Many-body optimization using an ab initio monte carlo method

Advances in computing power have made it possible to study solvated molecules using ab initio quantum chemistry. Inclusion of discrete solvent molecules is required to determine geometric information about solute/solvent clusters. Monte Carlo methods are well suited to finding minima in many-body systems, and ab initio methods are applicable to the widest range of systems. A first principles Monte Carlo (FPMC) method was developed to find minima in many-body systems, and emphasis was placed on implementing moves that increase the likelihood of finding minimum energy structures. Partial optimization and molecular interchange moves aid in finding minima and overcome the incomplete sampling that is unavoidable when using ab initio methods. FPMC was validated by studying the boron trifluoride-water system, and then the method was used to examine the methyl carbenium ion in water to demonstrate its application to solvation problems.     

Cages, baskets, ladders, and tubes: conformational studies of polyhedral oligomeric silsesquioxanes

The conformational flexibility of a series of cage, basket, ladder, and tube polyhedral oligomeric silsesquioxanes (POSS) has been examined using the Low Mode:Monte Carlo conformational search method in conjunction with the MM3/GBSA(CHCl3) surface. An ensemble of low energy structures was generated and used to explore the molecular shape and flexibility of each system. The results indicate that, except for the ladder molecule, the incompletely condensed systems that are studied are relatively rigid. Even in cases where the molecule is able to adopt numerous low energy conformations, the overall shape remains cage-like and the conformations differ only by small angles or substituent orientations. The ladder molecule is the most flexible and this ensemble clusters into two families: one that is cage-like and the other that is more open and ladder-like. The conformational flexibilities in the gas and solvent phases, as approximated using the GBSA continuum solvent model, are very similar.     

Structure and free energy of complex thermodynamic systems

Biological systems are intrinsically complex, involving many degrees of freedom, heterogeneity, and strong interactions among components. For the simplest of biological substances, e.g., biomolecules, which obey the laws of thermodynamics, we may try to investigate them by means of a statistical mechanical approach. Even for these simplest many-body systems, assuming all microscopic interactions are completely known, current physical and chemical methods of characterizing the overall structure and free energy face the fundamental challenge of an exponential amount of computations as the number of degrees of freedom of these systems increases. As a response to this problem, two general procedures have been developed to compute the structure (Monte Carlo-minimization method) and free energy (Monte Carlo recursion method) of a complex thermodynamic system. The Monte Carlo-minimization procedure has been applied to determine the structure of a pentapeptide Met-enkephalin, leading consistently to a stable β-bend structure, starting from random initial conformations. The Monte Carlo recursion method has been applied to a Lennard-Jones fluid, with results in agreement with previously published values of the free energy obtained from other procedures.     

DMC: A multifunctional hybrid dynamics/Monte Carlo simulation algorithm for the evaluation of conformational space

A hybrid conformational search algorithm (DMC) is described that combines a modified form of molecular dynamics with Metropolis Monte Carlo sampling, using the COSMIC(90) force field. Trial configurations are generated by short bursts of high-temperature dynamics in which the initial kinetic energy is focused into single bond rotations or alternatively into “corner-flapping” motions in ring systems. Constant temperature and simulated annealing search protocols have been applied to the conformational analysis of several model hydrocarbons (cyclopentane, cyclohexane, cycloheptane, cyclooctane, cycloheptadecane, decane, and tetradecane), and the performance compared with conventional molecular dynamics and Monte Carlo sampling methods. Optimum Metropolis sampling temperatures have been determined and range from 1000–2000 K for acyclic molecules to 3000 K for cyclic systems. Simulated annealing runs are most successful at locating the global minimum when cooling slowing from these optimum temperatures.     

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.     

Flexible polyelectrolyte simulations at the Poisson-Boltzmann level: a comparison of the kink-jump and multigrid configurational-bias Monte Carlo methods

We present a new approach for simulating the motions of flexible polyelectrolyte chains based on the continuous kink-jump Monte Carlo technique coupled to a lattice field theory based calculation of the Poisson-Boltzmann (PB) electrostatic free energy "on the fly." This approach is compared to the configurational-bias Monte Carlo technique, in which the chains are grown on a lattice and the PB equation is solved for each configuration with a linear scaling multigrid method to obtain the many-body free energy. The two approaches are used to calculate end-to-end distances of charged polymer chains in solutions with varying ionic strengths and give similar numerical results. The configurational-bias Monte Carlo/multigrid PB method is found to be more efficient, while the kink-jump Monte Carlo method shows potential utility for simulating nonequilibrium polyelectrolyte dynamics.     

