计算平均力势的理论方法

A universal theoretical framework is proposed for calculating potential of mean force (PMF) between two solute particles immersed in a solvent bath, the present method overcomes all of drawbacks of previous methods. The only input required to implement the recipe is solvent density distribution profile around a single solute particle. The universal framework is applied to calculate the PMF between two large spherical particles immersed in small hard sphere solvent bath. Comparison between the present predictions and existing simulation data shows reliability of the present recipe. Effects of solvent-solute interaction detail, solvent bulk density, and solute size on the excess PMF are investigated. The resultant conclusion is that depletion of solvent component by the solute particle induces attractive excess PMF, while gathering of solvent component by the solute particle induces repulsive excess PMF, high solvent bulk density and large solute size can strengthen the tendency of attraction or repulsion. Relevance of transition from depletion attraction to gathering repulsion with the biomolecular interaction, i.e. hydrophobic attraction and hydration repulsion, is discussed.     

Do computed crystal structures of nonpolar molecules depend on the electrostatic interactions? The case of tetracene

We present a strategy for comparing the global properties of competing potential models. By systematically sampling the potential energy surface of crystalline tetracene, we assess how the number, energy and structure of its minima are modified by switching on (or off) the Coulombic interactions. The increased complexity of the Coulombic potential leads to a more "rugged" potential energy surface with a larger number of minima, but the effect is not large. In fact, we find a subset of minima stable only in presence of the Coulombic interactions, a smaller subset stable only in their absence, and a large majority stable in both cases. Among these, there is a very good, but not perfect, correlation between the energies and the structures computed with and without the electrostatic interactions. Although electrostatic interactions play a role even in a rigid nonpolar molecule such as tetracene, they are not as crucial as often believed, because altering the electrostatic model (or switching it off completely) leads, in most cases, to equivalent results.     

Efficient calculation of relative binding free energies by umbrella sampling perturbation

An important task of biomolecular simulation is the calculation of relative binding free energies upon chemical modification of partner molecules in a biomolecular complex. The potential of mean force (PMF) along a reaction coordinate for association or dissociation of the complex can be used to estimate binding affinities. A free energy perturbation approach, termed umbrella sampling (US) perturbation, has been designed that allows an efficient calculation of the change of the PMF upon modification of a binding partner based on the trajectories obtained for the wild type reference complex. The approach was tested on the interaction of modified water molecules in aqueous solution and applied to in silico alanine scanning of a peptide‐protein complex. For the water interaction test case, excellent agreement with an explicit PMF calculation for each modification was obtained as long as no long range electrostatic perturbations were considered. For the alanine scanning, the experimentally determined ranking and binding affinity changes upon alanine substitutions could be reproduced within 0.1–2.0 kcal/mol. In addition, good agreement with explicitly calculated PMFs was obtained mostly within the sampling uncertainty. The combined US and perturbation approach yields, under the condition of sufficiently small system modifications, rigorously derived changes in free energy and is applicable to any PMF calculation. © 2014 Wiley Periodicals, Inc.     

A coupled reference interaction site model/molecular dynamics study of the potential of mean force curve of the SN2 Cl- + CH3Cl reaction in water

An application of the coupled reference interaction site model (RISM)/simulation methodology to the calculation of the potential of mean force (PMF) curve in aqueous solution for the identity nucleophilic substitution reaction Cl(-) + CH(3)Cl is performed. The free energy of activation is calculated to be 27.1 kcal/mol which compares very well with the experimentally determined barrier height of 26.6 kcal/mol. Furthermore, the calculated PMF is almost superimposed with that previously calculated using the computationally rigorous Monte Carlo with importance sampling method (Chandrasekhar, J.; Smith, S. F.; Jorgensen, W. L. J. Am. Chem. Soc. 1985, 107, 154). Using the calculated PMF, a crude estimate of the solvated kinetic transmission coefficient also compares well with that of previous more accurate simulations. These results indicate that the coupled RISM/simulation method provides a cost-effective methodology for studying reactions in solution.     

