The kinetics of phase separation in asymmetric membranes

Domain formation in bicomponent vesicles induced by composition-curvature coupling

Lipid vesicles composed of a mixture of two types of lipids are studied by intensive Monte Carlo numerical simulations. The coupling between the local composition and the membrane shape is induced by two different spontaneous curvatures of the components. We explore the various morphologies of these biphasic vesicles coupled to the observed patterns such as nano-domains or labyrinthine mesophases. The effect of the difference in curvatures, the surface tension, and the interaction parameter between components is thoroughly explored. Our numerical results quantitatively agree with the previous analytical results obtained by Gueguen et al. [Eur. Phys. J. E 37, 76 (2014)] in the disordered (high temperature) phase. Numerical simulations allow us to explore the full parameter space, especially close to and below the critical temperature, where analytical results are not accessible. Phase diagrams are constructed and domain morphologies are quantitatively studied by computing the structure factor and the domain size distribution. This mechanism likely explains the existence of nano-domains in cell membranes as observed by super-resolution fluorescence microscopy.

Dynamics of a multicomponent vesicle in shear flow

We study the fully nonlinear, nonlocal dynamics of two-dimensional multicomponent vesicles in a shear flow with matched viscosity of the inner and outer fluids. Using a nonstiff, pseudo-spectral boundary integral method, we investigate dynamical patterns induced by inhomogeneous bending for a two phase system. Numerical results reveal that there exist novel phase-treading and tumbling mechanisms that cannot be observed for a homogeneous vesicle. In particular, unlike the well-known steady tank-treading dynamics characterized by a fixed inclination angle, here the phase-treading mechanism leads to unsteady periodic dynamics with an oscillatory inclination angle. When the average phase concentration is around 1/2, we observe tumbling dynamics even for very low shear rate, and the excess length required for tumbling is significantly smaller than the value for the single phase case. We summarize our results in phase diagrams in terms of the excess length, shear rate, and concentration of the soft phase. These findings go beyond the well known dynamical regimes of a homogeneous vesicle and highlight the level of complexity of vesicle dynamics in a fluid due to heterogeneous material properties.

Thermal fluctuations and stiffening of symmetric heterogeneous fluid membranes

We study the effects of thermal fluctuations on symmetric tensionless heterogeneous (two-component) fluid membranes in a simple minimal model. Close to the critical point T(c) of the associated miscibility phase transition of the composition and for sufficiently strong curvature-composition interactions, mediated through a composition-dependent bending modulus, thermal fluctuations lead to enhancement of the effective bending modulus. Thus, the membrane conformation fluctuations will be suppressed near T(c), in comparison with a pure fluid membrane, for which thermal fluctuations are known to reduce the effective bending modulus at all nonzero temperatures.

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.

Phases and fluctuations in a model for asymmetric inhomogeneous fluid membranes

We propose and analyze a model for phase transitions in an inhomogeneous fluid membrane, that couples local composition with curvature nonlinearly. For asymmetric membranes, our model shows generic non-Ising behavior and the ensuing phase diagram displays either a first- or a second-order phase transition through a critical point (CP) or a tricritical point (TP), depending upon the bending modulus. It predicts generic nontrivial enhancement in fluctuations of asymmetric membranes that scales with system size in a power-law fashion at the CP and TP in two dimensions, not observed in symmetric membranes. It also yields two-dimensional Ising universality class for symmetric membranes, in agreement with experimental results.

Phase separation in biological membranes: integration of theory and experiment

Lipid bilayer model membranes that contain a single lipid species can undergo transitions between ordered and disordered phases, and membranes that contain a mixture of lipid species can undergo phase separations. Studies of these transformations are of interest for what they can tell us about the interaction energies of lipid molecules of different species and conformations. Nanoscopic phases (<200 nm) can provide a model for membrane rafts, specialized membrane domains enriched in cholesterol and sphingomyelin, which are believed to have essential biological functions in cell membranes. Crucial questions are whether lipid nanodomains can exist in stable equilibrium in membranes and what is the distribution of their sizes and lifetimes in membranes of different composition. Theoretical methods have supplied much information on these questions, but better experimental methods are needed to detect and characterize nanodomains under normal membrane conditions. This review summarizes linkages between theoretical and experimental studies of phase separation in lipid bilayer model membranes.

Dynamics of multicomponent vesicles in a viscous fluid

We develop and investigate numerically a thermodynamically consistent model of two-dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibil-ity of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. [81] for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, an initially elliptical vesicle tank-treads and attains a steady shape and surface phase distribution. A sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures yields a complex evolution of the vesicle morphology as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.

