Molecular modelling of liquid crystal systems: An internal coordinate Monte Carlo approach

A Monte Carlo scheme is presented which is designed to provide a convenient mechanism to model accurately the internal molecular structure of liquid crystalline molecules. The technique stores atomic positions in terms of bond lengths, bond angles and dihedral angles within a Z-matrix, and the Monte Carlo scheme involves generating trial configurations from changes to the Z-matrix using the MM2 molecular mechanics potential to describe energy changes between different molecular conformations. The technique is applied to the liquid crystal molecule 4-n-pentyl-4'-cyanobiphenyl (5CB), and results are presented for the conformational populations and dihedral angle distributions of 5CB in the gas phase at 300 K. The effect of a nematic mean field on the distribution of molecular conformations is also examined via the addition of a conformation-dependent potential of mean torque to the internal energy.     

Polypeptide folding using Monte Carlo sampling, concerted rotation, and continuum solvation

An efficient concerted rotation algorithm for use in Monte Carlo statistical mechanics simulations is applied to fold three polypeptides, U(1-17)T9D, alpha(1), and trpzip2, which exhibit native beta-hairpin and alpha-helix folds. The method includes flexible bond and dihedral angles, and a Gaussian bias is applied with driver bond and dihedral angles to optimize the sampling efficiency. Solvation in water is implemented with the generalized Born (GBSA) model. The computed lowest-energy manifolds for the folded structures of the two beta-hairpins agree closely with the corresponding NMR structures. In the case of the alpha(1) peptide, the folded alpha-helical state, which is observed as oligomers in concentrated solution and crystals, is not stable in isolation. The computed preference for random coil structures is in agreement with NMR experiments at low concentration. The fact that native states can be located on high dimensional energy surfaces starting from extended conformations shows that the present methodology samples all relevant parts of the conformational space. The OPLS-AA force field with the GBSA solvent model was also found to perform well in leading to clear energetic separation of the correctly folded structures from misfolded structures for the two peptides that form beta-turns.     

Efficient Monte Carlo trial moves for polypeptide simulations

A new move set for the Monte Carlo simulations of polypeptide chains is introduced. It consists of a rigid rotation along the (C(alpha)) ends of an arbitrary long segment of the backbone in such a way that the atoms outside this segment remain fixed. This fixed end move, or FEM, alters only the backbone dihedral angles phi and psi and the C(alpha) bond angles of the segment ends. Rotations are restricted to those who keep the alpha bond angles within their maximum natural range of approximately +/-10 degrees. The equations for the angular intervals (tau) of the allowed rigid rotations and the equations required for satisfying the detailed balance condition are presented in detail. One appealing property of the FEM is that the required number of calculations is minimal, as it is evident from the simplicity of the equations. In addition, the moving backbone atoms undergo considerable but limited displacements of up to 3 A. These properties, combined with the small number of backbone angles changed, lead to high acceptance rates for the new conformations and make the algorithm very efficient for sampling the conformational space. The FEMs, combined with pivot moves, are used in a test to fold a group of coarse-grained proteins with lengths of up to 200 residues.     

Monte Carlo sampling algorithm for searching a scale-transformed energy space of polypeptides

A Monte Carlo sampling algorithm for searching a scale-transformed conformational energy space of polypeptides is presented. This algorithm is based on the assumption that energy barriers can be overcome by a uniform sampling of the logarithmically transformed energy space. This algorithm is tested with Met-enkephalin. For comparison, the entropy sampling Monte Carlo (ESMC) simulation is performed. First, the global minimum is easily found by the optimization of a scale-transformed energy space. With a new Monte Carlo sampling, energy barriers of 3000 kcal/mol are frequently overcome, and low-energy conformations are sampled more efficiently than with ESMC simulations. Several thermodynamic quantities are calculated with good accuracy.     

