Molecular Dynamics Simulations With First Order Coupling to a Bath of Constant Chemical Potential

Abstract

In molecular dynamics simulations the temperature or pressure can be controlled by applying a weak first-order coupling to a bath of constant temperature or pressure. This weak coupling technique to control system properties using a first-order relaxation equation is analyzed from a statistical mechanics point of view. It is shown, how the weak coupling scheme can be generalized and applied to a bath of contstant chemical potential. The presented method, to which in the following will be referred to as chemical potential weak coupling, is applied and tested on a Lennard-Jones fluid. The thermodynamic quantities known from the literature are accuratly reproduced.

The temperature and chemical potential weak coupling methods aim to sample the canonical and grand canonical ensembles respectively. By analyzing the fluctuations in energy and number of particles, the tight relation between the ensembles and the distributions obtained from the weak coupling simulations is demonstrated. The influence of the choice of the coupling parameters on the quality of the approximation of the ensemble distribution is discussed.     

Free energy of formation of clusters of sulphuric acid and water molecules determined by guided disassembly

We evaluate the grand potential of a cluster of two molecular species, equivalent to its free energy of formation from a binary vapour phase, using a non-equilibrium molecular dynamics technique where guide particles, each tethered to a molecule by a harmonic force, move apart to disassemble a cluster into its components. The mechanical work performed in an ensemble of trajectories is analysed using the Jarzynski equality to obtain a free energy of disassembly, a contribution to the cluster grand potential. We study clusters of sulphuric acid and water at 300 K, using a classical interaction scheme, and contrast two modes of guided disassembly. In one, the cluster is broken apart through simple pulling by the guide particles, but we find the trajectories tend to be mechanically irreversible. In the second approach, the guide motion and strength of tethering are modified in a way that prises the cluster apart, a procedure that seems more reversible. We construct a surface representing the cluster grand potential, and identify a critical cluster for droplet nucleation under given vapour conditions. We compare the equilibrium populations of clusters with calculations reported by Henschel et al. [J. Phys. Chem. A 2014;118:2599] based on optimised quantum chemical structures.     

Blue Moon Approach to Rare Events

The "Blue Moon" ensemble is a computationally efficient molecular dynamics method to estimate the rate constants of rare activated events when the process can be described by a reaction coordinate ξ(r), a well-defined function in configuration space. By means of holonomic constraints a number of values of ξ(r) can be prescribed along the relevant path to identify the "bottleneck" region first and to sample an ensemble of starting conditions to generate activated trajectories. These MD trajectories sample phase space according to a biased configurational distribution. With a suitable re-weighting of averages from such ensemble of trajectories one can characterize completely rare events.     

Right‐ and left‐handed three‐helix proteins. II. Similarity and differences in mechanical unfolding of proteins

Here, we study mechanical properties of eight 3‐helix proteins (four right‐handed and four left‐handed ones), which are similar in size under stretching at a constant speed and at a constant force on the atomic level using molecular dynamics simulations. The analysis of 256 trajectories from molecular dynamics simulations with explicit water showed that the right‐handed three‐helix domains are more mechanically resistant than the left‐handed domains. Such results are observed at different extension velocities studied (192 trajectories obtained at the following conditions: v = 0.1, 0.05, and 0.01 Å ps^?1, T = 300 K) and under constant stretching force (64 trajectories, F = 800 pN, T = 300 K). We can explain this by the fact, at least in part, that the right‐handed domains have a larger number of contacts per residue and the radius of cross section than the left‐handed domains. Proteins 2014; 82:90–102. © 2013 Wiley Periodicals, Inc.     

