704], we develop a mean-field density functional theory to predict the orientationa…
Living cells employ excitable reaction-diffusion waves for internal cellular functions, in which curvature-inducing proteins are often involved. However, the role of their mechanochemical coupling is not well understood. Here, we report the membrane deformation induced by the excitable reaction-diffusion waves of curvature-inducing proteins and the alternation in the waves due to the deformation, using a coarse-grained simulation of tubular membranes with a modified FitzHugh--Nagumo model. Protein-propagating waves deform tubular membranes, and large deformations induce budding and erase waves…
Cellular wrapping of elastic particles by a supported lipid membrane
Université Paris Cité, CNRS, Institut Jacques Monod（巴黎西岱大学，法国国家科学研究中心，雅克·莫诺研究所）；Institut Curie, Université PSL, Sorbonne Université, CNRS UMR168（居里研究所，巴黎文理研究大学，索邦大学，法国国家科学研究中心联合研究实验室）；Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学研究高级学院物理系）；Department of Physics and Astronomy, University of Pennsylvania（宾夕法尼亚大学物理与天文学系）
Ordering configurations of a director field on a curved membrane induce stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes attraction-repulsion forces between defects; likewise, defects are attracted to patches with the same sign in gaussian curvature. These force…
In living cells, membrane morphology is regulated by various proteins. Many membrane reshaping proteins contain a Bin/Amphiphysin/Rvs (BAR) domain, which consists of a banana-shaped rod. The BAR domain bends the biomembrane along the rod axis and the features of this anisotropic bending have recently been studied. Here, we report on the role of the BAR protein rods in inducing membrane tubulation, using large-scale coarse-grained simulations. We reveal that a small spontaneous side curvature perpendicular to the rod can drastically alter the tubulation dynamics at high protein density, whereas…
Simulating Curved Lipid Membranes Using Anchored Frozen Patches
Department of Materials Science and Engineering, Grainger College of Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷恩工程学院材料科学与工程系）
In cell membranes, proteins and lipids are organized into sub-micrometric nanodomains of varying size, shape and composition, performing specific functions. Despite their biological importance, the detailed morphology of these nanodomains remains unknown. Not only can they hardly be observed by conventional microscopy due to their small size, but there is no full consensus on the theoretical models to describe their structuring and their shapes. Here, we use a combination of analytical calculations and Monte Carlo simulations based upon a model coupling membrane composition and shape to show t…
Particles embedded in a fluctuating interface experience forces and torques mediated by the deformations and by the thermal fluctuations of the medium. Considering a system of two cylinders bound to a fluid membrane we show that the entropic contribution enhances the curvature-mediated repulsion between the two cylinders. This is contrary to the usual attractive Casimir force in the absence of curvature-mediated interactions. For a large distance between the cylinders, we retrieve the renormalization of the surface tension of a flat membrane due to thermal fluctuations.
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 ves…
Membrane deformation inside living cells is crucial for the proper shaping of various intracellular organelles and is necessary during the fission/fusion processes that allow membrane recycling and transport (e.g. endocytosis). Proteins that induce membrane curvature play a key role in such processes, mostly by adsorbing to the membrane and forming a scaffold that deforms the membrane according to the curvature of the proteins. In this paper we explore the possibility of membrane tube destabilisation through a pearling mechanism enabled by the combined effects of the adsorbed curved proteins a…
We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent fluid, and (iii) intermonolayer slip between the two leaflets of the bilayer. We investigate relative to flat membranes how the membrane curvature and topology augment hydrodynamic responses. We show how both the translational and rotational mobility of protein inclusions are effecte…
Biological membranes typically contain a large number of different components dispersed in small concentrations in the main membrane phase, including proteins, sugars, and lipids of varying geometrical properties. Most of these components do not bind the cargo. Here, we show that such `inert' components can be crucial for precise control of cross-membrane trafficking. Using a statistical mechanics model and molecular dynamics simulations, we demonstrate that the presence of inert membrane components of small isotropic curvatures dramatically influences cargo endocytosis, even if the total spon…
Elucidating versatile configurations of spiral folding, and investigating the deployment performance is of relevant interest to extend the applicability of deployable membranes towards large-scale and functional configurations. In this paper we propose new schemes to package flat and curved membranes of finite thickness by using multiple spirals, whose governing equations render folding lines by juxtaposing spirals and by accommodating membrane thickness. Our experiments using a set of topologically distinct flat and curved membranes deployed by tensile forces applied in the radial and circumf…
Eukaryotic cells adhere to extracellular matrix during the normal development of the organism, forming static adhesion as well as during cell motility. We study this process by considering a simplified coarse-grained model of a vesicle that has uniform adhesion energy with a flat substrate, mobile curved membrane proteins and active forces. We find that a high concentration of curved proteins alone increases the spreading of the vesicle, by the self-organization of the curved proteins at the high curvature vesicle-substrate contact line, thereby reducing the bending energy penalty at the vesic…
Motivated to understand the behavior of biological filaments interacting with membranes of various types, we study a theoretical model for the shape and thermodynamics of intrinsically-helical filaments bound to curved membranes. We show filament-surface interactions lead to a host of non-uniform shape equilibria, in which filaments progressively unwind from their native twist with increasing surface interaction and surface curvature, ultimately adopting uniform-contact curved shapes. The latter effect is due to non-linear coupling between elastic twist and bending of filaments on anisotropica…
