Implicit solvent simulation models for biomembranes

Nonequilibrium Dynamic Phase Diagram for Transmembrane Transport of Active Particles

In recent years, there has been considerable push toward the biomedical applications with active particles, which have great potential to revolutionize disease diagnostics and therapy. The direct penetration of active particles through the cell membrane leads to more efficient intracellular delivery than previously considered endocytosis processes but may cause membrane disruption. Understanding fundamental behaviors of cell membranes in response to such extreme impacts by active particles is crucial to develop active particle-based biomedical technologies and manage health and safety issues in this emerging field. Unfortunately, the physical principles underlying the nonequilibrium behaviors from endocytosis to direct penetration remain elusive, and experiments are challenging. Here, we present a computed dynamic phase diagram for transmembrane transport of active particles and identify four characteristic dynamic phases in endocytosis and direct penetration according to the particle activity and membrane tension. The boundaries dividing these phases are analytically obtained with theoretical models, elucidating the nonequilibrium physics and criteria for the transition between different phases. Furthermore, we numerically and experimentally show three distinct dynamic regimes related to the interplay between necking and wrapping during the endocytosis process of active particles, which strikingly contrast the regimes for passive particles. Overall, these findings could be useful for sharpening the understanding of basic principles underlying biological issues related to the safe and efficient biomedical applications of such emerging matters.

Constructing 3-Dimensional Atomic-Resolution Models of Nonsulfated Glycosaminoglycans with Arbitrary Lengths Using Conformations from Molecular Dynamics

Glycosaminoglycans (GAGs) are the linear carbohydrate components of proteoglycans (PGs) and are key mediators in the bioactivity of PGs in animal tissue. GAGs are heterogeneous, conformationally complex, and polydisperse, containing up to 200 monosaccharide units. These complexities make studying GAG conformation a challenge for existing experimental and computational methods. We previously described an algorithm we developed that applies conformational parameters (i.e., all bond lengths, bond angles, and dihedral angles) from molecular dynamics (MD) simulations of nonsulfated chondroitin GAG 20-mers to construct 3-D atomic-resolution models of nonsulfated chondroitin GAGs of arbitrary length. In the current study, we applied our algorithm to other GAGs, including hyaluronan and nonsulfated forms of dermatan, keratan, and heparan and expanded our database of MD-generated GAG conformations. Here, we show that individual glycosidic linkages and monosaccharide rings in 10- and 20-mers of hyaluronan and nonsulfated dermatan, keratan, and heparan behave randomly and independently in MD simulation and, therefore, using a database of MD-generated 20-mer conformations, that our algorithm can construct conformational ensembles of 10- and 20-mers of various GAG types that accurately represent the backbone flexibility seen in MD simulations. Furthermore, our algorithm efficiently constructs conformational ensembles of GAG 200-mers that we would reasonably expect from MD simulations.

Efficient Construction of Atomic-Resolution Models of Non-Sulfated Chondroitin Glycosaminoglycan Using Molecular Dynamics Data

Glycosaminoglycans (GAGs) are linear, structurally diverse, conformationally complex carbohydrate polymers that may contain up to 200 monosaccharides. These characteristics present a challenge for studying GAG conformational thermodynamics at atomic resolution using existing experimental methods. Molecular dynamics (MD) simulations can overcome this challenge but are only feasible for short GAG polymers. To address this problem, we developed an algorithm that applies all conformational parameters contributing to GAG backbone flexibility (i.e., bond lengths, bond angles, and dihedral angles) from unbiased all-atom explicit-solvent MD simulations of short GAG polymers to rapidly construct models of GAGs of arbitrary length. The algorithm was used to generate non-sulfated chondroitin 10- and 20-mer ensembles which were compared to MD-generated ensembles for internal validation. End-to-end distance distributions in constructed and MD-generated ensembles have minimal differences, suggesting that our algorithm produces conformational ensembles that mimic the backbone flexibility seen in simulation. Non-sulfated chondroitin 100- and 200-mer ensembles were constructed within a day, demonstrating the efficiency of the algorithm and reduction in time and computational cost compared to simulation.