Diffusivity of CH4 In Model Silica Nanopores: Molecular Dynamics and Quasichemical Mean Field Theory

Equilibrium molecular dynamics and dual control volume grand canonical molecular dynamics experiments were carried out aiming at the investigation of the dependence of transport diffusivity upon the adsorbent pore size and sorbate concentration of CH₄ in cylindrical silica nanopores at 298?K, calibrated with respect to experimental data of zeolite VPI 5; the results of simulation were elaborated on the basis of the quasichemical mean field approximation via a theoretical model for surface diffusion. Our mapping procedure between simulation and quasichemical theory reveals that sorbate–sorbate energetics emerge as the physical reason for the variation of corrected (Darken) and hence transport diffusivity with respect to pore size and sorbed phase fractional occupancy.     

A New Monte Carlo Method for Direct Calculation of the Critical Size and the Formation Work of a Microdrop

New Monte Carlo procedures in open ensembles are proposed. Non-stationary Markov chain procedure in the μl;pT - ensemble provides a direct estimation for the critical size of a condensation nucleus at given p and T. A stationary procedure in the μlpT ensemble with two allowed particle numbers n and n + 1 provides the direct way to calculate the chemical potential and Gibbs free energy of a cluster; in the grand canonical (μlVT) ensemble the same approach gives μl and the Helmholtz free energy. The same procedures are readily applicable to periodic systems representing bulk phases.     

Approximate quantum trajectory dynamics for reactive processes in condensed phase

A method of molecular dynamics with quantum corrections, practical for studies of large molecular systems, is reviewed. The approach is based on the Bohmian formulation of the time-dependent Schrödinger equation in which a wavefunction is represented by an ensemble of interdependent trajectories. The quantum effects come from the quantum potential acting on trajectories on par with the usual classical potential. The quantum potential is determined from the evolving nuclear wavefunction, i.e. from the quantum trajectory (QT) ensemble itself. For practical and conceptual reasons the quantum potential and corresponding quantum nuclear effect are computed only for the selected light nuclei. For studies of reactive chemical processes, the classical potential is computed on-the-fly using the density functional tight binding method of electronic structure. A massively parallel implementation, based on the message passing interface allows for efficient simulations of ensembles of thousands of trajectories describing systems of up to 200 atoms. As a biochemical application, the approximate QT approach is used to model the tunnelling-dominated proton transfer in soybean-lipoxygenase-1. A materials science application is represented by a study of the nuclear quantum effect on adsorption of hydrogen and deuterium on a C₃₇H₁₅ molecule, which is a model ‘flake’ of graphene.     

Grand canonical ensemble Monte Carlo simulation of the dCpG/proflavine crystal hydrate.

The grand canonical ensemble Monte Carlo molecular simulation method is used to investigate hydration patterns in the crystal hydrate structure of the dCpG/proflavine intercalated complex. The objective of this study is to show by example that the recently advocated grand canonical ensemble simulation is a computationally efficient method for determining the positions of the hydrating water molecules in protein and nucleic acid structures. A detailed molecular simulation convergence analysis and an analogous comparison of the theoretical results with experiments clearly show that the grand ensemble simulations can be far more advantageous than the comparable canonical ensemble simulations.     

On using oscillating time-dependent restraints in MD simulation

The use of time-dependent restraints in molecular simulation in order to generate a conformational ensemble for molecules that is in accordance with measured ensemble averages for particular observable quantities is investigated. Using a model system consisting of liquid butane and the cyclic peptide antamanide the reproduction of particular average ³ J-coupling constant values in a molecular dynamics simulation is analysed. It is shown that the multiple-valuedness and the sizeable gradients of the Karplus curve relating ³ J-coupling constants measured in NMR experiments to the corresponding torsional-angle values cause severe problems when trying to restrain a ³ J-coupling constant to a value close to the extrema of the Karplus curve. The introduction of a factor oscillating with time into the restraining penalty function alleviates this problem and enhances the restrained conformational sampling.     

Ions Everywhere? Mg2+ in the μ-Opioid GPCR and Atomic Details of Their Impact on Function

Filizola and co-workers have applied a combination of long-time molecular dynamics and oscillating chemical potential grand canonical Monte Carlo/molecular dynamics to investigate the distribution of Mg²⁺ and Na⁺ in the μ-opioid receptor and their impact on its function. Results indicate atomic details of potential mechanisms by which Mg²⁺ leads to increased efficacy of opioid analgesics. The presence of information flow between the extracellular loops and the intracellular region of the G-protein-coupled receptors that interacts with G-proteins in the presence of Mg²⁺ may be a phenomenon occurring in other G-protein-coupled receptors and, therefore, potentially of broad impact.     