Analysis of the conformational dependence of mass-metric tensor determinants in serial polymers with constraints

It is well known that mass-metric tensor determinants det(G(s)) influence the equilibrium statistics and the rates of conformational transitions for polymers with constrained bond lengths and bond angles. It is now standard practice to include a Fixman-style compensating potential of the form U(c)(q(s)) proportional, variant(-k(B)T/2)ln[det(G(s))] as part of algorithms for torsional space molecular dynamics. This elegant strategy helps eliminate unwarranted biases that arise due to the imposition of holonomic constraints. However, the precise nature and extent of variation of det(G(s)) and hence ln[det(G(s))] with chain conformation and chain length has never been quantified. This type of analysis is crucial for understanding the nature of the conformational bias that the introduction of a Fixman potential aims to eliminate. Additionally, a detailed analysis of the conformational dependence of det(G(s)) will help resolve ambiguities regarding suggestions for incorporating terms related to det(G(s)) in the design of move sets in torsional space Monte Carlo simulations. In this work, we present results from a systematic study of the variation of det(G(s)) for a serial polymer with fixed bond lengths and bond angles as a function of chain conformation and chain length. This analysis requires an algorithm designed for rapid computation of det(G(s)) which simultaneously allows for a physical/geometric interpretation of the conformational dependence of det(G(s)). Consequently, we provide a detailed discussion of our adaptation of an O(n) algorithm from the robotics literature, which leads to simple recursion relations for direct evaluation of det(G(s)). Our analysis of the conformational dependence of det(G(s)) yields the following insights. (1) det(G(s)) is maximized for spatial conformers and minimized for planar conformations. (2) Previous work suggests that it is logical to expect that the conformational dependence of det(G(s)) becomes more pronounced with increase in chain length. Confirming this expectation, we provide systematic quantification of the nature of this dependency and show that the difference in det(G(s)) between spatial and planar conformers, i.e., between the maxima and minima of det(G(s)) grows systematically with chain length. Finally, we provide a brief discussion of implications of our analysis for the design of move sets in Monte Carlo simulations.     

Critical assessment of side-chain conformational space sampling procedures designed for quantifying the effect of side-chain environment

We introduce a family of procedures designed to sample side-chain conformational space at particular locations in protein structures. These procedures (CRSP) use intensive cycles of random assignment of side-chain conformations followed by minimization to determine all the conformations that a group of side-chains can adopt simultaneously. First, we consider a procedure evolving in the dihedral space (dCRSP). Our results suggest that it can accurately map low-energy conformations adopted by clusters of side-chains of a protein. dCRSP is relatively insensitive to various important parameters, and it is sufficiently accurate to capture efficiently the constraint induced by the environment on the conformations a particular side-chain can adopt. Our results show that dCRSP, compared with molecular dynamics (MD), can overcome the problem of the limited set of conformations reached in a reasonable amount of simulations. Next, we introduce procedures (vCRSP) in which valence angles are relaxed, and we assess how efficiently they quantify the conformational entropy of side-chains in the protein native state. For simple peptides, entropies obtained with vCRSP are fully compatible with those obtained with a Monte Carlo procedure. For side-chains in a protein environment, however, vCRSP appears of limited use. Finally, we consider a two-step procedure that combines dCRSP and vCRSP. Our tests suggest that it is able to overcome the limitations of vCRSP. We also note that dCRSP provides a reasonable initial approximation. This family of procedures offers promise in quantifying the contribution of conformational entropy to the energetics of protein structures.     

Conformational study of halogen- and hydrogen-substituted polythionylphosphazenes using density functional theory method

The structure and conformational stability of polythionylphosphazenes is investigated by modeling single polymer chains with small mimics. The model compounds are composed of repeat units of the corresponding polythionylphosphazenes. Two of the model compounds have hydrogens and two have chlorines as substituents on phosphorus atoms. The substituents on sulfur may be either fluorine or chlorine. Fully geometry-optimized structures and energies of the stable conformations involving rotations around the P? N bond near the sulfur are obtained using the density functional theory method. The structural and conformational analyses indicate that the rotation around the N? P bond leads to variations in the bond lengths, the SNP bond angle openings, as well as couplings between dihedral angles in different conformations in all model compounds. In addition, the conformational analysis suggests that the minima on the conformational potential energy surface in these compounds may be located in the vicinity of the following values of the NP? NS dihedral angle: -50°, 90° (or 60°), 180°, and 240°. It was found that the values of the conformational energy differences range between less than 1 to 5 kcal/mol. A comparison is made between the structural results obtained using the density functional theory and the ab initio molecular orbital theory for the global minimum structures. © 1995 John Wiley & Sons, Inc.     