Potential of mean force of hydrophobic association: dependence on solute size

The potentials of mean force (PMFs) were determined for systems involving formation of nonpolar dimers composed of methane, ethane, propane, isobutane, and neopentane, respectively, in water, using the TIP3P water model, and in vacuo. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field was carried out for each pair in either water or in vacuo. The PMFs were calculated by using the weighted histogram analysis method (WHAM). The shape of the PMFs for dimers of all five nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule, that is, for larger molecules they shift toward larger distances. The PMF of the neopentane dimer is similar to those of other small nonpolar molecules studied in this work, and hence the neopentane dimer is too small to be treated as a nanoscale hydrophobic object. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from the PMF in explicit solvent. The molecular surface area model correctly describes the solvent contribution to the PMF together with the changes of the height and positions of the desolvation barrier for all dimers investigated. The water molecules in the first solvation sphere of the dimer are more ordered compared to bulk water, with their dipole moments pointing away from the surface of the dimer. The average number of hydrogen bonds per water molecule in this first hydration shell is smaller compared to that in bulk water, which can be explained by coordination of water molecules to the hydrocarbon surface. In the second hydration shell, the average number of hydrogen bonds is greater compared to bulk water, which can be explained by increased ordering of water from the first hydration shell; the net effect is more efficient hydrogen bonding between the water molecules in the first and second hydration shells.     

Delineating Protein–Protein Curvilinear Dissociation Pathways and Energetics with Naïve Multiple-Walker Umbrella Sampling Simulations

The protein–protein interaction energetics can be obtained by calculating the potential of mean force (PMF) from umbrella sampling (US) simulations, in which samplings are often enhanced along a predefined vector as the reaction coordinate. However, any slight change in the vector may significantly vary the calculated PMF, and therefore the energetics using a random choice of vector may mislead. A non-predefined curve path-based sampling enhancement approach is a natural alternative, but was relatively less explored for protein–protein systems. In this work, dissociation of the barnase–barstar complex is simulated by implementing non-predefined curvilinear pathways in US simulations. A simple variational principle is applied to determine the lower bound PMF, which could be used to derive the standard free energy of binding. Two major dissociation pathways, which include interactions with the RNA-binding loop and the Val 36 to Gly 40 loop, are observed. Further, the proposed approach was used to discriminate the decoys from protein–protein docking studies. © 2019 Wiley Periodicals, Inc.     

Potential of mean force between hydrophobic solutes in the Jagla model of water and implications for cold denaturation of proteins

Using the Jagla model potential we calculate the potential of mean force (PMF) between hard sphere solutes immersed in a liquid displaying water-like properties. Consistent estimates of the PMF are obtained by (a) umbrella sampling, (b) calculating the work done by the mean force acting on the hard spheres as a function of their separation, and (c) determining the position dependent chemical potential after calculating the void space in the liquid. We calculate the PMF for an isobar along which cold denaturation of a model protein has previously been reported. We find that the PMF at contact varies non-monotonically, which is consistent with the observed cold denaturation. The Henry constant also varies non-monotonically with temperature. We find, on the other hand, that a second (solvent separated) minimum of the PMF becomes deeper as temperature decreases. We calculate the solvent-solvent pair correlation functions for solvents near the solute and in the bulk, and show that, as temperature decreases, the two pair correlation functions become indistinguishable, suggesting that the perturbation of solvent structure by the solute diminishes as temperature decreases. The solvent-solute pair correlation function at contact grows as the temperature decreases. We calculate the cavity correlation function and show the development of a solvent-separated peak upon decrease of temperature. These observations together suggest that cold denaturation occurs when the solvent penetrates between hydrophobic solutes in configurations with favorable free energy. Our results thus suggest that cold denatured proteins are structured and that cold denaturation arises from strong solvent-solute interactions, rather than from entropic considerations as in heat denaturation.     

