Modeling flexible amphiphilic bilayers: A solvent-free off-lattice Monte Carlo study

Frame tension governs the thermal fluctuations of a fluid membrane: new evidence

Two different tensions can be defined for a fluid membrane: the internal tension, γ, conjugated to the real membrane area in the Hamiltonian, and the frame tension, τ, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by γ. However, using rotational invariance arguments, several studies argued that the fluctuation spectrum must be governed by the frame tension τ instead. These studies disagree on the origin of the result obtained with the standard description yet: either a miscounting of configurations, quantified with the integration measure, or the use of a quadratic approximation of the Helfrich Hamiltonian. Analyzing the simplest case of a one-dimensional membrane, for which the arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by τ, and show that the origin of the issue with the standard description is a miscounting of membrane configurations. The origin itself of this miscounting depends on the thermodynamic ensemble in which calculations are made.

Scale-invariance in miniature coarse-grained red blood cells by fluctuation analysis

To accurately represent the morphological and elastic properties of a human red blood cell, Fu et al. [Fu et al., Lennard-Jones type pair-potential method for coarse-grained lipid bilayer membrane simulations in LAMMPS, 2017, 210, 193-203] recently developed a coarse-grained molecular dynamics model with particular detail in the membrane. However, such a model accrues an extremely high computational cost for whole-cell simulation when assuming an appropriate length scaling - that of the bilayer thickness. To date, the model has only simulated "miniature" cells in order to circumvent this, with the a priori assumption that these miniaturised cells correctly represent their full-sized counterparts. The present work assesses the validity of this approach, by testing the scale invariance of the model through simulating cells of various diameters; first qualitatively in their shape evolution, then quantitatively by measuring their bending rigidity through fluctuation analysis. Cells of diameter of at least 0.5 μm were able to form the characteristic biconcave shape of human red blood cells, though smaller cells instead equilibrated to bowl-shaped stomatocytes. Thermal fluctuation analysis showed the bending rigidity to be constant over all cell sizes tested, and consistent between measurements on the whole-cell and on a planar section of bilayer. This is as expected from the theory on both counts. Therefore, we confirm that the evaluated model is a good representation of a full-size RBC when the model diameter is ≥0.5 μm, in terms of the morphological and mechanical properties investigated.

Hydrodynamic coupling for particle-based solvent-free membrane models

The great challenge with biological membrane systems is the wide range of scales involved, from nanometers and picoseconds for individual lipids to the micrometers and beyond millisecond for cellular signaling processes. While solvent-free coarse-grained membrane models are convenient for large-scale simulations and promising to provide insight into slow processes involving membranes, these models usually have unrealistic kinetics. One major obstacle is the lack of an equally convenient way of introducing hydrodynamic coupling without significantly increasing the computational cost of the model. To address this, we introduce a framework based on anisotropic Langevin dynamics, for which major in-plane and out-of-plane hydrodynamic effects are modeled via friction and diffusion tensors from analytical or semi-analytical solutions to Stokes hydrodynamic equations. Using this framework, in conjunction with our recently developed membrane model, we obtain accurate dispersion relations for planar membrane patches, both free-standing and in the vicinity of a wall. We briefly discuss how non-equilibrium dynamics is affected by hydrodynamic interactions. We also measure the surface viscosity of the model membrane and discuss the affecting dissipative mechanisms.

Identifying systematic errors in a power spectral analysis of simulated lipid membranes

The elastic properties of lipid membranes can be measured by monitoring their thermal fluctuations. For instance, comparing the power spectra of membrane shape or lipid director fluctuations with predictions based on suitable continuum theories gives access to bending-, tilt-, and twist-moduli. However, to do so in a computer simulation, we must first define a continuum surface shape and lipid director field from the discrete configurations of lipid molecules in a typically fairly small box. Here, we show that the required mapping choices, as well as the details of the subsequent data analysis, can shift the measured values of these moduli by far more than their statistical uncertainties. We investigate the resulting systematic errors on the basis of atomistic simulation trajectories for 13 different lipids, previously published by Venable et al. [Chem. Phys. Lipids 192, 60-74 (2015)]. Specifically, we examine mapping choices for surface- and tilt-field definitions, normalizing and averaging lipid directors, accounting for wave vector dependent time autocorrelations, error propagation, and finding the right fitting range. We propose a set of criteria that may help to decide upon a particular combination of choices underlying the fluctuation analysis, and we make several recommendations based on these. While systematic shifts in observables that are extracted from large-wavelength limits vanish, in principle, for sufficiently large system size, no such exact limit exists for intrinsically local parameters, such as the twist modulus or the splay-tilt coupling, and so not all potential choices can be trivially verified.

