Hybrid simulations of lateral diffusion in fluctuating membranes

There and back again: bridging meso- and nano-scales to understand lipid vesicle patterning

We describe a complete methodology to bridge the scales between nanoscale molecular dynamics and (micrometer) mesoscale Monte Carlo simulations in lipid membranes and vesicles undergoing phase separation, in which curving molecular species are furthermore embedded. To go from the molecular to the mesoscale, we notably appeal to physical renormalization arguments enabling us to rigorously infer the mesoscale interaction parameters from its molecular counterpart. We also explain how to deal with the physical timescales at stake at the mesoscale. Simulating the as-obtained mesoscale system enables us to equilibrate the long wavelengths of the vesicles of interest, up to the vesicle size. Conversely, we then backmap from the meso- to the nano-scale, which enables us to equilibrate in turn the short wavelengths down to the molecular length-scales. By applying our approach to the specific situation of patterning a vesicle membrane, we show that macroscopic membranes can thus be equilibrated at all length-scales in achievable computational time offering an original strategy to address the fundamental challenge of timescale in simulations of large bio-membrane systems.

Diffusion coefficients and MSD measurements on curved membranes and porous media

We study some geometric aspects that influence the transport properties of particles that diffuse on curved surfaces. We compare different approaches to surface diffusion based on the Laplace-Beltrami operator adapted to predict concentration along entire membranes, confined subdomains along surfaces, or within porous media. Our goal is to summarize, firstly, how diffusion in these systems results in different types of diffusion coefficients and mean square displacement measurements, and secondly, how these two factors are affected by the concavity of the surface, the shape of the possible barriers or obstacles that form the available domains, the sinuosity, tortuosity, and constrictions of the trajectories and even how the observation plane affects the measurements of the diffusion. In addition to presenting a critical and organized comparison between different notions of MSD, in this review, we test the correspondence between theoretical predictions and numerical simulations by performing finite element simulations and illustrate some situations where diffusion theory can be applied. We briefly reviewed computational schemes for understanding surface diffusion and finally, discussed how this work contributes to understanding the role of surface diffusion transport properties in porous media and their relationship to other transport processes.

Dynamic correlations in lipid bilayer membranes over finite time intervals

Recent single-molecule measurements [Schoch et al., Proc. Natl. Acad. Sci. U. S. A. 118, e2113202118 (2021)] have observed dynamic lipid-lipid correlations in membranes with submicrometer spatial resolution and submillisecond temporal resolution. While short from an instrumentation standpoint, these length and time scales remain long compared to microscopic molecular motions. Theoretical expressions are derived to infer experimentally measurable correlations from the two-body diffusion matrix appropriate for membrane-bound bodies coupled by hydrodynamic interactions. The temporal (and associated spatial) averaging resulting from finite acquisition times has the effect of washing out correlations as compared to naive predictions (i.e., the bare elements of the diffusion matrix), which would be expected to hold for instantaneous measurements. The theoretical predictions are shown to be in excellent agreement with Brownian dynamics simulations of experimental measurements. Numerical results suggest that the experimental measurement of membrane protein diffusion, in complement to lipid diffusion measurements, might help to resolve the experimental ambiguities encountered for certain black lipid membranes.

Durotaxis of Passive Nanoparticles on Elastic Membranes

The transport of macromolecules and nanoscopic particles to a target cellular site is a crucial aspect in many physiological processes. This directional motion is generally controlled via active mechanical and chemical processes. Here we show, by means of molecular dynamics simulations and an analytical theory, that completely passive nanoparticles can exhibit directional motion when embedded in nonuniform mechanical environments. Specifically, we study the motion of a passive nanoparticle adhering to a mechanically nonuniform elastic membrane. We observe a nonmonotonic affinity of the particle to the membrane as a function of the membrane's rigidity, which results in the particle transport. This transport can be both up or down the rigidity gradient, depending on the absolute values of the rigidities that the gradient spans across. We conclude that rigidity gradients can be used to direct average motion of passive macromolecules and nanoparticles on deformable membranes, resulting in the preferential accumulation of the macromolecules in regions of certain mechanical properties.

