Solvent-free coarse-grained lipid model for large-scale simulations

Membrane domain formation induced by binding/unbinding of curvature-inducing molecules on both membrane surfaces

The domain formation of curvature-inducing molecules, such as peripheral or transmembrane proteins and conical surfactants, is studied in thermal equilibrium and nonequilibrium steady states using meshless membrane simulations. These molecules can bind to both surfaces of a bilayer membrane and also move to the opposite leaflet by a flip-flop. Under symmetric conditions for the two leaflets, the membrane domains form checkerboard patterns in addition to striped and spot patterns. The unbound membrane stabilizes the vertices of the checkerboard. Under asymmetric conditions, the domains form kagome-lattice and thread-like patterns. In the nonequilibrium steady states, a flow of the binding molecules between the upper and lower solutions can occur via flip-flop.

Membrane shape deformation induced by curvature-inducing proteins consisting of chiral crescent binding and intrinsically disordered domains

Curvature-inducing proteins containing a bin/amphiphysin/Rvs domain often have intrinsically disordered domains. Recent experiments have shown that these disordered chains enhance curvature sensing and generation. Here, we report on the modification of protein–membrane interactions by disordered chains using meshless membrane simulations. The protein and bound membrane are modeled together as a chiral crescent protein rod with two excluded-volume chains. As the chain length increases, the repulsion between them reduces the cluster size of the proteins. It induces spindle-shaped vesicles and a transition between arc-shaped and circular protein assemblies in a disk-shaped vesicle. For flat membranes, an intermediate chain length induces many tubules owing to the repulsion between the protein assemblies, whereas longer chains promote perpendicular elongation of tubules. Moreover, protein rods with zero rod curvature and sufficiently long chains stabilize the spherical buds. For proteins with a negative rod curvature, an intermediate chain length induces a rugged membrane with branched protein assemblies, whereas longer chains induce the formation of tubules with periodic concave-ring structures.

Binding of anisotropic curvature-inducing proteins onto membrane tubes

Bin/Amphiphysin/Rvs superfamily proteins and other curvature-inducing proteins have anisotropic shapes and anisotropically bend biomembranes. Here, we report how the anisotropic proteins bind the membrane tube and are orientationally ordered using mean-field theory including an orientation-dependent excluded volume. The proteins exhibit a second-order or first-order nematic transition with increasing protein density depending on the radius of the membrane tube. The tube curvatures for the maximum protein binding and orientational order are different and varied by the protein density and rigidity. As the external force along the tube axis increases, a first-order transition from a large tube radius with low protein density to a small radius with high density occurs once, and subsequently, the protein orientation tilts to the tube-axis direction. When an isotropic bending energy is used for the proteins with an elliptic shape, the force-dependence curves become symmetric and the first-order transition occurs twice. This theory quantitatively reproduces the results of meshless membrane simulation for short proteins, whereas deviations are seen for long proteins owing to the formation of protein clusters.

Binding of curvature-inducing proteins onto tethered vesicles

A tethered vesicle, which consists of a cylindrical membrane tube and a spherical vesicle, is produced by a mechanical force that is experimentally imposed by optical tweezers and a micropipette. This tethered vesicle is employed for examining the curvature sensing of curvature-inducing proteins. In this study, we clarify how the binding of proteins with a laterally isotropic spontaneous curvature senses and generates the membrane curvatures of the tethered vesicle using mean-field theory and meshless membrane simulation. The force-dependence curves of the protein density in the membrane tube and the tube curvature are reflection symmetric and point symmetric, respectively, from the force point, in which the tube has a sensing curvature. The bending rigidity and spontaneous curvature of the bound proteins can be estimated from these force-dependence curves. First-order transitions can occur between low and high protein densities in the tube at both low and high force amplitudes. The simulation results of the homogeneous phases agree very well with the theoretical predictions. In addition, beaded-necklace-like tubes with microphase separation are found in the simulation.