Enhanced sampling method in molecular simulations using genetic algorithm for biomolecular systems

We propose a molecular simulation method using genetic algorithm (GA) for biomolecular systems to obtain ensemble averages efficiently. In this method, we incorporate the genetic crossover, which is one of the operations of GA, to any simulation method such as conventional molecular dynamics (MD), Monte Carlo, and other simulation methods. The genetic crossover proposes candidate conformations by exchanging parts of conformations of a target molecule between a pair of conformations during the simulation. If the candidate conformations are accepted, the simulation resumes from the accepted ones. While conventional simulations are based on local update of conformations, the genetic crossover introduces global update of conformations. As an example of the present approach, we incorporated genetic crossover to MD simulations. We tested the validity of the method by calculating ensemble averages and the sampling efficiency by using two kinds of peptides, ALA3 and (AAQAA)₃. The results show that for ALA3 system, the distribution probabilities of backbone dihedral angles are in good agreement with those of the conventional MD and replica-exchange MD simulations. In the case of (AAQAA)₃ system, our method showed lower structural correlation of α-helix structures than the other two methods and more flexibility in the backbone ψ angles than the conventional MD simulation. These results suggest that our method gives more efficient conformational sampling than conventional simulation methods based on local update of conformations. © 2018 Wiley Periodicals, Inc.     

Conformation‐Family Monte Carlo (CFMC): An Efficient Computational Method for Identifying the Low‐Energy States of a Macromolecule

A highly efficient method, Conformation‐Family Monte Carlo (CFMC), has been developed for searching the conformational space of a macromolecule and identifying its low‐energy conformations. This method maintains a database of low‐energy conformations that are clustered into families. The conformations in this database are improved iteratively by a Metropolis‐type Monte Carlo procedure, together with energy minimization, in which the search is biased towards investigating the regions of the lowest‐energy families. The CFMC method has the advantages of our earlier potential‐smoothing methods (in that it `coarse‐grains' the conformational space and exploits information about nearby low‐energy states), but avoids their disadvantages (such as the displacement of the global minimum at large smoothings). The CFMC method is applied to a test protein, domain B of Staphylococcal protein A. Independent CFMC runs yielded the same low‐energy families of conformations from random starts, indicating that the thermodynamically relevant conformational space of this protein has been explored thoroughly. The CFMC method is highly efficient, performing as well as or better than competing methods, such as Monte Carlo with minimization, conformational‐space annealing, and the self‐consistent basin‐to‐deformed‐basin method.     

Conformational analysis of a stereochemically complete set of cis-enediol peptide analogues

A conformational analysis of a stereochemically complete set of peptide analogues based on a cis-enediol unit is presented. The cis-enediol unit, which can replace a two or a three amino acid segment of a peptide, contains two "side chains", four asymmetrical carbon atoms, and six free dihedral angles. To determine the accessible conformational space, the molecules are divided into three fragments, each containing two free dihedral angles. The energy surfaces are computed for all dihedral angle values, and the possible conformations of the cis-enediol unit analogues are built using all combinations of the surface minima. Such a "build-up" procedure, which is very fast, is able to reproduce 75% of the minima obtained from a full dihedral angle exploration of the conformational space. The cis-enediol unit minima are compared with the corresponding di- and tripeptide minima; all peptide minima can be closely matched by a cis-enediol unit minimum of low energy (less than 2.2 kcal/mol above the lowest energy conformer). However, there are low energy minima of the cis-enediol unit that have no corresponding minima in peptides. The results are shown to depend strongly on the chirality of the analogues. The ability of each of the stereoisomers to mimic natural peptides, evaluated by the present approach, is correlated with its experimental activity in a renin inhibition assay.     