Binding of a Brownian nanoparticle to a thermally fluctuating membrane surface

We investigate the Brownian dynamics of a nanoparticle bound to a thermally undulating elastic membrane. The ligand-functionalized nanoparticle is assumed to interact monovalently with the receptor expressed on the membrane. In order to resolve the nanoparticle transient motion subject to the instantaneous membrane configuration in a consistent manner, we employ a set of coupled Langevin equations that simultaneously incorporate the hydrodynamic effects, ligand-receptor binding interaction, intramembrane elastic forces, and thermal fluctuations. We show that the presence of a deformable, elastic fluid membrane not only affects the dynamics of a bound nanoparticle but also alters the effective binding potential felt by the nanoparticle. In contrast to a nanoparticle bound to a flat surface, the oscillatory characteristics of the nanoparticle velocity autocorrelation function are suppressed and transition to an anticorrelated long-time tail. Moreover, the nanoparticle position fluctuation becomes more coherent with that of the membrane binding site, and the width of the distribution of the nanoparticle distance from the membrane decreases with increasing membrane bending rigidity. By introducing a locally harmonic, bistable potential as an effective potential for the ligand-receptor pair, the rate of nanoparticle transitioning between two bound states is facilitated by membrane undulations as a result of stronger positional variations associated with the nanoparticle.

Simple differences in the protein-membrane attachment mechanism have functional consequences for surface mechanics

This paper describes two methods for propagating coupled membrane and embedded particle dynamics with ensembles that are valid to second order in the deformation of the membrane. Proteins and functional lipids associate with cellular membranes, and their attachments influence membrane physical and dynamical properties. Therefore, it is necessary to accurately model the coupled dynamics of the membrane and any associated material of interest. We have developed two methods for coupling membrane and particle dynamics that differ in the binding mechanism of the particle to the surface. The "on-surface" mechanism should be used for particles that slide along the membrane; this description leads to an effective reduction in the membrane surface tension. The "in-surface" mechanism treats the particles as tightly bound to the lipidic binding sites; the method avoids double counting lateral entropy of implicitly modeled lipids. We emphasize the differences between these two mechanisms, when it is appropriate to use them, and how the methods differ from previously used dynamic methods.

An implicit lipid model for efficient reaction-diffusion simulations of protein binding to surfaces of arbitrary topology

Localization of proteins to a membrane is an essential step in a broad range of biological processes such as signaling, virion formation, and clathrin-mediated endocytosis. The strength and specificity of proteins binding to a membrane depend on the lipid composition. Single-particle reaction-diffusion methods offer a powerful tool for capturing lipid-specific binding to membrane surfaces by treating lipids explicitly as individual diffusible binding sites. However, modeling lipid particle populations is expensive. Here, we present an algorithm for reversible binding of proteins to continuum surfaces with implicit lipids, providing dramatic speed-ups to many body simulations. Our algorithm can be readily integrated into most reaction-diffusion software packages. We characterize changes to kinetics that emerge from explicit vs implicit lipids as well as surface adsorption models, showing excellent agreement between our method and the full explicit lipid model. Compared to models of surface adsorption, which couple together binding affinity and lipid concentration, our implicit lipid model decouples them to provide more flexibility for controlling surface binding properties and lipid inhomogeneity, thus reproducing binding kinetics and equilibria. Crucially, we demonstrate our method's application to membranes of arbitrary curvature and topology, modeled via a subdivision limit surface, again showing excellent agreement with explicit lipid simulations. Unlike adsorption models, our method retains the ability to bind lipids after proteins are localized to the surface (through, e.g., a protein-protein interaction), which can greatly increase the stability of multiprotein complexes on the surface. Our method will enable efficient cell-scale simulations involving proteins localizing to realistic membrane models, which is a critical step for predictive modeling and quantification of in vitro and in vivo dynamics.