Modeling lipid-protein interactions for coarse-grained lipid and Cα protein models

Biological membranes that play major roles in diverse functions are composed of numerous lipids and proteins, making them an important target for coarse-grained (CG) molecular dynamics (MD) simulations. Recently, we have developed the CG implicit solvent lipid force field (iSoLF) that has a resolution compatible with the widely used Cα protein representation [D. Ugarte La Torre and S. Takada, J. Chem. Phys. 153, 205101 (2020)]. In this study, we extended it and developed a lipid-protein interaction model that allows the combination of the iSoLF and the Cα protein force field, AICG2+. The hydrophobic-hydrophilic interaction is modeled as a modified Lennard-Jones potential in which parameters were tuned partly to reproduce the experimental transfer free energy and partly based on the free energy profile normal to the membrane surface from previous all-atom MD simulations. Then, the obtained lipid-protein interaction is tested for the configuration and placement of transmembrane proteins, water-soluble proteins, and peripheral proteins, showing good agreement with prior knowledge. The interaction is generally applicable and is implemented in the publicly available software, CafeMol.

Binding of thermalized and active membrane curvature-inducing proteins

The phase behavior of a membrane induced by the binding of curvature-inducing proteins is studied by a combination of analytical and numerical approaches. In thermal equilibrium under the detailed balance between binding and unbinding, the membrane exhibits three phases: an unbound uniform flat phase (U), a bound uniform flat phase (B), and a separated/corrugated phase (SC). In the SC phase, the bound proteins form hexagonally-ordered bowl-shaped domains. The transitions between the U and SC phases and between the B and SC phases are second order and first order, respectively. At a small spontaneous curvature of the protein or high surface tension, the transition between B and SC phases becomes continuous. Moreover, a first-order transition between the U and B phases is found at zero spontaneous curvature driven by the Casimir-like interactions between rigid proteins. Furthermore, nonequilibrium dynamics is investigated by the addition of active binding and unbinding at a constant rate. The active binding and unbinding processes alter the stability of the SC phase.

Formation and Properties of a Self-Assembled Nanoparticle-Supported Lipid Bilayer Probed through Molecular Dynamics Simulations

We have carried out coarse-grained molecular dynamics (MD) simulations to study the self-assembly procedure of a system of randomly placed lipid molecules, water beads, and a nanoparticle (NP). The self-assembly results in the formation of the nanoparticle-supported lipid bilayer (NPSLBL), with the self-assembly mechanism being driven by events such as the formation of small lipid clusters, merging of the lipid clusters in the vicinity of the NP to form NP-embedded vesicle with a pore, and collapsing of that pore to eventually form the equilibrated NPSLBL system overcoming a large free-energy barrier. Subsequently, we quantify the properties and the configurations of this NPSLBL system. We reveal that unlike our proposition of an equal number of lipid molecules occupying the inner and outer leaflets in a recent report studying the properties of a preassembled lipid bilayer, the equilibrated self-assembled NPSLBL system demonstrates a much larger number of lipid molecules occupying the outer leaflet as compared to the inner leaflet. Second, the thickness of the water layer entrapped between the NP and the inner leaflet shows similar values as predicted by experiments and our previous study. Finally, we reveal that, similar to our previous study, the diffusivity of the lipid molecules in the outer leaflet is larger than that in the inner leaflet but, due to higher temperature employed during our simulations, are even larger than that predicted by our previous study.

Detachment of a fluid membrane from a substrate and vesiculation

The detachment dynamics of a fluid membrane with isotropic spontaneous curvature from a flat substrate are studied by using meshless membrane simulations. The membrane is detached from an open edge leading to vesicle formation. With strong adhesion, the competition between the bending and adhesion energies determines the minimum value of the spontaneous curvature for the detachment. In contrast, with weak adhesion, detachment occurs at smaller spontaneous curvatures due to the membrane thermal undulation. When parts of the membrane are pinned on the substrate, the detachment becomes slower and a remaining membrane patch forms straight or concave membrane edges. The edge undulation induces vesiculation of long strips and disk-shaped patches. Therefore, membrane rolling is obtained only for membrane strips shorter than the wavelength for deformation into unduloids. This suggests that the rolling observed for Ca²⁺-dependent membrane-binding proteins annexins A3, A4, A5, and A13 results from the anisotropic spontaneous curvature induced by the proteins.

