Dynamic simulations of multicomponent lipid membranes over long length and time scales

Dynamic swarms regulate the morphology and distribution of soft membrane domains

We study the dynamic structure of lipid domain inclusions embedded within a phase-separated reconstituted lipid bilayer in contact with a swarming flow of gliding filamentous actin. Passive circular domains transition into highly deformed morphologies that continuously elongate, rotate, and pinch off into smaller fragments, leading to a dynamic steady state with ≈23× speedup in the relaxation of the intermediate scattering function compared with passive membrane domains driven by purely thermal forces. To corroborate experimental results, we develop a phase-field model of the lipid domains with two-way coupling to the Toner-Tu equations. We report phase domains that become entrained in the chaotic eddy patterns, with oscillating waves of domains that correlate with the dominant wavelengths of the Toner-Tu flow fields.

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.

Experimentally Probing the Effect of Confinement Geometry on Lipid Diffusion

The lateral mobility of molecules within the cell membrane is ultimately governed by the local environment of the membrane. Confined regions induced by membrane structures, such as protein aggregates or the actin meshwork, occur over a wide range of length scales and can impede or steer the diffusion of membrane components. However, a detailed picture of the origins and nature of these confinement effects remains elusive. Here, we prepare model lipid systems on substrates patterned with confined domains of varying geometries constructed with different materials to explore the influences of physical boundary conditions and specific molecular interactions on diffusion. We demonstrate a platform that is capable of significantly altering and steering the long-range diffusion of lipids by using simple oxide deposition approaches, enabling us to systematically explore how confinement size and shape impact diffusion over multiple length scales. While we find that a "boundary condition" description of the system captures underlying trends in some cases, we are also able to directly compare our systems to analytical models, revealing the unexpected breakdown of several approximate solutions. Our results highlight the importance of considering the length scale dependence when discussing properties such as diffusion.

Physical limits to membrane curvature sensing by a single protein

Membrane curvature sensing is essential for a diverse range of biological processes. Recent experiments have revealed that a single nanometer-sized septin protein has different binding rates to membrane-coated glass beads of 1-µm and 3-µm diameters, even though the septin is orders of magnitude smaller than the beads. This sensing ability is especially surprising since curvature-sensing proteins must deal with persistent thermal fluctuations of the membrane, leading to discrepancies between the bead's curvature and the local membrane curvature sensed instantaneously by a protein. Using continuum models of fluctuating membranes, we investigate whether it is feasible for a protein acting as a perfect observer of the membrane to sense micron-scale curvature either by measuring local membrane curvature or by using bilayer lipid densities as a proxy. To do this, we develop algorithms to simulate lipid density and membrane shape fluctuations. We derive physical limits to the sensing efficacy of a protein in terms of protein size, membrane thickness, membrane bending modulus, membrane-substrate adhesion strength, and bead size. To explain the experimental protein-bead association rates, we develop two classes of predictive models: (i) for proteins that maximally associate to a preferred curvature and (ii) for proteins with enhanced association rates above a threshold curvature. We find that the experimentally observed sensing efficacy is close to the theoretical sensing limits imposed on a septin-sized protein. Protein-membrane association rates may depend on the curvature of the bead, but the strength of this dependence is limited by the fluctuations in membrane height and density.

Active Surface Flows Accelerate the Coarsening of Lipid Membrane Domains

Phase separation of multicomponent lipid membranes is characterized by the nucleation and coarsening of circular membrane domains that grow slowly in time as ∼t^{1/3}, following classical theories of coalescence and Ostwald ripening. In this Letter, we study the coarsening kinetics of phase-separating lipid membranes subjected to nonequilibrium forces and flows transmitted by motor-driven gliding actin filaments. We experimentally observe that the activity-induced surface flows trigger rapid coarsening of noncircular membrane domains that grow as ∼t^{2/3}, a 2x acceleration in the growth exponent compared to passive coalescence and Ostwald ripening. We analyze these results by developing analytical theories based on the Smoluchowski coagulation model and the phase field model to predict the domain growth in the presence of active flows. Our Letter demonstrates that active matter forces may be used to control the growth and morphology of membrane domains driven out of equilibrium.

Inhomogeneous Canham-Helfrich Abscission in Catenoid Necks under Critical Membrane Mosaicity

The mechanical effects of membrane compositional inhomogeneities are analyzed in a process analogous to neck formation in cellular membranes. We cast on the Canham-Helfrich model of fluid membranes with both the spontaneous curvature and the surface tension being non-homogeneous functions along the cell membrane. The inhomogeneous distribution of necking forces is determined by the equilibrium mechanical equations and the boundary conditions as considered in the axisymmetric setting compatible with the necking process. To establish the role played by mechanical inhomogeneity, we focus on the catenoid, a surface of zero mean curvature. Analytic solutions are shown to exist for the spontaneous curvature and the constrictive forces in terms of the border radii. Our theoretical analysis shows that the inhomogeneous distribution of spontaneous curvature in a mosaic-like neck constrictional forces potentially contributes to the membrane scission under minimized work in living cells.

