Phase separation in three-component lipid membranes: From Monte Carlo simulations to Ginzburg-Landau equations

Lattice-based mesoscale simulations and mean-field theory of cell membrane adhesion

Adhesion of cell membranes involves multi-scale phenomena, ranging from specific molecular binding at Angstrom scale all the way up to membrane deformations and phase separation at micrometer scale. Consequently, theory and simulations of cell membrane adhesion require multi-scale modeling and suitable approximations that capture the essential physics of these phenomena. Here, we present a mesoscale model for membrane adhesion which we have employed in a series of our recent studies. This model quantifies, in particular, how nanoscale lipid clusters physically affect and respond to the intercellular receptor-ligand binding that mediates membrane adhesion. The goal of this Chapter is to present all details and subtleties of the mean-field theory and Monte Carlo simulations of this mesoscale model, which can be used to further explore physical phenomena related to cell membrane adhesion.

A lattice model of ternary mixtures of lipids and cholesterol with tunable domain sizes

Much of our understanding of the physical properties of raft domains in biological membranes, and some insight into the mechanisms underlying their formation stem from atomistic simulations of simple model systems, especially ternary mixtures consisting of saturated and unsaturated lipids, and cholesterol (Chol). To explore the properties of such systems at large spatial scales, we here present a simple ternary mixture lattice model, involving a small number of nearest neighbor interaction terms. Monte Carlo simulations of mixtures with different compositions show excellent agreement with experimental and atomistic simulation observations across multiple scales, ranging from the local distributions of lipids to the phase diagram of the system. The simplicity of the model allows us to identify the roles played by the different interactions between components, and the interplay between them. Importantly, by changing the value of one of the model parameters, we can tune the size of the liquid-ordered domains, thereby simulating both Type II mixtures exhibiting macroscopic phase separation and Type I mixtures with nanoscopic domains. The Type II mixture simulation results fit well to the experimentally determined phase diagram of mixtures containing saturated DPPC/unsaturated DOPC/Chol. When the tunable parameter is changed, we obtain the Type I version of DPPC/DOPC/Chol, i.e., a mixture not showing thermodynamic phase transitions but one that may be fitted to the same phase diagram if local measures are used to distinguish between the different states. Our model results suggest that short range packing is likely to be a key regulator of the stability and size distribution of biological rafts.

Spatial extent of a single lipid's influence on bilayer mechanics

To what spatial extent does a single lipid affect the mechanical properties of the membrane that surrounds it? The lipid composition of a membrane determines its mechanical properties. The shapes available to the membrane depend on its compositional material properties, and therefore, the lipid environment. Because each individual lipid species' chemistry is different, it is important to know its range of influence on membrane mechanical properties. This is defined herein as the lipid's mechanical extent. Here, a lipid's mechanical extent is determined by quantifying lipid redistribution and the average curvature that lipid species experience on fluctuating membrane surfaces. A surprising finding is that, unlike unsaturated lipids, saturated lipids have a complicated, nonlocal effect on the surrounding surface, with the interaction strength maximal at a finite length-scale. The methodology provides the means to substantially enrich curvature-energy models of membrane structures, quantifying what was previously only conjecture.

Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling

We investigate a generic two-field Cahn-Hilliard model with variational and nonvariational coupling. It describes, for instance, passive and active ternary mixtures, respectively. Already a linear stability analysis of the homogeneous mixed state shows that activity not only allows for the usual large-scale stationary (Cahn-Hilliard) instability of the well-known passive case but also for small-scale stationary (Turing) and large-scale oscillatory (Hopf) instabilities. In consequence of the Turing instability, activity may completely suppress the usual coarsening dynamics. In a fully nonlinear analysis, we first briefly discuss the passive case before focusing on the active case. Bifurcation diagrams and selected direct time simulations are presented that allow us to establish that nonvariational coupling (i) can partially or completely suppress coarsening and (ii) may lead to the emergence of drifting and oscillatory states. Throughout, we emphasize the relevance of conservation laws and related symmetries for the encountered intricate bifurcation behavior.

Critical size dependence of domain formation observed in coarse-grained simulations of bilayers composed of ternary lipid mixtures

Model cellular membranes are known to form micro- and macroscale lipid domains dependent on molecular composition. The formation of macroscopic lipid domains by lipid mixtures has been the subject of many simulation investigations. We present a critical study of system size impact on lipid domain phase separation into liquid-ordered and liquid-disordered macroscale domains in ternary lipid mixtures. In the popular di-C16:0 PC:di-C18:2 PC:cholesterol at 35:35:30 ratio mixture, we find systems with a minimum of 1480 lipids to be necessary for the formation of macroscopic phase separated domains and systems of 10 000 lipids to achieve structurally converged conformations similar to the thermodynamic limit. To understand these results and predict the behavior of any mixture forming two phases, we develop and investigate an analytical Flory-Huggins model which is recursively validated using simulation and experimental data. We find that micro- and macroscale domains can coexist in ternary mixtures. Additionally, we analyze the distributions of specific lipid-lipid interactions in each phase, characterizing domain structures proposed based on past experimental studies. These findings offer guidance in selecting appropriate system sizes for the study of phase separations and provide new insights into the nature of domain structure for a popular ternary lipid mixture.