Dual-potential approach for coarse-grained implicit solvent models with accurate, internally consistent energetics and predictive transferability

The dual-potential approach promises coarse-grained (CG) models that accurately reproduce both structural and energetic properties, while simultaneously providing predictive estimates for the temperature-dependence of the effective CG potentials. In this work, we examine the dual-potential approach for implicit solvent CG models that reflect large entropic effects from the eliminated solvent. Specifically, we construct implicit solvent models at various resolutions, R, by retaining a fraction 0.10 ≤ R ≤ 0.95 of the molecules from a simple fluid of Lennard-Jones spheres. We consider the dual-potential approach in both the constant volume and constant pressure ensembles across a relatively wide range of temperatures. We approximate the many-body potential of mean force for the remaining solutes with pair and volume potentials, which we determine via multiscale coarse-graining and self-consistent pressure-matching, respectively. Interestingly, with increasing temperature, the pair potentials appear increasingly attractive, while the volume potentials become increasingly repulsive. The dual-potential approach not only reproduces the atomic energetics but also quite accurately predicts this temperature-dependence. We also derive an exact relationship between the thermodynamic specific heat of an atomic model and the energetic fluctuations that are observable at the CG resolution. With this generalized fluctuation relationship, the approximate CG models quite accurately reproduce the thermodynamic specific heat of the underlying atomic model.

Simulation Best Practices for Lipid Membranes [Article v1.0]

We establish a reliable and robust standardization of settings for practical molecular dynamics (MD) simulations of pure and mixed (single- and multi-component) lipid bilayer membranes. In lipid membranes research, particle-based molecular simulations are a powerful tool alongside continuum theory, lipidomics, and model, in vitro, and in vivo experiments. Molecular simulations can provide precise and reproducible spatiotemporal (atomic- and femtosecond-level) information about membrane structure, mechanics, thermodynamics, kinetics, and dynamics. Yet the simulation of lipid membranes can be a daunting task, given the uniqueness of lipid membranes relative to conventional liquid-liquid and solid-liquid interfaces, the immense and complex thermodynamic and statistical mechanical theory, the diversity of multiscale lipid models, limitations of modern computing power, the difficulty and ambiguity of simulation controls, finite size effects, competitive continuum simulation alternatives, and the desired application, including vesicle experiments and biological membranes. These issues can complicate an essential understanding of the field of lipid membranes, and create major bottlenecks to simulation advancement. In this article, we clarify these issues and present a consistent, thorough, and user-friendly framework for the design of state-of-the-art lipid membrane MD simulations. We hope to allow early-career researchers to quickly overcome common obstacles in the field of lipid membranes and reach maximal impact in their simulations.

Cell Membrane Disruption by Vertical Micro-/Nanopillars: Role of Membrane Bending and Traction Forces

Gaining access to the cell interior is fundamental for many applications, such as electrical recording and drug and biomolecular delivery. A very promising technique consists of culturing cells on micro-/nanopillars. The tight adhesion and high local deformation of cells in contact with nanostructures can promote the permeabilization of lipids at the plasma membrane, providing access to the internal compartment. However, there is still much experimental controversy regarding when and how the intracellular environment is targeted and the role of the geometry and interactions with surfaces. Consequently, we investigated, by coarse-grained molecular dynamics simulations of the cell membrane, the mechanical properties of the lipid bilayer under high strain and bending conditions. We found out that a high curvature of the lipid bilayer dramatically lowers the traction force necessary to achieve membrane rupture. Afterward, we experimentally studied the permeabilization rate of the cell membrane by pillars with comparable aspect ratios but different sharpness values at the edges. The experimental data support the simulation results: even pillars with diameters in the micron range may cause local membrane disruption when their edges are sufficiently sharp. Therefore, the permeabilization likelihood is connected to the local geometric features of the pillars rather than diameter or aspect ratio. The present study can also provide significant contributions to the design of three-dimensional biointerfaces for tissue engineering and cellular growth.