Mode‐coupling Smoluchowski dynamics of a double‐stranded DNA oligomer

The local dynamics of a double‐stranded DNA d(TpCpGpCpG)₂ is obtained to second order in the mode‐coupling expansion of the Smoluchowski diffusion theory. The time correlation functions of bond variables are derived and the ¹³C‐nmr spin–lattice relaxation times T₁ of different ¹³C along the chains are calculated and compared to experimental data from the literature at three frequencies. The DNA is considered as a fluctuating three‐dimensional structure undergoing rotational diffusion. The fluctuations are evaluated using molecular dynamics simulations, with the ensemble averages approximated by time averages along a trajectory of length 1 ns. Any technique for sampling the configurational space can be used as an alternative. For a fluctuating three‐dimensional (3D) structure using the three first‐order vector modes of lower rates, higher order basis sets of second‐rank tensor are built to give the required mode coupling dynamics. Second‐ and even first‐order theories are found to be in close agreement with the experimental results, especially at high frequency, where the differences in T₁ for ¹³C in the base pairs, sugar, and backbone are well described. These atomistic calculations are of general application for studying, on a molecular basis, the local dynamics of fluctuating 3D structures such as double‐helix DNA fragments, proteins, and protein–DNA complexes. © 1999 John Wiley & Sons, Inc. Biopoly 50: 613–629, 1999     

Gibbs-Ensemble Molecular Dynamics: Liquid-Gas Equilibria for Lennard-Jones Spheres and n-Hexane

Abstract

We present a novel method to simulate phase equilibria in atomic and molecular systems. The method is a Molecular Dynamics version of the Gibbs-Ensemble Monte Carlo technique, which has been developed some years ago for the direct simulation of phase equilibria in fluid systems. The idea is to have two separate simulation boxes, which can exchange particles (or molecules) in a thermodynamically consistent fashion. Here we pres the derivation of the generalized equations of motion and discuss the relation of the resulting trajectory averages to the relevant ensemble. We test this Gibbs-Ensemble Molecular Dynamics algorithm by applying it to an atomic and a molecular system, i.e. to the liquid-gas coexistence in a Lennard-Jones fluid and in n-hexane. In both cases our results are in good accord with previous mean field and Gibbs-Ensemble Monte Carlo results as well as with the experimental data in the case of hexane. We also show that our Gibbs-Ensemble Molecular Dynamics algorithm like other Molecular Dynamics techniques can be used to study the dynamics of the system. Self-diffusion coefficients calculated with this method are in agreement with the result of conventional constant temperature Molecular Dynamics.     

Combined Diffusive and Viscous Transport of Methane in a Carbon Slit Pore

Abstract

The transport of mass through porous materials can occur by essentially two different mechanisms: (1) diffusion and (2) viscous flow. The former occurs when there is a gradient in chemical potential of the pore fluid, while the latter occurs in the presence of a pressure gradient. In general, fluid transport occurs by both of these mechanisms and their respective contributions to the total intra-pore flux are approximately additive. Experimentally, there is no unambiguous way of determining the individual contributions to the total flux of these two modes of transport. Fortunately, molecular simulations does provide a solution.

We present a novel simulation method in which the separate contributions to the total flux are determined. The method involves the use of two non-equilibrium molecular dynamics techniques: dual control volume grand canonical molecular dynamics (DCV GCMD) and an algorithm for simulating planar Poiseuille flow. We apply this technique to study the combined (viscous and diffusive) transport of methane through single slit-shaped graphite pores of width 2.5, 5.0 and 10.0 methane diameters. We find that the viscous contribution to the total intrapore flux through each of these pores is 10%, 15% and 34%, respectively.     