Conformational search of peptides and proteins: Monte Carlo minimization with an adaptive bias method applied to the heptapeptide deltorphin

The energy function of a protein consists of a tremendous number of minima. Locating the global energy minimum (GEM) structure, which corresponds approximately to the native structure, is a severe problem in global optimization. Recently we have proposed a conformational search technique based on the Monte Carlo minimization (MCM) method of Li and Scheraga, where trial dihedral angles are not selected at random within the range [-180 degrees,180 degrees ] (as with MCM) but with biased probabilities depending on the increased structure-energy correlations as the GEM is approached during the search. This method, called the Monte Carlo minimization with an adaptive bias (MCMAB), was applied initially to the pentapeptide Leu-enkephalin. Here we study its properties further by applying it to the larger peptide with bulky side chains, deltorphin (H-Tyr-D-Met-Phe-His-Leu-Met-Asp-NH(2)). We find that on average the number of energy minimizations required by MCMAB to locate the GEM for the first time is smaller by a factor of approximately three than the number required by MCM-in accord with results obtained for Leu-enkephalin.     

Conformational search using a molecular dynamics–minimization procedure: Applications to clusters of coulombic charges,Lennard–Jones particles,and waters

A new conformational search method, molecular dynamics–minimization (MDM), is proposed, which combines a molecular dynamics sampling strategy with energy minimizations in the search for low-energy molecular structures. This new method is applied to the search for low energy configurations of clusters of coulombic charges on a unit sphere, Lennard–Jones clusters, and water clusters. The MDM method is shown to be efficient in finding the lowest energy configurations of these clusters. A closer comparison of MDM with alternative conformational search methods on Lennard–Jones clusters shows that, although MDM is not as efficient as the Monte Carlo–minimization method in locating the global energy minima, it is more efficient than the diffusion equation method and the method of minimization from randomly generated structures. Given the versatility of the molecular dynamics sampling strategy in comparison to Monte Carlo in treating molecular complexes or molecules in explicit solution, one anticipates that the MDM method could be profitably applied to conformational search problems where the number of degrees of freedom is much greater. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 60–70, 1998     

Torsional flexing: Conformational searching of cyclic molecules in biased internal coordinate space

A new stochastic (Monte Carlo) procedure, termed torsional flexing, has been devised for searching the conformational space of cyclic molecules. Torsional flexing causes a local, torsion angle-biased, distortion of a ring bond in a cyclic molecule. Because torsional flexing does not cause large atomic movements, even when it is applied to several bonds simultaneously, subsequent energy minimization generally proceeds rapidly. Nevertheless, the torsional flexing method is prone to generate structures that cross energy barriers so that the structure resulting after energy minimization is frequently a different conformer of the cyclic molecule. Conformational searches on cycloheptadecane, oxobrefeldin A, cyclopenta-L -alanine, and rifamycin SV based upon torsional flexing indicated that torsional flexing is among the best methods yet devised for searching the conformational space of flexible cyclic molecules. © 1993 John Wiley & Sons, Inc.     

Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations

An efficient method for the calculation of minimum free energy pathways and free energy profiles for conformational transitions is presented. Short restricted perturbation-targeted molecular dynamics trajectories are used to generate an approximate free energy surface. Approximate reaction pathways for the conformational change are constructed from one-dimensional line segments on this surface using a Monte Carlo optimization. Accurate free energy profiles are then determined along the pathways by means of one-dimensional adaptive umbrella sampling simulations. The method is illustrated by its application to the alanine "dipeptide." Due to the low computational cost and memory demands, the method is expected to be useful for the treatment of large biomolecular systems.     

Order parameters and molecular shapes of phosphonic acid dialkyl esters in a uniaxial hydrophobic environment. A theoretical study

The conformational behaviour of phosphonic acid dialkyl ester (PAE) molecules in a uniaxial hydrophobic environment was studied using the Monte Carlo method with the well-known Metropolis algorithm. The influence of the surrounding molecules on ordering processes of the intrinsic PAE molecules was taken into account by a molecular field. The intramolecular energy was calculated using 6–12 atom-atom and torsion potential functions. Conformations were analysed using torsion angle distributions, segment and bond order parameters. The order parameters of the C-H bonds are compared with experimental results of²H NMR. Further a general method of determining the effective shape of a molecule is presented. The shape is defined by probability distribution for finding atomic coordinates within volume elements during a Monte Carlo run. Thus the asymmetry of a molecule can be visualized. PAE molecules studied show a positive asymmetry in all cases.     

Efficiency of simulated annealing for peptides with increasing geometrical restrictions

Simulated annealing (SA) is a popular global minimizer that can conveniently be applied to complex macromolecular systems. Thus, a molecular dynamics or a Monte Carlo simulation starts at high temperature, which is decreased gradually, and the system is expected to reach the low-energy region on the potential energy surface of the molecule. However, in many cases this process is not efficient. Alternatively, the low-energy region can be reached more effectively by minimizing the energy of selected molecular structures generated along the simulation pathway. The efficiency of SA to locate energy-minimized structures within 5 kcal/mol above the global energy minimum is studied as applied to three peptide models with increasing geometrical restrictions: (1) The linear pentapeptide Leu-enkephalin described by the ECEPP potential, (2) a cyclic hexapeptide described by the GROMOS force field energy E_GRO alone, and (3) the same cyclic peptide with E_GRO combined with a restraining potential based on 31 proton–proton restraints obtained from nuclear magnetic resonance (NMR) experiments. The efficiency of SA is compared to that of the Monte Carlo minimization (MCM) method of Li and Scheraga, and to our local torsional deformations (LTD) method for the conformational search of cyclic molecules. The results for the linear peptide show that SA provides a relatively weak guidance towards the most stable energy region; as expected, this guidance increases for the cyclic peptide and the cyclic peptide with NMR restraints. However, in general, MCM and LTD are significantly more efficient than SA as generators of low-energy minimized structures. This suggests that LTD might provide a better search tool than SA in structure determination of protein regions for which a relatively small number of restraints are provided by NMR. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1659–1670, 1999     