Intramolecular structural parameters are key modulators of the gel-liquid transition in coarse grained simulations of DPPC and DOPC lipid bilayers

The capability of coarse-grained models based on the MARTINI mapping to reproduce the gel-liquid phase transition in saturated and unsaturated model lipids was investigated. We found that the model is able to reproduce a lower critical temperature for 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) with respect to 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC). Nonetheless, the appearance of a gel phase for DOPC is strictly dependent on the intramolecular parameters chosen to model its molecular structure. In particular, we show that the bending angle at the coarse-grained bead corresponding to the unsaturated carbon-carbon bond acts as an order parameter determining the temperature of the phase transition. Structural analysis of the molecular dynamics simulations runs evidences that in the gel phase, the packing of the lipophilic tails of DOPC assume a different conformation than in the liquid phase. In the latter phase, the DOPC geometry resembles that of the relaxed free molecule. DPPC:DOPC mixtures show a single phase transition temperature, indicating that the observation of a phase separation between the two lipids requires the simulation of systems with sizes much larger than the ones used here.

Predicting ligand binding poses for low-resolution membrane protein models: Perspectives from multiscale simulations

Membrane receptors constitute major targets for pharmaceutical intervention. Drug design efforts rely on the identification of ligand binding poses. However, the limited experimental structural information available may make this extremely challenging, especially when only low-resolution homology models are accessible. In these cases, the predictions may be improved by molecular dynamics simulation approaches. Here we review recent developments of multiscale, hybrid molecular mechanics/coarse-grained (MM/CG) methods applied to membrane proteins. In particular, we focus on our in-house MM/CG approach. It is especially tailored for G-protein coupled receptors, the largest membrane receptor family in humans. We show that our MM/CG approach is able to capture the atomistic details of the receptor/ligand binding interactions, while keeping the computational cost low by representing the protein frame and the membrane environment in a highly simplified manner. We close this review by discussing ongoing improvements and challenges of the current implementation of our MM/CG code.

Molecular dynamics simulations of cholesterol-rich membranes using a coarse-grained force field for cyclic alkanes

The architecture of a biological membrane hinges upon the fundamental fact that its properties are determined by more than the sum of its individual components. Studies on model membranes have shown the need to characterize in molecular detail how properties such as thickness, fluidity, and macroscopic bending rigidity are regulated by the interactions between individual molecules in a non-trivial fashion. Simulation-based approaches are invaluable to this purpose but are typically limited to short sampling times and model systems that are often smaller than the required properties. To alleviate both limitations, the use of coarse-grained (CG) models is nowadays an established computational strategy. We here present a new CG force field for cholesterol, which was developed by using measured properties of small molecules, and can be used in combination with our previously developed force field for phospholipids. The new model performs with precision comparable to atomistic force fields in predicting the properties of cholesterol-rich phospholipid bilayers, including area per lipid, bilayer thickness, tail order parameter, increase in bending rigidity, and propensity to form liquid-ordered domains in ternary mixtures. We suggest the use of this model to quantify the impact of cholesterol on macroscopic properties and on microscopic phenomena involving localization and trafficking of lipids and proteins on cellular membranes.

Strong influence of periodic boundary conditions on lateral diffusion in lipid bilayer membranes

The Saffman-Delbrück hydrodynamic model for lipid-bilayer membranes is modified to account for the periodic boundary conditions commonly imposed in molecular simulations. Predicted lateral diffusion coefficients for membrane-embedded solid bodies are sensitive to box shape and converge slowly to the limit of infinite box size, raising serious doubts for the prospects of using detailed simulations to accurately predict membrane-protein diffusivities and related transport properties. Estimates for the relative error associated with periodic boundary artifacts are 50% and higher for fully atomistic models in currently feasible simulation boxes. MARTINI simulations of LacY membrane protein diffusion and LacY dimer diffusion in DPPC membranes and lipid diffusion in pure DPPC bilayers support the underlying hydrodynamic model.

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.