Coarse-Grained Modeling of Pore Dynamics on the Red Blood Cell Membrane under Large Deformations

Transient pore formation on the membrane of red blood cells (RBCs) under high mechanical tensions is of great importance in many biomedical applications, such as RBC damage (hemolysis) and mechanoporation-based drug delivery. The dynamic process of pore formation, growth, and resealing is hard to visualize in experiments. We developed a mesoscale coarse-grained model to study the characteristics of transient pores on a patch of the lipid bilayer that is strengthened by an elastic meshwork representing the cytoskeleton. Unsteady molecular dynamics was used to study the pore formation and reseal at high strain rates close to the physiological ranges. The critical strain for pore formation, pore characteristics, and cytoskeleton effects were studied. Results show that the presence of the cytoskeleton increases the critical strain of pore formation and confines the pore growth. Moreover, the pore recovery process under negative strain rates (compression) is analyzed. Simulations show that pores can remain open for a long time during the high-speed tank-treading induced stretching and compression process that a patch of the RBC membrane usually experiences under high shear flow. Furthermore, complex loading conditions can affect the pore characteristics and result in denser pores. Finally, the effects of strain rate on pore formation are analyzed. Higher rate stretching of membrane patch can result in a significant increase in the critical areal strain and density of pores. Such a model reveals the dynamic molecular process of RBC damage in biomedical devices and mechanoporation that, to our knowledge, has not been reported before.

Physical properties of phospholipids and integral proteins and their biofunctional roles in pulmonary surfactant from molecular dynamics simulation

This work deals with a quantitative investigation of the physical properties of pulmonary surfactant near melting temperature. To this end, we make use of molecular dynamics simulations, using the MARTINI coarse-grained model, for determining the physical properties of the system, such as the potential energy, the specific heat, the microstructure, the diffusion laws, and the elastic properties of the surfactant. The microstructure is studied by computation of the radial-distribution-function upon varying the distance between constituents (lipid molecules or proteins). The diffusion phenomenon is investigated by determination of the mean-squared-displacement and the time dependent velocity-autocorrelation-function for various values of temperature. We show that the dynamics of lipids and proteins exhibit a subdiffusion regime (slow movement) due to the cage effect within pulmonary surfactant. From the obtained mean-squared-displacement, we get the values of the self-diffusion-coefficients and the anomalous exponents at different temperatures close to the melting temperature. For the mathematical description of the cage effect, we make use of the scale relations in terms of the waiting time probability distribution. The last study is concerned with determination of the dependence of the lateral stress upon the strain of pulmonary surfactant, which is found to be linear, and from which we deduce the lateral-elastic-modulus.

A consistent quadratic curvature-tilt theory for fluid lipid membranes

The tilt of a lipid molecule describes the deviation of its orientation away from the local normal of its embedding membrane. Tilt is the subleading degree of freedom after a membrane's geometry, and it becomes relevant at scales comparable to lipid bilayer thickness. Building on earlier work by Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)], who envisioned lipid membranes as thin prestressed fluid elastic films, and Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)], who discovered a new coupling term between splay and tilt divergence, we construct a theory of membrane elasticity that is quadratic in geometry and tilt and complete at order 1/length². We show that a general and consistent treatment of both lateral and transverse depth-dependent shear stresses creates several contributions to the elastic energy density, of which only a subset had previously been identified. Apart from the well-known penalty of lipid twist (the curl of tilt), these terms generate no qualitatively new phenomenology, but they quantitatively revise the connections between the moduli of a tilt-curvature theory and its underlying microscopic foundation. In particular, we argue that the monolayer Gaussian curvature modulus κ¯_m, widely believed to be equal to the second moment of the transmonolayer stress profile, acquires a second contribution from lipid twist, which is always negative. This could resolve the long-standing conundrum that many measured values of κ¯_m appeared to have a sign that violates basic stability considerations. We also show that the previously discovered novel coupling between splay and tilt divergence is not simply proportional to κ¯_m but acquires its own splay-tilt coupling modulus, κ_st,m. We explore the predictions of our theory for various elastic moduli and their mutual interrelations and use an extensive set of existing atomistic molecular dynamics simulations for 12 different lipid types to collectively reason about such predictions. We find that bending rigidities are captured fairly well by existing theories, while reliable predictions for local moduli, especially the splay-tilt coupling modulus, remain challenging.

Novel tilt-curvature coupling in lipid membranes

On mesoscopic scales, lipid membranes are well described by continuum theories whose main ingredients are the curvature of a membrane's reference surface and the tilt of its lipid constituents. In particular, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] have shown how to systematically derive such a tilt-curvature Hamiltonian based on the elementary assumption of a thin fluid elastic sheet experiencing internal lateral pre-stress. Performing a dimensional reduction, they not only derive the basic form of the effective surface Hamiltonian but also express its emergent elastic couplings as trans-membrane moments of lower-level material parameters. In the present paper, we argue, though, that their derivation unfortunately missed a coupling term between curvature and tilt. This term arises because, as one moves along the membrane, the curvature-induced change of transverse distances contributes to the area strain-an effect that was believed to be small but nevertheless ends up contributing at the same (quadratic) order as all other terms in their Hamiltonian. We illustrate the consequences of this amendment by deriving the monolayer and bilayer Euler-Lagrange equations for the tilt, as well as the power spectra of shape, tilt, and director fluctuations. A particularly curious aspect of our new term is that its associated coupling constant is the second moment of the lipid monolayer's lateral stress profile-which within this framework is equal to the monolayer Gaussian curvature modulus, κ¯_m. On the one hand, this implies that many theoretical predictions now contain a parameter that is poorly known (because the Gauss-Bonnet theorem limits access to the integrated Gaussian curvature); on the other hand, the appearance of κ¯_m outside of its Gaussian curvature provenance opens opportunities for measuring it by more conventional means, for instance by monitoring a membrane's undulation spectrum at short scales.