Shape transition from elliptical to cylindrical membrane tubes induced by chiral crescent-shaped protein rods

Proteins often form chiral assembly structures on a biomembrane. However, the role of the chirality in the interaction with an achiral membrane is poorly understood. Here, we report how chirality of crescent-shaped protein rods changes their assembly and tubulation using meshless membrane simulations. The achiral rods deformed the membrane tube into an elliptical shape by stabilizing the edges of the ellipse. In contrast, the chiral rods formed a helical assembly that generated a cylindrical membrane tube with a constant radius in addition to the elliptical tube. This helical assembly could be further stabilized by the direct side-to-side attraction between the protein rods. The chirality also promotes the tubulation from a flat membrane. These results agree with experimental findings of the constant radius of membrane tubules induced by the Bin/Amphiphysin/Rvs (BAR) superfamily proteins.

Mechanics of the Formation, Interaction, and Evolution of Membrane Tubular Structures

Membrane nanotubes, also known as membrane tethers, play important functional roles in many cellular processes, such as trafficking and signaling. Although considerable progresses have been made in understanding the physics regulating the mechanical behaviors of individual membrane nanotubes, relatively little is known about the formation of multiple membrane nanotubes due to the rapid occurring process involving strong cooperative effects and complex configurational transitions. By exerting a pair of external extraction upon two separate membrane regions, here, we combine molecular dynamics simulations and theoretical analysis to investigate how the membrane nanotube formation and pulling behaviors are regulated by the separation between the pulling forces and how the membrane protrusions interact with each other. As the force separation increases, different membrane configurations are observed, including an individual tubular protrusion, a relatively less deformed protrusion with two nanotubes on its top forming a V shape, a Y-shaped configuration through nanotube coalescence via a zipper-like mechanism, and two weakly interacting tubular protrusions. The energy profile as a function of the separation is determined. Moreover, the directional flow of lipid molecules accompanying the membrane shape transition is analyzed. Our results provide new, to our knowledge, insights at a molecular level into the interaction between membrane protrusions and help in understanding the formation and evolution of intra- and intercellular membrane tubular networks involved in numerous cell activities.

Entropic elasticity based coarse-grained model of lipid membranes

Various models for lipid bilayer membranes have been presented to investigate their morphologies. Among them, the aggressive coarse-grained models, where the membrane is represented by a single layer of particles, are computationally efficient and of practical importance for simulating membrane dynamics at the microscopic scale. In these models, soft potentials between particle pairs are used to maintain the fluidity of membranes, but the underlying mechanism of the softening requires further clarification. We have analyzed the membrane area decrease due to thermal fluctuations, and the results demonstrate that the intraparticle part of entropic elasticity is responsible for the softening of the potential. Based on the stretching response of the membrane, a bottom-up model is developed with an entropic effect explicitly involved. The model reproduces several essential properties of the lipid membrane, including the fluid state and a plateau in the stretching curve. In addition, the area compressibility modulus, bending rigidity, and spontaneous curvature display linear dependence on model parameters. As a demonstration, we have investigated the closure and morphology evolution of membrane systems driven by spontaneous curvature, and vesicle shapes observed experimentally are faithfully reproduced.

Acceleration and suppression of banana-shaped-protein-induced tubulation by addition of small membrane inclusions of isotropic spontaneous curvatures

Membrane tubulation induced by banana-shaped protein rods is investigated by using coarse-grained meshless membrane simulations. It is found that tubulation is promoted by laterally isotropic membrane inclusions that generate the same sign of spontaneous curvature as the adsorbed protein rods. The inclusions are concentrated in the tubules and reduce the bending energy of the tip of the tubules. On the other hand, inclusions with an opposite curvature suppress tubulation by percolated-network formation at a high protein-rod density while they induce the formation of a spherical membrane bud at a low rod density. When equal amounts of the two types of inclusions (with positive and negative curvatures) are added, their effects cancel each other for the first short period but later the tubulation is slowly accelerated. Positive surface tension suppresses tubulation. Our results suggest that the cooperation of scaffolding of BAR (Bin/Amphiphysin/Rvs) domains and isotropic membrane inclusions is important for tubulation.