Gaussian curvature and the budding kinetics of enveloped viruses

The formation of a membrane-enveloped virus starts with the assembly of a curved layer of capsid proteins lining the interior of the plasma membrane (PM) of the host cell. This layer develops into a spherical shell (capsid) enveloped by a lipid-rich membrane. In many cases, the budding process stalls prior to the release of the virus. Recently, Brownian dynamics simulations of a coarse-grained model system reproduced protracted pausing and stalling, which suggests that the origin of pausing/stalling is to be found in the physics of the budding process. Here, we propose that the pausing/stalling observed in the simulations can be understood as a purely kinetic phenomenon associated with the neck geometry. A geometrical potential energy barrier develops during the budding that must be overcome by capsid proteins diffusing along the membrane prior to incorporation into the capsid. The barrier is generated by a conflict between the positive Gauss curvature of the assembling capsid and the negative Gauss curvature of the neck region. A continuum theory description is proposed and is compared with the Brownian simulations of the budding of enveloped viruses.

Despite intense study, the life-cycle of the HIV-1 virus continues to pose mysteries. One of these is the fact that the assembly of an HIV-1 virus along the plasma membrane (PM) of the host cell—the budding process—stalls prior to release of the virus. Many other important viral pathogens with a surrounding lipid membrane envelope display similar stalling. Combining numerical and analytical methods, we demonstrate that the neck-like shape of the membrane that forms prior to release of the virus creates a barrier that blocks the proteins required for the assembly process from reaching the budding virus. An improved understanding of the physics of the blocking process could enable new strategies to combat enveloped viruses.

Brownian Translational Dynamics on a Flexible Surface: Nuclear Spin Relaxation of Fluid Membrane Phases

A general model for nuclear magnetic resonance (NMR) relaxation studies of fluid bilayer systems is introduced, combining a mesoscopic Brownian dynamics description of the bilayer with atomistic molecular dynamics (MD) simulations. An example is given for dipalmitoylphosphatidylcholine in 2H2O solvent and compared with the experiment. Experimental agreement is within a factor of 2 in the water relaxation rates, based on a postulated model with fixed parameters, which are largely available from the MD simulation. Relaxation rates are particularly sensitive to the translational diffusion of water perturbed by the interface dynamics and structure. Simulation results suggest that a notable deviation in the relaxation rates may follow from the commonly used small-angle approximation of bilayer undulation. The method has the potential to overcome the temporal and spatial limitations in computing NMR relaxation with atomistic MD, as well as the shortcomings of continuum models enabling a consistent description of experiments performed on a solvent lipid and added spin probes. This work opens for possibilities to understand relaxation processes involving systems such as micelles, multilamellar vesicles, red blood cells, and so forth at biologically relevant timescales in great detail.

Lipid and Peptide Diffusion in Bilayers: The Saffman-Delbrück Model and Periodic Boundary Conditions

The periodic Saffman-Delbrück (PSD) model, an extension of the Saffman-Delbrück model developed to describe the effects of periodic boundary conditions on the diffusion constants of lipids and proteins obtained from simulation, is tested using the coarse-grained Martini and all-atom CHARMM36 (C36) force fields. Simulations of pure Martini dipalmitoylphosphatidylcholine (DPPC) bilayers and those with one embedded gramicidin A (gA) dimer or one gA monomer with sizes ranging from 512 to 2048 lipids support the PSD model. Underestimates of D^∞ (the value of the diffusion constant for an infinite system) from the 512-lipid system are 35% for DPPC, 45% for the gA monomer, and 70% for the gA dimer. Simulations of all-atom DPPC and dioleoylphosphatidylcholine (DOPC) bilayers yield diffusion constants not far from experiment. However, the PSD model predicts that diffusion constants at the sizes of the simulation should underestimate experiment by approximately a factor of 3 for DPPC and 2 for DOPC. This likely implies a deficiency in the C36 force field. A Bayesian method for extrapolating diffusion constants of lipids and proteins in membranes obtained from simulation to infinite system size is provided.