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.

Protein drift-diffusion dynamics and phase separation in curved cell membranes and dendritic spines: Hybrid discrete-continuum methods

We develop methods for investigating protein drift-diffusion dynamics in heterogeneous cell membranes and the roles played by geometry, diffusion, chemical kinetics, and phase separation. Our hybrid stochastic numerical methods combine discrete particle descriptions with continuum-level models for tracking the individual protein drift-diffusion dynamics when coupled to continuum fields. We show how our approaches can be used to investigate phenomena motivated by protein kinetics within dendritic spines. The spine geometry is hypothesized to play an important biological role regulating synaptic strength, protein kinetics, and self-assembly of clusters. We perform simulation studies for model spine geometries varying the neck size to investigate how phase-separation and protein organization is influenced by different shapes. We also show how our methods can be used to study the roles of geometry in reaction-diffusion systems including Turing instabilities. Our methods provide general approaches for investigating protein kinetics and drift-diffusion dynamics within curved membrane structures.

Sensing the shape of a cell with reaction diffusion and energy minimization

Some dividing cells sense their shape by becoming polarized along their long axis. Cell polarity is controlled in part by polarity proteins, like Rho GTPases, cycling between active membrane-bound forms and inactive cytosolic forms, modeled as a "wave-pinning" reaction-diffusion process. Does shape sensing emerge from wave pinning? We show that wave pinning senses the cell's long axis. Simulating wave pinning on a curved surface, we find that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, when we introduce roughness to our surfaces, shape sensing can be disrupted, and high-activity domains can become localized to locations other than the global peaks and valleys of the surface. On rough surfaces, the domains of the wave-pinning model are more robust in finding the peaks and troughs than the minimization rule, although both can become trapped in steady states away from the peaks and valleys. We can control the robustness of shape sensing by altering the Rho GTPase diffusivity and the domain size. We also find that the shape-sensing properties of cell polarity models can explain how domains localize to curved regions of deformed cells. Our results help to understand the factors that allow cells to sense their shape-and the limits that membrane roughness can place on this process.

Boundary fluctuation dynamics of a phase-separated domain in planar geometry

Using phase-ordering kinetics and of renormalization group theories, we derive analytically the relaxation times of the long wavelength fluctuations of a phase-separated domain boundary in the vicinity of (and below) the critical temperature in the planar Ising universality class. For a conserved order parameter, the relaxation time grows like Λ^{3} at wavelength Λ and can be expressed in terms of parameters relevant at the microscopic scale: lattice spacing, bulk diffusion coefficient of the minority phase, and temperature. These results are supported by numerical simulations of 2D Ising models, enabling in addition calculating the nonuniversal numerical prefactor. We discuss the applications of these findings to the determination of the real timescale associated with elementary Monte Carlo moves from the measurement of long wavelength relaxation times on experimental systems or molecular dynamics simulations.

Mechanochemical coupling of lipid organization and protein function through membrane thickness deformations

Cell membranes are composed of a great variety of protein and lipid species with distinct unperturbed hydrophobic thicknesses. To achieve hydrophobic matching, the lipid bilayer tends to deform around membrane proteins so as to match the protein hydrophobic thickness at bilayer-protein interfaces. Such protein-induced distortions of the lipid bilayer hydrophobic thickness incur a substantial energy cost that depends critically on the bilayer-protein hydrophobic mismatch, while distinct conformational states of membrane proteins often show distinct hydrophobic thicknesses. As a result, hydrophobic interactions between membrane proteins and lipids can yield a rich interplay of lipid-protein organization and transitions in protein conformational state. We combine here the membrane elasticity theory of protein-induced lipid bilayer thickness deformations with the Landau-Ginzburg theory of lipid domain formation to systematically explore the coupling between local lipid organization, lipid and protein hydrophobic thickness, and protein-induced lipid bilayer thickness deformations in membranes with heterogeneous lipid composition. We allow for a purely mechanical coupling of lipid and protein composition through the energetics of protein-induced lipid bilayer thickness deformations as well as a chemical coupling driven by preferential interactions between particular lipid and protein species. We find that the resulting lipid-protein organization can endow membrane proteins with diverse and controlled mechanical environments that, via protein-induced lipid bilayer thickness deformations, can strongly influence protein function. The theoretical approach employed here provides a general framework for the quantitative prediction of how membrane thickness deformations influence the joint organization and function of lipids and proteins in cell membranes.