A comparison of methods to compute the potential of mean force.

Most processes occurring in a system are determined by the relative free energy between two or more states because the free energy is a measure of the probability of finding the system in a given state. When the two states of interest are connected by a pathway, usually called reaction coordinate, along which the free-energy profile is determined, this profile or potential of mean force (PMF) will also yield the relative free energy of the two states. Twelve different methods to compute a PMF are reviewed and compared, with regard to their precision, for a system consisting of a pair of methane molecules in aqueous solution. We analyze all combinations of the type of sampling (unbiased, umbrella-biased or constraint-biased), how to compute free energies (from density of states or force averaging) and the type of coordinate system (internal or Cartesian) used for the PMF degree of freedom. The method of choice is constraint-bias simulation combined with force averaging for either an internal or a Cartesian PMF degree of freedom.     

Quantum mechanics/molecular mechanics minimum free-energy path for accurate reaction energetics in solution and enzymes: sequential sampling and optimization on the potential of mean force surface

To accurately determine the reaction path and its energetics for enzymatic and solution-phase reactions, we present a sequential sampling and optimization approach that greatly enhances the efficiency of the ab initio quantum mechanics/molecular mechanics minimum free-energy path (QM/MM-MFEP) method. In the QM/MM-MFEP method, the thermodynamics of a complex reaction system is described by the potential of mean force (PMF) surface of the quantum mechanical (QM) subsystem with a small number of degrees of freedom, somewhat like describing a reaction process in the gas phase. The main computational cost of the QM/MM-MFEP method comes from the statistical sampling of conformations of the molecular mechanical (MM) subsystem required for the calculation of the QM PMF and its gradient. In our new sequential sampling and optimization approach, we aim to reduce the amount of MM sampling while still retaining the accuracy of the results by first carrying out MM phase-space sampling and then optimizing the QM subsystem in the fixed-size ensemble of MM conformations. The resulting QM optimized structures are then used to obtain more accurate sampling of the MM subsystem. This process of sequential MM sampling and QM optimization is iterated until convergence. The use of a fixed-size, finite MM conformational ensemble enables the precise evaluation of the QM potential of mean force and its gradient within the ensemble, thus circumventing the challenges associated with statistical averaging and significantly speeding up the convergence of the optimization process. To further improve the accuracy of the QM/MM-MFEP method, the reaction path potential method developed by Lu and Yang [Z. Lu and W. Yang, J. Chem. Phys. 121, 89 (2004)] is employed to describe the QM/MM electrostatic interactions in an approximate yet accurate way with a computational cost that is comparable to classical MM simulations. The new method was successfully applied to two example reaction processes, the classical SN2 reaction of Cl-+CH3Cl in solution and the second proton transfer step of the reaction catalyzed by the enzyme 4-oxalocrotonate tautomerase. The activation free energies calculated with this new sequential sampling and optimization approach to the QM/MM-MFEP method agree well with results from other simulation approaches such as the umbrella sampling technique with direct QM/MM dynamics sampling, demonstrating the accuracy of the iterative QM/MM-MFEP method.     

Molecular Dynamics Simulations of Water-Mediated Cholesterol Capture within an Open-Ended Single-Walled Carbon Nanotube

The excess concentration of cholesterol in the bloodstream can be brought down to a safer level by utilizing a potential cholesterol-binding agent such as a carbon nanotube (CNT). Here, we have probed solvent-mediated interactions between cholesterol and CNT by performing molecular dynamics simulations and potential-of-mean force (PMF) calculations. Simulations predict favorable interactions between water-mediated cholesterol and CNT owing to strong mutual interactions between them, whereas water plays an opposing role in the association. The breakdown of PMF into its enthalpic and entropic contributions indicates that contrary to traditional entropy-driven hydrophobic association, the cholesterol encapsulation within a CNT is primarily driven by enthalpy.     

