Simple estimation of absolute free energies for biomolecules

One reason that free energy difference calculations are notoriously difficult in molecular systems is due to insufficient conformational overlap, or similarity, between the two states or systems of interest. The degree of overlap is irrelevant, however, if the absolute free energy of each state can be computed. We present a method for calculating the absolute free energy that employs a simple construction of an exactly computable reference system which possesses high overlap with the state of interest. The approach requires only a physical ensemble of conformations generated via simulation and an auxiliary calculation of approximately equal central-processing-unit cost. Moreover, the calculations can converge to the correct free energy value even when the physical ensemble is incomplete or improperly distributed. As a "proof of principle," we use the approach to correctly predict free energies for test systems where the absolute values can be calculated exactly and also to predict the conformational equilibrium for leucine dipeptide in implicit solvent.     

Synergistic approach to improve "alchemical" free energy calculation in rugged energy surface

The authors present an integrated approach to "alchemical" free energy simulation, which permits efficient calculation of the free energy difference on rugged energy surface. The method is designed to obtain efficient canonical sampling for rapid free energy convergence. The proposal is motivated by the insight that both the exchange efficiency in the presently designed dual-topology alchemical Hamiltonian replica exchange method (HREM), and the confidence of the free energy determination using the overlap histogramming method, depend on the same criterion, viz., the overlaps of the energy difference histograms between all pairs of neighboring states. Hence, integrating these two techniques can produce a joint solution to the problems of the free energy convergence and conformational sampling in the free energy simulations, in which lambda parameter plays two roles to simultaneously facilitate the conformational sampling and improve the phase space overlap for the free energy determination. Specifically, in contrast with other alchemical HREM based free energy simulation methods, the dual-topology approach can ensure robust conformational sampling. Due to these features (a synergistic solution to the free energy convergence and canonical sampling, and the improvement of the sampling efficiency with the dual-topology treatment), the present approach, as demonstrated in the model studies of the authors, is highly efficient in obtaining accurate free energy differences, especially for the systems with rough energy landscapes.     

Metadynamics in essential coordinates: free energy simulation of conformational changes

We propose an approach that combines an extraction of collective motions of a molecular system with a sampling of its free energy surface. A recently introduced method of metadynamics allows exploration of the free energy surface of a molecular system by means of coarse-grained dynamics with flooding of free energy minima. This free energy surface is defined as a function of a set of collective variables (e.g., interatomic distances, angles, torsions, and others). In this study, essential coordinates determined by essential dynamics (principle component analysis) were used as collective variables in metadynamics. First, dynamics of the model system (explicitly solvated alanine dipeptide, Ace-Ala-Nme) was simulated by a classical molecular dynamics simulation. The trajectory (1 ns) was then analyzed by essential dynamics to obtain essential coordinates. The free energy surface as a function of the first and second essential coordinates was then explored by metadynamics. The resulting free energy surface is in agreement with other studies of this system. We propose that a combination of these two methods (metadynamics and essential dynamics) has great potential in studies of conformational changes in peptides and proteins.     

Multiple free energies from a single simulation: extending enveloping distribution sampling to nonoverlapping phase-space distributions

A recently proposed method to obtain free energy differences for multiple end states from a single simulation of a reference state which was called enveloping distribution sampling (EDS) [J. Chem. Phys. 126, 184110 (2007)] is expanded to situations where the end state configuration space densities do not show overlap. It uses a reference state Hamiltonian suggested by Han in 1992 [Phys. Lett. A 165, 28 (1992)] in a molecular dynamics implementation. The method allows us to calculate multiple free energy differences "on the fly" from a single molecular dynamics simulation. The influence of the parameters on the accuracy and precision of the obtained free energy differences is investigated. A connection is established between the presented method and the Bennett acceptance ratio method. The method is applied to four two-state test systems (dipole inversion, van der Waals perturbation, charge inversion, and water to methanol conversion) and two multiple-state test systems [dipole inversion with five charging states and five (dis-)appearing water molecules]. Accurate results could be obtained for all test applications if the parameters of the reference state Hamiltonian were optimized according to a given algorithm. The deviations from the exact result or from an independent calculation were at most 0.6 kJ/mol. An accurate estimation of the free energy difference is always possible, independent of how different the end states are. However, the convergence times of the free energy differences are longer in cases where the end state configuration space densities do not show overlap [charge inversion, water to methanol conversion, (dis-)appearing water molecules] than in cases where the configuration space densities do show some overlap [(multiple) dipole inversion and van der Waals perturbation].     