A rapidly convergent simulation method: Mixed Monte Carlo/stochastic dynamics

Although Monte Carlo and molecular dynamics are the primary methods used for free energy simulations of molecular systems, their application to molecules that have multiple conformations separated by energy barriers of ≥ 3 kcal/mol is problematic because of slow rates of convergence. In this article we introduce a hybrid simulation method termed MC-SD which mixes Monte Carlo (MC) and stochastic dynamics (SD). This new method generates a canonical ensemble via alternating MC and SD steps and combines the local exploration strengths of dynamics with the barrier-crossing ability of large-step Monte Carlo. Using calculations on double-well potentials and long simulations (10⁸ steps of MC and 1 μs of SD) of the simple, conformationally flexible molecule n-pentane, we find that MC-SD simulations converage faster than either MC or SD alone and generate ensembles which are equivalent to those created by classical MC or SD. Using pure SD at 300 K, the conformational populations of n-pentane are shown to be poorly converged even after a full microsecond of simulation. © 1994 by John Wiley & Sons, Inc.     

Bond dissocation and conformational energetics of tetrasulfur: a quantum Monte Carlo study

Variational Monte Carlo (VMC) and fixed-node diffusion Monte Carlo (DMC) calculations are performed for S4. The effect of single- and multireference trial functions, as well as choice of orbitals, is investigated for its effect on the quality of the Monte Carlo estimates. Estimates of symmetric (two S2 molecules) and asymmetric (S atom and S3 molecule) bond dissociation are reported. The conformational change of S4 from C2v to D2h defines a double-well potential and is also estimated. Multireference DMC with natural orbitals (DMC/NO) estimates the energy of the conformational change as 1.20(20) kcal/mol; the dissociation of the long S-S single bond is estimated at 21.1(1.3) kcal/mol, and the asymmetric bond energy is estimated as 53.2(2.4) kcal/mol. An estimate of the total atomization energy using multireference DMC/NO gives a value of 219.5(2.2) kcal/mol. The relative quality of result and implications for simplified trial function design are discussed.     

Torsional anharmonicity in the conformational analysis of beta-D-galactose

Schemes to include a treatment of torsional anharmonicity in the conformational analysis of biological molecules are introduced. The approaches combine ab initio electronic energies and harmonic frequencies with anharmonic torsional partition functions calculated using the torsional path integral Monte Carlo method on affordable potential energy surfaces. The schemes are applied to the conformational study of the monosaccharide beta-d-galactose in the gas phase. The global minimum structure is almost exclusively populated at 100 K, but a large number of conformers are present at ambient and higher temperatures. Both quantum mechanical and anharmonic effects in the torsional modes have little effect on the populations at all temperatures considered, and it is, therefore, expected that standard harmonic treatments are satisfactory for the conformational study of monosaccharides.     

Spatial delocalization in para-H2 clusters

We applied the quantum path integral Monte Carlo method for the study of (para-H)N (N = 5-33) clusters at T = 2 K, exploring static and dynamic order, which originates from the effects of zero-point energy, kinetic energy, and thermal fluctuations in quantum clusters. Information on dynamic structure was inferred from the asymptotic tails of the cage correlation function calculated from the centroid Monte Carlo trajectory. The centroid cage correlation function decays to zero for large clusters (N = 15-33), manifesting the interchange of molecules between different solvation shells, with statistically diminishing back interchange. Further evidence for the floppiness of para-hydrogen clusters emerges from the Monte Carlo evolution of the centroid of a tagged molecule, which exhibits significant changes in the list of its first and second solvation shells due to the interchange of molecules between these shells.     

Effect of hydrophilic walls on the hydration of sodium cations in planar nanopores

A computer simulation of the structure of Na⁺ ion hydration shells with sizes in the range of 1 to 100 molecules in a planar model nanopore 0.7 nm wide with structureless hydrophilic walls is performed using the Monte Carlo method at a temperature of 298 K. A detailed model of many-body intermolecular interactions, calibrated with reference to experimental data on the free energy and enthalpy of reactions after gaseous water molecules are added to a hydration shell, is used. It is found that perturbations produced by hydrophilic walls cause the hydration shell to decay into two components that differ in their spatial arrangement and molecular orientational order.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.