Suppressing membrane height fluctuations leads to a membrane-mediated interaction among proteins

Membrane-induced interactions can play a significant role in the spatial distribution of membrane-bound proteins. We develop a model that combines a continuum description of lipid bilayers with a discrete particle model of proteins to probe the emerging structure of the combined membrane-protein system. Our model takes into account the membrane's elastic behavior, the steric repulsion between proteins, and the quenching of membrane shape fluctuations due to the presence of the proteins. We employ coupled Langevin equations to describe the dynamics of the system. We show that coupling to the membrane induces an attractive interaction among proteins, which may contribute to the clustering of proteins in biological membranes. We investigate the lateral protein diffusion and find that it is reduced due to transient fluctuations in membrane shape.

Binding constants of membrane-anchored receptors and ligands: A general theory corroborated by Monte Carlo simulations

Adhesion processes of biological membranes that enclose cells and cellular organelles are essential for immune responses, tissue formation, and signaling. These processes depend sensitively on the binding constant K2D of the membrane-anchored receptor and ligand proteins that mediate adhesion, which is difficult to measure in the "two-dimensional" (2D) membrane environment of the proteins. An important problem therefore is to relate K2D to the binding constant K3D of soluble variants of the receptors and ligands that lack the membrane anchors and are free to diffuse in three dimensions (3D). In this article, we present a general theory for the binding constants K2D and K3D of rather stiff proteins whose main degrees of freedom are translation and rotation, along membranes and around anchor points "in 2D," or unconstrained "in 3D." The theory generalizes previous results by describing how K2D depends both on the average separation and thermal nanoscale roughness of the apposing membranes, and on the length and anchoring flexibility of the receptors and ligands. Our theoretical results for the ratio K2D/K3D of the binding constants agree with detailed results from Monte Carlo simulations without any data fitting, which indicates that the theory captures the essential features of the "dimensionality reduction" due to membrane anchoring. In our Monte Carlo simulations, we consider a novel coarse-grained model of biomembrane adhesion in which the membranes are represented as discretized elastic surfaces, and the receptors and ligands as anchored molecules that diffuse continuously along the membranes and rotate at their anchor points.

Seeing the Forest in Lieu of the Trees: Continuum Simulations of Cell Membranes at Large Length Scales

Biological membranes exhibit long-range spatial structure in both chemical composition and geometric shape, which gives rise to remarkable physical phenomena and important biological functions. Continuum models that describe these effects play an important role in our understanding of membrane biophysics at large length scales. We review the mathematical framework used to describe both composition and shape degrees of freedom, and present best practices to implement such models in a computer simulation. We discuss in detail two applications of continuum models of cell membranes: the formation of microemulsion and modulated phases, and the effect of membrane-mediated interactions on the assembly of membrane proteins.

Mesoscale computational studies of membrane bilayer remodeling by curvature-inducing proteins