J-coupling constants for a trialanine peptide as a function of dihedral angles calculated by density functional theory over the full Ramachandran space

We present 13 (3)J, seven (2)J and four (1)J coupling constants (24 in all) calculated using B3LYP/D95** as a function of the φ and ψ Ramachandran dihedral angles of the acetyl(Ala)(3)NH(2) capped trialanine peptide over the entire Ramachandran space. With the exception of three of these J couplings, all show significant dependence upon both dihedral angles. For each J coupling considered, a two dimensional grid with respect to φ and ψ angles can be used to interpolate the values for any pair of φ and ψ values. Such simple interpolation is shown to be very accurate. Most of these calculated J couplings should prove useful for improving the accuracy of the determination of peptide and protein structures from NMR measurements in solution over that provided by the common procedure of treating the J couplings as functions of a single dihedral angle by means of Karplus-type fittings.     

Efficient evaluation of sampling quality of molecular dynamics simulations by clustering of dihedral torsion angles and Sammon mapping

A direct conformational clustering and mapping approach for peptide conformations based on backbone dihedral angles has been developed and applied to compare conformational sampling of Met-enkephalin using two molecular dynamics (MD) methods. Efficient clustering in dihedrals has been achieved by evaluating all combinations resulting from independent clustering of each dihedral angle distribution, thus resolving all conformational substates. In contrast, Cartesian clustering was unable to accurately distinguish between all substates. Projection of clusters on dihedral principal component (PCA) subspaces did not result in efficient separation of highly populated clusters. However, representation in a nonlinear metric by Sammon mapping was able to separate well the 48 highest populated clusters in just two dimensions. In addition, this approach also allowed us to visualize the transition frequencies between clusters efficiently. Significantly, higher transition frequencies between more distinct conformational substates were found for a recently developed biasing-potential replica exchange MD simulation method allowing faster sampling of possible substates compared to conventional MD simulations. Although the number of theoretically possible clusters grows exponentially with peptide length, in practice, the number of clusters is only limited by the sampling size (typically much smaller), and therefore the method is well suited also for large systems. The approach could be useful to rapidly and accurately evaluate conformational sampling during MD simulations, to compare different sampling strategies and eventually to detect kinetic bottlenecks in folding pathways.     

Realistic sampling of amino acid geometries for a multipolar polarizable force field

The Quantum Chemical Topological Force Field (QCTFF) uses the machine learning method kriging to map atomic multipole moments to the coordinates of all atoms in the molecular system. It is important that kriging operates on relevant and realistic training sets of molecular geometries. Therefore, we sampled single amino acid geometries directly from protein crystal structures stored in the Protein Databank (PDB). This sampling enhances the conformational realism (in terms of dihedral angles) of the training geometries. However, these geometries can be fraught with inaccurate bond lengths and valence angles due to artefacts of the refinement process of the X‐ray diffraction patterns, combined with experimentally invisible hydrogen atoms. This is why we developed a hybrid PDB/nonstationary normal modes (NM) sampling approach called PDB/NM. This method is superior over standard NM sampling, which captures only geometries optimized from the stationary points of single amino acids in the gas phase. Indeed, PDB/NM combines the sampling of relevant dihedral angles with chemically correct local geometries. Geometries sampled using PDB/NM were used to build kriging models for alanine and lysine, and their prediction accuracy was compared to models built from geometries sampled from three other sampling approaches. Bond length variation, as opposed to variation in dihedral angles, puts pressure on prediction accuracy, potentially lowering it. Hence, the larger coverage of dihedral angles of the PDB/NM method does not deteriorate the predictive accuracy of kriging models, compared to the NM sampling around local energetic minima used so far in the development of QCTFF. © 2015 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Conformational analysis of the eight-membered ring of the oxidized cysteinyl-cysteine unit implicated in nicotinic acetylcholine receptor ligand recognition.