Hyperuniformity and phase enrichment in vortex and rotor assemblies

Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Such rotors are found in the natural world spanning vastly disparate length scales — from the rotor proteins in cellular membranes to models of atmospheric dynamics. Here we show that an initially random distribution of either driven rotors in a viscous membrane, or ideal vortices with minute perturbations, spontaneously self assemble into a distinct arrangement. Despite arising from drastically different physics, these systems share a Hamiltonian structure that sets geometrical conservation laws resulting in prominent structural states. We find that the rotationally invariant interactions isotropically suppress long-wavelength fluctuations — a hallmark of a disordered hyperuniform material. With increasing area fraction, the system orders into a hexagonal lattice. In mixtures of two co-rotating populations, the stronger population will gain order from the other and both will become phase enriched. Finally, we show that classical 2D point vortex systems arise as exact limits of the experimentally accessible microscopic membrane rotors, yielding a new system through which to study topological defects.

Rotor-like dynamics is observed in many natural systems, from the rotor proteins in cellular membranes to atmospheric models. Here, the authors uncover geometrical conservation laws that limit distribution of driven rotors in a membrane or a soap film and allow to predict their structural states.

Persistent collective motion of a dispersing membrane domain

We study the Brownian motion of an assembly of mobile inclusions embedded in a fluid membrane. The motion includes the dispersal of the assembly, accompanied by the diffusion of its center of mass. Usually, the former process is much faster than the latter because the diffusion coefficient of the center of mass is inversely proportional to the number of particles. However, in the case of membrane inclusions, we find that the two processes occur on the same timescale, thus significantly prolonging the lifetime of the assembly as a collectively moving object. This effect is caused by the quasi-two-dimensional membrane flows, which couple the motions of even the most remote inclusions in the assembly. The same correlations also cause the diffusion coefficient of the center of mass to decay slowly with time, resulting in weak subdiffusion. We confirm our analytical results by Brownian dynamics simulations with flow-mediated correlations. The effect reported here should have implications for the stability of nanoscale membrane heterogeneities.

Regulation of membrane proteins through local heterogeneity in lipid bilayer thickness

Cell membranes show an intricate organization of lipids and membrane proteins into domains with distinct composition and hydrophobic thickness. Using mechanosensitive ion channels as a model system, we employ the membrane elasticity theory of lipid-protein interactions together with the Landau-Ginzburg theory of lipid domain formation to quantify protein-induced lipid bilayer thickness deformations in lipid bilayers with heterogeneous hydrophobic thickness. We show that protein-induced lipid bilayer thickness deformations yield, without any assumptions about preferential interactions between particular lipid and protein species, organization of lipids and membrane proteins according to their preferred hydrophobic thickness, and couple the conformational states of membrane proteins to the local membrane composition. Our calculations suggest that protein-induced lipid bilayer thickness deformations endow proteins in cell membranes with diverse and controlled mechanical environments that, in turn, allow targeted regulation of membrane proteins.

Lateral heterogeneity and domain formation in cellular membranes

As early as the development of the fluid mosaic model for cellular membranes, researchers began observing the telltale signs of lateral heterogeneity. Over the decades this has led to the development of the lipid raft hypothesis and the ensuing controversy that has unfolded, as a result. Here, we review the physical concepts behind domain formation in lipid membranes, both of their structural and dynamic origins. This, then leads into a discussion of coarse-grained, phenomenological approaches that describe the wide range of phases associated with lipid lateral heterogeneity. We use these physical concepts to describe the interaction between raft-lipid species, such as long-chain saturated lipids, sphingomyelin, and cholesterol, and non-raft forming lipids, such as those with short acyl chains or unsaturated fatty acids. While debate has persisted on the biological relevance of lipid domains, recent research, described here, continues to identify biological roles for rafts and new experimental approaches have revealed the existence of lipid domains in living systems. Given the recent progress on both the biological and structural aspects of raft formation, the research area of membrane lateral heterogeneity will not only expand, but will continue to produce exciting results.

A computational study of lateral phase separation in biological membranes

Conservative and non-conservative phase-field models are considered for the numerical simulation of lateral phase separation and coarsening in biological membranes. An unfitted finite element method is proposed to allow for a flexible treatment of complex shapes in the absence of an explicit surface parametrization. For a set of biologically relevant shapes and parameter values, the paper compares the dynamic coarsening produced by conservative and non-conservative numerical models, its dependence on certain geometric characteristics and convergence to the final equilibrium.