Biological membranes constitute boundaries of cells and cell organelles. These membranes are soft fluid interfaces whose thermodynamic states are dictated by bending moduli, induced curvature fields, and thermal fluctuations. Recently, there has been a flood of experimental evidence highlighting active roles for these structures in many cellular processes ranging from trafficking of cargo to cell motility. It is believed that the local membrane curvature, which is continuously altered due to its interactions with myriad proteins and other macromolecules attached to its surface, holds the key to the emergent functionality in these cellular processes. Mechanisms at the atomic scale are dictated by protein-lipid interaction strength, lipid composition, lipid distribution in the vicinity of the protein, shape and amino acid composition of the protein, and its amino acid contents. The specificity of molecular interactions together with the cooperativity of multiple proteins induce and stabilize complex membrane shapes at the mesoscale. These shapes span a wide spectrum ranging from the spherical plasma membrane to the complex cisternae of the Golgi apparatus. Mapping the relation between the protein-induced deformations at the molecular scale and the resulting mesoscale morphologies is key to bridging cellular experiments across the various length scales. In this review, we focus on the theoretical and computational methods used to understand the phenomenology underlying protein-driven membrane remodeling. Interactions at the molecular scale can be computationally probed by all atom and coarse grained molecular dynamics (MD, CGMD), as well as dissipative particle dynamics (DPD) simulations, which we only describe in passing. We choose to focus on several continuum approaches extending the Canham - Helfrich elastic energy model for membranes to include the effect of curvature-inducing proteins and explore the conformational phase space of such systems. In this description, the protein is expressed in the form of a spontaneous curvature field. The approaches include field theoretical methods limited to the small deformation regime, triangulated surfaces and particle-based computational models to investigate the large-deformation regimes observed in the natural state of many biological membranes. Applications of these methods to understand the properties of biological membranes in homogeneous and inhomogeneous environments of proteins, whose underlying curvature fields are either isotropic or anisotropic, are discussed. The diversity in the curvature fields elicits a rich variety of morphological states, including tubes, discs, branched tubes, and caveola. Mapping the thermodynamic stability of these states as a function of tuning parameters such as concentration and strength of curvature induction of the proteins is discussed. The relative stabilities of these self-organized shapes are examined through free-energy calculations. The suite of methods discussed here can be tailored to applications in specific cellular settings such as endocytosis during cargo trafficking and tubulation of filopodial structures in migrating cells, which makes these methods a powerful complement to experimental studies.

Diffusion of a particle quadratically coupled to a thermally fluctuating field

We study the diffusion of a Brownian particle quadratically coupled to a thermally fluctuating field. In the weak-coupling limit, a path-integral formulation allows us to compute the effective diffusion coefficient in the cases of an active particle, which tends to suppress field fluctuations, and of a passive particle, which only undergoes field fluctuations. We show that the behavior is similar to what was previously found for a linear coupling: an active particle is always slowed down, whereas a passive particle is slowed down in a slow field and accelerated in a fast field. Numerical simulations show a good agreement with the analytical calculations. The examples of a membrane protein coupled to the curvature or composition of the membrane are discussed, with a focus on the room for anomalous diffusion.

Contributions to membrane-embedded-protein diffusion beyond hydrodynamic theories

The diffusion coefficients of proteins embedded in a lipid membrane are traditionally described by the hydrodynamic Saffman-Delbrück theory, which predicts a weak dependence of the diffusion coefficient on protein radius, D∼lnR. Recent experiments have observed a stronger dependence, D∼1/R. This has led to speculation that the primary sources of drag on the protein are not hydrodynamic, but originate in coupling to other fields, such as lipid chain stretching or tilt. We discuss a generic model of a protein coupled to a nonconserved scalar order parameter (e.g., chain stretching), and show that earlier results may not be as universal as previously believed. In particular, we note that the drag depends on the way the protein-order parameter coupling is imposed. In this model, D∼1/R can be obtained if the protein is much larger than the order parameter correlation length. However, if we modify the model to include advection of the order parameter, which is a more appropriate assumption for a fluid membrane, we find that the entrainment of the order parameter by the protein's motion significantly changes the scaling of the diffusion coefficient. For parameters appropriate to protein diffusion, the Saffman-Delbrück-like scaling is restored, but with an effective radius for the protein that depends on the order parameter's correlation length. This qualitative difference suggests that hydrodynamic effects cannot be neglected in the computation of drag on a protein interacting with the membrane.

Systems biology and physical biology of clathrin-mediated endocytosis

In this review, we describe the application of experimental data and modeling of intracellular endocytic trafficking mechanisms with a focus on the process of clathrin-mediated endocytosis. A detailed parts-list for the protein-protein interactions in clathrin-mediated endocytosis has been available for some time. However, recent experimental, theoretical, and computational tools have proved to be critical in establishing a sequence of events, cooperative dynamics, and energetics of the intracellular process. On the experimental front, total internal reflection fluorescence microscopy, photo-activated localization microscopy, and spinning-disk confocal microscopy have focused on assembly and patterning of endocytic proteins at the membrane, while on the theory front, minimal theoretical models for clathrin nucleation, biophysical models for membrane curvature and bending elasticity, as well as methods from computational structural and systems biology, have proved insightful in describing membrane topologies, curvature mechanisms, and energetics.

