High-Dimensional Parameter Search Method to Determine Force Field Mixing Terms in Molecular Simulations

Molecular dynamics (MD) force fields for lipids and ions are typically developed independently of one another. In simulations consisting of both lipids and ions, lipid-ion interaction energies are estimated using a predefined set of mixing rules for Lennard-Jones (LJ) interactions. This, however, does not guarantee their reliability. In fact, compared to the quantum mechanical reference data, Lorentz-Berthelot mixing rules substantially underestimate the binding energies of Na⁺ ions with small-molecule analogues of lipid headgroups, yielding errors on the order of 80 and 130 kJ/mol, respectively, for methyl acetate and diethyl phosphate. Previously, errors associated with mixing force fields have been reduced using approaches such as "NB-fix" in which LJ interactions are computed using explicit cross terms rather than those from mixing rules. Building on this idea, we derive explicit lipid-ion cross terms that also may implicitly include many-body cooperativity effects. Additionally, to account for the interdependency between cross terms, we optimize all cross terms simultaneously by performing high-dimensional searches using our ParOpt software. The cross terms we obtain reduce the errors due to mixing rules to below 10 kJ/mol. MD simulation of the lipid bilayer conducted using these optimized cross terms resolves the structural discrepancies between our previous simulations and small-angle X-ray and neutron scattering experiments. These results demonstrate that simulations of lipid bilayers with ions that are accurate up to structural data from scattering experiments can be performed without explicit polarization terms. However, it is worth noting that such NB-fix cross terms are not based on any physical principle; a polarizable lipid model would be more realistic and is still desired. Our approach is generic and can be applied to improve the accuracies of simulations employing mixed force fields.

Why Class Formation Occurs in Humans but Not among Other Primates : A Primate Coalitions Model

Most human societies exhibit a distinct class structure, with an elite, middle classes, and a bottom class, whereas animals form simple dominance hierarchies in which individuals with higher fighting ability do not appear to form coalitions to "oppress" weaker individuals. Here, we extend our model of primate coalitions and find that a division into a bottom class and an upper class is inevitable whenever fitness-enhancing resources, such as food or real estate, are exploitable or tradable and the members of the bottom class cannot easily leave the group. The model predicts that the bottom class has a near flat, low payoff and always comprises at least half the society. The upper class may subdivide into one or more middle class(es), resulting in improved payoff for the topmost members (elite). The model predicts that the bottom class on its own is incapable of mounting effective counter-coalitions against the upper class, except when receiving support from dissatisfied members of the middle class(es). Such counter-coalitions can be prevented by keeping the payoff to the lowest-ranked members of the middle classes (through concessions) well above that of the bottom class. This simple model explains why classes are also absent in nomadic hunter-gatherers and predominate in (though are not limited to) societies that produce and store food. Its results also agree well with various other known features of societies with classes.

Molecular basis for higher affinity of SARS-CoV-2 spike RBD for human ACE2 receptor

Severe acute respiratory syndrome coronavirus‐2 (SARS‐CoV‐2) has caused substantially more infections, deaths, and economic disruptions than the 2002‐2003 SARS‐CoV. The key to understanding SARS‐CoV‐2's higher infectivity lies partly in its host receptor recognition mechanism. Experiments show that the human angiotensin converting enzyme 2 (ACE2) protein, which serves as the primary receptor for both CoVs, binds to the receptor binding domain (RBD) of CoV‐2's spike protein stronger than SARS‐CoV's spike RBD. The molecular basis for this difference in binding affinity, however, remains unexplained from X‐ray structures. To go beyond insights gained from X‐ray structures and investigate the role of thermal fluctuations in structure, we employ all‐atom molecular dynamics simulations. Microseconds‐long simulations reveal that while CoV and CoV‐2 spike‐ACE2 interfaces have similar conformational binding modes, CoV‐2 spike interacts with ACE2 via a larger combinatorics of polar contacts, and on average, makes 45% more polar contacts. Correlation analysis and thermodynamic calculations indicate that these differences in the density and dynamics of polar contacts arise from differences in spatial arrangements of interfacial residues, and dynamical coupling between interfacial and non‐interfacial residues. These results recommend that ongoing efforts to design spike‐ACE2 peptide blockers will benefit from incorporating dynamical information as well as allosteric coupling effects.