Particle-based membrane model for mesoscopic simulation of cellular dynamics

We present a simple and computationally efficient coarse-grained and solvent-free model for simulating lipid bilayer membranes. In order to be used in concert with particle-based reaction-diffusion simulations, the model is purely based on interacting and reacting particles, each representing a coarse patch of a lipid monolayer. Particle interactions include nearest-neighbor bond-stretching and angle-bending and are parameterized so as to reproduce the local membrane mechanics given by the Helfrich energy density over a range of relevant curvatures. In-plane fluidity is implemented with Monte Carlo bond-flipping moves. The physical accuracy of the model is verified by five tests: (i) Power spectrum analysis of equilibrium thermal undulations is used to verify that the particle-based representation correctly captures the dynamics predicted by the continuum model of fluid membranes. (ii) It is verified that the input bending stiffness, against which the potential parameters are optimized, is accurately recovered. (iii) Isothermal area compressibility modulus of the membrane is calculated and is shown to be tunable to reproduce available values for different lipid bilayers, independent of the bending rigidity. (iv) Simulation of two-dimensional shear flow under a gravity force is employed to measure the effective in-plane viscosity of the membrane model and show the possibility of modeling membranes with specified viscosities. (v) Interaction of the bilayer membrane with a spherical nanoparticle is modeled as a test case for large membrane deformations and budding involved in cellular processes such as endocytosis. The results are shown to coincide well with the predicted behavior of continuum models, and the membrane model successfully mimics the expected budding behavior. We expect our model to be of high practical usability for ultra coarse-grained molecular dynamics or particle-based reaction-diffusion simulations of biological systems.

On the dynamics of molecular self-assembly and the structural analysis of bilayer membranes using coarse-grained molecular dynamics simulations

We present a molecular dynamics simulation study of the self-assembly of coarse-grained lipid molecules from unbiased random initial configurations. Our lipid model is based on a well-tried CG polymer model with an additional potential that mimics the hydrophobic properties of lipid tails. We find that several stages of self-organization of lipid clusters are involved in the dynamics of bilayer formation and that the resulting equilibrium structures sensitively depend on the strength of hydrophobic interactions hc of the lipid tails and on temperature T. The obtained stable lipid membranes are quantitatively analyzed with respect to their local structure and their degree of order. At equilibrium, we obtain self-stabilizing bilayer membrane structures that exhibit a bending stiffness κB and compression modulus KC comparable to experimental measurements under physiological conditions. We present a phase diagram of our lipid model which covers a sol-gel transition, a liquid (or gel-like) phase including stable bilayer structures and vesicle formation, as well as a quasi-crystalline phase. We also determine the exact conditions for temperature T and degree of hydrophobicity hc for stable bilayer formation including closed vesicles.

Modeling of band-3 protein diffusion in the normal and defective red blood cell membrane

We employ a two-component red blood cell (RBC) membrane model to simulate lateral diffusion of band-3 proteins in the normal RBC and in the RBC with defective membrane proteins. The defects reduce the connectivity between the lipid bilayer and the membrane skeleton (vertical connectivity), or the connectivity of the membrane skeleton itself (horizontal connectivity), and are associated with the blood disorders of hereditary spherocytosis (HS) and hereditary elliptocytosis (HE) respectively. Initially, we demonstrate that the cytoskeleton limits band-3 lateral mobility by measuring the band-3 macroscopic diffusion coefficients in the normal RBC membrane and in a lipid bilayer without the cytoskeleton. Then, we study band-3 diffusion in the defective RBC membrane and quantify the relation between band-3 diffusion coefficients and percentage of protein defects in HE RBCs. In addition, we illustrate that at low spectrin network connectivity (horizontal connectivity) band-3 subdiffusion can be approximated as anomalous diffusion, while at high horizontal connectivity band-3 diffusion is characterized as confined diffusion. Our simulations show that the band-3 anomalous diffusion exponent depends on the percentage of protein defects in the membrane cytoskeleton. We also confirm that the introduction of attraction between the lipid bilayer and the spectrin network reduces band-3 diffusion, but we show that this reduction is lower than predicted by the percolation theory. Furthermore, we predict that the attractive force between the spectrin filament and the lipid bilayer is at least 20 times smaller than the binding forces at band-3 and glycophorin C, the two major membrane binding sites. Finally, we explore diffusion of band-3 particles in the RBC membrane with defects related to vertical connectivity. We demonstrate that in this case band-3 diffusion can be approximated as confined diffusion for all attraction levels between the spectrin network and the lipid bilayer. By comparing the diffusion coefficients measured in horizontal vs. vertical defects, we conclude that band-3 mobility is primarily controlled by the horizontal connectivity.