Continuum simulations of biomembrane dynamics and the importance of hydrodynamic effects

Traditional particle-based simulation strategies are impractical for the study of lipid bilayers and biological membranes over the longest length and time scales (microns, seconds and longer) relevant to cellular biology. Continuum-based models developed within the frameworks of elasticity theory, fluid dynamics and statistical mechanics provide a framework for studying membrane biophysics over a range of mesoscopic to macroscopic length and time regimes, but the application of such ideas to simulation studies has occurred only relatively recently. We review some of our efforts in this direction with emphasis on the dynamics in model membrane systems. Several examples are presented that highlight the prominent role of hydrodynamics in membrane dynamics and we argue that careful consideration of fluid dynamics is key to understanding membrane biophysics at the cellular scale.

Diffusive motion of C60 on a graphene sheet

The motion of a C60 molecule over a graphene sheet at finite temperature is investigated both theoretically and computationally. We show that a graphene sheet generates a van der Waals laterally periodic potential, which directly influences the motion of external objects in its proximity. The translational motion of a C60 molecule near a graphene sheet is found to be diffusive in the lateral directions, while in the perpendicular direction, the motion may be described as diffusion in an effective harmonic potential which is determined from the distribution function of the position of the C60 molecule. We also examine the rotational diffusion of C60 and show that its motion over the graphene sheet is not a rolling motion.

Specific adhesion of membranes: Mapping to an effective bond lattice gas

We theoretically consider specific adhesion of a fluctuating membrane to a hard substrate via the formation of bonds between receptors attached to the substrate and ligands in the membrane. By integrating out the degrees of freedom of the membrane shape, we show that in the biologically relevant limit specific adhesion is well described by a lattice gas model, where lattice sites correspond to bond sites. We derive an explicit expression for the effective bond interactions induced by the thermal undulations of the membrane. Furthermore, we compare kinetic Monte Carlo simulations for our lattice gas model with full dynamic simulations that take into account both the shape fluctuations of the membrane and reactions between receptors and ligands at bond sites. We demonstrate that an appropriate mapping of the height dependent binding and unbinding rates in the full scheme to rates in the lattice gas model leads to good agreement.

Main phase transition in lipid bilayers: Phase coexistence and line tension in a soft, solvent-free, coarse-grained model

We devise a soft, solvent-free, coarse-grained model for lipid bilayer membranes. The nonbonded interactions take the form of a weighted-density functional, which allows us to describe the thermodynamics of self-assembly and packing effects of the coarse-grained beads in terms of a density expansion of the equation of state and weighting functions that regularize the microscopic bead densities, respectively. Identifying the length and energy scales via the bilayer thickness and the thermal energy scale, k(B)T, the model qualitatively reproduces key characteristics (e.g., bending rigidity, area per molecule, and compressibility) of lipid membranes. We employ this model to study the main phase transition between the fluid and the gel phase of the bilayer membrane. We accurately locate the phase coexistence using free energy calculations and also obtain estimates for the bare and the thermodynamic line tension.

Coarse-grained simulations of membranes under tension

We investigate the properties of membranes under tension by Monte Carlo simulations of a generic coarse-grained model for lipid bilayers. We give a comprising overview of the behavior of several membrane characteristics, such as the area per lipid, the monolayer overlap, the nematic order, and pressure profiles. Both the low-temperature regime, where the membranes are in a gel L(beta(')) phase, and the high-temperature regime, where they are in the fluid L(alpha) phase, are considered. In the L(beta(')) state, the membrane is hardly influenced by tension. In the fluid state, high tensions lead to structural changes in the membrane, which result in different compressibility regimes. The ripple state P(beta(')), which is found at tension zero in the transition regime between L(alpha) and L(beta(')), disappears under tension and gives way to an interdigitated phase. We also study the membrane fluctuations in the fluid phase. In the low-tension regime the data can be fitted nicely to a suitably extended elastic theory. At higher tensions the elastic fit consistently underestimates the strength of long-wavelength fluctuations. Finally, we investigate the influence of tension on the effective interaction between simple transmembrane inclusions and show that tension can be used to tune the hydrophobic mismatch interaction between membrane proteins.

