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.     

A Monte Carlo study of the excluded-volume effect on the amylosic chain conformation

A Monte Carlo procedure was used to determine the effect of excluded volume on the conformational dimensions of amylosic chains. The excluded volume was introduced into the model by assuming that hard spheres, which cannot overlap each other, exist at the center of mass of each glucose unit in the chain sequence. Monte Carlo chains, which were generated to be distributed consistent with the potential energy of nonbonded nearest-neighbor interactions, underwent self-intersections, and the attrition rate in the generation of self-avoiding chains was found to obey an exponential decay law with increasing chain length x. Thus, in order to generate effectively a large number of self-avoiding chains with long sequences, we used the Wall–Erpenbeck s-p method of chain enrichment [F. T. Wall and J. J. Erpenbeck (1959) J. Chem. Phys. 30 , 634–637]. By examination of the radial distribution of the end-to-end distance and the chain-length dependence of the quantity 〈r²〉/xl² (〈r²〉 is the mean square end-to-end distance and l is the virtual bond length), it was found that unperturbed amylosic chains change in overall conformation from a non-Gaussian chain having a helical character to Gaussian as x is increased, whereas perturbed chains do not show Gaussian behavior even at x = 500. For the perturbed chains, 〈r²〉 can be expressed by the equation 〈r²〉 = ax^b in the range of 100 ≤ x ≤ 500, where a and b > 1 are constants. From comparisons of the persistence vectors and perspective drawings of representative unperturbed and perturbed chains, we felt the local conformation of the amylosic chains, i.e., the local helical character, is also affected by the long-range excluded-volume interaction.     

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.     

Modeling Multivalent Ligand-Receptor Interactions with Steric Constraints on Configurations of Cell-Surface Receptor Aggregates

We use flow cytometry to characterize equilibrium binding of a fluorophore-labeled trivalent model antigen to bivalent IgE-FcεRI complexes on RBL cells. We find that flow cytometric measurements are consistent with an equilibrium model for ligand-receptor binding in which binding sites are assumed to be equivalent and ligand-induced receptor aggregates are assumed to be acyclic. However, this model predicts extensive receptor aggregation at antigen concentrations that yield strong cellular secretory responses, which is inconsistent with the expectation that large receptor aggregates should inhibit such responses. To investigate possible explanations for this discrepancy, we evaluate four rule-based models for interaction of a trivalent ligand with a bivalent cell-surface receptor that relax simplifying assumptions of the equilibrium model. These models are simulated using a rule-based kinetic Monte Carlo approach to investigate the kinetics of ligand-induced receptor aggregation and to study how the kinetics and equilibria of ligand-receptor interaction are affected by steric constraints on receptor aggregate configurations and by the formation of cyclic receptor aggregates. The results suggest that formation of linear chains of cyclic receptor dimers may be important for generating secretory signals. Steric effects that limit receptor aggregation and transient formation of small receptor aggregates may also be important.     

Monte Carlo generation of polypeptide random coil conformations; the persistence vector and the chain vector distribution

Polypeptide random coil conformations of various chain lenghts (N = 5, 10, 20, 40, 80 peptide units) are generated by a Monte Carlo procedure. The characteristic ratio obtained for the sets of generated conformations is identical with the exact value calculated with the average transformation matrix procedure, indicating the equivalence of the two treatments. On the basic of the generated sets of conformations the length and direction of the persistence vector (the averaged chain vector expressed in the reference frame of the first two skeletal bonds) are investigated for various chain lengths. The radial distribution function for the chain vector shows the length of the chain vector for small polypeptides (N = 5, 10) not to deviate far from its most probable value. Also for larger chains up to chains of 80 peptide units very significant deviations from a gaussian distribution are observed.The distribution of the length of the vector connecting the remote end of the chain with the end of the persistence vector exhibited behavior much doser to the gaussian approximation, an improvement especially significant for the short chains.     

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.     

A general synthesis of long chain ω- and (ω-1)-hydroxy fatty acids

A method for the synthesis of long chain fatty acids substituted at the ω and ω-1 positions has been developed. The key step is the isomerization of the triple bond of an alkyn-1-ol from an internal position in the chain to the free terminus with a new, convenient reagent, sodium aminopropylamide (NaAPA). Standard functional group manipulations i.e., Jones oxidation, esterification and hydroboration of the triple bond are used to prepare ω-hydroxy fatty esters. The generality of the method is illustrated with syntheses of ω-hydroxy fatty esters with 24, 26, 28 and 30 carbon chains.In the 24 carbon series, hydration of the terminal triple bond of alkynoic ester 4a followed by reduction gave the (ω-1)-hydroxy ester.     