Membrane structure formation induced by two types of banana-shaped proteins

The assembly of banana-shaped rodlike proteins on membranes and the associated membrane shape transformations are investigated by analytical theory and coarse-grained simulations. The membrane-mediated interactions between two banana-shaped inclusions are derived theoretically using a point-like formalism based on fixed anisotropic curvatures, both for zero surface tension and for finite surface tension. On a larger scale, the interactions between the assemblies of such rodlike inclusions are determined analytically. Meshless membrane simulations are performed in the presence of a large number of inclusions of two types, corresponding to the curved rods of opposite curvatures, both for flat membranes and vesicles. Rods of the same type aggregate into linear assemblies perpendicular to the rod axis, leading to membrane tubulation. However, rods of the other type, those of opposite curvature, are attracted to the lateral sides of these assemblies, and stabilize a straight bump structure that prevents tubulation. When the two types of rods have almost opposite curvatures, the bumps attract one another, forming a striped structure. Positive surface tension is found to stabilize stripe formation. The simulation results agree well with the theoretical predictions provided the point-like curvatures of the model are scaled-down to account for the effective flexibility of the simulated rods.

Solvent-free, molecular-level modeling of self-assembling amphiphiles in water

Aggregation mesophases of self-assembling amphiphiles in water are highly important in the context of biology (biomembranes), therapy (liposomes), industry (polymer surfactants), and condensed-matter physics (lyotropic liquid crystals). Besides helping to increase fundamental understanding of collective molecular behavior, simulations of these lyotropic phases are pivotal to technological and medical developments such as smart drug carriers for gene therapy. Implicit-solvent, coarse-grained, low resolution modeling with a simple pair potential is the key to realizing the larger length and time scales associated with such mesoscopic phenomena during a computer simulation. Modeling amphiphiles by directed, soft, ellipsoidal cores interacting via a computationally simple yet tunable anisotropic pair potential, we have come to such a single-site model amphiphile that can rapidly self-assemble to give diverse lyotropic phases (such as fluid bilayers, micelles, etc.) without requiring the explicit incorporation of solvent particles. The model directly represents a tunable packing parameter that manifests in the spontaneous curvature of the amphiphile aggregates. Besides the all-important hydrophobic interaction, the hydration force is also treated implicitly. Thanks to the efficient solvent-free molecular-level coarse graining, this model is suitable for generic mesoscale studies of phenomena such as self-assembly, amphiphile mixing, domain formation, fusion, elasticity, etc., in amphiphile aggregates.

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.

Shape deformation of lipid membranes by banana-shaped protein rods: Comparison with isotropic inclusions and membrane rupture

The assembly of curved protein rods on fluid membranes is studied using implicit-solvent meshless membrane simulations. As the rod curvature increases, the rods on a membrane tube assemble along the azimuthal direction first and subsequently along the longitudinal direction. Here, we show that both transition curvatures decrease with increasing rod stiffness. For comparison, curvature-inducing isotropic inclusions are also simulated. When the isotropic inclusions have the same bending rigidity as the other membrane regions, the inclusions are uniformly distributed on the membrane tubes and vesicles even for large spontaneous curvature of the inclusions. However, the isotropic inclusions with much larger bending rigidity induce shape deformation and are concentrated on the region of a preferred curvature. For high rod density, high rod stiffness, and/or low line tension of the membrane edge, the rod assembly induces vesicle rupture, resulting in the formation of a high-genus vesicle. A gradual change in the curvature suppresses this rupture. Hence, large stress, compared to the edge tension, induced by the rod assembly is the key factor determining rupture. For rod curvature with the opposite sign to the vesicle curvature, membrane rupture induces inversion of the membrane, leading to division into multiple vesicles as well as formation of a high-genus vesicle.