Transferable interactions of Li+ and Mg2+ ions in polarizable models

Therapeutic implications of Li⁺, in many cases, stem from its ability to inhibit certain Mg²⁺-dependent enzymes, where it interacts with or substitutes for Mg²⁺. The underlying details of its action are, however, unknown. Molecular simulations can provide insights, but their reliability depends on how well they describe relative interactions of Li⁺ and Mg²⁺ with water and other biochemical groups. Here, we explore, benchmark, and recommend improvements to two simulation approaches: the one that employs an all-atom polarizable molecular mechanics (MM) model and the other that uses a hybrid quantum and MM implementation of the quasi-chemical theory (QCT). The strength of the former is that it describes thermal motions explicitly and that of the latter is that it derives local contributions from electron densities. Reference data are taken from the experiment, and also obtained systematically from CCSD(T) theory, followed by a benchmarked vdW-inclusive density functional theory. We find that the QCT model predicts relative hydration energies and structures in agreement with the experiment and without the need for additional parameterization. This implies that accurate descriptions of local interactions are essential. Consistent with this observation, recalibration of local interactions in the MM model, which reduces errors from 10.0 kcal/mol to 1.4 kcal/mol, also fixes aqueous phase properties. Finally, we show that ion-ligand transferability errors in the MM model can be reduced significantly from 10.3 kcal/mol to 1.2 kcal/mol by correcting the ligand's polarization term and by introducing Lennard-Jones cross-terms. In general, this work sets up systematic approaches to evaluate and improve molecular models of ions binding to proteins.

Interactions of Monovalent and Divalent Cations at Palmitoyl-Oleoyl-Phosphatidylcholine Interface

Li⁺ is a biologically active and medically important cation. Experiments show that Li⁺ modulates some phospholipid bilayer properties in a manner similar to divalent cations, rather than other monovalent cations. We previously performed a comparative simulation study of the interaction of several monovalent cations with palmitoyl-oleoyl-phosphatidylcholine bilayers and reported that Li⁺ exhibited the highest association with lipids and formed a unique tetrahedral coordinated structure with lipid head groups. Here we extend these studies to two biologically important divalent cations, Mg²⁺ and Ca²⁺, and observe that, just like monovalent cations, Mg²⁺ and Ca²⁺ reduce bilayer areas and increase chain order. Bilayer area changes induced by cations are strongly correlated with the amount of charge inside the headgroup region; however, Mg²⁺ and Li⁺ are clear outliers. At the same time though, Mg²⁺ adsorption in the bilayer is the smallest among all cations, which is in contrast to Li⁺ that binds strongly to lipids. In fact, in contrast to all other cations, Mg²⁺ remains fully hydrated in the lipid headgroup region. However, Li⁺ and Mg²⁺ share high overlap between their inner-shell coordination topologies. This suggests that Li⁺ can structurally replace Mg²⁺, which is bound to other biomolecules with up to fourfold coordination, provided such replacement is energetically feasible. We compute structural topologies and compare them quantitatively using a new weighted-graphs-based method. Finally, we find that the specificity of cation interaction with lipid head groups exhibit consistent trend with the solvation shell energetics of ions in lipid headgroup and bulk water regions.

Interaction of salt with ether- and ester-linked phospholipid bilayers

A distinguishing feature of Archaeal plasma membranes is that their phospholipids contain ether-links, as opposed to bacterial and eukaryotic plasma membranes where phospholipids primarily contain ester-links. Experiments show that this chemical difference in headgroup-tail linkage does produce distinct differences in model bilayer properties. Here we examine the effects of salt on bilayer structure in the case of an ether-linked lipid bilayer. We use molecular dynamics simulations and compare equilibrium properties of two model lipid bilayers in NaCl salt solution - POPC and its ether-linked analog that we refer to as HOPC. We make the following key observations. The headgroup region of HOPC "adsorbs" fewer ions compared to the headgroup region of POPC. Consistent with this, we note that the Debye screening length in the HOPC system is ∼ 10% shorter than that in the POPC system. Herein, we introduce a protocol to identify the lipid-water interfacial boundary that reproduces the bulk salt distribution consistent with Gouy-Chapman theory. We also note that the HOPC bilayer has excess solvent in the headgroup region when compared to POPC, coinciding with a trough in the electrostatic potential. Waters in this region have longer autocorrelation times and smaller lateral diffusion rates compared to the corresponding region in the POPC bilayer, suggesting that the waters in HOPC are more strongly coordinated to the lipid headgroups. Furthermore, we note that it is this region of tightly coordinated waters in the HOPC system that has a lower density of Na⁺ ions. Based on these observations we conclude that an ether-linked lipid bilayer has a lower binding affinity for Na⁺ compared to an ester-linked lipid bilayer.