Critical and phase-equilibrium properties of an ab initio based potential model of methanol and 1-propanol using two-phase molecular dynamics simulations

Two-phase molecular dynamics simulations employing a Monte Carlo volume sampling method were performed using an ab initio based force field model parameterized to reproduce quantum-mechanical dimer energies for methanol and 1-propanol at temperatures approaching the critical temperature. The intermolecular potential models were used to obtain the binodal vapor-liquid phase dome at temperatures to within about 10 K of the critical temperature. The efficacy of two all-atom, site-site pair potential models, developed solely from the energy landscape obtained from high-level ab initio pair interactions, was tested for the first time. The first model was regressed from the ab initio landscape without point charges using a modified Morse potential to model the complete interactions; the second model included point charges to separate Coulombic and dispersion interactions. Both models produced equivalent phase domes and critical loci. The model results for the critical temperature, density, and pressure, in addition to the sub-critical equilibrium vapor and liquid densities and vapor pressures, are compared to experimental data. The model's critical temperature for methanol is 77 K too high while that for 1-propanol is 80 K too low, but the critical densities are in good agreement. These differences are likely attributable to the lack of multi-body interactions in the true pair potential models used here.     

Modifying the OPLS-AA force field to improve hydration free energies for several amino acid side chains using new atomic charges and an off-plane charge model for aromatic residues

The hydration free energies of amino acid side chains are an important determinant of processes that involve partitioning between different environments, including protein folding, protein complex formation, and protein-membrane interactions. Several recent papers have shown that calculated hydration free energies for polar and aromatic residues (Trp, His, Tyr, Asn, Gln, Asp, Glu) in several common molecular dynamics force fields differ significantly from experimentally measured values. We have attempted to improve the hydration energies for these residues by modifying the partial charges of the OPLS-AA force field based on natural population analysis of density functional theory calculations. The resulting differences between calculated hydration free energies and experimental results for the seven side chain analogs are less than 0.1 kcal/mol. Simulations of the synthetic Trp-rich peptide Trpzip2 show that the new charges lead to significantly improved geometries for interacting Trp-side chains. We also investigated an off-plane charge model for aromatic rings that more closely mimics their electronic configuration. This model results in an improved free energy of hydration for Trp and a somewhat altered benzene-sodium potential of mean force with a more favorable energy for direct benzene-sodium contact.     

Binding free energy, energy and entropy calculations using simple model systems

Free energy differences are calculated for a set of two model host molecules, binding acetone and methanol. Two active sites of different characteristics were constructed based on an artificially extended C60 fullerene molecule, possibly functionalised to include polar interactions in an otherwise apolar, spherical cavity. The model host systems minimise the necessary sampling of conformational space while still capturing key aspects of ligand binding. The estimates of the free energies are split up into energetic and entropic contributions, using three different approaches investigating the convergence behaviour. For these systems, a direct calculation of the total energy and entropy is more efficient than calculating the entropy from the temperature dependence of the free energy or from a direct thermodynamic integration formulation. Furthermore, the compensating surrounding–surrounding energies and entropies are split off by calculating reduced ligand-surrounding energies and entropies. These converge much more readily and lead to properties that are more straightforwardly interpreted in terms of molecular interactions and configurations. Even though not experimentally accessible, the reduced thermodynamic properties may prove highly relevant for computational drug design, as they may give direct insights into possibilities to further optimise ligand binding while optimisation in the surrounding–surrounding energy or entropy will exactly cancel and not lead to improved affinity.     