Formation and regulation of lipid microdomains in cell membranes: theory, modeling, and speculation

Compositional lipid microdomains ("lipid rafts") in plasma membranes are believed to be important components of many cellular processes. The biophysical mechanisms by which cells regulate the size, lifetime, and spatial localization of these domains are rather poorly understood at the moment. Over the years, experimental studies of raft formation have inspired several phenomenological theories and speculations incorporating a wide variety of thermodynamic assumptions regarding lipid-lipid and lipid-protein interactions, and the potential role of active cellular processes on membrane structure. Here we critically review and discuss these theories, models, and speculations, and present our view on future directions.

Reversible formation of nanodomains in monolayers of DPPC studied by kinetic modeling

Dipalmitoylphosphatidylcholine (DPPC) is the most abundant component in pulmonary surfactants and is believed to be responsible for maintaining low surface tension in alveoli during breathing. In this work, a kinetic model is introduced that describes the phase separation in DPPC films that produces the liquid-condensed (LC) and liquid-expanded (LE) fractions, which differ according to the area density of DPPC. The phase separation in an initially homogeneous film has been investigated numerically. Furthermore, explicit simulations of periodic compression-expansion cycles are reported. In this process, a moderate change of the surface area resulted in a dramatic change in the total amount of LC fraction, as well as in the surface morphology. Depending on the extent of the film's compression, the simulated surface morphologies comprised individual nanosized LC domains embedded in the LE fraction, interconnected networks of such domains, or continuous LC films with nanopores. Equilibration of the total area of the LC nanodomains occurred over a few milliseconds, indicating that the rate of the LE-LC phase transformation is sufficient for maintaining low surface tension during breathing, and that nanoscale LC domains are likely to play a major role in this process. Unlike the total content of the LC fraction, which stabilized quickly, the average size of LC nanodomains showed a tendency to increase slowly, at a rate determined by the diffusivity of DPPC. The computed average domain size seems to be compatible with published experiments for DPPC films. The numeric results also elucidate the distinction between thermodynamically determined and kinetically limited structural properties during phase separation in the major structure-forming component of pulmonary surfactants.

Equilibrium theory and geometrical constraint equation for two-component lipid bilayer vesicles

This paper aims at the general mathematical framework for the equilibrium theory of two-component lipid bilayer vesicles. To take into account the influences of the local compositions together with the mean curvature and Gaussian curvature of the membrane surface, a general potential functional is constructed. We introduce two kinds of virtual displacement modes: the normal one and the tangential one. By minimizing the potential functional, the equilibrium differential equations and the boundary conditions of two-component lipid vesicles are derived. Additionally, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are presented. The physical, mathematical, and biological meanings of the equilibrium differential equations and the geometrical constraint equations are discussed. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. Numerical results can be used to explain recent experiments.

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.

Numerical study of the phase separation in binary lipid membrane containing protein inclusions under stationary shear flow

The phase separation of lipids is believed to be responsible for the formation of lipid rafts in biological cell membrane. In the present work, a continuum model and a particle model are constructed to study the phase separation in binary lipid membrane containing inclusions under stationary shear flow. In each model, employing the cell dynamical system (CDS) approach, the kinetic equations of the confusion-advection process are numerically solved. Snapshot figures of the phase morphology are performed to intuitively display such phase evolving process. Considering the effects from both the inclusions and the shear flow, the time growth law of the characteristic domain size is discussed.

Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission

We develop a thermodynamically consistent phase-field model to simulate the dynamics of multicomponent vesicles. The model accounts for bending stiffness, spontaneous curvature, excess (surface) energy, and a line tension between the coexisting surface phases. Our approach is similar to that recently used by Wang and Du [J. Math. Biol. 56, 347 (2008)] with a key difference. Here, we concentrate on the dynamic evolution and solve the surface mass conservation equation explicitly; this equation was not considered by Wang and Du. The resulting fourth-order strongly coupled system of nonlinear nonlocal equations are solved numerically using an adaptive finite element numerical method. Although the system is valid for three dimensions, we limit our studies here to two dimensions where the vesicle is a curve. Differences between the spontaneous curvatures and the bending rigidities of the surface phases are found numerically to lead to the formation of buds, asymmetric vesicle shapes and vesicle fission even in two dimensions. In addition, simulations of configurations far from equilibrium indicate that phase separation via spinodal decomposition and coarsening not only affect the vesicle shape but also that the vesicle shape affects the phase separation dynamics, especially the coarsening and may lead to lower energy states than might be achieved by evolving initially phase-separated configurations.