Membrane tubule formation by banana-shaped proteins with or without transient network structure

In living cells, membrane morphology is regulated by various proteins. Many membrane reshaping proteins contain a Bin/Amphiphysin/Rvs (BAR) domain, which consists of a banana-shaped rod. The BAR domain bends the biomembrane along the rod axis and the features of this anisotropic bending have recently been studied. Here, we report on the role of the BAR protein rods in inducing membrane tubulation, using large-scale coarse-grained simulations. We reveal that a small spontaneous side curvature perpendicular to the rod can drastically alter the tubulation dynamics at high protein density, whereas no significant difference is obtained at low density. A percolated network is intermediately formed depending on the side curvature. This network suppresses tubule protrusion, leading to the slow formation of fewer tubules. Thus, the side curvature, which is generated by protein–protein and membrane–protein interactions, plays a significant role in tubulation dynamics. We also find that positive surface tensions and the vesicle membrane curvature can stabilize this network structure by suppressing the tubulation.

Force Field Development for Lipid Membrane Simulations

With the rapid development of computer power and wide availability of modelling software computer simulations of realistic models of lipid membranes, including their interactions with various molecular species, polypeptides and membrane proteins have become feasible for many research groups. The crucial issue of the reliability of such simulations is the quality of the force field, and many efforts, especially in the latest several years, have been devoted to parametrization and optimization of the force fields for biomembrane modelling. In this review, we give account of the recent development in this area, covering different classes of force fields, principles of the force field parametrization, comparison of the force fields, and their experimental validation. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

Biomembranes in atomistic and coarse-grained simulations

The architecture of biological membranes is tightly coupled to the localization, organization, and function of membrane proteins. The organelle-specific distribution of lipids allows for the formation of functional microdomains (also called rafts) that facilitate the segregation and aggregation of membrane proteins and thus shape their function. Molecular dynamics simulations enable to directly access the formation, structure, and dynamics of membrane microdomains at the molecular scale and the specific interactions among lipids and proteins on timescales from picoseconds to microseconds. This review focuses on the latest developments of biomembrane force fields for both atomistic and coarse-grained molecular dynamics (MD) simulations, and the different levels of coarsening of biomolecular structures. It also briefly introduces scale-bridging methods applicable to biomembrane studies, and highlights selected recent applications.

Morphological changes of amphiphilic molecular assemblies induced by chemical reactions

Shape transformations of amphiphilic molecular assemblies induced by chemical reactions are studied using coarse-grained molecular simulations. A binding reaction between hydrophilic and hydrophobic molecules is considered. It is found that the reaction induces transformation of an oil droplet to a tubular vesicle via bicelles and vesicles with discoidal arms. The discoidal arms close into vesicles, which are subsequently fused into the tubular vesicle. Under the chemical reaction, the bicelle-to-vesicle transition occurs at smaller sizes than in the absence of the hydrophobic molecules. It is revealed that the enhancement of this transition is due to embedded hydrophobic particles that reduce the membrane bending rigidity.

Destruction of cancer cells by laser-induced shock waves: recent developments in experimental treatments and multiscale computer simulations

In this emerging area article we review recent progress in the mechanical destruction of cancer cells using laser-induced shock waves. The pure mechanical damaging and destruction of cancer cells associated with this technique possibly opens up a new route to tumor treatments and the corresponding therapies. At the same time progress in multiscale simulation techniques makes it possible to simulate mechanical properties of soft biological matter such as membranes, cytoskeletal networks and even whole cells and tissue. In this way an interdisciplinary approach to understanding key mechanisms in shock wave interactions with biological matter has become accessible. Mechanical properties of biological materials are also critical for many physiological processes and cover length scales ranging from the atomistic to the macroscopic scale. We argue that the latest developments and progress in experimentation enable the investigation of the shock wave interaction with cancer cells on multiple time- and length-scales. In this way the integrated use of experiment and simulation can address key challenges in this field. The exploration of the biological effects of laser-generated shock waves on a fundamental level constitutes an emerging multidisciplinary research area combining scientific methods from the areas of physics, biology, medicine and computer science.