Mesoscopic Simulations on the Aggregate Behavior of Oligomeric Adamantane Surfactants in Aqueous Solutions

Oligomeric adamantane surfactants are novel surfactants using rigid adamantane as spacer group, which can form a variety of scales and shapes of micelles and aggregates in aqueous solution. In this paper, the self-assembling aggregate behavior of N,N-dimethyl dodecyl ammonium bromide adamantine [Ad-(NC12Br)], 1,3-di(N,N-dimethyl dodecyl ammonium bromide) adamantine [Ad-(NC12Br)2] and 1,3,5-tri-(N,N-dimethyl dodecyl ammonium bromide) adamantine [Ad-(NC12Br)3] in aqueous solution is studied by mesoscopic simulations of dissipative particle dynamics (DPD). The simulation results show that the critical micelle concentrations of the three adamantane oligomeric surfactants is in the order of [Ad-(NC12Br)3] < [Ad-(NC12Br)2] < [Ad-(NC12Br)]. The three oligomeric adamantane surfactants are able to form aggregates with a variety of spatial structures, such as spherical micelles, vesicles, lamellas, reticular lamellas, column lamellas or steric reticules in response to different surfactant concentrations in the aqueous solution.

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.

A Nonradial Coarse-Grained Potential for Proteins Produces Naturally Stable Secondary Structure Elements

We introduce a nonradial potential term for coarse-grained (CG) molecular simulations of proteins. This term mimics the backbone dipole-dipole interactions and accounts for the needed directionality to form stable folded secondary structure elements. We show that α-helical and β-sheet peptide chains are correctly described in dynamics without the need of introducing any a priori bias potentials or ad hoc parametrizations, which limit broader applicability of CG simulations for proteins. Moreover, our model is able to catch the formation of supersecondary structural motifs, like transitions from long single α-helices to helix-coil-helix or β-hairpin assemblies. This novel scheme requires the structural information of Cα beads only; it does not introduce any additional degrees of freedom to the system and has a general formulation, which allows it to be used in synergy with various CG protocols, leading to an improved description of the structural and dynamic properties of protein assemblies and networks.

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.

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.

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.

Computer simulations of phase separation in lipid bilayers and monolayers

Studying phase coexistence in lipid bilayers and monolayers is important for understanding lipid-lipid interactions underlying lateral organization in biological membranes. Computer simulations follow experimental approaches and use model lipid mixtures of simplified composition. Atomistic simulations give detailed information on the specificity of intermolecular interactions, while coarse-grained simulations achieve large time and length scales and provide a bridge towards state-of-the-art experimental techniques. Computer simulations allow characterizing the structure and composition of domains during phase transformations at Angstrom and picosecond resolution, and bring new insights into phase behavior of lipid membranes.

The ratio of the lateral correlation length and particle radius determines the density profile of spherical molecules near a fluctuating membrane

Interactions between membranes and molecules are important for many biological processes, e.g., transport of molecules across cell membranes. However, the detailed physical description of the membrane-biomolecule system remains a challenge and simplified schemes allow capturing its main intrinsic features. In this work, by means of Monte Carlo computer simulations, we systematically study the distribution of uncharged spherical molecules in contact with a flexible surface. Our results show that the distribution for finite size particles has the same simple functional form as the one obtained for point-like particles and depends only on the ratio of the lateral correlation length of the membrane and the radius of the molecules.

Modeling the electrostatic potential of asymmetric lipopolysaccharide membranes: the MEMPOT algorithm implemented in DelPhi