Effect of cut-off distance used in molecular dynamics simulations on fluid properties

The effect of cut-off distance used in molecular dynamics (MD) simulations on fluid properties was studied systematically in both canonical (NVT) and isothermal–isobaric (NPT) ensembles. Results show that the cut-off distance in the NVT ensemble plays little role in determining the equilibrium structure of fluid if the ensemble has a high density. However, pressures calculated in the same NVT ensembles strongly depend on the cut-off distance used. In the NPT ensemble, cut-off distance plays a key role in determining fluid equilibrium structure, density and self-diffusion coefficient. The characteristic of the radial distribution function of fluid in NPT ensembles depending on the cut-off distance used in MD simulations means that the WCA theory (a perturbation theory developed by Weeks, Chandler and Andersen) is not suitable for NPT ensembles because the assumption (the effect of the attractive force in determining the liquid structure is negligible) used in the WCA theory is not valid. The dependence of fluid properties on the cut-off distance also indicates that using the WCA potential (the repulsive part of the intermolecular potential proposed in the WCA theory) to calculate fluid transport in heterogeneous systems could lead to significant errors or incorrect results.     

On the Sampling Method for Grand Canonical Monte Carlo Simulations

Abstract

A bulk Lennard-Jones fluid was simulated using the grand canonical Monte Carlo method. Three different sampling methods were used in the transition matrix, namely the Metropolis, Barker and a third novel method. While it can be shown that the Metropolis method will give the most accurate ensemble averages in the limit of an infinitely long run, the new method termed “Modified Barker Sampling” (MBS), is shown to be superior for the runs of practical length for the particular system studied.     

Recent advances in maximum entropy biasing techniques for molecular dynamics

ABSTRACT

This review describes recent advances by the authors and others on the topic of incorporating experimental data into molecular simulations through maximum entropy methods. Methods which incorporate experimental data improve accuracy in molecular simulation by minimally modifying the thermodynamic ensemble. This is especially important where force fields are approximate, such as when employing coarse-grain models, or where high accuracy is required, such as when attempting to mimic a multiscale self-assembly process. The authors review here the experiment directed simulation (EDS) and experiment directed metadynamics (EDM) methods that allow matching averages and distributions in simulations, respectively. Important system-specific considerations are discussed such as using enhanced sampling simultaneously, the role of pressure, treating uncertainty, and implementations of these methods. Recent examples of EDS and EDM are reviewed including applications to ab initio molecular dynamics of water, incorporating environmental fluctuations inside of a macromolecular protein complex, improving RNA force fields, and the combination of enhanced sampling with minimal biasing to model peptides     

Folding energetics of ligand binding proteins. I. Theoretical model

Heat capacity curves as obtained from differential scanning calorimetry are an outstanding source for molecular information on protein folding and ligand-binding energetics. However, deconvolution of C(p) data of proteins in the presence of ligands can be compromised by indeterminacies concerning the correct choice of the statistical thermodynamic ensemble. By convent, the assumption of constant free ligand concentration has been used to derive formulae for the enthalpy. Unless the ligand occurs at large excess, this assumption is incorrect. Still the relevant ensemble is the grand canonical ensemble. We derive formulae for both constraints, constancy of total or free ligand concentration and illustrate the equations by application to the typical equilibrium Nx <=> N + x <=> D + x. It is demonstrated that as long as the thermodynamic properties of the ligand can be completely corrected for by performing a reference measurement, the grand canonical approach provides the proper and mathematically significantly simpler choice. We demonstrate on the two cases of sequential or independent ligand-binding the fact, that similar binding mechanisms result in different and distinguishable heat capacity equations. Finally, we propose adequate strategies for DSC experiments as well as for obtaining first estimates of the characteristic thermodynamic parameters, which can be used as starting values in a global fit of DSC data.     