Monte Carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area

Three types of surface tensions can be defined for lipid membranes: the internal tension, σ, conjugated to the real membrane area in the Hamiltonian, the mechanical frame tension, τ, conjugated to the projected area, and the "fluctuation tension", r, obtained from the fluctuation spectrum of the membrane height. We investigate these surface tensions by means of a Monge gauge lattice Monte Carlo simulation involving the exact, nonlinear, Helfrich Hamiltonian and a measure correction for the excess entropy of the Monge gauge. Our results for the relation between σ and τ agrees well with the theoretical prediction of [J.-B. Fournier and C. Barbetta, Phys. Rev. Lett., 2008, 100, 078103] based on a Gaussian approximation. This provides a valuable knowledge of τ in the standard Gaussian models where the tension is controlled by σ. However, contrary to the conjecture in the above paper, we find that r exhibits no significant difference from τ over more than five decades of tension. Our results appear to be valid in the thermodynamic limit and are robust to changing the ensemble in which the membrane area is controlled.

Partial wrapping and spontaneous endocytosis of spherical nanoparticles by tensionless lipid membranes

Computer simulations of an implicit-solvent particle-based model are performed to investigate the interactions between small spherical nanoparticles and tensionless lipid bilayers. We found that nanoparticles are either unbound, wrapped by the bilayer, or endocytosed. The degree of wrapping increases with increasing the adhesion strength. The transition adhesion strength between the unbound and partially wrapped states decreases as the nanoparticle diameter is increased. We also observed that the transition adhesion strength between the wrapped states and endocytosis state decreases with increasing the nanoparticle diameter. The partial wrapping of the nanoparticles by the tensionless bilayer is explained by an elastic theory which accounts for the fact that the interaction between the nanoparticle and the bilayer extends beyond the contact region. The theory predicts that for small nanoparticles, the wrapping angle increases continuously with increasing the adhesion strength. However, for relatively large nanoparticles, the wrapping angle exhibits a discontinuity between weakly and strongly wrapped states. The size of the gap in the wrapping angle between the weakly wrapped and strongly wrapped states increases with decreasing the range of nanoparticle-bilayer interaction.

Simulating Membrane Dynamics in Nonhomogeneous Hydrodynamic Environments

Two previously introduced simulation algorithms for the dynamics of elastic membrane sheets embedded in a fluid medium are extended to account for inhomogeneous hydrodynamic environments. We calculate the height autocorrelation function for a lipid bilayer randomly pinned to a flat substrate and the influence of fluid confinement by the spectrin cytoskeleton on short wavelength membrane undulations of the human red blood cell. Altering the hydrodynamic environment of the membrane leads to significant changes in dynamics, and we discuss these effects in the context of recent experiments.

Erythrocyte membrane model with explicit description of the lipid bilayer and the spectrin network

The membrane of the red blood cell (RBC) consists of spectrin tetramers connected at actin junctional complexes, forming a two-dimensional (2D) sixfold triangular network anchored to the lipid bilayer. Better understanding of the erythrocyte mechanics in hereditary blood disorders such as spherocytosis, elliptocytosis, and especially, sickle cell disease requires the development of a detailed membrane model. In this study, we introduce a mesoscale implicit-solvent coarse-grained molecular dynamics (CGMD) model of the erythrocyte membrane that explicitly describes the phospholipid bilayer and the cytoskeleton, by extending a previously developed two-component RBC membrane model. We show that the proposed model represents RBC membrane with the appropriate bending stiffness and shear modulus. The timescale and self-consistency of the model are established by comparing our results with experimentally measured viscosity and thermal fluctuations of the RBC membrane. Furthermore, we measure the pressure exerted by the cytoskeleton on the lipid bilayer. We find that defects at the anchoring points of the cytoskeleton to the lipid bilayer (as in spherocytes) cause a reduction in the pressure compared with an intact membrane, whereas defects in the dimer-dimer association of a spectrin filament (as in elliptocytes) cause an even larger decrease in the pressure. We conjecture that this finding may explain why the experimentally measured diffusion coefficients of band-3 proteins are higher in elliptocytes than in spherocytes, and higher than in normal RBCs. Finally, we study the effects that possible attractive forces between the spectrin filaments and the lipid bilayer have on the pressure applied on the lipid bilayer by the filaments. We discover that the attractive forces cause an increase in the pressure as they diminish the effect of membrane protein defects. As this finding contradicts with experimental results, we conclude that the attractive forces are moderate and do not impose a complete attachment of the filaments to the lipid bilayer.