Four chemotypes of the rough lipopolysaccharides (LPS) membrane from Pseudomonas aeruginosa were investigated by a combined approach of explicit water molecular dynamics (MD) simulations and Poisson-Boltzmann continuum electrostatics with the goal to deliver the distribution of the electrostatic potential across the membrane. For the purpose of this investigation, a new tool for modeling the electrostatic potential profile along the axis normal to the membrane, MEMbrane POTential (MEMPOT), was developed and implemented in DelPhi. Applying MEMPOT on the snapshots obtained by MD simulations, two observations were made: (a) the average electrostatic potential has a complex profile but is mostly positive inside the membrane due to the presence of Ca(2+) ions, which overcompensate for the negative potential created by lipid phosphate groups; and (b) correct modeling of the electrostatic potential profile across the membrane requires taking into account the water phase, while neglecting it (vacuum calculations) results in dramatic changes including a reversal of the sign of the potential inside the membrane. Furthermore, using DelPhi to assign different dielectric constants for different regions of the LPS membranes, it was investigated whether a single frame structure before MD simulations with appropriate dielectric constants for the lipid tails, inner, and the external leaflet regions, can deliver the same average electrostatic potential distribution as obtained from the MD-generated ensemble of structures. Indeed, this can be attained by using smaller dielectric constant for the tail and inner leaflet regions (mostly hydrophobic) than for the external leaflet region (hydrophilic) and the optimal dielectric constant values are chemotype-specific.

The power of coarse graining in biomolecular simulations

Computational modeling of biological systems is challenging because of the multitude of spatial and temporal scales involved. Replacing atomistic detail with lower resolution, coarse grained (CG), beads has opened the way to simulate large-scale biomolecular processes on time scales inaccessible to all-atom models. We provide an overview of some of the more popular CG models used in biomolecular applications to date, focusing on models that retain chemical specificity. A few state-of-the-art examples of protein folding, membrane protein gating and self-assembly, DNA hybridization, and modeling of carbohydrate fibers are used to illustrate the power and diversity of current CG modeling.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

A guiding potential method for evaluating the bending rigidity of tensionless lipid membranes from molecular simulation

A new method is proposed to estimate the bending rigidity of lipid membranes from molecular dynamics simulations. An external cylindrical guiding potential is used to impose a sinusoidal deformation to a planar membrane. The bending rigidity is obtained from the mean force acting on the cylinder by calibrating against a discretized Helfrich model that accounts for thermal fluctuations of the membrane surface. The method has been successfully applied to a dimyristoyl phosphatidylcholine bilayer simulated with a coarse-grained model. A well-converged bending rigidity was obtained for the tension-free membrane and showed reasonable agreement with that obtained from the height fluctuation spectrum.

Measuring the shapes of macromolecules - and why it matters

The molecular basis of life rests on the activity of biological macromolecules, mostly nucleic acids and proteins. A perhaps surprising finding that crystallized over the last handful of decades is that geometric reasoning plays a major role in our attempt to understand these activities. In this paper, we address this connection between geometry and biology, focusing on methods for measuring and characterizing the shapes of macromolecules. We briefly review existing numerical and analytical approaches that solve these problems. We cover in more details our own work in this field, focusing on the alpha shape theory as it provides a unifying mathematical framework that enable the analytical calculations of the surface area and volume of a macromolecule represented as a union of balls, the detection of pockets and cavities in the molecule, and the quantification of contacts between the atomic balls. We have shown that each of these quantities can be related to physical properties of the molecule under study and ultimately provides insight on its activity. We conclude with a brief description of new challenges for the alpha shape theory in modern structural biology.

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.

Coarse-Grained Models for Protein-Cell Membrane Interactions

The physiological properties of biological soft matter are the product of collective interactions, which span many time and length scales. Recent computational modeling efforts have helped illuminate experiments that characterize the ways in which proteins modulate membrane physics. Linking these models across time and length scales in a multiscale model explains how atomistic information propagates to larger scales. This paper reviews continuum modeling and coarse-grained molecular dynamics methods, which connect atomistic simulations and single-molecule experiments with the observed microscopic or mesoscale properties of soft-matter systems essential to our understanding of cells, particularly those involved in sculpting and remodeling cell membranes.

Systematic methods for structurally consistent coarse-grained models

This chapter provides a primer on theories for coarse-grained (CG) modeling and, in particular, reviews several systematic methods for determining effective potentials for CG models. The chapter first reviews a statistical mechanics framework for relating atomistic and CG models. This framework naturally leads to a quantitative criterion for CG models that are "consistent" with a particular atomistic model for the same system. This consistency criterion is equivalent to minimizing the relative entropy between the two models. This criterion implies that a many-body PMF is the appropriate potential for a CG model that is consistent with a particular atomistic model. This chapter then presents a unified exposition of the theory and numerical methods for several approaches for approximating this many-body PMF. Finally, this chapter closes with a brief discussion of a few of the outstanding challenges facing the field of systematic coarse-graining.

