Monte Carlo calculations of order parameter profiles in models of lipid-protein interactions in bilayers

The Monte Carlo method has been utilized to calculate lipid chain order parameters in model monomolecular layers (half-bilayers) containing several different model polypeptides. The systems all consist of a periodic array of identical cells, each containing 35 hydrocarbon chains and 1 "perturbant" (a small model polypeptide or protein). The lipid chains are each 10 CH2 subunits long, have one end constrained to lie in the bilayer plane, and interact via van der Waals forces between all subunits. The chains also interact with the perturbant via van der Waals forces. With standard Monte Carlo procedures order parameter profiles are calculated for chains that are close to the perturbant and for the nonneighboring chains. In order to examine a wide range of possibilities, several different model polypeptides are considered: (i) a rigid smooth cylinder, (ii) a cylinder with identical side chains at alpha-helical positions, (iii) a cylinder with nonidentical side chains at alpha-helical positions, and (iv) a cylinder identical with (ii) but which only extends about halfway through the monolayer. Although results differ for the different systems studied, in all cases only slight conformational differences between the bulk chains and the chains that are nearest the perturbants are found, and it is not possible to characterize the boundary chains as "more ordered" or "less ordered" than the nonboundary chains.     

Lipid-cholesterol interactions. Monte Carlo simulations and theory.

Results of Monte Carlo calculations of order parameter profiles of lipid chains interacting with cholesterol are presented. Cholesterol concentrations in the simulations are sufficiently large that it is possible to analyze profiles for chains which are near neighbors of two or more cholesterol molecules, chains which are neighbors to a single cholesterol, and chains which are not near any cholesterol molecules. The profiles, show that cholesterol acts to significantly decrease the ability of neighboring chains to undergo trans-gauche isomeric rotations, although these chains are not all forced into all-trans conformations. The effect is significantly greater for chains which are neighbors to more than one cholesterol. The Monte Carlo results are next used as a guide to develop a theoretical model for lipid-cholesterol mixtures. The properties of this model and the phase diagram which it predicts are described. The phase diagram is then compared with experimentally determined phase diagrams. The model calculations and the computer simulations upon which they are based yield a molecular mechanism for several of the observed phases exhibited by lipid-cholesterol mixtures. The theoretical model predicts that at low temperatures the system should exhibit solid phase immiscibility.     

Lipid chains and cholesterol in model membranes: a Monte Carlo Study

The Monte Carlo method has been employed to study the equilibrium properties of a planar array of hydrocarbon chains interacting with a cholesterol molecule. The chains are arranged to model one monolayer of a lipid bilayer and within this monolayer are allowed to move laterally and change conformations by gauche rotations. In the simulation cell there are 90 lipid chains and a single cholesterol molecule. Periodic boundary conditions are imposed upon the cell. The primary results of the calculations are order parameter profiles for the C-C bonds. These are calculated for (i) all chains, (ii) the 6 chains which are nearest neighbors to the cholesterol, and (iii) the 12 chains which are next-nearest neighbors to the cholesterol. Calculations are carried out for C-14, C-16, and C-18 chains. The results show that cholesterol strongly affects the upper portions of the chains, leaving them less able to change conformations. For C-16 and C-18 chains, the chain termini of the cholesterol neighbors are more disordered than the bulk chain termini. The magnitude of the effect depends strongly on the chain length. The results suggest that the changes in the lipid phase transition caused by cholesterol are a consequence of each cholesterol hindering the rotameric freedom of five to seven lipid chains.     

Monte Carlo studies of lipid chains and gramicidin A in a model membrane

The Monte Carlo method has been used to simulate the equilibrium properties of a planar array of 94 saturated lipid chains and one monomer of Gramicidin A. Chains are free to move laterally in the layer plane and to change conformation via gauche rotations and long axis rotations in a continuum. All non-hydrogen atoms on chains and on the Gramicidin A monomer interact via 6-12 potentials, and periodic boundary conditions are imposed. Calculated results consist of order parameter profiles for C-14 and C-16 chains. Profiles are calculated for chains which are neighbors to the Gramicidin A molecule and for chains which are not neighbors to the peptide. The main conclusion is that the average conformations of the chains neighboring the Gramicidin A monomer are very similar to those of the bulk chains.     