Molecular dynamics simulations of ether- and ester-linked phospholipids

Dissimilarities in the bulk structure of bilayers composed of ether- vs ester-linked lipids are well-established; however, the atomistic interactions responsible for these differences are not well known. These differences are important in understanding of why archaea have a different bilayer composition than the other domains of life and why humans have larger concentrations of plasmalogens in specialized membranes? In this paper, we simulate two lipid bilayers, the ester linked dipalmitoylphosphatidylcholine (DPPC) and the ether lined dihexadecylphosphatidylcholine (DHPC), to study these variations. The structural analysis of the bilayers reveals that DPPC is more compressible than DHPC. A closer examination of dipole potential shows DHPC, despite having a smaller dipole potential of the bilayer, has a higher potential barrier than DPPC at the surface. Analysis of water order and dynamics suggests DHPC has a more ordered, less mobile layer of water in the headgroup. These results seem to resolve the issue as to whether the decrease in permeability of DHPC is due to of differences in minimum area per lipid (A₀) or diffusion coefficient of water in the headgroup region (D_head) (Guler et al., 2009) since we have shown significant changes in the order and mobility of water in that region.

Effects of Lithium and Other Monovalent Ions on Palmitoyl Oleoyl Phosphatidylcholine Bilayer

Interactions of monovalent salts with lipid membranes are explored with molecular dynamics (MD) simulations. The simulations included the monovalent ions Na⁺ and K⁺, for their importance in physiology, Li⁺ for its small size and importance in several medical conditions including bipolar disorder, and Rb⁺ for its large size. All simulations included Cl^- as counterions. One bilayer was simulated without salt as a control. Palmitoyl oleoyl phosphatidylcholine (POPC) bilayers experienced reductions in area per lipid with the addition of salt; the smaller the ion the smaller the area, with the exception of Li⁺. Li⁺ exhibited unique binding affinities between phosphates and sn-2 carbonyls that lowered the order of the top part of sn-2 chain, which increased the area per lipid, compared to other ionic simulations. Further, we observe that monovalent salts alter bilayer properties through structural changes and not so much through the changes in surface potential.

DCMS: A data analytics and management system for molecular simulation

Molecular Simulation (MS) is a powerful tool for studying physical/chemical features of large systems and has seen applications in many scientific and engineering domains. During the simulation process, the experiments generate a very large number of atoms and intend to observe their spatial and temporal relationships for scientific analysis. The sheer data volumes and their intensive interactions impose significant challenges for data accessing, managing, and analysis. To date, existing MS software systems fall short on storage and handling of MS data, mainly because of the missing of a platform to support applications that involve intensive data access and analytical process. In this paper, we present the database-centric molecular simulation (DCMS) system our team developed in the past few years. The main idea behind DCMS is to store MS data in a relational database management system (DBMS) to take advantage of the declarative query interface (i.e., SQL), data access methods, query processing, and optimization mechanisms of modern DBMSs. A unique challenge is to handle the analytical queries that are often compute-intensive. For that, we developed novel indexing and query processing strategies (including algorithms running on modern co-processors) as integrated components of the DBMS. As a result, researchers can upload and analyze their data using efficient functions implemented inside the DBMS. Index structures are generated to store analysis results that may be interesting to other users, so that the results are readily available without duplicating the analysis. We have developed a prototype of DCMS based on the PostgreSQL system and experiments using real MS data and workload show that DCMS significantly outperforms existing MS software systems. We also used it as a platform to test other data management issues such as security and compression.

Atomically detailed lipid bilayer models for the interpretation of small angle neutron and X-ray scattering data

We present a new atom density profile (ADP) model and a statistical approach for extracting structural characteristics of lipid bilayers from X-ray and neutron scattering data. Models for five lipids with varying head and tail chemical composition in the fluid phase, 1,2-dioleoyl-sn-glycero-3-phosphatidylcholine (DOPC), 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine (POPC), 1,2-dipalmitoyl-sn-glycero-3-phosphatidylcholine (DPPC), 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylserine (POPS), and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylglycerol (POPG), are optimized using a simplex based method to simultaneously reproduce both neutron and X-ray scattering data. Structural properties are determined using statistical analysis of multiple optimal model structures. The method and models presented make minimal assumptions regarding the atomic configuration, while taking into account the underlying physical properties of the system. The more general model and statistical approach yield data with well defined uncertainties, indicating the precision in determining density profiles, atomic locations, and bilayer structural characteristics. Resulting bilayer structures include regions exhibiting large conformational variation. Due to the increased detail in the model, the results demonstrate the possibility of a distinct hydration layer within the interfacial (backbone) region.

Automated optimization of water-water interaction parameters for a coarse-grained model

We have developed an automated parameter optimization software framework (ParOpt) that implements the Nelder-Mead simplex algorithm and applied it to a coarse-grained polarizable water model. The model employs a tabulated, modified Morse potential with decoupled short- and long-range interactions incorporating four water molecules per interaction site. Polarizability is introduced by the addition of a harmonic angle term defined among three charged points within each bead. The target function for parameter optimization was based on the experimental density, surface tension, electric field permittivity, and diffusion coefficient. The model was validated by comparison of statistical quantities with experimental observation. We found very good performance of the optimization procedure and good agreement of the model with experiment.