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.

Effective tension and fluctuations in active membranes

We calculate the fluctuation spectrum of the shape of a lipid vesicle or cell exposed to a nonthermal source of noise. In particular, we take constraints on the membrane area and the volume of fluid that it encapsulates into account when obtaining expressions for the dependency of the membrane tension on the noise. We then investigate three possible origins of the nonthermal noise taken from the literature: A direct force, which models an external medium pushing on the membrane, a curvature force, which models a fluctuating spontaneous curvature, and a permeation force coming from an active transport of fluid through the membrane. For the direct force and curvature force cases, we compare our results to existing experiments on active membranes.

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.

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.

Multiscale computer simulation of the immature HIV-1 virion

Multiscale computer simulations, employing a combination of experimental data and coarse-graining methods, are used to explore the structure of the immature HIV-1 virion. A coarse-grained (CG) representation is developed for the virion membrane shell and Gag polypeptides using molecular level information. Building on the results from electron cryotomography experiments, the simulations under certain conditions reveal the existence of an incomplete p6 hexameric lattice formed from hexameric bundles of the Gag CA domains. In particular, the formation and stability of the immature Gag lattice at the CG level requires enhanced interfacial interactions of the CA protein C-terminal domains (CTDs). An exact mapping of the CG representation back to the molecular level then allows for detailed atomistic molecular dynamics studies to confirm the existence of these enhanced CA(CTD) interactions and to probe their possible origin. The multiscale simulations further provide insight into potential CA(CTD) mutations that may disrupt or modify the Gag immature lattice assembly process in the immature HIV-1 virion.

A comparative study on the ability of two implicit solvent lipid models to predict transmembrane helix tilt angles

Free-energy profiles describing the relative orientation of membrane proteins along predefined coordinates can be efficiently calculated by means of umbrella simulations. Such simulations generate reliable orientational distributions but are difficult to converge because of the very long equilibration times of the solvent and the lipid bilayer in explicit representation. Two implicit lipid membrane models are here applied in combination with the umbrella sampling strategy to the simulation of the transmembrane (TM) helical segment from virus protein U (Vpu). The models are used to study both orientation and energetics of this α-helical peptide as a function of hydrophobic mismatch. We observe that increasing the degree of positive hydrophobic mismatch increased the tilt angle of Vpu. These findings agree well with experimental data and as such validate the solvation models used in this study.

Reference state for the generalized Yvon-Born-Green theory: application for coarse-grained model of hydrophobic hydration

Coarse-grained (CG) models provide a computationally efficient means for investigating phenomena that remain beyond the scope of atomically detailed models. Although CG models are often parametrized to reproduce the results of atomistic simulations, it is highly desirable to determine accurate CG models from experimental data. Recently, we have introduced a generalized Yvon-Born-Green (g-YBG) theory for directly (i.e., noniteratively) determining variationally optimized CG potentials from structural correlation functions. In principle, these correlation functions can be determined from experiment. In the present work, we introduce a reference state potential into the g-YBG framework. The reference state defines a fixed contribution to the CG potential. The remaining terms in the potential are then determined, such that the combined potential provides an optimal approximation to the many-body potential of mean force. By specifying a fixed contribution to the potential, the reference state significantly reduces the computational complexity and structural information necessary for determining the remaining potentials. We also validate the quantitative accuracy of the proposed method and numerically demonstrate that the reference state provides a convenient framework for transferring CG potentials from neat liquids to more complex systems. The resulting CG model provides a surprisingly accurate description of the two- and three-particle solvation structures of a hydrophobic solute in methanol. This work represents a significant step in developing the g-YBG theory as a useful computational framework for determining accurate CG models from limited experimental data.