Nicotinic acetylcholine receptors (nAChRs) are membrane-bound, pentameric ligand-gated ion channels associated with a variety of human disorders such as Alzheimer's disease, Parkinson's disease, schizophrenia, and pain. Most known nAChRs contain an unusual eight-membered disulfide-containing cysteinyl-cysteine ring, ox-[Cys-Cys], as does the soluble acetylcholine binding protein (AChBP) found in the snail Lymnaea stagnalis. The cysteinyl-cysteine ring is located in a region implicated in ligand binding, and conformational changes involving this ring may be important for modulation of nAChR function. We have studied the preferred conformations of Ac-ox-[Cys-Cys]-NH2 by NMR in water and computationally by Monte Carlo simulations using the OPLS-AA force field and GB/SA water model. ox-[Cys-Cys] adopts four distinct low-energy conformers at slightly above 0 degrees C in water. Two populations are dependent on the peptide omega2 dihedral angle, with the trans amide favored over the cis amide by a ratio of ca. 60:40. Two ox-[Cys-Cys] conformers with a cis amide bond (C+ and C-) differ from each other primarily by variation of the chi3 dihedral angle, which defines the orientation of the helicity about the S-S bond (+/- 90 degrees ). Two trans amide conformers have the same S-S helicity (chi3 approximately -90 degrees ), but are distinguished by a backbone rotation about phi2 and psi1 (T- and T'-). The ratio of T-/T'-/C+/C- is 47:15:29:9. The orientation of the pendant moieties from the eight-membered ring is more compact for the major trans conformer (T-) than for the extended conformations adopted by T'-, C+, and C-. These conformational preferences are also observed in tetrapeptide and undecapeptide fragments of the human alpha7 subtype of the nAChR that contains the ox-[Cys-Cys] unit. Conformer T- is nearly identical to the conformation seen in the X-ray structure of ox-[Cys(187)-Cys(188)] found in the unliganded AChBP, and is a Type VIII beta-turn.     

Using internal and collective variables in Monte Carlo simulations of nucleic acid structures: chain breakage/closure algorithm and associated Jacobians

This article describes a method for solving the geometric closure problem for simplified models of nucleic acid structures by using the constant bond lengths approximation. The resulting chain breakage/closure equations, formulated in the space of variable torsion and bond angles, are easy to solve, and have only two solutions. The analytical simplicity is in contrast with the high complexity of the closure problem in the torsion angle space with at most 16 solutions, which has been dealt with by several authors and was solved analytically by Wu and Deem (J. Chem. Phys. 1999, 111, 6625). The discussion on the choice of variables and associated Jacobians is focussed on the question of how conformational equilibration is affected in Monte Carlo simulations of molecular systems. In addition to the closure of the phosphate backbone, it is necessary to also solve the closure problem for the five-membered flexible furanose sugar ring. Explicit closure equations and the resulting Jacobians are given both for the complete four-variable model of the furanose ring and simulations in the phase-amplitude space of the five-membered ring, which are based on the approximate two-variable model of furanose introduced by Gabb et al. (J. Comput. Chem. 1995, 16, 667). The suggested closure algorithm can be combined with collective variables defined by translations and rotations of the monomeric nucleotide units. In comparison with simple internal coordinate moves, the resulting concerted moves describe local structural changes that have high acceptance rates and enable fast conformational equilibration. Appropriate molecular models are put forward for prospective Monte Carlo simulations of nucleic acids, but can be easily adapted to other biomolecular systems, such as proteins and lipid structures in biological membranes.     

Effect of torsional potential on the predicted phase behavior of n-alkanes

The effect of torsional potential on the predictions of simulation for vapor–liquid equilibria of n-alkanes is determined. Calculations are performed with histogram-reweighting Monte Carlo simulations in the grand canonical ensemble. Decreasing the magnitude of energy barriers to dihedral rotation or allowing free rotation is found to have no effect on the predicted vapor–liquid equilibria. Restriction of the dihedral angles to a Gaussian distribution around the minimum energy conformation causes an under-prediction of the liquid densities and critical temperatures by a maximum of 7% and 2%, respectively, with discrepancies increasing monotonically with the number of dihedral angles present in a molecule. No significant deviation in vapor pressure is observed for any compound, regardless of torsional potential used. An analysis of the conformational behavior reveals that restriction of the dihedral sampling has a measurable effect on excluded volume of the molecule, and this change of conformational behavior is responsible for the reduction in the predicted saturated liquid densities observed in this work. Similar calculations for force fields employing reduced dihedral potentials or freely jointed chains show little change in the predicted excluded volume compared to the reference force field.     