Phase behavior and morphology of multicomponent liquid mixtures

Multicomponent systems are ubiquitous in nature and industry. While the physics of few-component liquid mixtures (i.e., binary and ternary ones) is well-understood and routinely taught in undergraduate courses, the thermodynamic and kinetic properties of N-component mixtures with N > 3 have remained relatively unexplored. An example of such a mixture is provided by the intracellular fluid, in which protein-rich droplets phase separate into distinct membraneless organelles. In this work, we investigate equilibrium phase behavior and morphology of N-component liquid mixtures within the Flory-Huggins theory of regular solutions. In order to determine the number of coexisting phases and their compositions, we developed a new algorithm for constructing complete phase diagrams, based on numerical convexification of the discretized free energy landscape. Together with a Cahn-Hilliard approach for kinetics, we employ this method to study mixtures with N = 4 and 5 components. We report on both the coarsening behavior of such systems, as well as the resulting morphologies in three spatial dimensions. We discuss how the number of coexisting phases and their compositions can be extracted with Principal Component Analysis (PCA) and K-means clustering algorithms. Finally, we discuss how one can reverse engineer the interaction parameters and volume fractions of components in order to achieve a range of desired packing structures, such as nested "Russian dolls" and encapsulated Janus droplets.

Cholesterol Effects on the Physical Properties of Lipid Membranes Viewed by Solid-state NMR Spectroscopy

In this chapter, we review the physical properties of lipid/cholesterol mixtures involving studies of model membranes using solid-state NMR spectroscopy. The approach allows one to quantify the average membrane structure, fluctuations, and elastic deformation upon cholesterol interaction. Emphasis is placed on understanding the membrane structural deformation and emergent fluctuations at an atomistic level. Lineshape measurements using solid-state NMR spectroscopy give equilibrium structural properties, while relaxation time measurements study the molecular dynamics over a wide timescale range. The equilibrium properties of glycerophospholipids, sphingolipids, and their binary and tertiary mixtures with cholesterol are accessible. Nonideal mixing of cholesterol with other lipids explains the occurrence of liquid-ordered domains. The entropic loss upon addition of cholesterol to sphingolipids is less than for glycerophospholipids, and may drive formation of lipid rafts. The functional dependence of ²H NMR spin-lattice relaxation (R _1Z) rates on segmental order parameters (S _CD) for lipid membranes is indicative of emergent viscoelastic properties. Addition of cholesterol shows stiffening of the bilayer relative to the pure lipids and this effect is diminished for lanosterol. Opposite influences of cholesterol and detergents on collective dynamics and elasticity at an atomistic scale can potentially affect lipid raft formation in cellular membranes.

Biophysics of membrane curvature remodeling at molecular and mesoscopic lengthscales

At the micron scale, where cell organelles display an amazing complexity in their shape and organization, the physical properties of a biological membrane can be better-understood using continuum models subject to thermal (stochastic) undulations. Yet, the chief orchestrators of these complex and intriguing shapes are a specialized class of membrane associating often peripheral proteins called curvature remodeling proteins (CRPs) that operate at the molecular level through specific protein-lipid interactions. We review multiscale methodologies to model these systems at the molecular as well as at the mesoscopic and cellular scales, and also present a free energy perspective of membrane remodeling through the organization and assembly of CRPs. We discuss the morphological space of nearly planar to highly curved membranes, methods to include thermal fluctuations, and review studies that model such proteins as curvature fields to describe the emergent curved morphologies. We also discuss several mesoscale models applied to a variety of cellular processes, where the phenomenological parameters (such as curvature field strength) are often mapped to models of real systems based on molecular simulations. Much insight can be gained from the calculation of free energies of membranes states with protein fields, which enable accurate mapping of the state and parameter values at which the membrane undergoes morphological transformations such as vesiculation or tubulation. By tuning the strength, anisotropy, and spatial organization of the curvature-field, one can generate a rich array of membrane morphologies that are highly relevant to shapes of several cellular organelles. We review applications of these models to budding of vesicles commonly seen in cellular signaling and trafficking processes such as clathrin mediated endocytosis, sorting by the ESCRT protein complexes, and cellular exocytosis regulated by the exocyst complex. We discuss future prospects where such models can be combined with other models for cytoskeletal assembly, and discuss their role in understanding the effects of cell membrane tension and the mechanics of the extracellular microenvironment on cellular processes.

Lipid-protein interaction induced domains: Kinetics and conformational changes in multicomponent vesicles

The spatio-temporal organization of proteins and the associated morphological changes in membranes are of importance in cell signaling. Several mechanisms that promote the aggregation of proteins at low cell surface concentrations have been investigated in the past. We show, using Monte Carlo simulations, that the affinity of proteins for specific lipids can hasten their aggregation kinetics. The lipid membrane is modeled as a dynamically triangulated surface with the proteins defined as in-plane fields at the vertices. We show that, even at low protein concentrations, strong lipid-protein interactions can result in large protein clusters indicating a route to lipid mediated signal amplification. At high protein concentrations, the domains form buds similar to that seen in lipid-lipid interaction induced phase separation. Protein interaction induced domain budding is suppressed when proteins act as anisotropic inclusions and exhibit nematic orientational order. The kinetics of protein clustering and resulting conformational changes are shown to be significantly different for the isotropic and anisotropic curvature inducing proteins.