Vesiculation of healthy and defective red blood cells

Vesiculation of mature red blood cells (RBCs) contributes to removal of defective patches of the erythrocyte membrane. In blood disorders, which are related to defects in proteins of the RBC membrane, vesiculation of the plasma membrane is intensified. Several hypotheses have been proposed to explain RBC vesiculation but the exact underlying mechanisms and what determines the sizes of the vesicles are still not completely understood. In this work, we apply a two-component coarse-grained molecular dynamics RBC membrane model to study how RBC vesiculation is controlled by the membrane spontaneous curvature and by lateral compression of the membrane. Our simulation results show that the formation of small homogeneous vesicles with a diameter less than 40 nm can be attributed to a large spontaneous curvature of membrane domains. On the other hand, compression on the membrane can cause the formation of vesicles with heterogeneous composition and with sizes comparable with the size of the cytoskeleton corral. When spontaneous curvature and lateral compression are simultaneously considered, the compression on the membrane tends to facilitate formation of vesicles originating from curved membrane domains. We also simulate vesiculation of RBCs with membrane defects connected to hereditary elliptocytosis (HE) and to hereditary spherocytosis (HS). When the vertical connectivity between the lipid bilayer and the membrane skeleton is elevated, as in normal RBCs, multiple vesicles are shed from the compressed membrane with diameters similar to the cytoskeleton corral size. In HS RBCs, where the connectivity between the lipid bilayer and the cytoskeleton is reduced, larger-size vesicles are released under the same compression ratio as in normal RBCs. Lastly, we find that vesicles released from HE RBCs can contain cytoskeletal filaments due to fragmentation of the membrane skeleton while vesicles released from the HS RBCs are depleted of cytoskeletal filaments.

Efficient solvent-free dissipative particle dynamics for lipid bilayers

We rigorously derived effective potentials for solvent-free DPD simulation of lipid bilayers. The derivation relies on an earlier developed hybrid particle/field method and is based on the idea that the solvent is always in local equilibrium on a coarse-grained time scale, given the instantaneous templates set by the self-assembly structure. By relating the parameters in the effective implicit-solvent potentials directly to the lipid-solvent interactions and membrane properties for the explicit solvent DPD model, we constitute an efficient and general procedure for reformulating any DPD membrane model in an implicit-solvent form. Here, we determined these membrane properties for two existing DPD models, via an analysis of membrane fluctuation spectra. Equivalent single-processor implicit- and explicit-solvent calculations show the trade-mark of implicit solvent simulation: a 20-fold reduction of the total simulation time for a system containing 92% solvent. This increased efficiency enabled us to realistically simulate the spontaneous formation of a ∼20 nm diameter vesicle on a single processor overnight. We believe that this work will contribute to an enhanced computational study of large vesicles and thus a better understanding of experimental liposome dynamics.

FLUCTUATIONS IN LIPID BILAYERS: ARE THEY UNDERSTOOD?

We review recent computer simulation studies of undulating lipid bilayers. Theoretical interpretations of such fluctuating membranes are most commonly based on generalized Helfrich-type elastic models, with additional contributions of local "protrusions" and/or density fluctuations. Such models provide an excellent basis for describing the fluctuations of tensionless bilayers in the fluid Lαphase at a quantitative level.

However, this description is found to fail for membranes in the gel phase and for membranes subject to high tensions. The fluctuations of tilted gel membranes (Lβ′phase) show a signature of the modulated ripple structure Pβ′, which is a nearby phase observed in the pretransition regime between the Lαand Lβ′state. This complicates a quantitative analysis on mesoscopic length scales. In the case of fluid membranes under tension, the large-wavelength fluctuation modes are found to be significantly softer than predicted by theory.

In the latter context, we also address the general problem of the relation between frame tension and the fluctuation tension, which has been discussed somewhat controversially in recent years. Simulations of very simple model membranes with fixed area show that the fluctuations should be controlled by the frame tension, and not by the internal tension.

Dynamic implicit-solvent coarse-grained models of lipid bilayer membranes: fluctuating hydrodynamics thermostat

We introduce a thermostat based on fluctuating hydrodynamics for dynamic simulations of implicit-solvent coarse-grained models of lipid bilayer membranes. We show our fluctuating hydrodynamics approach captures interesting correlations in the dynamics of lipid bilayer membranes that are missing in simulations performed using standard Langevin dynamics. Our momentum conserving thermostat accounts for solvent-mediated momentum transfer by coupling coarse-grained degrees of freedom to stochastic continuum fields that account for both the solvent hydrodynamics and thermal fluctuations. We present both a general framework and specific methods to couple the particle and continuum degrees of freedom in a manner consistent with statistical mechanics and amenable to efficient computational simulation. For self-assembled vesicles, we study the diffusivity of lipids and their spatial correlations. We find the hydrodynamic coupling yields within the bilayer interesting correlations between diffusing lipids that manifest as a vortex-like structure similar to those observed in explicit-solvent simulations. We expect the introduced fluctuating hydrodynamics methods to provide a way to extend implicit-solvent models for use in a wide variety of dynamic studies.

Elastic properties and line tension of self-assembled bilayer membranes

The elastic properties of a self-assembled bilayer membrane are studied using the self-consistent field theory, applied to a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents. Examining the free energy of bilayer membranes with different geometries allows us to calculate their bending modulus, Gaussian modulus, two fourth-order membrane moduli, and the line tension. The dependence of these parameters on the microscopic characteristics of the amphiphilic chain, characterized by the volume fraction of the hydrophilic component, is systematically studied. The theoretical predictions are compared with the results from a simple monolayer model, which approximates a bilayer membrane by two monolayers. The region of validity of the linear elasticity theory is analyzed by examining the higher-order contributions.