Modeling the phase separation in binary lipid membrane under externally imposed oscillatory shear flow

By adding external velocity terms, the two-dimensional time-dependent Ginzburg-Landau (TDGL) equations are modified. Based on this, the phase separation in binary lipid membrane under externally imposed oscillatory shear flow is numerically modeled employing the Cell Dynamical System (CDS) approach. Considering shear flows with different frequencies and amplitudes, several aspects of such a phase evolving process are studied. Firstly, visualized results are shown via snapshot figures of the membrane shape. And then, the simulated scattering patterns at typical moments are presented. Furthermore, in order to more quantitatively discuss this phase-separation process, the time growth laws of the characteristic domain sizes in both directions parallel and perpendicular to the flow are investigated for each case. Finally, the peculiar rheological properties of such binary lipid membrane system have been discussed, mainly the normal stress difference and the viscoelastic complex shear moduli.

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.

Influence of monolayer-monolayer coupling on the phase behavior of a fluid lipid bilayer

We suggest a minimal model for the coupling of the lateral phase behavior in an asymmetric lipid membrane across its two monolayers. Our model employs one single order parameter for each monolayer leaflet, namely its composition. Regular solution theory on the mean-field level is used to describe the free energy in each individual leaflet. Coupling between monolayers entails an energy penalty for any local compositional differences across the membrane. We calculate and analyze the phase behavior of this model. It predicts a range of possible scenarios. A monolayer with a propensity for phase separation is able to induce phase separation in the apposed monolayer. Conversely, a monolayer without this propensity is able to prevent phase separation in the apposed monolayer. If there is phase separation in the membrane, it may lead to either complete or partial registration of the monolayer domains across the membrane. The latter case which corresponds to a three-phase coexistence is only found below a critical coupling strength. We calculate that critical coupling strength. Above the critical coupling strength, the membrane adopts a uniform compositional difference between its two monolayers everywhere in the membrane, implying phase coexistence between only two phases and thus perfect spatial registration of all domains on the apposed membrane leafs. We use the lattice Boltzmann simulation method to also study the morphologies that form during phase separation within the three-phase coexistence region. Generally, domains in one monolayer diffuse but remain fully enclosed within domains in the other monolayer.

Finite-size domains in membranes with active two-state inclusions

The distribution of inclusion-rich domains in membranes with active two-state inclusions is studied by simulations. Our study shows that typical size of inclusion-rich domains (L) can be controlled by inclusion activities in several ways. When there is effective attraction between state-1 inclusions, we find: (i) Small domains with only several inclusions are observed for inclusions with time scales (approximately 10(-3)s) and interaction energy [approximately Omicron(kBT)] comparable to motor proteins. (ii) L scales as 13 power of the lifetime of state-1 for a wide range of parameters. (iii) L shows a switch-like dependence on state-2 lifetime k12(-1). That is, L depends weakly on k12 when k12<k12* but increases rapidly with k12 when k12>k12*, the crossover k12* occurs when the diffusion length of a typical state-2 inclusion within its lifetime is comparable to L. (iv) Inclusion-curvature coupling provides another length scale that competes with the effects of transition rates.

Shrinkage dynamics of a vesicle in surfactant solutions

We develop a theory for shrinkage dynamics of a vesicle interacting with surfactant molecules. A stepwise shrinkage is formulated in such a way that it consists of two processes. One is a nucleation process of a pore under increasing of the membrane tension. The other is a closure process of the pore due to the line tension of the pore edge after leakage of the inner fluid. We carry out numerical simulations and show that the results agree with experiments semi-quantitatively. An analytical study is also carried out to understand the periodic shrinkage.

Anomalously slow domain growth in fluid membranes with asymmetric transbilayer lipid distribution

The effect of asymmetry in the transbilayer lipid distribution on the dynamics of phase separation in fluid vesicles is investigated numerically. This asymmetry is shown to set a spontaneous curvature for the domains that alter the morphology and dynamics considerably. For moderate tension, the domains are capped and the spontaneous curvature leads to anomalously slow dynamics, as compared to the case of symmetric bilayers. In contrast, in the limiting cases of high and low tensions, the dynamics proceeds toward full phase separation.

'KMC-TDGL'-a coarse-grained methodology for simulating interfacial dynamics in complex fluids: application to protein-mediated membrane processes

In this article, we describe a new multiscale simulation algorithm (which we term the 'KMC-TDGL' method) applicable for the description of equilibrium and dynamic processes associated with a particular class of complex fluids with nanoscale inclusions, namely, biological membranes mediated by membrane-associating and membrane-bound proteins. We adopt a novel strategy of integrating two different phenomenological approaches, namely, a field theoretic (continuum) description for the membrane dynamics given by the time-dependent Ginzburg-Landau equation and a random walk on a discretized lattice description for protein diffusion dynamics. We illustrate that this integrated approach results in a unified description of protein-mediated membrane dynamics.