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.

Modeling the interplay between protein and lipid aggregation in supported membranes

We present a theoretical model that deals with the complex interplay between lipid segregation and the self-aggregation of lipid-attached proteins. The model, in contrast to previous ones that consider proteins only as passive elements affecting the lipid distribution, describes the system including three terms: the dynamic interactions between protein monomers, the interactions between lipid components, and a mixed term considering protein-lipid interactions. It is used to explain experimental results performed on a well-defined system in which a self-aggregating soluble bacterial cytoskeletal protein polymerizes on a lipid bilayer containing two lipid components. All the elements considered in a previously described protein model, including torsion of the monomers within the filament, are needed to account for the observed filament shapes. The model also points out that lipid segregation can affect the length and curvature of the filaments and that the dynamic behavior of the lipids and proteins can have different time scales, giving rise to memory effects. This simple model that considers a dynamic protein assembly on a fluid and active lipid surface can be easily extended to other biologically relevant situations in which the interplay between protein and lipid aggregation is needed to fully describe the system.

Lateral dynamics of charged lipids and peripheral proteins in spatially heterogeneous membranes: comparison of continuous and Monte Carlo approaches

Biological membranes are complex environments whose physico-chemical properties are of utmost importance for the understanding of many crucial biological processes. Much attention has been given in the literature to the description of membranes along the z-axis perpendicular to the membrane. Here, we instead consider the lateral dynamics of lipids and peripheral proteins due to their electrostatic interaction. Previously, we constructed a Monte Carlo automaton capable of simulating mutual diffusive dynamics of charged lipids and associated positively charged peptides. Here, we derive and numerically analyze a system of Poisson-Boltzmann-Nernst-Planck (PBNP) equations that provide a mean-field approximation compatible with our Monte Carlo model. The thorough comparison between the mean-field PBNP equations and Monte Carlo simulations demonstrates that both the approaches are in a good qualitative agreement in all tested scenarios. We find that the two methods quantitatively deviate when the local charge density is high, presumably because the Poisson-Boltzmann formalism is applicable in the so-called weak coupling limit, whose validity is restricted to low charge densities. Nevertheless, we conclude that the mean-field PBNP approach provides a good approximation for the considerably more detailed Monte Carlo model at only a fraction of the associated computational cost and allows simulation of the membrane lateral dynamics on the space and time scales relevant for the realistic biological problems.

MONTE CARLO STUDY OF LIPID NANOSCALE ORGANIZATION IN CELL MEMBRANES: TUNING DOMAIN SIZE AND STABILITY NEAR THE PHASE BOUNDARY

Lipids in cell membranes are organized in cholesterol-rich domains that are involved in many important cellular functions. Depending on the cell state, such structures are believed to develop covering a wide range of submicrometric sizes and different levels of stability. Using a simple Monte Carlo approach I demonstrate that when the membrane lipid mixture approaches a phase boundary, the cell has several mechanisms to tune both the size and stability of these domains. Cholesterol levels variation, presence of ceramides and insertion of proteins provide plausible mechanisms for the control of the nanoscale lipid organization in cell membranes.

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.

A simple thermodynamic model of the liquid-ordered state and the interactions between phospholipids and cholesterol

A theoretical model is proposed to describe the heat capacity function and the phase behavior of binary mixtures of phospholipids and cholesterol. The central idea is that the liquid-ordered state (L(o)) is a thermodynamic state or an ensemble of conformations of the phospholipid, characterized by enthalpy and entropy functions that are intermediate between those of the solid and the liquid-disordered (L(d)) states. The values of those thermodynamic functions are such that the L(o) state is not appreciably populated in the pure phospholipid, at any temperature, because either the solid or the L(d) state have much lower free energies. Cholesterol stabilizes the L(o) state by nearest-neighbor interactions, giving rise to the appearance of the L(o) phase. The model is studied by Monte Carlo simulations on a lattice with nearest-neighbor interactions, which are derived from experiment as much as possible. The calculated heat capacity function closely resembles that obtained by calorimetry. The phase behavior produced by the model is also in agreement with experimental data. The simulations indicate that separation between solid and L(o) phases occurs below the melting temperature of the phospholipid (T(m)). Above T(m), small L(d) and L(o) domains do exist, but there is no phase separation.