Monte Carlo studies of phospholipid lamellae. Effects of proteins, cholesterol, bilayer curvature, and lateral mobility on order parameters

In this paper we present the results of a Monte Carlo study of the effects of protein, cholesterol, bilayer curvature, and mobility on the chain order parameters of a lipid layer. The Monte Carlo method used is identical to the version developed earlier (Scott, Jr., H.L. (1977) Biochim. Biophys. Acta 469, 264–271). Simulations of protein and cholesterol effects are accomplished by insertion of a rigid stationary cylinder into the lipid matrix. The protein studies show the presence of boundary lipid (Jost, P., Griffith, O.H., Capaldi, R.H. and Vanderkooi, G. (1973) Biochim. Biophys. Acta 311, 141–152). The effect of cholesterol is dependent upon the length of the lipid hydrocarbon chains relative to the cholesterol depth of penetration. Our computer studies of bilayer curvature show the manner in which this curvature disrupts chain packing and are consistent with experimental results (Chrzeszczyk, A., Wishnia, A. and Springer, C.S. (1977) Biochim. Biophys. Acta, 470, 161–171). We also find that restricting lateral motion in chains, the simplest manner in which head group interactions can affect hydrocarbon chain order, does not measurably alter the order parameters. We argue that this provides some support for an earlier hypothesis by Scott (Scott, Jr., H.L. (1975) Biochim. Biophys. Acta 406, 329–346) regarding head group-chain interaction in monolayer experiments.     

Monte Carlo Simulation of Tetrahedral Chains II: Properties of “First Self-avoiding Walks” and their Usability as Starting Configurations for Dynamic Relaxation Mechanisms

Very long model chains may be produced in a highly efficient manner using dynamic Monte Carlo methods. As any dynamic Monte Carlo procedure transforms one chain into another one, some starting configuration is necessary. This might be an unbiased self-avoiding walk (SAW) obtained by any static method, or an arbitrary configuration, e.g. a rodlike chain, equilibrated by a sufficiently large number of relaxations, the corresponding chains not being used for data sampling. An alternative method is to start with a non reversal random walk (NRRW) and to apply a dynamic Monte Carlo procedure under the constraint that the new chain must have a smaller (or at least an equal) number of double occupancies than the old one. The properties of those chains that are free of overlaps for the first time (FSAWs) are strongly dependent on the relaxation mechanism chosen. Whereas FSAWs obtained by local motions are very similar to the (initial) NRRWs on a macroscopic scale, pivot algorithms and reptation yield configurations with properties comparable to unbiased self-avoiding chains. When reptation is used and the relaxation is continued until each bond of the initial NRRW is replaced by a new bond (if the chain is self-avoiding earlier) no further equilibration is necessary prior to data sampling.     

A rigorous mathematical treatment for the excluded volume effect in Monte Carlo simulations of polymeric chains

In Monte Carlo simulations of polymeric chains, the chains are most often represented as spheres, or cylinders with flat ends. In this methodological paper, we adopt a representation of the chains as spherocylinders (continuous cylinders ending in semispheres). With such a representation the testing for chain overlap, which is the crucial step for the inclusion of the excluded volume effect in the simulations, can be defined in a rigorous geometrical framework. The treatment we then derive fulfills the following features: it allows a very simple, automatic, and exhaustive classification of all the possible configurations; and it provides a physical representation for steric hindrance effects more natural than the flat-ended cylinders. Notably, this representation avoids the introduction of artificial anisotropies in the treatments. This spherocylindrical representation is also well suited for several types of calculations that can be involved in elaborate Monte Carlo simulations.     

Experimental and Monte Carlo simulation studies of the thermodynamics of polyethyleneglycol chains grafted to lipid bilayers.