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.

The crystal structure of cholesteryl 17-bromoheptadecanoate

Crystals of cholesteryl-17-bromoheptadecanoate (C₄₄H₇₇BrO₂) are monoclinic (P2₁) with a = 7.663(2), b = 10.311(5), c = 55.96(2) A and β = 103.10(3°). These are two molecules in the asymmetric unit which have different conformations of the cholesterol side chain and about the ester bond. The molecules pack with regions of only steroid skeleta alternating with regions of hydrocarbon chains. Due to the packing requirements of the skeleta the carbon chains are forced into a hybrid type packing which contains features of the earlier known O⊥ and T∥ subcells. The subcell (HS1) is orthorhombic with a_s = 10.3, b_s = 7.5 and c_s = 2.54A. The molecular packing is such that the ω-bromine atoms do not continue the trans-carbon chains but adopt a gauche conformation.     

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.     

pK(a) values for the unfolded state under native conditions explain the pH-dependent stability of PGB1

Understanding the role of electrostatics in protein stability requires knowledge of these interactions in both the folded and unfolded states. Electrostatic interactions can be probed experimentally by characterizing ionization equilibria of titrating groups, parameterized as pK_a values. However, pK_a values of the unfolded state are rarely accessible under native conditions, where the unfolded state has a very low population. Here, we report pK_a values under nondenaturing conditions for two unfolded fragments of the protein G B1 domain that mimic the unfolded state of the intact protein. pK_a values were determined for carboxyl groups by monitoring their pH-dependent ¹³C chemical shifts. Monte Carlo simulations using a Gaussian chain model provide corrections for changes in electrostatic interactions that arise from fragmentation of the protein. Most pK_a values for the unfolded state agree well with model values, but some residues show significant perturbations that can be rationalized by local electrostatic interactions. The pH-dependent stability was calculated from the experimental pK_a values of the folded and unfolded states and compared to experimental stability data. The use of experimental pK_a values for the unfolded state results in significantly improved agreement with experimental data, as compared to calculations based on model data alone.     

Side-chain entropy and packing in proteins.

What role does side-chain packing play in protein stability and structure? To address this question, we compare a lattice model with side chains (SCM) to a linear lattice model without side chains (LCM). Self-avoiding configurations are enumerated in 2 and 3 dimensions exhaustively for short chains and by Monte Carlo sampling for chains up to 50 main-chain monomers long. This comparison shows that (1) side-chain degrees of freedom increase the entropy of open conformations, but side-chain steric exclusion decreases the entropy of compact conformations, thus producing a substantial entropy that opposes folding; (2) there is a side-chain “freezing” or ordering, i.e., a sharp decrease in entropy, near maximum compactness; and (3) the different types of contacts among side chains (s) and main-chain elements (m) have different frequencies, and the frequencies have different dependencies on compactness. mm contacts contribute significantly only at high densities, suggesting that main-chain hydrogen bonding in proteins may be promoted by compactness. The distributions of mm, ms, and ss contacts in compact SCM configurations are similar to the distributions in protein structures in the Brookhaven Protein Data Bank. We propose that packing in proteins is more like the packing of nuts and bolts in a jar than like the pairwise matching of jigsaw puzzle pieces.     

Monte Carlo Simulation of Polyampholyte Chains

Abstract

Polyampholyte copolymers containing both positive and negative monomers regularly dispersed along the chain were studied. The Monte Carlo method was used to simulate chains with charged monomers interacting by screened Coulomb potential. The neutral polyampholyte chains collapse due to the attractive electrostatic interactions. The nonneutral chains are in extended conformations due to the repulsive polyelectrolyte effects that dominate the attractive polyampholyte interactions. The results are in good agreement with experiment.     