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

We present a stochastic phase-field model for multicomponent lipid bilayers that explicitly accounts for the quasi-two-dimensional hydrodynamic environment unique to a thin fluid membrane immersed in aqueous solution. Dynamics over a wide range of length scales (from nanometers to microns) for durations up to seconds and longer are readily accessed and provide a direct comparison to fluorescence microscopy measurements in ternary lipid-cholesterol mixtures. Simulations of phase separation kinetics agree with experiment and elucidate the importance of hydrodynamics in the coarsening process.

Effect of integral proteins in the phase stability of a lipid bilayer: application to raft formation in cell membranes

The existence of lipid rafts is a controversial issue. The affinity of cholesterol for saturated lipids is manifested in macroscopic phase separation in model membranes, and is believed to be the thermodynamic driving force for raft formation. However, there is no clear reason to explain the small (nanometric) size of raft domains in cell membranes. In a recent paper Yethiraj and Weisshaar [Biophys. J. 93, 3113 (2007)] proposed that the effect of neutral integral membrane proteins may prevent from the formation of large lipid domains. In this paper we extend this approach by studying the effect of the protein size, as well as the lipid-protein interaction. Depending on these factors, two different mechanisms for nanodomain stabilization are shown to be possible for static proteins. The application of these results to a biological context is discussed.

Stability of lipid domains

Membranes with simple lipid composition exhibit complex phase behavior. Ordered and disordered liquid phases can coexist in cholesterol-containing membranes with lipid compositions resembling biological membranes and at physiological temperatures. Research during the last years suggests that these lipid domains play a role in the organization of biological membranes. Understanding the principles that govern the formation and stability of lipid domains is of great importance to build a model that properly describes membrane structure and function. In this review, we describe the current knowledge of the chemical and physical basis of lipid domains and its application to biological membranes.

Nonequilibrium patterns in phase-separating ternary membranes

We present a nonequilibrium approach for the study of a two-dimensional phase-separating ternary mixture. When the component that promotes phase separation is dynamically exchanged with the medium, the separation process is halted and actively maintained finite-size segregation domains appear in the system. In addition to this effect, already reported in our earlier work [J. Gómez, F. Sagués, and R. Reigada, Phys. Rev. E 77, 021907 (2008)], the use of a generic Ginzburg-Landau formalism and the inclusion of thermal fluctuations provide a more dynamic description of the resulting domain organization. Its size, shape, and stability properties are studied. Larger and more circular and stable domains are formed when decreasing the recycling rate, increasing the mobility of the exchanged component, and the mixture is quenched deeper. We expect this outcome to be of applicability in raft phenomenology in plasmatic cell membranes.

Use of an enhanced bulk diffusion-based algorithm for phase separation of a ternary mixture

Phase separation kinetics of two-dimensional ternary mixtures have been studied by means of a Monte Carlo approach. Standard Kawasaki kinetics are impractical to study the late stages of the segregation process at deep quenches. An extension of the accelerated algorithm for binary mixtures proposed by Marko and Barkema [Phys. Rev. E 52, 2522 (1995)] is presented to overcome this problem in a three-component system discretized with two coupled lattices. We study the domain growth and the scaling behavior over a wide range of quench depths. Computer performances of the Kawasaki and the accelerated schemes are compared.

Thermodynamics of lipid interactions in complex bilayers

The mutual interactions between lipids in bilayers are reviewed, including mixtures of phospholipids, and mixtures of phospholipids and cholesterol (Chol). Binary mixtures and ternary mixtures are considered, with special emphasis on membranes containing Chol, an ordered phospholipid, and a disordered phospholipid. Typically the ordered phospholipid is a sphingomyelin (SM) or a long-chain saturated phosphatidylcholine (PC), both of which have high phase transitions temperatures; the disordered phospholipid is 1-palmitoyl-2-oleoylphosphatidylcholine (POPC) or dioleoylphosphatidylcholine (DOPC). The unlike nearest-neighbor interaction free energies (omega(AB)) between lipids (including Chol), obtained by an variety of unrelated methods, are typically in the range of 0-400 cal/mol in absolute value. Most are positive, meaning that the interaction is unfavorable, but some are negative, meaning it is favorable. It is of special interest that favorable interactions occur mainly between ordered phospholipids and Chol. The interpretation of domain formation in complex mixtures of Chol and phospholipids in terms of phase separation or condensed complexes is discussed in the light of the values of lipid mutual interactions.