Experimental measurements of the affinity of binding of fluorescent acylated polyethyleneglycol (PEG) conjugates to bilayers containing varying levels of phosphatidylethanolamine-PEGs (PE-PEGs) have been combined with Monte Carlo simulations to investigate the properties of the polymer chains at a PEG-grafted lipid interface. The affinity of binding of such conjugates to large unilamellar phosphatidylcholine/phosphatidylethanolamine (9:1) vesicles decreases 27-fold as the size of the coupled PEG chain increases from 1 to 114 monomer units. Incorporation of increasing amounts of PE-PEG2000 or PE-PEG5000 into the vesicles progressively reduces the affinity of binding of acylpeptide-PEG2000 or -PEG5000 conjugates. Monte Carlo simulations of surfaces with grafted PEG chains revealed no significant dependence of several characteristic properties of the polymer chains, including the average internal energy per polymer and the radii of gyration, on the grafting density in the range examined experimentally. The average conformation of a surface-grafted PEG2000 or PEG5000 chain was calculated to be fairly extended even at low grafting densities, and the projected cross-sectional areas of the grafted PEG chains are considerably smaller than those predicted on the basis of the estimated Flory radius. The experimental variation of the binding affinity of acylated conjugates for bilayers containing varying mole fractions of PE-PEG2000 or -PEG5000 is well explained by expressions treating the surface-grafted PEG polymers either as a van der Waals gas or as a system of rigid discs described by scaled particle theory. From the combined results of our experimental and simulation studies we conclude that the grafted PEG chains exist in a "mushroom" regime throughout the range of polymer densities examined experimentally and that the diminished affinity of binding of acylated-PEG conjugates to bilayers containing PE-PEGs results from occlusion of the surface area accessible for conjugate binding by the mobile PE-PEG polymer chains.     

Temperature dependence of conformational properties of natural lipid oligomer chains: computer simulation

The conformational properties of several oligomeric chain molecules at T = 278 and T = 403 K have been studied using Monte Carlo computer simulations. Hydrocarbon oligomers with methylene-interrupted cis double bonds in the main chain were considered. These oligomers are typical constituents of natural lipid molecules. The characteristics of the shape of C-H and C-C bond orientation distribution functions with respect to the principal axis of inertia of the chains and their temperature dependences were studied. It was found that the temperature sensitivity of not only the common geometric characteristics of the polyunsaturated chain is significantly reduced compared with the saturated one, but also that of local characteristics, i. e., the shape of each bond orientation distribution function of the polyunsaturated chain. The relationship between the properties of lipid polyunsaturated hydrocarbon chains and their functions in natural membrane systems, in particular their possible role in the stabilization or optimization of lipid-protein interactions, was discussed.     

Monte carlo study of the percolation in two-dimensional polymer systems

The structure of a two-dimensional film formed by adsorbed polymer chains was studied by means of Monte Carlo simulations. The polymer chains were represented by linear sequences of lattice beads and positions of these beads were restricted to vertices of a two-dimensional square lattice. Two different Monte Carlo methods were employed to determine the properties of the model system. The first was the random sequential adsorption (RSA) and the second one was based on Monte Carlo simulations with a Verdier-Stockmayer sampling algorithm. The methodology concerning the determination of the percolation thresholds for an infinite chain system was discussed. The influence of the chain length on both thresholds was presented and discussed. It was shown that the RSA method gave considerably lower thresholds for longer chains. This behavior can be explained by a different pool of chain conformations used in the calculations in both methods under consideration.

Parallel Markov chain Monte Carlo - bridging the gap to high-performance Bayesian computation in animal breeding and genetics

Background

Most Bayesian models for the analysis of complex traits are not analytically tractable and inferences are based on computationally intensive techniques. This is true of Bayesian models for genome-enabled selection, which uses whole-genome molecular data to predict the genetic merit of candidate animals for breeding purposes. In this regard, parallel computing can overcome the bottlenecks that can arise from series computing. Hence, a major goal of the present study is to bridge the gap to high-performance Bayesian computation in the context of animal breeding and genetics.

Results

Parallel Monte Carlo Markov chain algorithms and strategies are described in the context of animal breeding and genetics. Parallel Monte Carlo algorithms are introduced as a starting point including their applications to computing single-parameter and certain multiple-parameter models. Then, two basic approaches for parallel Markov chain Monte Carlo are described: one aims at parallelization within a single chain; the other is based on running multiple chains, yet some variants are discussed as well. Features and strategies of the parallel Markov chain Monte Carlo are illustrated using real data, including a large beef cattle dataset with 50K SNP genotypes.

Conclusions

Parallel Markov chain Monte Carlo algorithms are useful for computing complex Bayesian models, which does not only lead to a dramatic speedup in computing but can also be used to optimize model parameters in complex Bayesian models. Hence, we anticipate that use of parallel Markov chain Monte Carlo will have a profound impact on revolutionizing the computational tools for genomic selection programs.     