The effect of cross-link distributions in axially-ordered, cross-linked networks

Cross-linking between the constituent chains of biopolymers has a marked effect on their materials' properties. In certain of these materials, such as fibrillar collagen, increases in cross-linking lead to an increase in the melting temperature. Fibrillar collagen is an axially-ordered network of cross-linked polymer chains exhibiting a broadened denaturation transition, which has been explained in terms of the successive denaturation with temperature of multiple species. We model axially-ordered, cross-linked materials as stiff chains with distinct arrangements of cross-link-forming sites. Simulations suggest that systems composed of chains with identical arrangements of cross-link-forming sites exhibit critical behavior. In contrast, systems composed of non-identical chains undergo a crossover. This model suggests that the arrangement of cross-link-forming sites may contribute to the broadening of the denaturation transition in fibrillar collagen.

Mixing properties of sphingomyelin ceramide bilayers: a simulation study

Ceramide is the simplest molecule in the class of glycosphingolipids composed of a sphingosine backbone and acyl moiety. It plays significant roles in cell signaling; apoptosis; binding of hormones, toxins, and viruses; and many other biologically important functions. Sphingomyelin, ceramide with a phosphotidylcholine headgroup, is another biologically vital lipid present in the myelin sheath of nerve cell axons. Regions with high concentrations of ceramide can be formed in biological membranes composed of sphingomyelin by enzymatic catalysis with sphingomyelinase. To better understand the biophysical and thermodynamic properties of these molecules and their mixtures, we have preformed NPT molecular dynamics simulations of hydrated 16:0 sphingomyelin bilayers with increasing concentrations of 16:0 ceramide at 323, 332, 340, and 358 K. From analyses of electron densities, hydrogen bonding, NMR order parameters, partial molecular volume, and partial molecular area, we have identified possible structural changes corresponding to liquid ordered and liquid disordered phases. These structural changes are the results of changes in intra- and intermolecular hydrogen bonds between SM and Cer molecules. Our results correspond to DSC experiments for sphingomyelin bilayer concentrations up to 50% Cer. Above 50% concentration, we observe conformational changes in the SM headgroup similar to that of the umbrella model for lipid cholesterol mixtures.

Mating strategies in primates: a game theoretical approach to infanticide

Infanticide by newly immigrated or newly dominant males is reported among a variety of taxa, such as birds, rodents, carnivores and primates. Here we present a game theoretical model to explain the presence and prevalence of infanticide in primate groups. We have formulated a three-player game involving two males and one female and show that the strategies of infanticide on the males' part and polyandrous mating on the females' part emerge as Nash equilibria that are stable under certain conditions. Moreover, we have identified all the Nash equilibria of the game and arranged them in a novel hierarchical scheme. Only in the subspace spanned by the males are the Nash equilibria found to be strict, and hence evolutionarily stable. We have therefore proposed a selection mechanism informed by adaptive dynamics to permit the females to transition to, and remain in, optimal equilibria after successive generations. Our model concludes that polyandrous mating by females is an optimal strategy for the females that minimizes infanticide and that infanticide confers advantage to the males only in certain regions of parameter space. We have shown that infanticide occurs during turbulent changes accompanying male immigration into the group. For changes in the dominance hierarchy within the group, we have shown that infanticide occurs only in primate groups where the chance for the killer to sire the next infant is high. These conclusions are confirmed by observations in the wild. This model thus has enabled us to pinpoint the fundamental processes behind the reproductive decisions of the players involved, which was not possible using earlier theoretical studies.

Self-consistent mean-field model for palmitoyloleoylphosphatidylcholine-palmitoyl sphingomyelin-cholesterol lipid bilayers

The connection between membrane inhomogeneity and the structural basis of lipid rafts has sparked interest in the lateral organization of model lipid bilayers of two and three components. In an effort to investigate anisotropic lipid distribution in mixed bilayers, a self-consistent mean-field theoretical model is applied to palmitoyloleoylphosphatidylcholine (POPC)--palmitoyl sphingomyelin (PSM)--cholesterol mixtures. The compositional dependence of lateral organization in these mixtures is mapped onto a ternary plot. The model utilizes molecular dynamics simulations to estimate interaction parameters and to construct chain conformation libraries. We find that at some concentration ratios the bilayers separate spatially into regions of higher and lower chain order coinciding with areas enriched with PSM and POPC, respectively. To examine the effect of the asymmetric chain structure of POPC on bilayer lateral inhomogeneity, we consider POPC-lipid interactions with and without angular dependence. Results are compared with experimental data and with results from a similar model for mixtures of dioleoylphosphatidylcholine, steroyl sphingomyelin, and cholesterol.