Physical principles of intracellular organization via active and passive phase transitions

Exciting recent developments suggest that phase transitions represent an important and ubiquitous mechanism underlying intracellular organization. We describe key experimental findings in this area of study, as well as the application of classical theoretical approaches for quantitatively understanding these data. We also discuss the way in which equilibrium thermodynamic driving forces may interface with the fundamentally out-of-equilibrium nature of living cells. In particular, time and/or space-dependent concentration profiles may modulate the phase behavior of biomolecules in living cells. We suggest future directions for both theoretical and experimental work that will shed light on the way in which biological activity modulates the assembly, properties, and function of viscoelastic states of living matter.

Geodesic curvature driven surface microdomain formation

Lipid bilayer membranes are not uniform and clusters of lipids in a more ordered state exist within the generally disorder lipid milieu of the membrane. These clusters of ordered lipids microdomains are now referred to as lipid rafts. Recent reports attribute the formation of these microdomains to the geometrical and molecular mechanical mismatch of lipids of different species on the boundary. Here we introduce the geodesic curvature to characterize the geometry of the domain boundary, and develop a geodesic curvature energy model to describe the formation of these microdomains as a result of energy minimization. Our model accepts the intrinsic geodesic curvature of any binary lipid mixture as an input, and will produce microdomains of the given geodesic curvature as demonstrated by three sets of numerical simulations. Our results are in contrast to the surface phase separation predicted by the classical surface Cahn-Hilliard equation, which tends to generate large domains as a result of the minimizing line tension. Our model provides a direct and quantified description of the structure inhomogeneity of lipid bilayer membrane, and can be coupled to the investigations of biological processes on membranes for which such inhomogeneity plays essential roles.

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.

Cholesterol-induced suppression of membrane elastic fluctuations at the atomistic level

Applications of solid-state NMR spectroscopy for investigating the influences of lipid-cholesterol interactions on membrane fluctuations are reviewed in this paper. Emphasis is placed on understanding the energy landscapes and fluctuations at an emergent atomistic level. Solid-state (2)H NMR spectroscopy directly measures residual quadrupolar couplings (RQCs) due to individual C-(2)H labeled segments of the lipid molecules. Moreover, residual dipolar couplings (RDCs) of (13)C-(1)H bonds are obtained in separated local-field NMR spectroscopy. The distributions of RQC or RDC values give nearly complete profiles of the order parameters as a function of acyl segment position. Measured equilibrium properties of glycerophospholipids and sphingolipids including their binary and tertiary mixtures with cholesterol show unequal mixing associated with liquid-ordered domains. The entropic loss upon addition of cholesterol to sphingolipids is less than for glycerophospholipids and may drive the formation of lipid rafts. In addition relaxation time measurements enable one to study the molecular dynamics over a wide time-scale range. For (2)H NMR the experimental spin-lattice (R1Z) relaxation rates follow a theoretical square-law dependence on segmental order parameters (SCD) due to collective slow dynamics over mesoscopic length scales. The functional dependence for the liquid-crystalline lipid membranes is indicative of viscoelastic properties as they emerge from atomistic-level interactions. A striking decrease in square-law slope upon addition of cholesterol denotes stiffening relative to the pure lipid bilayers that is diminished in the case of lanosterol. Measured equilibrium properties and relaxation rates infer opposite influences of cholesterol and detergents on collective dynamics and elasticity at an atomistic scale that potentially affects lipid raft formation in cellular membranes.

Fluctuating hydrodynamics of multicomponent membranes with embedded proteins

A simulation method for the dynamics of inhomogeneous lipid bilayer membranes is presented. The membrane is treated using stochastic Saffman-Delbrück hydrodynamics, coupled to a phase-field description of lipid composition and discrete membrane proteins. Multiple applications are considered to validate and parameterize the model. The dynamics of membrane composition fluctuations above the critical point and phase separation dynamics below the critical point are studied in some detail, including the effects of adding proteins to the mixture.