Diffusing proteins on a fluctuating membrane: analytical theory and simulations

Using analytical calculations and computer simulations, we consider both the lateral diffusion of a membrane protein and the fluctuation spectrum of the membrane in which the protein is embedded. The membrane protein interacts with the membrane shape through its spontaneous curvature and bending rigidity. The lateral motion of the protein may be viewed as diffusion in an effective potential, hence, the effective mobility is always reduced compared to the case of free diffusion. Using a rigorous path-integral approach, we derive an analytical expression for the effective diffusion coefficient for small ratios of temperature and bending rigidity, which is the biologically relevant limit. Simulations show very good quantitative agreement with our analytical result. The analysis of the correlation functions contributing to the diffusion coefficient shows that the correlations between the stochastic force of the protein and the response in the membrane shape are responsible for the reduction. Our quantitative analysis of the membrane height correlation spectrum shows an influence of the protein-membrane interaction causing a distinctly altered wave-vector dependence compared to a free membrane. Furthermore, the time correlations exhibit the two relevant time scales of the system: that of membrane fluctuations and that of lateral protein diffusion with the latter typically much longer than the former. We argue that the analysis of the long-time decay of membrane height correlations can thus provide a new means to determine the effective diffusion coefficient of proteins in the membrane.

Calculation of free energies in fluid membranes subject to heterogeneous curvature fields

We present a computational methodology for incorporating thermal effects and calculating relative free energies for elastic fluid membranes subject to spatially dependent intrinsic curvature fields using the method of thermodynamic integration. Based on a simple model for the intrinsic curvature imposed only in a localized region of the membrane, we employ thermodynamic integration to calculate the free-energy change as a function of increasing strength of the intrinsic curvature field and a thermodynamic cycle to compute free-energy changes for different sizes of the localized region. By explicitly computing the free-energy changes and by quantifying the loss of entropy accompanied with increasing membrane deformation, we show that the membrane stiffness increases with increasing intrinsic field, thereby, renormalizing the membrane bending rigidity. The second main conclusion of this work is that the entropy of the membrane decreases with increasing size of the localized region subject to the curvature field. Our results help to quantify the free-energy change when a planar membrane deforms under the influence of curvature-inducing proteins at a finite temperature.

Correlated diffusion of membrane proteins and their effect on membrane viscosity

We extend the Saffman theory of membrane hydrodynamics to account for the correlated motion of membrane proteins, along with the effect of protein concentration on that correlation and on the response of the membrane to stresses. Expressions for the coupling diffusion coefficients of protein pairs and their concentration dependence are derived in the limit of small protein size relative to the interprotein separation. The additional role of membrane viscosity as determining the characteristic length scale for membrane response leads to unusual concentration effects at large separation-the transverse coupling increases with protein concentration, whereas the longitudinal one becomes concentration-independent.

Hybrid elastic and discrete-particle approach to biomembrane dynamics with application to the mobility of curved integral membrane proteins

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.