Lateral organization of complex lipid mixtures from multiscale modeling

The organizational properties of complex lipid mixtures can give rise to functionally important structures in cell membranes. In model membranes, ternary lipid-cholesterol (CHOL) mixtures are often used as representative systems to investigate the formation and stabilization of localized structural domains ("rafts"). In this work, we describe a self-consistent mean-field model that builds on molecular dynamics simulations to incorporate multiple lipid components and to investigate the lateral organization of such mixtures. The model predictions reveal regions of bimodal order on ternary plots that are in good agreement with experiment. Specifically, we have applied the model to ternary mixtures composed of dioleoylphosphatidylcholine:18:0 sphingomyelin:CHOL. This work provides insight into the specific intermolecular interactions that drive the formation of localized domains in these mixtures. The model makes use of molecular dynamics simulations to extract interaction parameters and to provide chain configuration order parameter libraries.

An improved united atom force field for simulation of mixed lipid bilayers

We introduce a new force field (43A1-S3) for simulation of membranes by the Gromacs simulation package. Construction of the force fields is by standard methods of electronic structure computations for bond parameters and charge distribution and specific volumes and heats of vaporization for small-molecule components of the larger lipid molecules for van der Waals parameters. Some parameters from the earlier 43A1 force field are found to be correct in the context of these calculations, while others are modified. The validity of the force fields is demonstrated by correct replication of X-ray form factors and NMR order parameters over a wide range of membrane compositions in semi-isotropic NTP 1 atm simulations. 43-A1-S3 compares favorably with other force fields used in conjunction with the Gromacs simulation package with respect to the breadth of phenomena that it accurately reproduces.

Multiscale simulations of heterogeneous model membranes

This review will focus on computer modeling aimed at providing insights into the existence, structure, size, and thermodynamic stability of localized domains in membranes of heterogeneous composition. Modeling the lateral organization within a membrane is problematic due to the relatively slow lateral diffusion rate for lipid molecules so that microsecond or longer time scales are needed to fully model the formation and stability of a raft in a membrane. Although atomistic simulations currently are not able to reach this scale, they can provide data on the intermolecular forces and correlations that are involved in lateral organization. These data can be used to define coarse grained models that are capable of predictions of lateral organization in membranes. In this paper, we review modeling efforts that use interaction data from MD simulations to construct coarse grained models for heterogeneous bilayers. In this review we will discuss MD simulations done with the aim of gaining the information needed to build accurate coarse-grained models. We will then review some of the coarse-graining work, emphasizing modeling that has resulted from or has a basis in atomistic simulations.

Cholesterol packing around lipids with saturated and unsaturated chains: a simulation study

The fundamental role of cholesterol in the regulation of eukaryotic membrane structure is well-established. However the manner in which atomic level interactions between cholesterol and lipids, with varying degrees of chain unsaturation and polar groups, affect the overall structure and organization of the bilayer is only beginning to be understood. In this paper we describe a series of Molecular Dynamics simulations designed to provide new insights into lipid-cholesterol interactions as a function of chain unsaturation. We have run simulations of varying concentrations of cholesterol in dipalmitoyl phosphatidylcholine (DPPC), palmitoyl-oleyol phosphatidylcholine (POPC), and dioleyol phosphatidylcholine (DOPC) bilayers. Structural analysis of the simulations reveals both atomistic and systemic details of the interactions and are presented here. In particular, we find that the minimum partial molecular area of cholesterol occurs in POPC-Chol mixtures implying the most favorable packing. Physically, this appears to be related to the fact that the two faces of the cholesterol molecule are different from each other and that the steric cross section of cholesterol molecules drops sharply near the small chain tails.

Atomistic and coarse-grained computer simulations of raft-like lipid mixtures

Computer modeling can provide insights into the existence, structure, size, and thermodynamic stability of localized raft-like regions in membranes. However, the challenges in the construction and simulation of accurate models of heterogeneous membranes are great. The primary obstacle in modeling the lateral organization within a membrane is the relatively slow lateral diffusion rate for lipid molecules. Microsecond or longer time-scales are needed to fully model the formation and stability of a raft in a membra ne. Atomistic simulations currently are not able to reach this scale, but they do provide quantitative information on the intermolecular forces and correlations that are involved in lateral organization. In this chapter, the steps needed to carry out and analyze atomistic simulations of hydrated lipid bilayers having heterogeneous composition are outlined. It is then shown how the data from a molecular dynamics simulation can be used to construct a coarse-grained model for the heterogeneous bilayer that can predict the lateral organization and stability of rafts at up to millisecond time-scales.