RNA transcription modulates phase transition-driven nuclear body assembly

Nuclear bodies are RNA and protein-rich, membraneless organelles that play important roles in gene regulation. The largest and most well-known nuclear body is the nucleolus, an organelle whose primary function in ribosome biogenesis makes it key for cell growth and size homeostasis. The nucleolus and other nuclear bodies behave like liquid-phase droplets and appear to condense from the nucleoplasm by concentration-dependent phase separation. However, nucleoli actively consume chemical energy, and it is unclear how such nonequilibrium activity might impact classical liquid-liquid phase separation. Here, we combine in vivo and in vitro experiments with theory and simulation to characterize the assembly and disassembly dynamics of nucleoli in early Caenorhabditis elegans embryos. In addition to classical nucleoli that assemble at the transcriptionally active nucleolar organizing regions, we observe dozens of "extranucleolar droplets" (ENDs) that condense in the nucleoplasm in a transcription-independent manner. We show that growth of nucleoli and ENDs is consistent with a first-order phase transition in which late-stage coarsening dynamics are mediated by Brownian coalescence and, to a lesser degree, Ostwald ripening. By manipulating C. elegans cell size, we change nucleolar component concentration and confirm several key model predictions. Our results show that rRNA transcription and other nonequilibrium biological activity can modulate the effective thermodynamic parameters governing nucleolar and END assembly, but do not appear to fundamentally alter the passive phase separation mechanism.

Two-Point Microrheology of Phase-Separated Domains in Lipid Bilayers

Though the importance of membrane fluidity for cellular function has been well established for decades, methods for measuring lipid bilayer viscosity remain challenging to devise and implement. Recently, approaches based on characterizing the Brownian dynamics of individual tracers such as colloidal particles or lipid domains have provided insights into bilayer viscosity. For fluids in general, however, methods based on single-particle trajectories provide a limited view of hydrodynamic response. The technique of two-point microrheology, in which correlations between the Brownian dynamics of pairs of tracers report on the properties of the intervening medium, characterizes viscosity at length-scales that are larger than that of individual tracers and has less sensitivity to tracer-induced distortions, but has never been applied to lipid membranes. We present, to our knowledge, the first two-point microrheological study of lipid bilayers, examining the correlated motion of domains in phase-separated lipid vesicles and comparing one- and two-point results. We measure two-point correlation functions in excellent agreement with the forms predicted by two-dimensional hydrodynamic models, analysis of which reveals a viscosity intermediate between those of the two lipid phases, indicative of global fluid properties rather than the viscosity of the local neighborhood of the tracer.

Calculating hydrodynamic interactions for membrane-embedded objects

A recently introduced numerical scheme for calculating self-diffusion coefficients of solid objects embedded in lipid bilayer membranes is extended to enable calculation of hydrodynamic interactions between multiple objects. The method is used to validate recent analytical predictions by Oppenheimer and Diamant [Biophys. J. 96, 3041 2009] related to the coupled diffusion of membrane embedded proteins and is shown to converge to known near-field lubrication results as objects closely approach one another; however, the present methodology also applies outside of the limiting regimes where analytical results are available. Multiple different examples involving pairs of disk-like objects with various constraints imposed on their relative motions demonstrate the importance of hydrodynamic interactions in the dynamics of proteins and lipid domains on membrane surfaces. It is demonstrated that the relative change in self-diffusion of a membrane embedded object upon perturbation by a similar proximal solid object displays a maximum for object sizes comparable to the Saffman-Delbrück length of the membrane.

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.

Measuring lipid membrane viscosity using rotational and translational probe diffusion

The two-dimensional fluidity of lipid bilayers enables the motion of membrane-bound macromolecules and is therefore crucial to biological function. Microrheological methods that measure fluid viscosity via the translational diffusion of tracer particles are challenging to apply and interpret for membranes, due to uncertainty about the local environment of the tracers. Here, we demonstrate a new technique in which determination of both the rotational and translational diffusion coefficients of membrane-linked particles enables quantification of viscosity, measurement of the effective radii of the tracers, and assessment of theoretical models of membrane hydrodynamics. Surprisingly, we find a wide distribution of effective tracer radii, presumably due to a variable number of lipids linked to each tracer particle. Furthermore, we show for the first time that a protein involved in generating membrane curvature, the vesicle trafficking protein Sar1p, dramatically increases membrane viscosity. Using the rheological method presented here, therefore, we are able to reveal a class of previously unknown couplings between protein activity and membrane mechanics.

Transient domain formation in membrane-bound organelles undergoing maturation

The membrane components of cellular organelles have been shown to segregate into domains as the result of biochemical maturation. We propose that the dynamical competition between maturation and lateral segregation of membrane components regulates domain formation. We study a two-component fluid membrane in which enzymatic reaction irreversibly converts one component into another and phase separation triggers the formation of transient membrane domains. The maximum domain size is shown to depend on the maturation rate as a power law similar to the one observed for domain growth with time in the absence of maturation, despite this time dependence not being verified in the case of irreversible maturation. This control of domain size by enzymatic activity could play a critical role in regulating exchange between organelles or within compartmentalized organelles such as the Golgi apparatus.