A Monte Carlo method for generating structures of short single-stranded DNA sequences

A Monte Carlo method has been developed for generating the conformations of short single-stranded DNAs from arbitrary starting states. The chain conformers are constructed from energetically favorable arrangements of the constituent mononucleotides. Minimum energy states of individual dinucleotide monophosphate molecules are identified using a torsion angle minimizer. The glycosyl and acyclic backbone torsions of the dimers are allowed to vary, while the sugar rings are held fixed in one of the two preferred puckered forms. A total of 108 conformationally distinct states per dimer are considered in this first stage of minimization. The torsion angles within 5 kcal/mole of the global minimum in the resulting optimized states are then allowed to vary by ±10° in an effort to estimate the breadth of the different local minima. The energies of a total of 2187 (3⁷) angle combinations are examined per local conformational minimum. Finally, the energies of all dinucleotide conformers are scaled so that the populations of differently puckered sugar rings in the theoretical sample match those found in nmr solution studies. This last step is necessitated by limitations in the theoretical methods to predict DNA sugar puckering accurately. The conformer populations of the individual acyclic torsion angles in the composite dimer ensembles are found to be in good agreement with the distributions of backbone conformations deduced from nmr coupling constants and the frequencies of glycosyl conformations in x-ray crystal structures, suggesting that the low energy states are reasonable. The low energy dimer forms (consisting of 150–325 conformational states per dimer step) are next used as variables in a Monte Carlo algorithm, which generates the conformations of single-stranded d(CX_nG) chains, where X = A, T and n = 3, 4, 5. The oligonucleotides are built sequentially from the 5′ end of the chain using random numbers to select the conformations of overlapping dimer units. The simulations are very fast, involving a total of 10⁶ conformations per chain sequence. The potential errors in the buildup procedure are minimized by taking advantage of known rotational interdependences in the sugar–phosphate backbone. The distributions of oligonucleotide conformations are examined in terms of the magnitudes, positions, and orientations of the end-to-end vectors of the chains. The differences in overall flexibility and extension of the oligomers are discussed in terms of the conformations of the constituent dinucleotide steps, while the general methodology is discussed and compared with other nucleic acid model building techniques. © 1993 John Wiley & Sons, Inc.     

Monte Carlo studies of a model for lipid-gramicidin A bilayers.

This paper presents results of Monte Carlo simulations of a full bilayer of 200 lipid chains and one gramicidin A dimer. Simulations are described for systems with lipid chains of 14, 16, and 18 carbons, respectively. Using accepted potential functions to calculate interactions between all non-hydrogen atoms a Monte Carlo configuration sampling is generated from which order parameter profiles are calculated and specific configurations are displayed. Results are compared with experimental data for lipid-gramicidin bilayers.     

An analysis of membrane phase transition theories based on kink and jog defects

We derive the geometric properties of isomeric defects referred to as kinks and jogs. With the aid of these geometric properties we are able to list in tabular form according to energy and area all possible kink and jog configurations. This information is the kink and jog single chain density of states in discrete form. We relate the properties of the single chain density of states to the thermodynamic bilayer system. Based on the single chain information, we calculate an upper bound on the entropy change of the membrane phase transition and show that it is less than one half the experimental entropy change of the transition. Finally we show that the discrepancy in the macroscopic value of the entropy results from the fact that the total number of single chain kink and jog defects is orders of magnitude too small.     

Translocation of non-interacting heteropolymer protein chains in terms of single helical propensity and size

In this work we present an analytical framework to calculate the average translocation time τ required for an ideal proteinogenic polypeptide chain to cross over a small pore on a membrane. Translocation is considered to proceed as a chain of non-interacting amino acid residues of sequence {X_j} diffuses through the pore against an energy barrier Δℱ, set by chain entropy and unfolding-folding energetics. We analyze the effect of sequence heterogeneity on the dynamics of translocation by means of helical propensity of amino acid residues. In our calculations we use sequences of fifteen well-known proteins that are translocated which span two orders of magnitude in size according to the number of residues N. Results show non-symmetric free energy barriers as a consequence of sequence heterogeneity, such asymmetry in energy may be useful in differentiated directions of translocation. For the fifteen polypeptide chains considered we found conditions when sequence heterogeneity has not a significant effect on the time scale of translocation leading to a scaling law τ ∝ N^ν, where ν ∼ 1.6 is an exponent that holds for most ground state energies. We also identify conditions when sequence heterogeneity has a great impact on the time scale of translocation, in consequence, no more scaling laws for τ there exist.     

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.     

Monte Carlo simulations of structure and entanglements in polymer melts

Atomistic models of short chain branched (SCB) polyethylene melts have been equilibrated at 450 K using a connectivity altering Monte Carlo method. Quantities related to the chain dimensions and entanglements have been determined. The simulated tube diameters, 〈a_pp〉, of SCB melts are found to scale with the backbone weight fraction, ?, as 〈a_pp〉～?^? 0.46, close to the scaling predicted by the binary contact model, 〈a_pp〉～?^? 0.5. Similar relationships are observed experimentally for polymer solutions, and reproduced by the present methods.