Lateral organization in lipid-cholesterol mixed bilayers

Interactions between lipid and cholesterol molecules in membranes play an important role in the structural and functional properties of cell membranes. Although structural properties of lipid-cholesterol mixtures have been extensively studied, an understanding of the role of cholesterol in the lateral organization of bilayers has been elusive. In this article, we propose a simple yet powerful model, based on self-consistent mean-field theory and molecular dynamics simulations, for lipid bilayers containing cholesterol. Properties predicted by our model are shown to be in excellent agreement with experimental data. Our model predicts that cholesterol induces structural changes in the bilayer through the formation of regions of ordered lipids surrounding each cholesterol molecule. We find that the "smooth" and "rough" sides of cholesterol play crucial roles in formation and distribution of the ordered regions. Our model is predictive in that input parameters are obtained from independent atomistic molecular dynamics simulations. The model and method are general enough to describe other heterogeneous lipid bilayers, including lipid rafts.

Cholesterol surrogates: a comparison of cholesterol and 16:0 ceramide in POPC bilayers

Experimental evidence indicates that, under some circumstances, "surrogate" molecules may play the same role as cholesterol in ordering membrane lipids. The simplest molecule in this class is Ceramide. In this article, we describe atomic-level molecular dynamics simulations designed to shed light on this phenomenon. We run simulations of hydrated phosphoryl-oleoyl phosphatidylcholine (POPC) bilayers containing cholesterol, and containing ceramide, in concentrations ranging from 5% to 33%. We also perform a simulation of a pure POPC bilayer to verify the simulation force fields against experimental structural data for POPC. Our simulation data are in good agreement with experimental data for the partial molecular volumes, areas, form factors, and order parameters. These simulations suggest that ceramide and cholesterol have a very similar effect on the POPC bilayer, although ceramide is less effective in inducing order in the bilayer compared with cholesterol at the same concentrations.

Imaging alcohol cue exposure in alcohol dependence using a PET 15O-H2O paradigm: results from a pilot study

Craving is a commonly used term to describe an intense desire for a substance or behaviour; however, its underlying neurobiology is not fully characterized. We have successfully used a cue exposure paradigm with functional neuro-imaging (H2 15O PET; PET, positron emission tomography) in abstinent opiate addicts. This study showed that salient cue exposure results in activation in the left anterior cingulate/mediofrontal cortex and elicited craving correlated with activity in the left orbitofrontal cortex. We therefore aimed to replicate this study in alcohol dependence to see if a similar pattern of neural activation occurred. We recruited six abstinent alcohol-dependent and six non-dependent subjects who each underwent a 12-run PET scan using H2 15O to measure changes in regional blood flow during exposure to an alcoholic drink or its visually matched non-alcoholic drink. Physiological data and subjective ratings were also recorded. Statistical parametric mapping (SPM99) was used to analyse the PET images. Compared with control subjects, abstinent alcohol-dependent subjects rated their alcohol craving higher at baseline and throughout the study, but there was no significant change in the scores in response to the cues in either group. SPM analysis across all subjects showed significant activation in the occipital cortex in response to the alcohol cue as compared with the neutral one. Analysis of the same regions that were activated in the opiate study, revealed significant increases in signal activation in the left medial prefrontal area, but only in abstinent alcohol-dependent subjects. In conclusion, in abstinent alcohol dependence we suggest that a simple cue exposure paradigm is not sufficiently powerful in functional imaging studies to determine the underlying neurobiology of subjective craving. Comparisons with the finding in opiate dependence suggest a shared region, the anterior cingulate/left medial prefrontal cortex is involved in the cue response in dependent subjects but not controls.

Molecular-dynamics simulation of a ceramide bilayer

Ceramide is the simplest lipid in the biologically important class of glycosphingolipids. Ceramide is an important signaling molecule and a major component of the strateum corneum layer in the skin. In order to begin to understand the biophysical properties of ceramide, we have carried out a molecular-dynamics simulation of a hydrated 16:0 ceramide lipid bilayer at 368 K (5 degrees above the main phase transition). In this paper we describe the simulation and present the resulting properties of the bilayer. We compare the properties of the simulated ceramide bilayer to an earlier simulation of 18:0 sphingomyelin, and we discuss the results as they relate to experimental data for ceramide and other sphingolipids. The most significant differences arise at the lipid/water interface, where the lack of a large ceramide polar group leads to a different electron density and a different electrostatic potential but, surprisingly, not a different overall "dipole potential," when ceramide is compared to sphingomyelin.