Formation of double glass in binary mixtures of anisotropic particles: dynamic heterogeneities in rotations and displacements

We study glass behavior in a mixture of elliptic and circular particles in two dimensions at low temperatures using an orientation-dependent Lennard-Jones potential. The ellipses have a mild aspect ratio (∼1.2) and tend to align at low temperatures, while the circular particles play the role of impurities disturbing the ellipse orientations at a concentration of 20%. These impurities have a size smaller than that of the ellipses and attract them in the homeotropic alignment. As a result, the coordination number around each impurity is mostly 5 or 4 in glassy states. We realize double glass, where both the orientations and the positions are disordered but still hold mesoscopic order. We find a strong heterogeneity in the flip motions of the ellipses, which sensitively depends on the impurity clustering. In our model, a small fraction of the ellipses still undergo flip motions relatively rapidly even at low temperatures. In contrast, the nonflip rotations (with angle changes not close to ±π) are mainly caused by the cooperative configuration changes involving many particles. Then, there arises a long-time heterogeneity in the nonflip rotations closely correlated with the dynamic heterogeneity in displacements.

A Hybrid Approach for Highly Coarse-grained Lipid Bilayer Models

We present a systematic methodology to develop highly coarse-grained (CG) lipid models for large scale bio-membrane simulations, in which we derive CG interactions using a powerful combination of the multiscale coarse-graining (MS-CG) method, and an analytical form of the CG potential to model interactions at short range. The resulting hybrid coarse-graining (HCG) methodology is used to develop a three-site solvent-free model for 1,2-dilauroyl-sn-glycero-3-phosphocholine (DLPC), 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), and a 1:1 mixture of 1,2-dioleoyl-sn-glycero-3-phospho-L-serine (DOPS) and DOPC. In addition, we developed a four-site model of DOPC, demonstrating the capability of the HCG methodology in designing model lipid systems of a desired resolution. We carried out microsecond-scale molecular dynamics (MD) simulations of large vesicles, highlighting the ability of the model to study systems at mesoscopic length and time scales. The models of DLPC, DOPC and DOPC-DOPS have elastic properties consistent with experiment and structural properties such as the radial distribution functions (RDF), bond and angle distributions, and the z-density distributions that compare well with reference all-atom systems.

Two-dimensional lattice liquid models

A family of models of liquid on a two-dimensional lattice (2D lattice liquid models) have been proposed as primitive models of soft-material membrane. As a first step, we have formulated them as single-component, single-layered, classical particle systems on a two-dimensional surface with no explicit viscosity. Among the family of the models, we have shown and constructed two stochastic models, a vicious walk model and a flow model, on an isotropic regular lattice and on some honeycomb lattices of various sizes. In both cases, the dynamics is governed by the nature of the frustration of the particle movements. By simulations, we have found the approximate functional form of the frustration probability and peculiar anomalous diffusions in their time-averaged mean-square displacements in the flow model. The relations to other existing statistical models and possible extensions of the models are also discussed.

Estimation of the bending rigidity and spontaneous curvature of fluid membranes in simulations

Several numerical methods for measuring the bending rigidity and the spontaneous curvature of fluid membranes are studied using two types of meshless membrane models. The bending rigidity is estimated from the thermal undulations of planar and tubular membranes and the axial force of tubular membranes. We found a large dependence of its estimate value from the thermal undulation analysis on the upper-cutoff frequency q(cut) of the least-squares fit. The inverse power-spectrum fit with an extrapolation to q(cut)→0 yields the smallest estimation error among the investigated methods. The spontaneous curvature is estimated from the axial force of tubular membranes and the average curvature of bent membrane strips. The results of these methods show good agreement with each other.

Anisotropic surface tension of buckled fluid membranes

Thin solid sheets and fluid membranes exhibit buckling under lateral compression. Here it is revealed that buckled fluid membranes have anisotropic mechanical surface tension contrary to solid sheets. Surprisingly, the surface tension perpendicular to the buckling direction shows stronger dependence on the projected area than that parallel to it. Our theoretical predictions are supported by numerical simulations of a meshless membrane model. This anisotropic tension can be used to measure the membrane bending rigidity. It is also found that phase synchronization occurs between multilayered buckled membranes.