Investigation of the influence of external factors on the conformational dynamics of rhodopsin-like receptors by means of molecular dynamics simulation

Abstract

The study reports about the influence of binding of orthosteric ligands on the conformational dynamics of β-2-adrenoreceptor. Using molecular dynamics (MD) simulation, we found that there was a little fraction of active states of the receptor in its apo (ligand-free) ensemble. Analysis of MD trajectories indicated that such spontaneous activation of the receptor is accompanied by the motion in intracellular part of its alpha-helices. Thus, receptor’s constitutive activity directly results from its conformational dynamics. On the other hand, the binding of a full agonist resulted in a significant shift of the initial equilibrium towards its active state. Finally, the binding of the inverse agonist stabilized the receptor in its inactive state. It is likely that the binding of inverse agonists might be a universal way of constitutive activity inhibition in vivo. Our results indicate that ligand binding redistribute pre-existing conformational degrees of freedom (in accordance to the Monod–Wyman–Changeux Model) of the receptor rather than cause induced fit in it. Therefore, the ensemble of biologically relevant receptor conformations is encoded in its spatial structure, and individual conformations from that ensemble might be used by the cell in conformity with the physiological behavior.     

Development of a grand canonical-kinetic Monte Carlo scheme for simulation of mixtures

A rejection-free methodology-based kinetic Monte Carlo (kMC) method has been developed in the grand canonical ensemble to simulate fluid mixtures. It comprises two different moves: entropic displacement of a selected molecule (based on the Rosenbluth algorithm) in the volume space of the system, and exchange of molecules with the surroundings (insertion or deletion). These two moves are made sequentially with M displacement moves followed by one exchange. The displacement moves are treated as sub-NVT sequences within a grand canonical ensemble. The procedure for deletion or insertion of a molecule is either, based on the Rosenbluth algorithm, or on a direct comparison, in which the average activity of one component is compared with its specified activity. The components are chosen either with equal probability or with a probability proportional to their density. The implementation of rejection-free kMC is much simpler than the Metropolis importance sampling MC procedure, which requires three different types of move, all of which must be tested for acceptance or rejection. The new scheme has been evaluated by applying it to fluid argon and to an equimolar mixture of methane, ethane and propane.     

NpH-MD-Simulations of the Elastic Moduli of Cellulose II at Room Temperatue

We have used molecular dynamics modeling to investigate the stucture and mechanical properties of regenerated cellulose fibres. This work is motivated by continued interest in replacing the environmentally hazardous viscose process by alternative spinning methods. An important input parameter for any realistic model of the elastic properties is the stiffness tensor of the crystalline constituent, cellulose II. Conventional molecular mechanics techniques can be used to estimate the elastic reaction of a material with respect to small external stresses or strains, i.e. the compliance and stiffness tensors, and the elastic moduli derived therefrom, at zero temperature. In order to access non-zero temperatures, it is necessary to use either the quasi-harmonic approximation for the vibrational free energy or molecular dynamics (MD) simulations. In the present work, Parrinello-Rahman constant-stress MD was performed to generate trajectories in constant particle number (N), constant external stress tensor (p or t) and constant enthalpy H (NpH or HtN) ensemble. This was found to be less time consuming than working with isothermal conditions, as done by other authors. The fluctuations in kinetic energy and MD cell vectors were then used to calculate adiabatic elastic constants, thermal expansion coefficients and heat capacity. The isothermal elastic constants were found by applying a standard thermodynamic relation. The Youngs modulus along the chain direction, E_l, was determined to be 155 GPa, whereas the values in the perpendicular directions vary between 51 and 24 GPa. These results are of the same order of magnitude as those obtained by Tashiro and Kobayashi [1] with the static (T = 0K) method, but our value of E_l is 5% lower and, unexpectedly, the lateral values are up to six times higher. A strong anisotropy is found for shear along the chains in planes containing the chain axis, the shear modulus ranging from 5 to 20 GPa. Convergence was achieved in the simulations, to the extend that the elastic constants become stationary, but significant internal stresses remain, pointing to shortcomings in the software used. Further work is necessary to resolve these problems, although the major conclusions should be unaffected.