Self-consistent mean-field model based on molecular dynamics: application to lipid-cholesterol bilayers

We have developed a dynamic self-consistent mean-field model, based on molecular-dynamics simulations, to study lipid-cholesterol bilayers. In this model the lipid bilayer is represented as a two-dimensional lattice field in the lipid chain order parameters, while cholesterol molecules are represented by hard rods. The motion of rods in the system is continuous and is not confined to lattice cells. The statistical mechanics of chain ordering is described by a mean field derived from an extension of a model due to Marcelja. The time evolution of the system is governed by stochastic equations. The ensemble of chain configurations required in partition sums, and the energies of interaction, are taken from atomistic level molecular-dynamics simulations of lipid bilayers. The model allows us to simulate systems 500 nm in lateral size for 20 micros time scales, or greater. We have applied the model to dipalmitoyl-phosphatidylcholine-cholesterol (Chol) bilayers at 50 degrees C for Chol concentrations between 2% and 33%. At low concentrations of Chol (2%-4%), the model predicts the formation of isolated clusters of Chol surrounded by relatively ordered lipid chains, randomly dispersed in the disordered bilayer. With increasing Chol composition, regions of Chol-induced order begin to overlap. Starting from about 11% Chol this ordering effect becomes system wide and regions unaffected by Chol are no longer detectable. From the analysis of properties of the model we conclude that the change in lipid chain order with increasing Chol concentration is continuous over the 20-mus scale of the simulations. We also conclude that at 50 degrees C no large-scale Chol-rich and Chol-depleted coexisting phase-separated regions form at any concentration. At no point in any of the simulations do we observe a higher degree of lateral organization, such as Chol-based superlattice structures.

Simulation of the early stages of nano-domain formation in mixed bilayers of sphingomyelin, cholesterol, and dioleylphosphatidylcholine

It is known from experimental studies that lipid bilayers composed of unsaturated phospholipids, sphingomyelin, and cholesterol contain microdomains rich in sphingomyelin and cholesterol. These domains are similar to "rafts" isolated from cell membranes, although the latter are much smaller in lateral size. Such domain formation can be a result of very specific and subtle lipid-lipid interactions. To identify and study these interactions, we have performed two molecular dynamics simulations, of 200-ns duration, of dioleylphosphatidylcholine (DOPC), sphingomyelin (SM), and cholesterol (Chol) systems, a 1:1:1 mixture of DOPC/SM/Chol, and a 1:1 mixture of DOPC/SM. The simulations show initial stages of the onset of spontaneous phase-separated domains in the systems. On the simulation timescale cholesterol favors a position at the interface between the ordered SM region and the disordered DOPC region in the ternary system and accelerates the process of domain formation. We find that the smooth alpha-face of Chol preferentially packs next to SM molecules. Based on a comparative analysis of interaction energies, we find that Chol molecules do not show a preference for SM or DOPC. We conclude that Chol molecules assist in the process of domain formation and the process is driven by entropic factors rather than differences in interaction energies.

Complexation of phosphatidylcholine lipids with cholesterol

It is postulated that the specific interactions between cholesterol and lipids in biological membranes are crucial in the formation of complexes leading subsequently to membrane domains (so-called rafts). These interactions are studied in molecular dynamics simulations performed on a dipalmitoylphosphatidylcholine (DPPC)-cholesterol bilayer mixture and a dilauroylphosphatidylcholine (DLPC)-cholesterol bilayer mixture, both having a cholesterol concentration of 40 mol %. Complexation of the simulated phospholipids with cholesterol is observed and visualized, exhibiting 2:1 and 1:1 stoichiometries. The most popular complex is found to be 1:1 in the case of DLPC, whereas the DPPC system carries a larger population of 2:1 complexes. This difference in the observed populations of complexes is shown to be a result of differences in packing geometry and phospholipid conformation due to the differing tail length of the two phosphatidylcholine lipids. Furthermore, aggregation of these complexes appears to form hydrogen-bonded networks in the system containing a mixture of cholesterol and DPPC. The CH...O hydrogen bond plays a crucial role in the formation of these complexes as well as the hydrogen bonded aggregates. The aggregation and extension of such a network implies a possible means by which phospholipid:cholesterol domains form.

Sphingomyelin-cholesterol domains in phospholipid membranes: atomistic simulation

We have carried out an atomic-level molecular dynamics simulation of a system of nanoscopic size containing a domain of 18:0 sphingomyelin and cholesterol embedded in a fully hydrated dioleylposphatidylcholine (DOPC) bilayer. To analyze the interaction between the domain and the surrounding phospholipid, we calculate order parameters and area per molecule as a function of molecule type and proximity to the domain. We propose an algorithm based on Voronoi tessellation for the calculation of the area per molecule of various constituents in this ternary mixture. The calculated areas per sphingomyelin and cholesterol are in agreement with previous simulations. The simulation reveals that the presence of the liquid-ordered domain changes the packing properties of DOPC bilayer at a distance as large as approximately 8 nm. We calculate electron density profiles and also calculate the difference in the thickness between the domain and the surrounding DOPC bilayer. The calculated difference in thickness is consistent with data obtained in atomic force microscopy experiments.