Mode specific elastic constants for the gel, liquid-ordered, and liquid-disordered phases of DPPC/DOPC/cholesterol model lipid bilayers

Using microscopic molecular theory, we determine the bending and saddle-splay constants of three-component lipid bilayers. The membrane contains cholesterol, dipalmitoyl-phosphatidylcholine (DPPC) and dioleoylphosphatidylcholine (DOPC) and the predictions of the theory have been shown to qualitatively reproduce phase diagrams of giant unilamellar vesicles (GUVs) of the same three components. The bending and saddle-splay constants were calculated for the gel, liquid-ordered (lo) and liquid-disordered (ld) phases. By proper expansion of the free energy, the molecular theory enables us to determine the effects of the mode of membrane bending deformation on the value of the elastic constants for different phases. In particular, we refer to the ability of the molecules to arrange the composition between the two monolayers upon deformation. The bending and saddle-splay constants obtained from the free energy expansion can be expressed in terms of moments of the local lateral pressures and their derivatives, all evaluated for a symmetric planar bilayer. The effect of blocked vs. free exchange of lipids across the two monolayers on the values of the bending constant is as high as 50 k(B)Tin the ld phase to as high as 200 k(B)T in the lo phase. These results show that one must strongly consider the mode of deformation in determining the mechanical properties of lipid bilayers. We discuss how the different contributions to the lateral pressures affect the values of the elastic constants, including the effects of the cholesterol concentration and temperature on the membrane elastic constants. We also calculate the equilibrium binding concentrations of lipid tail anchors as a function of membrane curvature by explicitly determining the chemical potential difference of species across a curved bilayer. Our results are in excellent agreement with recent experimental results.

Determining the Gaussian curvature modulus of lipid membranes in simulations

The Gaussian curvature modulus κ¯ of lipid bilayers likely contributes more than 100 kcal/mol to every cellular fission or fusion event. This huge impact on membrane remodeling energetics might be a factor that codetermines the complex lipid composition of biomembranes through tuning of κ¯. Yet, its value has been measured only for a handful of simple lipids, and no simulation has so far determined it better than a factor of two, rendering a systematic investigation of such enticing speculations impossible. Here we propose a highly accurate method to determine κ¯ in computer simulations. It relies on the interplay between curvature stress and edge tension of partially curved axisymmetric membrane disks and requires determining their closing probability. For a simplified lipid model we obtain κ¯ and its relation to the normal bending modulus κ for membranes differing both in stiffness and spontaneous lipid curvature. The elastic ratio κ¯/κ can be determined with a few percent statistical accuracy. Its value agrees with the scarce experimental data, and its change with spontaneous lipid curvature is compatible with theoretical expectations, thereby granting additional information on monolayer properties. We also show that an alternative determination of these elastic parameters based on moments of the lateral stress profile gives markedly different and unphysical values.

Two-component coarse-grained molecular-dynamics model for the human erythrocyte membrane

We present a two-component coarse-grained molecular-dynamics model for simulating the erythrocyte membrane. The proposed model possesses the key feature of combing the lipid bilayer and the erythrocyte cytoskeleton, thus showing both the fluidic behavior of the lipid bilayer and the elastic properties of the erythrocyte cytoskeleton. In this model, three types of coarse-grained particles are introduced to represent clusters of lipid molecules, actin junctions, and band-3 complexes, respectively. The proposed model facilitates simulations that span large length scales (approximately micrometers) and timescales (approximately milliseconds). By tuning the interaction potential parameters, we were able to control the diffusivity and bending rigidity of the membrane model. We studied the membrane under shearing and found that at a low shear strain rate, the developed shear stress was due mainly to the spectrin network, whereas the viscosity of the lipid bilayer contributed to the resulting shear stress at higher strain rates. In addition, we investigated the effects of a reduced spectrin network connectivity on the shear modulus of the membrane.

Mechanical surface tension governs membrane thermal fluctuations

We use analytical considerations and computer simulations to show that the membrane spectrum of thermal fluctuations is governed by the mechanical and not the intrinsic tension. Our study highlights the fact that the commonly used quadratic approximation of the Helfrich effective Hamiltonian is not rotationally invariant. We demonstrate that this nonphysical feature leads to a calculated mechanical tension that differs dramatically from the correct mechanical tension. Specifically, our results suggest that the mechanical and intrinsic tensions vanish simultaneously, which contradicts recent theoretical predictions derived for the approximated Hamiltonian.