Converging free energies of binding in cucurbit[7]uril and octa-acid host–guest systems from SAMPL4 using expanded ensemble simulations

Molecular containers such as cucurbit[7]uril (CB7) and the octa-acid (OA) host are ideal simplified model test systems for optimizing and analyzing methods for computing free energies of binding intended for use with biologically relevant protein–ligand complexes. To this end, we have performed initially blind free energy calculations to determine the free energies of binding for ligands of both the CB7 and OA hosts. A subset of the selected guest molecules were those included in the SAMPL4 prediction challenge. Using expanded ensemble simulations in the dimension of coupling host–guest intermolecular interactions, we are able to show that our estimates in most cases can be demonstrated to fully converge and that the errors in our estimates are due almost entirely to the assigned force field parameters and the choice of environmental conditions used to model experiment. We confirm the convergence through the use of alternative simulation methodologies and thermodynamic pathways, analyzing sampled conformations, and directly observing changes of the free energy with respect to simulation time. Our results demonstrate the benefits of enhanced sampling of multiple local free energy minima made possible by the use of expanded ensemble molecular dynamics and may indicate the presence of significant problems with current transferable force fields for organic molecules when used for calculating binding affinities, especially in non-protein chemistries.     

The Role of Binary and Many-centre Molecular Interactions in Spin Crossover in the Solid State. Part I. Equation for Free Energy

A formalism has been developed that describes spin crossover equilibrium in the solid state by taking into account the effects of n nearest neighbours of a given molecule on its partition function. In this way binary and many-body interactions of the order n + 1 are included into the theoretical model and represented by non-ideality parameters connected with the splitting of free energy levels. Binary interactions are characterised by the main splittings whereas higher order interactions manifest themselves in asymmetries of splittings within multiplets. The contribution of molecular interactions can also be written in terms of formal excess free energies of the second, third, fourth and higher orders. Simple relationships between excess free energies and parameters of multiplets have been found for binary, ternary and quaternary interactions. This formalism is reduced to that of the model of binary interactions when effects of surroundings are additive leading to equidistant free energy multiplets. Higher order interactions may cause an abrupt spin crossover but in a limited range of compositions around the transition point. The regression of experimental transition curves of one-step spin crossover may yield estimates of excess energies up to the fifth order.     

The Role of Binary and Many-centre Molecular Interactions in Spin Crossover in the Solid State. Part I. Equation for Free Energy

Summary. A formalism has been developed that describes spin crossover equilibrium in the solid state by taking into account the effects of n nearest neighbours of a given molecule on its partition function. In this way binary and many-body interactions of the order n + 1 are included into the theoretical model and represented by non-ideality parameters connected with the splitting of free energy levels. Binary interactions are characterised by the main splittings whereas higher order interactions manifest themselves in asymmetries of splittings within multiplets. The contribution of molecular interactions can also be written in terms of formal excess free energies of the second, third, fourth and higher orders. Simple relationships between excess free energies and parameters of multiplets have been found for binary, ternary and quaternary interactions. This formalism is reduced to that of the model of binary interactions when effects of surroundings are additive leading to equidistant free energy multiplets. Higher order interactions may cause an abrupt spin crossover but in a limited range of compositions around the transition point. The regression of experimental transition curves of one-step spin crossover may yield estimates of excess energies up to the fifth order.     

The incorporation of hydration forces determined by continuum electrostatics into molecular mechanics simulations

A new technique is presented for incorporating hydration forces into molecular mechanics simulations. The method assumes the classical continuum approximation, where a solvated molecule is represented as a low-dielectric cavity of arbitrary shape embedded in a continuous region of high dielectric constant. Electrostatic effects are computed by first calculating the distribution of polarization charge (induced by the configuration of solute fixed charges) at the molecular surface. The hydration force at a particular atom is then found as the sum of the coulombic interaction with the induced surface charge, plus a purely mechanical contribution that arises from the pressure of the polarized solvent as it is pulled toward the solute. A procedure is developed to use the computed hydration forces in conjunction with the CHARMM molecular mechanics package to carry out energy minimizations in which the effects of solvation are explicitly included. This new technique also allows a detailed analysis of the relative balance of coulombic, hydration, and steric energies as a function of molecular conformation. The method is applied to the test case of a zwitterionic tripeptide (LYS-GLY-GLU), and the computational results suggest that hydration effects can play a significant role in determining a stable conformation for a solvated polar molecule. The future application to larger molecules is discussed.     