Fluctuating hydrodynamics simulations of coarse-grained lipid membranes under steady-state conditions and in shear flow

The stochastic Eulerian-Lagrangian method (SELM) is used to simulate coarse-grained lipid membrane models under steady-state conditions and in shear flow. SELM is an immersed boundary method which combines the efficiency of particle-based simulations with the realistic solvent dynamics provided by fluctuating hydrodynamics. Membrane simulations in SELM are shown to give structural properties in accordance with equilibrium statistical mechanics and dynamic properties in agreement with previous simulations of highly detailed membrane models in explicit solvent. Simulations of sheared membranes are used to calculate surface shear viscosities and inter-monolayer friction coefficients. The membrane models are shown to be shear thinning under a wide range of applied shear rates.

Microfluidic methods for forming liposomes

Liposome structures have a wide range of applications in biology, biochemistry, and biophysics. As a result, several methods for forming liposomes have been developed. This review provides a critical comparison of existing microfluidic technologies for forming liposomes and, when applicable, a comparison with their analogous macroscale counterparts. The properties of the generated liposomes, including size, size distribution, lamellarity, membrane composition, and encapsulation efficiency, form the basis for comparison. We hope that this critique will allow the reader to make an informed decision as to which method should be used for a given biological application.

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.

Dynamic scaling in phase separation kinetics for quasi-two-dimensional membranes

We consider the dynamics of phase separation in lipid bilayer membranes, modeled as flat two-dimensional liquid sheets within a bulk fluid, both in the creeping flow approximation. We present scaling arguments that suggest asymptotic coarsening in these systems is characterized by a length scale R(t) ~ t(1/2) for critical (bicontinuous) phase separation and R(t) ~t(1/3) for off-critical concentrations (droplet morphology). In this limit, the bulk fluid is the primary source of dissipation. We also address these questions with continuum stochastic hydrodynamic simulations. We see evidence of scaling violation in critical phase separation, where isolated circular domains coarsen slower than elongated ones. However, we also find a region of apparent scaling where R(t) ~ t(1/2) is observed. This appears to be due to the competition of thermal and hydrodynamic effects. We argue that the diversity of scaling exponents measured in experiment and prior simulations can in part be attributed to certain measurements lying outside the asymptotic long-length-scale regime, and provide a framework to help understand these results. We also discuss a few simple generalizations to confined membranes and membranes in which inertia is relevant.

The intermediate scattering function for lipid bilayer membranes: from nanometers to microns

A numerical scheme based upon established hydrodynamic and elastic considerations is introduced and used to predict the intermediate scattering function for lipid bilayer membranes. The predictions span multiple wavelength regimes, including those studied by dynamic light scattering (DLS; microns) and neutron spin-echo (NSE) spectroscopy (10-100 nm). The results validate a recent theory specific to the NSE regime and expose slight inaccuracies associated with the theoretical results available in the DLS regime. The assumptions that underlie both our numerical methods and the related theoretical predictions are reviewed in detail to explain when certain results can be applied to experiment and where caution must be exercised.

Making a tool of an artifact: the application of photoinduced Lo domains in giant unilamellar vesicles to the study of Lo/Ld phase spinodal decomposition and its modulation by the ganglioside GM1

Electroformed giant unilamellar vesicles containing liquid-ordered Lo domains are important tools for the modeling of the physicochemical properties and biological functions of lipid rafts. Lo domains are usually imaged using fluorescence microscopy of differentially phase-partionioning membrane-embedded probes. Recently, it has been shown that these probes also have a photosensitizing effect that leads to lipid chemical modification during the fluorescence microscopy experiments. Moreover, the lipid reaction products are able as such to promote Lo microdomain formation, leading to potential artifacts. We show here that this photoinduced effect can also purposely be used as a new approach to study Lo microdomain formation in giant unilamellar vesicles. Photosensitized lipid modification can promote Lo microdomain appearance and growth uniformly and on a faster time scale, thereby yielding new information on such processes. For instance, in egg phosphatidylcholine/egg sphingomyelin/cholesterol 50:30:20 (mol/mol) giant unilamellar vesicles, photoinduced Lo microdomain formation appears to occur by the rarely observed spinodal decomposition process rather than by the common nucleation process usually observed for Lo domain formation in bilayers. Moreover, temperature and the presence of the ganglioside GM1 have a profound effect on the morphological outcome of the photoinduced phase separation, eventually leading to features such as bicontinuous phases, phase percolation inversions, and patterns evoking double phase separations. GM1 also has the effect of destabilizing Lo microdomains. These properties may have consequences for Lo nanodomains stability and therefore for raft dynamics in biomembranes. Our data show that photoinduced Lo microdomains can be used to obtain new data on fast raft-mimicking processes in giant unilamellar vesicles.