An implicit solvent coarse-grained lipid model with correct stress profile

We develop a coarse-grained parametrization strategy for lipid membranes that we illustrate for a dipalmitoylphosphatidylcholine bilayer. Our coarse-graining approach eliminates the high cost of explicit solvent but maintains more lipid interaction sites. We use a broad attractive tail-tail potential and extract realistic bonded potentials of mean force from all-atom simulations, resulting in a model with a sharp gel to fluid transition, a correct bending modulus, and overall very reasonable dynamics when compared with experiment. We also determine a quantitative stress profile and correct breakdown of contributions from lipid components when compared with detailed all-atom simulation benchmarks, which has been difficult to achieve for implicit membrane models. Such a coarse-grained lipid model will be necessary for efficiently simulating complex constructs of the membrane, such as protein assembly and lipid raft formation, within these nonaqueous chemical environments.

A systematically coarse-grained solvent-free model for quantitative phospholipid bilayer simulations

We present an implicit solvent coarse-grained (CG) model for quantitative simulations of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) bilayers. The absence of explicit solvent enables membrane simulations on large length and time scales at moderate computational expense. Despite improved computational efficiency, the model preserves chemical specificity and quantitative accuracy. The bonded and nonbonded interactions together with the effective cohesion mimicking the hydrophobic effect were systematically tuned by matching structural and mechanical properties from experiments and all-atom bilayer simulations, such as saturated area per lipid, radial distribution functions, density and pressure profiles across the bilayer, P(2) order, etc. The CG lipid model is shown to self-assemble into a bilayer starting from a random dispersion. Its line tension and elastic properties, such as bending and stretching modulus, are semiquantitatively consistent with experiments. The effects of (i) reduced molecular friction and (ii) more efficient integration combine to an overall speed-up of 3-4 orders of magnitude compared to all-atom bilayer simulations. Our CG lipid model is especially useful for studies of large-scale phenomena in membranes that nevertheless require a fair description of chemical specificity, e.g., membrane patches interacting with movable and transformable membrane proteins and peptides.

Systematic implicit solvent coarse-graining of bilayer membranes: lipid and phase transferability of the force field

We study the lipid and phase transferability of our recently developed systematically coarse-grained solvent-free membrane model. The force field was explicitly parameterized to describe a fluid 1-palmitoyl-2-oleoyl-phosphatidylcholine (POPC) bilayer at 310 K with correct structure and area per lipid, while gaining at least three orders of magnitude in computational efficiency (see Wang and Deserno 2010 J. Phys. Chem. B 114 11207-20). Here, we show that exchanging CG tails, without any subsequent re-parameterization, creates reliable models of 1,2-dioleoylphosphatidylcholine (DOPC) and 1,2-dipalmitoylphosphatidylcholine (DPPC) lipids in terms of structure and area per lipid. Furthermore, all CG lipids undergo a liquid-gel transition upon cooling, with characteristics like those observed in experiments and all-atom simulations during phase transformation. These studies suggest a promising transferability of our force field parameters to different lipid species and thermodynamic state points, properties that are a prerequisite for even more complex systems, such as mixtures.

One-particle-thick, solvent-free, coarse-grained model for biological and biomimetic fluid membranes

Biological membranes are involved in numerous intriguing biophysical and biological cellular phenomena of different length scales, ranging from nanoscale raft formation, vesiculation, to microscale shape transformations. With extended length and time scales as compared to atomistic simulations, solvent-free coarse-grained membrane models have been exploited in mesoscopic membrane simulations. In this study, we present a one-particle-thick fluid membrane model, where each particle represents a cluster of lipid molecules. The model features an anisotropic interparticle pair potential with the interaction strength weighed by the relative particle orientations. With the anisotropic pair potential, particles can robustly self-assemble into fluid membranes with experimentally relevant bending rigidity. Despite its simple mathematical form, the model is highly tunable. Three potential parameters separately and effectively control diffusivity, bending rigidity, and spontaneous curvature of the model membrane. As demonstrated by selected examples, our model can naturally simulate dynamics of phase separation in multicomponent membranes and the topological change of fluid vesicles.

Molecular dynamics simulations of rupture in lipid bilayers

Magnetic fluid hyperthermia is a promising cancer therapy in which magnetic nanoparticles are acted upon by a high-frequency oscillating magnetic field. While the accepted mechanism is localized hyperthermia, it is plausible that shear stresses due to nanoparticles rotating near a cell membrane may induce rupture, enhancing the effectiveness of the treatment. With the goal of understanding this further, molecular dynamics simulations were carried out on a model cell membrane. A bilayer composed of dipalmitoylphosphatidylcholine lipids was subjected to an incremental tension as well as an incremental shear stress. In both cases, it was found that the bilayer could withstand a surface tension of approximately 90 mN/m prior to rupture. Under tension, the bilayer ruptured at double its initial area, whereas under shear, the bilayer ruptured at 1.8 times its initial area. The results show that both incremental tension and incremental shearing are able to produce bilayer rupture, with shear being more injurious, yielding a larger surface tension for a smaller deformation. This information allows for comparison between the estimated energy required to rupture a cell membrane and the energy that a magnetic nanoparticle would be able to generate while rotating in a cellular environment. Our estimates indicate that magnetically blocked nanoparticles with diameters larger than 50 nm may result in rupture due to shear.