Computation of methodology-independent ionic solvation free energies from molecular simulations. I. The electrostatic potential in molecular liquids

The computation of ionic solvation free energies from atomistic simulations is a surprisingly difficult problem that has found no satisfactory solution for more than 15 years. The reason is that the charging free energies evaluated from such simulations are affected by very large errors. One of these is related to the choice of a specific convention for summing up the contributions of solvent charges to the electrostatic potential in the ionic cavity, namely, on the basis of point charges within entire solvent molecules (M scheme) or on the basis of individual point charges (P scheme). The use of an inappropriate convention may lead to a charge-independent offset in the calculated potential, which depends on the details of the summation scheme, on the quadrupole-moment trace of the solvent molecule, and on the approximate form used to represent electrostatic interactions in the system. However, whether the M or P scheme (if any) represents the appropriate convention is still a matter of on-going debate. The goal of the present article is to settle this long-standing controversy by carefully analyzing (both analytically and numerically) the properties of the electrostatic potential in molecular liquids (and inside cavities within them). Restricting the discussion to real liquids of "spherical" solvent molecules (represented by a classical solvent model with a single van der Waals interaction site), it is concluded that (i) for Coulombic (or straight-cutoff truncated) electrostatic interactions, the M scheme is the appropriate way of calculating the electrostatic potential; (ii) for non-Coulombic interactions deriving from a continuously differentiable function, both M and P schemes generally deliver an incorrect result (for which an analytical correction must be applied); and (iii) finite-temperature effects, including intermolecular orientation correlations and a preferential orientational structure in the neighborhood of a liquid-vacuum interface, must be taken into account. Applications of these results to the computation methodology-independent ionic solvation free energies from molecular simulations will be the scope of a forthcoming article.     

The multiscale coarse-graining method. VII. Free energy decomposition of coarse-grained effective potentials

The potential of mean force (PMF) with respect to coarse-grained (CG) coordinates is often calculated in order to study the molecular interactions in atomistic molecular dynamics (MD) simulations. The multiscale coarse-graining (MS-CG) approach enables the computation of the many-body PMF of an atomistic system in terms of the CG coordinates, which can be used to parameterize CG models based on all-atom configurations. We demonstrate here that the MS-CG method can also be used to analyze the CG interactions from atomistic MD trajectories via PMF calculations. In addition, MS-CG calculations at different temperatures are performed to decompose the PMF values into energetic and entropic contributions as a function of the CG coordinates, which provides more thermodynamic information regarding the atomistic system. Two numerical examples, liquid methanol and a dimyristoylphosphatidylcholine lipid bilayer, are presented. The results show that MS-CG can be used as an analysis tool, comparable to various free energy computation methods. The differences between the MS-CG approach and other PMF calculation methods, as well as the characteristics and advantages of MS-CG, are also discussed.     

Role of attractive methane-water interactions in the potential of mean force between methane molecules in water

On the basis of a Gaussian quasichemical model of hydration, a model of non-van der Waals character, we explore the role of attractive methane-water interactions in the hydration of methane and in the potential of mean force between two methane molecules in water. We find that the hydration of methane is dominated by packing and a mean-field energetic contribution. Contributions beyond the mean-field term are unimportant in the hydration phenomena for a hydrophobic solute such as methane. Attractive solute-water interactions make a net repulsive contribution to these pair potentials of mean force. With no conditioning, the observed distributions of binding energies are super-Gaussian and can be effectively modeled by a Gumbel (extreme value) distribution. This further supports the view that the characteristic form of the unconditioned distribution in the high-epsilon tail is due to energetic interactions with a small number of molecules. Generalized extreme value distributions also effectively model the results with minimal conditioning, but in those cases the distributions are sufficiently narrow that the details of their shape are not significant.