Diffusion in a fluid membrane with a flexible cortical cytoskeleton

We calculate the influence of a flexible network of long-chain proteins, which is anchored to a fluid membrane, on protein diffusion in this membrane. This is a model for the cortical cytoskeleton and the lipid bilayer of the red blood cell, which we apply to predict the influence of the cytoskeleton on the diffusion coefficient of a mobile band 3 protein. Using the pressure field that the cytoskeleton exerts on the membrane, from the steric repulsion between the diffusing protein and the cytoskeletal filaments, we define a potential landscape for the diffusion within the bilayer. We study the changes to the diffusion coefficient on removal of one type of anchor proteins, e.g., in several hemolytic anemias, as well as for isotropic and anisotropic stretching of the cytoskeleton. We predict an overall increase of the diffusion for a smaller number of anchor proteins and increased diffusion for anisotropic stretching in the direction of the stretch, because of the decrease in the spatial frequency as well as in the height of the potential barriers.

Dynamics of specific vesicle-substrate adhesion: from local events to global dynamics

We present a synergistic combination of simulations and experimental data on the dynamics of membrane adhesion. We show that a change in either the density or the strength of the bonds results in very different dynamics. Such behavior is explained by introducing an effective binding affinity that emerges as a result of the competition between the strength of the chemical bonds and the environment defined by the fluctuating membrane.

Diffusion of macromolecules on lipid vesicles

Diffusion of macromolecules on a surface of lipid vesicles of different reduced volume and geometry is investigated. It is assumed that the macromolecules deform the surface of the vesicles by inducing the spontaneous curvature which is proportional to their concentration. We study how nonuniform distribution of macromolecules is reflected in the shape of the vesicles and how the shape of the vesicles influences the diffusion process.

Landscape of finite-temperature equilibrium behaviour of curvature-inducing proteins on a bilayer membrane explored using a linearized elastic free energy model

Using a recently developed multiscale simulation methodology, we describe the equilibrium behaviour of bilayer membranes under the influence of curvature-inducing proteins using a linearized elastic free energy model. In particular, we describe how the cooperativity associated with a multitude of protein-membrane interactions and protein diffusion on a membrane-mediated energy landscape elicits emergent behaviour in the membrane phase. Based on our model simulations, we predict that, depending on the density of membrane-bound proteins and the degree to which a single protein molecule can induce intrinsic mean curvature in the membrane, a range of membrane phase behaviour can be observed including two different modes of vesicle-bud nucleation and repressed membrane undulations. A state diagram as a function of experimentally tunable parameters to classify the underlying states is proposed.

Curvature coupling dependence of membrane protein diffusion coefficients

We consider the lateral diffusion of a protein interacting with the curvature of the membrane. The interaction energy is minimized if the particle is at a membrane position with a certain curvature that agrees with the spontaneous curvature of the particle. We employ stochastic simulations that take into account both the thermal fluctuations of the membrane and the diffusive behavior of the particle. In this study, we neglect the influence of the particle on the membrane dynamics, thus the membrane dynamics agrees with that of a freely fluctuating membrane. Overall, we find that this curvature coupling substantially enhances the diffusion coefficient. We compare the ratio of the projected or measured diffusion coefficient and the free intramembrane diffusion coefficient, which is a parameter of the simulations, with analytical results that rely on several approximations. We find that the simulations always lead to a somewhat smaller diffusion coefficient than that from our analytical approach. A detailed study of the correlations of the forces acting on the particle indicates that the diffusing inclusion tries to follow favorable positions on the membrane such that forces along the trajectory are on average smaller than they would be for random particle positions.

Diffusion on ruffled membrane surfaces

We present a position Langevin equation for overdamped particle motion on rough two-dimensional surfaces. A Brownian dynamics algorithm is suggested to evolve this equation numerically, allowing for the prediction of effective (projected) diffusion coefficients over corrugated surfaces. In the case of static surface roughness, we find that a simple area-scaling prediction for the projected diffusion coefficient leads to seemingly quantitative agreement with numerical results. To study the effect of dynamic surface evolution on the diffusive process, we consider particle diffusion over a thermally fluctuating elastic membrane. Surface fluctuation has the effect of increasing the effective diffusivity toward a limiting annealed-surface value discussed previously. We argue that protein motion over cell surfaces spans a variety of physical regimes, making it impossible to identify a single approximation scheme appropriate to all measurements of interest.