Mixed bilayer containing dipalmitoylphosphatidylcholine and dipalmitoylphosphatidylserine: lipid complexation, ion binding, and electrostatics

Two mixed bilayers containing dipalmitoylphosphatidylcholine and dipalmitoylphosphatidylserine at a ratio of 5:1 are simulated in NaCl electrolyte solutions of different concentration using the molecular dynamics technique. Direct NH.O and CH.O hydrogen bonding between lipids was observed to serve as the basis of interlipid complexation. It is deduced from our results and previous studies that dipalmitoylphosphatidylcholine alone is less likely to form interlipid complexes than in the presence of bound ions or other bilayer "impurities" such as dipalmitoylphosphatidylserine. The binding of counterions is observed and quantitated. Based upon the calculated ion binding constants, the Gouy-Chapman surface potential (theta) is calculated. In addition we calculated the electrostatic potential profile (Phi) by twice integrating the system charge distribution. A large discrepancy between and the value of Phi at the membrane surface is observed. However, at "larger" distance from the bilayer surface, a qualitative similarity in the z-profiles of Phi and psi(GC) is seen. The discrepancy between the two potential profiles near the bilayer surface is attributed to the discrete and nonbulk-like nature of water in the interfacial region and to the complex geometry of this region.

Molecular dynamics simulation of a dipalmitoylphosphatidylcholine bilayer with NaCl

Molecular dynamics simulations are performed on two hydrated dipalmitoylphosphatidylcholine bilayer systems: one with pure water and one with added NaCl. Due to the rugged nature of the membrane/electrolyte interface, ion binding to the membrane surface is characterized by the loss of ion hydration. Using this structural characterization, binding of Na(+) and Cl(-) ions to the membrane is observed, although the binding of Cl(-) is seen to be slightly weaker than that of Na(+). Dehydration is seen to occur to a different extent for each type of ion. In addition, the excess binding of Na(+) gives rise to a net positive surface charge density just outside the bilayer. The positive density produces a positive electrostatic potential in this region, whereas the system without salt shows an electrostatic potential of zero.

Shape anisotropy of lipid molecules and voids

Biological polymers, viz., proteins, membranes and micelles exhibit structural discontinuities in terms of spaces unfilled by the polymeric phase, termed voids. These voids exhibit dynamics and lead to interesting properties which are experimentally demonstrable. In the specific case of phospholipid membranes, numerical simulations on a two-dimensional model system showed that voids are induced primarily due to the shape anisotropy in binary mixtures of interacting disks. The results offer a minimal description required to explain the unusually large permeation seen in liposomes made up of specific lipid mixtures (Mathai & Sitaramam, 1994). The results are of wider interest, voids being ubiquitous in biopolymers.

Molecular dynamics simulation of dipalmitoylphosphatidylserine bilayer with Na+ counterions

We performed a molecular dynamics simulation of dipalmitoylphosphatidylserine (DPPS) bilayer with Na+ counterions. We found that hydrogen bonding between the NH group and the phosphate group leads to a reduction in the area per headgroup when compared to the area in dipalmitoylphosphatidylcholine bilayer. The Na+ ions bind to the oxygen in the carboxyl group of serine, thus giving rise to a dipolar bilayer similar to dipalmitoylphosphatidylethanolamine bilayer. The results of the simulation show that counterions play a crucial role in determining the structural and electrostatic properties of DPPS bilayer.

Random spread on the family of small-world networks

We present analytical and numerical results of a random walk on the family of small-world graphs. The average access time shows a crossover from regular to random behavior with increasing distance from the starting point of the random walk. We introduce an independent step approximation, which enables us to obtain analytic results for the average access time. We observe a scaling relation for the average access time in the degree of the nodes. The behavior of the average access time as a function of p shows striking similarity with that of the characteristic length of the graph. This observation may have important applications in routing and switching in networks with a large number of nodes.

Characterization and control of small-world networks

Recently, Watts and Strogatz [Nature (London) 393, 440 (1998)] offered an interesting model of small-world networks. Here we concretize the concept of a "faraway" connection in a network by defining a far edge. Our definition is algorithmic and independent of any external parameters such as topology of the underlying space of the network. We show that it is possible to control the spread of an epidemic by using the knowledge of far edges. We also suggest a model for better product advertisement using the far edges. Our findings indicate that the number of far edges can be a good intrinsic parameter to characterize small-world phenomena.