Coarse-grained modeling of lipids

Molecular modeling of phospholipids on many scales has progressed significantly over the last years. Here we review several membrane models on intermediate to large length scales restricting ourselves to particle based coarse-grained models with implicit and explicit solvent. We explain similarities and differences as well as their connection to experiments and fine-grained models. We neglect any field descriptions on larger scales. We discuss then a few examples where we focus on studies of lipid phase behavior as well as supported lipid bilayers as these examples can only be meaningfully studied using large-scale models to date.

Multiscale modeling of emergent materials: biological and soft matter

In this review, we focus on four current related issues in multiscale modeling of soft and biological matter. First, we discuss how to use structural information from detailed models (or experiments) to construct coarse-grained ones in a hierarchical and systematic way. This is discussed in the context of the so-called Henderson theorem and the inverse Monte Carlo method of Lyubartsev and Laaksonen. In the second part, we take a different look at coarse graining by analyzing conformations of molecules. This is done by the application of self-organizing maps, i.e., a neural network type approach. Such an approach can be used to guide the selection of the relevant degrees of freedom. Then, we discuss technical issues related to the popular dissipative particle dynamics (DPD) method. Importantly, the potentials derived using the inverse Monte Carlo method can be used together with the DPD thermostat. In the final part we focus on solvent-free modeling which offers a different route to coarse graining by integrating out the degrees of freedom associated with solvent.

The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations

We present an improved and extended version of our coarse grained lipid model. The new version, coined the MARTINI force field, is parametrized in a systematic way, based on the reproduction of partitioning free energies between polar and apolar phases of a large number of chemical compounds. To reproduce the free energies of these chemical building blocks, the number of possible interaction levels of the coarse-grained sites has increased compared to those of the previous model. Application of the new model to lipid bilayers shows an improved behavior in terms of the stress profile across the bilayer and the tendency to form pores. An extension of the force field now also allows the simulation of planar (ring) compounds, including sterols. Application to a bilayer/cholesterol system at various concentrations shows the typical cholesterol condensation effect similar to that observed in all atom representations.

The computational route from bilayer membranes to vesicle fusion

Biological membranes are examples of 'smart' materials whose properties and behaviour emerge from the propagation across many scales of the molecular characteristics of their constituents. Artificial smart materials, such as drug delivery vehicles and biosensors, often rely on modifying naturally occurring soft matter, such as polymers and lipid vesicles, so that they possess useful behaviour. However, the complexity of natural membranes, both in their static properties, exemplified in their phase behaviour, and in their dynamic properties, as in the kinetics of their formation and interactions, hinders their rational modification. Mesoscopic simulations, such as dissipative particle dynamics (DPD), allow in silico experiments to be easily and cheaply performed on complex, soft materials requiring as input only the molecular structure of the constituents at a coarse-grained level. They can therefore act as a guide to experimenters prior to performing costly assays. Additionally, mesoscopic simulations provide the only currently feasible window on the length- and timescales relevant to important biophysical processes such as vesicle fusion. We review here the development of computational models of bilayer membranes, and in particular the use of mesoscopic simulations to follow the molecular rearrangements that occur during membrane fusion.

Meshless membrane model based on the moving least-squares method

A meshless particle-based membrane model is proposed. The particles possess no internal degree of freedom and interact via a potential, which has three different contributions: a short-range repulsive pair potential, an attractive multibody potential, and a curvature potential based on the moving least-squares method. Brownian dynamics simulations are employed to demonstrate that the particles self-assemble into a membrane and to study equilibrium properties, such as bending rigidity, surface tension, line tension, and diffusion constant. The bending rigidity and line tension are shown to depend on different potential parameters and can therefore be varied independently. The finite-size effects of nearly planar membranes are investigated. This model is well suited to study the membrane dynamics with topological changes.

Pore nucleation in mechanically stretched bilayer membranes

We report a computer-simulation study of the free-energy barrier for the nucleation of pores in the bilayer membrane under constant stretching lateral pressure. We find that incipient pores are hydrophobic but as the lateral size of the pore nucleus becomes comparable with the molecular length, the pore becomes hydrophilic. In agreement with previous investigations, we find that the dynamical process of growth and closure of hydrophilic pores is controlled by the competition between the surface tension of the membrane and the line tension associated with the rim of the pore. We estimate the line tension of a hydrophilic pore from the shape of the computed free-energy barriers. The line tension thus computed is in a good agreement with available experimental data. We also estimate the line tension of hydrophobic pores at both macroscopic and microscopic levels. The comparison of line tensions at these two different levels indicates that the "microscopic" line tension should be carefully distinguished from the "macroscopic" effective line tension used in the theoretical analysis of pore nucleation. The overall shape of the free-energy barrier for pore nucleation shows no indication for the existence of a metastable intermediate during pore nucleation.

Implicit solvent simulation models for biomembranes

Fully atomic simulation strategies are infeasible for the study of many processes of interest to membrane biology, biophysics and biochemistry. We review various coarse-grained simulation methodologies with special emphasis on methods and models that do not require the explicit simulation of water. Examples from our own research demonstrate that such models have potential for simulating a variety of biologically relevant phenomena at the membrane surface.