Excluded volume of an intermediate-molecular-weight DNA. A Monte Carlo analysis

A Monte Carlo procedure wasused to determine the effect of excluded volume on the dimensions of an intermediate-molecular-weight DNA for different Na⁺ concentrations. The calculation of α, the parameter for the linear expansion due to excluded volume, was accomplished by generating sets of chains and, for each set, comparing the average radius of gyration for the set of chains that do not overlap to that averaged over the entire set of chains. Each chain was defined by cylinders linked with free rotation and with bend angles generated according to a weighted Gaussian distribution. The chain parameters—contour lenght, cylinder lenght and diameter—were fixed in order to resemble published light-scattering experiments on Col E₁ DNA. Values for α were less than 1.08. for Na⁺ concentrations between 0.007 and 1.0M. A previously reported analytical calculation of the excluded-volume correction of intermediate-sized DNA gave results that are closely similar to those from the Monte Carlo analysis.     

Monte Carlo simulation of protein folding in the presence of residue-specific binding sites

Ensemble growth Monte Carlo (EGMC) and dynamic Monte Carlo (DMC) simulations are used to study sequential folding and thermodynamic stability of hydrophobic-polar (HP) chains that fold to a compact structure. Molecularly imprinted cavities are modeled as hard walls having sites that are attractive to specific polar residues on the chain. Using EGMC simulation, we find that the folded conformation can be stabilized using a small number of carefully selected residue-specific sites while a random selection of surface-bound residues may only slightly contribute toward stabilizing the folded conformation, and in some cases may hinder the folding of the chain. DMC simulations of the surface-bound chain confirm increased stability of the folded conformation over a free chain. However, a different trend of the equilibrium population of folded chains as a function of residue-external site interactions is predicted with the two simulation methods.     

Melting of individual lipid components in binary lipid mixtures studied by FTIR spectroscopy, DSC and Monte Carlo simulations

Monte Carlo (MC) simulations, Differential Scanning Calorimetry (DSC) and Fourier Transform InfraRed (FTIR) spectroscopy were used to study the melting behavior of individual lipid components in two-component membranes made of DMPC and DSPC. We employed Monte Carlo simulations based on parameters obtained from DSC profiles to simulate the melting of the different lipids as a function of temperature. The simulations show good agreement with the FTIR data recorded for deuterated and non-deuterated lipids, which demonstrates that the information on the differential melting of the individual components is already contained in the calorimetric profiles. In mixtures, both lipids melt over a wide temperature range. As expected, the lipid melting events of the lipid with the lower melting temperature occur on average at lower temperatures. The simulations also yield information on the lateral distribution of the lipids that is neither directly contained in the DSC nor in the FTIR data. In the phase coexistence region, liquid disordered domains are typically richer in the lower-melting-temperature lipid species.     

A Monte Carlo study of the amylosic chain conformation

Monte Carlo studies of the unperturbed amylosic chain conformation have been carried out in the approximation of separable chain configuration energies. Sample chains of arbitrary chain length have been generated so as to be distributed consistent with refined estimates of the configuration energy and thus suitable for evaluation of averages of the desired configuration-dependent properties. Perspective drawings of representative chains from the Monte Carlo sample have been made for comparison with standard idealizations of amylosic chain conformation. He molecular model employed generates a randomly coiling chain possessing perceptible regions of left-handed pseudohelical backbone trajectory. Distribution functions for the end-to-end distance of short amylosic chains disclose some propensity for the chain to suffer self-intersections at sort range in the chain sequence, which may vitiate the usual amylosic chain models based on the assumed independence of sets of glycosidic linkage torsion angles. The amylosic persistence vector and persistence length have been calculated as a function of chain length for the chain model employed.     

General model for lipid-mediated two-dimensional array formation of membrane proteins: application to bacteriorhodopsin.

Based on experimental evidence for 2D array formation of bacteriorhodopsin, we propose a general model for lipid-mediated 2D array formation of membrane proteins in lipid bilayers. The model includes two different lipid species, "annular" lipids and "neutral" lipids, and one protein species. The central assumption of the model is that the annular lipids interact more strongly with the protein than with the neutral lipids. Monte Carlo simulations performed on this model show that 2D arrays of proteins only form when there are annular lipids present. In addition, no arrays form if all of the lipids present are annular lipids. The geometry of the observed arrays is for the most part hexagonal. However, for a certain range of low annular lipid/protein ratios, arrays form that have geometries other than hexagonal. Using the assumption that the hydrocarbon chains of the annular lipids are restricted in motion when close to a protein, we expand the model to include a ground state and an excited state of the annular lipids. The main result from the extended model is that within a certain temperature range, increasing the temperature will lead to larger and more regular protein arrays.     