Conformational searching of transition metal compounds

To date, no conformational search of inorganic complexes has been reported that searches for all the conformations and configurations available to the complex. This is due to the various coordination geometries that transition metal ions can adopt and the difficulties in conducting conformational searches with systems that have connected ring systems, such as the ones formed when a metal ion binds a multidentate ligand. Using three test complexes {[Co(dien)₂]³⁺, [Co(dien)(dpt)]³⁺, and [Co(hexamethylcyclam)(Cl)} the ability of the random kick (Cartesian stochastic Monte Carlo search) method and the Monte Carlo dihedral and positional method to find all conformations and geometric isomers was tested (dien, diethylenetriamine; dpt, di(3-aminopropyl)amine; hexamethylcyclam: tet-a, meso-5,5,7,12,12,14-hexamethyl-1,4,8,11-tetraazacyclotetradecane; tet-b, racemic-5,5,7,12,12,14-hexamethyl-1,4,8,11-tetraazacyclotetradecane). Both methods are significant improvements on the current method by which all possible isomers are entered graphically and minimized individually. The major difficulty that was encountered was how to differentiate between the large number of similar conformations found. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1549–1558, 1999     

Finite-state and reduced-parameter representations of protein backbone conformations

This article studies the representation of protein backbone conformations using a finite number of values for the backbone dihedral angles. We develop a combinatorial search algorithm that guarantees finding the global minima of functions over the configuration space of discrete protein conformations, and use this procedure to fit finite-state models to the backbones of globular proteins. It is demonstrated that a finite-state representation with a reasonably small number of states yields either a small root-mean-square error or a small dihedral angle deviation from the native structure, but not both at the same time. The problem can be resolved by introducing limited local optimization in each step of the combinatorial search. In addition, it is shown that acceptable approximation is achieved using a single dihedral angle as an independent variable in local optimization. Results for 11 proteins demonstrate the advantages and shortcomings of both the finite-state and reduced-parameter approximations of protein backbone conformations. © 1994 by John Wiley & Sons, Inc.     

Knowledge-based potential for the polypeptide backbone

A knowledge-based potential for the polypeptide backbone as a function of the dihedral angles is developed and tested. The potential includes correlations due to the conformations and compositions of adjacent residues. Its purpose is to serve as a major component of a coarse-grained protein potential by including the most relevant local interactions while averaging out nonbonded ones. A probability density estimation algorithm and a multi-resolution probability combination procedure are developed to produce smooth probability distributions and dihedral angle potentials. The potential is described by a set of two-dimensional dihedral angle surfaces involving the various combinations of amino acid triplets and duplets. Several tests are carried out to evaluate the quality of the potential. Monte Carlo simulations, using only the dihedral angle potential and a coarse-grained excluded volume potential, show that the resulting dihedral angle distributions and correlations are consistent with those extracted from protein structures. Additional simulations of unfolded proteins are carried out to measure the NMR residual dipolar coupling (RDC). Significant correlations are obtained between the simulations and the corresponding experiments consistent with other simulations in the literature. Finally, the dihedral angle entropies are calculated for the 20 amino acids. In particular, the entropy difference between alanine and glycine agrees with the ones computed from molecular dynamics simulations ( approximately 0.4 kcal/mol).     

Optimization of Monte Carlo trial moves for protein simulations

Closed rigid-body rotations of residue segments under bond-angle restraints are simple and effective Monte Carlo moves for searching the conformational space of proteins. The efficiency of these moves is examined here as a function of the number of moving residues and the magnitude of their displacement. It is found that the efficiency of folding and equilibrium simulations can be significantly improved by tailoring the distribution of the number of moving residues to the simulation temperature. In general, simulations exploring compact conformations are more efficient when the average number of moving residues is smaller. It is also demonstrated that the moves do not require additional restrictions on the magnitude of the rotation displacements and perform much better than other rotation moves that do not restrict the bond angles a priori. As an example, these results are applied to the replica exchange method. By assigning distributions that generate a smaller number of moving residues to lower temperature replicas, the simulation times are decreased as long as the higher temperature replicas are effective.