In-plane dynamics of membranes with immobile inclusions

Cell membranes are anchored to the cytoskeleton via immobile inclusions. We investigate the effect of such anchors on the in-plane dynamics of a fluid membrane and mobile inclusions (proteins) embedded in it. The immobile particles lead to a decreased diffusion coefficient of mobile ones and suppress the correlated diffusion of particle pairs. Because of the long-range, quasi-two-dimensional nature of membrane flows, these effects become significant at a low area fraction (below 1%) of immobile inclusions.

Comprehensive analysis of compositional interface fluctuations in planar lipid bilayer membranes

In this paper we present a comprehensive analysis of line tension-driven compositional interface fluctuations in planar lipid bilayer membranes. Our starting point is the advective Cahn-Hilliard equation for the local lipid composition in symmetric membranes, which explicitly incorporates both advective and diffusive lipid transport processes, and which is coupled to the continuum hydrodynamic equations governing the flow behavior of the membrane and surrounding solvent with finite subphase thickness. In order to extract the interface dynamics from the continuum phase-field formalism, we first derive the appropriate sharp-interface limit equations. We then carry out a linear perturbation analysis for the relaxational dynamics of small-amplitude sinusoidal interface fluctuations to yield the general dispersion relation ω(k) as a function of perturbation wave number k. The resulting expression incorporates the effects of diffusive and advective lipid transport processes within the membrane, viscous or viscoelastic membrane properties, coupling between membrane and solvent, and inertial effects within the membrane and solvent. It is shown that previously considered scenarios naturally emerge as limiting cases of the general result. Furthermore, we discuss two additional scenarios amenable to analysis, one in which the inertia of the solvent is relevant, and another one in which the membrane displays significant viscoelastic properties. Finally, we numerically evaluate the general dispersion relation for three representative model membrane systems.

Beyond the creeping viscous flow limit for lipid bilayer membranes: theory of single-particle microrheology, domain flicker spectroscopy, and long-time tails

Recent experiments suggest that lipid bilayer membranes may be viscoelastic. We present a generalized "Saffman-Einstein" relation that may be used to determine the linear viscoelastic shear modulus from single-bead microrheology experiments on membranes. We show that viscoelastic parameters can also be extracted from membrane domain flicker spectroscopy experiments. Contributions from fluid inertia are expected to be negligible in both microrheology and domain flicker spectroscopy experiments, but can create a "long-time tail" in the membrane velocity autocorrelation function. In a viscous membrane, this tail crosses over from t(-1) at intermediate times, as in a two-dimensional fluid, to t(-3/2) at long times, as in a three-dimensional fluid. If the membrane is viscoelastic, the velocity autocorrelation function may be negative at intermediate times.

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.

Hydrodynamic effects on spinodal decomposition kinetics in planar lipid bilayer membranes

The formation and dynamics of spatially extended compositional domains in multicomponent lipid membranes lie at the heart of many important biological and biophysical phenomena. While the thermodynamic basis for domain formation has been explored extensively in the past, domain growth in the presence of hydrodynamic interactions both within the (effectively) two-dimensional membrane and in the three-dimensional solvent in which the membrane is immersed has received little attention. In this work, we explore the role of hydrodynamic effects on spinodal decomposition kinetics via continuum simulations of a convective Cahn-Hilliard equation for membrane composition coupled to the Stokes equation. Our approach explicitly includes hydrodynamics both within the planar membrane and in the three-dimensional solvent in the viscously dominated flow regime. Numerical simulations reveal that dynamical scaling breaks down for critical lipid mixtures due to distinct coarsening mechanisms for elongated versus more isotropic compositional lipid domains. The breakdown in scaling should be readily observable in experiments on model membrane systems.

Correlated dynamics of inclusions in a supported membrane

The hydrodynamic theory of heterogeneous fluid membranes is extended to the case of a membrane adjacent to a solid substrate. We derive the coupling diffusion coefficients of pairs of membrane inclusions in the limit of large separation compared to the inclusion size. Two-dimensional compressive stresses in the membrane make the coupling coefficients decay asymptotically as 1/r(2) with interparticle distance r. For the common case, where the distance to the substrate is of submicrometer scale, we present expressions for the coupling between distant disklike inclusions, which are valid for arbitrary inclusion size. We calculate the effect of inclusions on the response of the membrane and the associated corrections to the coupling diffusion coefficients to leading order in the concentration of inclusions. While at short distances the response is modified as if the membrane were a two-dimensional suspension, the large-distance response is not renormalized by the inclusions.