Monte Carlo study of cycloamylose: chain conformation, radius of gyration, and diffusion coefficient

Cyclic (1 --> 4)-alpha-D-glucan chains with or without excluded volume have been collected from a huge number (about 10(7)) of linear amylosic chains generated by the Monte Carlo method with a conformational energy map for maltose, and their mean-square radii of gyration and translational diffusion coefficients D (based on the Kirkwood formula) have been computed as functions of x (the number of glucose residues in a range from 7 to 300) and the excluded-volume strength represented by the effective hard-core radius. Both /x and D in the unperturbed state weakly oscillate for x < 30 and the helical nature of amylose appears more pronouncedly in cyclic chains than in linear chains. As x increases, these properties approach the values expected for Gaussian rings. Though excluded-volume effects on them are always larger in cycloamylose than in the corresponding linear amylose, the ratios of and the hydrodynamic radius of the former to the respective properties of the latter in good solvents can be slightly lower than or comparable to the (asymptotic) Gaussian-chain values when x is not sufficiently large. An interpolation expression is constructed for the relation between the gyration-radius expansion factors for linear and cyclic chains from the present Monte Carlo data and the early proposed asymptotic relation with the aid of the first-order perturbation theories.     

Ring closure probabilities for DNA fragments by Monte Carlo simulation

The rate of ligation of DNA molecules into circular forms depends on the ring closure probability, commonly called the j-factor, which is a sensitive measure of the extent to which thermal fluctuations contribute to bending and twisting of DNA molecules in solution. We present a theoretical treatment of the cyclization equilibria of DNA that employs a special Monte Carlo method for generating large ensembles of model DNA chains. Using this method, the chain length dependence of the j-factor was calculated for molecules. in the size range 250 to 2000 base-pairs. The Monte Carlo results are compared with recent analytical theory and experimental data. We show that a value of 475 A for the persistence length of DNA, close to values measured by a number of other methods, is in excellent agreement with the cyclization results. Preliminary applications of the Monte Carlo method to the problem of systematically bent DNA molecules are presented. The calculated j-factor is shown to be very sensitive to the amount of bending in these fragments. This fact suggests that ligase closure measurements of systematically bent DNA molecules should be a useful method for studying sequence-directed bending in DNA.     

A theoretical model for the association of amphiphilic transmembrane peptides in lipid bilayers

A theoretical model is proposed for the association of trans-bilayer peptides in lipid bilayers. The model is based on a lattice model for the pure lipid bilayer, which accounts accurately for the most important conformational states of the lipids and their mutual interactions and statistics. Within the lattice formulation the bilayer is formed by two independent monolayers, each represented by a triangular lattice, on which sites the lipid chains are arrayed. The peptides are represented by regular objects, with no internal flexibility, and with a projected area on the bilayer plane corresponding to a hexagon with seven lattice sites. In addition, it is assumed that each peptide surface at the interface with the lipid chains is partially hydrophilic, and therefore interacts with the surrounding lipid matrix via selective anisotropic forces. The peptides would therefore assemble in order to shield their hydrophilic residues from the hydrophobic surroundings. The model describes the self-association of peptides in lipid bilayers via lateral and rotational diffusion, anisotropic lipid-peptide interactions, and peptide-peptide interactions involving the peptide hydrophilic regions. The intent of this model study is to analyse the conditions under which the association of trans-bilayer and partially hydrophilic peptides (or their dispersion in the lipid matrix) is lipid-mediated, and to what extent it is induced by direct interactions between the hydrophilic regions of the peptides. The model properties are calculated by a Monte Carlo computer simulation technique within the canonical ensemble. The results from the model study indicate that direct interactions between the hydrophilic regions of the peptides are necessary to induce peptide association in the lipid bilayer in the fluid phase. Furthermore, peptides within each aggregate are oriented in such a way as to shield their hydrophilic regions from the hydrophobic environment. The average number of peptides present in the aggregates formed depends on the degree of mismatch between the peptide hydrophobic length and the lipid bilayer hydrophobic thickness: The lower the degree of mismatch is the higher this number is. Received: 30 December 1996 / Accepted: 9 May 1997