Evaluation of configurational entropy methods from peptide folding-unfolding simulation

A 4-micros molecular dynamics simulation of the second beta-hairpin of the B1 domain of streptococcal protein G is used to characterize the free energy surface and to evaluate different configurational entropy estimators. From the equilibrium folding-unfolding trajectory, 200 000 conformers are clustered according to their root-mean-square deviation (RMSD). The height of the free energy barrier between pairs of clusters is found to be significantly correlated with their pairwise RMSD. Relative free energies and relative configurational entropies of the clusters are determined by explicit evaluation of the partition functions of the different clusters. These entropies are used to evaluate different entropy estimators for the largest 20 clusters as well as a subensemble comprising exclusively extended conformers. It is found that the quasi-harmonic entropy estimator operating in dihedral angle space performs better than the one using Cartesian coordinates. A recent generalization of the quasi-harmonic approach that computes Shannon entropies of probability distributions obtained by projecting the conformers along the eigenvectors of the covariance matrix performs similarly well. For the best entropy estimators, a linear correlation coefficient between 0.92 and 0.97 is found. Unexpectedly, when correlations between dihedral angles are neglected, the agreement with the reference entropies improved.     

Dihedral angle principal component analysis of molecular dynamics simulations

It has recently been suggested by Mu et al. [Proteins 58, 45 (2005)] to use backbone dihedral angles instead of Cartesian coordinates in a principal component analysis of molecular dynamics simulations. Dihedral angles may be advantageous because internal coordinates naturally provide a correct separation of internal and overall motion, which was found to be essential for the construction and interpretation of the free energy landscape of a biomolecule undergoing large structural rearrangements. To account for the circular statistics of angular variables, a transformation from the space of dihedral angles {phi(n)} to the metric coordinate space {x(n)=cos phi(n),y(n)=sin phi(n)} was employed. To study the validity and the applicability of the approach, in this work the theoretical foundations underlying the dihedral angle principal component analysis (dPCA) are discussed. It is shown that the dPCA amounts to a one-to-one representation of the original angle distribution and that its principal components can readily be characterized by the corresponding conformational changes of the peptide. Furthermore, a complex version of the dPCA is introduced, in which N angular variables naturally lead to N eigenvalues and eigenvectors. Applying the methodology to the construction of the free energy landscape of decaalanine from a 300 ns molecular dynamics simulation, a critical comparison of the various methods is given.     

A hybrid recursion method to robustly ensure convergence efficiencies in the simulated scaling based free energy simulations

Recently, we developed an efficient free energy simulation technique, the simulated scaling (SS) method [H. Li et al., J. Chem. Phys. 126, 024106 (2007)], in the framework of generalized ensemble simulations. In the SS simulations, random walks in the scaling parameter space are realized so that both phase space overlap sampling and conformational space sampling can be simultaneously enhanced. To flatten the distribution in the scaling parameter space, in the original SS implementation, the Wang-Landau recursion was employed due to its well-known recursion capability. In the Wang-Landau recursion based SS free energy simulation scheme, at the early stage, recursion efficiencies are high and free energy regions are quickly located, although at this stage, the errors of estimated free energy values are large; at the later stage, the errors of estimated free energy values become smaller, however, recursions become increasingly slow and free energy refinements require very long simulation time. In order to robustly resolve this efficiency problem during free energy refinements, a hybrid recursion strategy is presented in this paper. Specifically, we let the Wang-Landau update method take care of the early stage recursion: the location of target free energy regions, and let the adaptive reweighting method take care of the late stage recursion: the refinements of free energy values. As comparably studied in the model systems, among three possible recursion procedures, the adaptive reweighting recursion approach is the least favorable one because of its low recursion efficiency during free energy region locations; and compared to the original Wang-Landau recursion approach, the proposed hybrid recursion technique can be more robust to guarantee free energy simulation efficiencies.     

Construction of the free energy landscape of biomolecules via dihedral angle principal component analysis

A systematic approach to construct a low-dimensional free energy landscape from a classical molecular dynamics (MD) simulation is presented. The approach is based on the recently proposed dihedral angle principal component analysis (dPCA), which avoids artifacts due to the mixing of internal and overall motions in Cartesian coordinates and circumvents problems associated with the circularity of angular variables. Requiring that the energy landscape reproduces the correct number, energy, and location of the system's metastable states and barriers, the dimensionality of the free energy landscape (i.e., the number of essential components) is obtained. This dimensionality can be determined from the distribution and autocorrelation of the principal components. By performing an 800 ns MD simulation of the folding of hepta-alanine in explicit water and using geometric and kinetic clustering techniques, it is shown that a five-dimensional dPCA energy landscape is a suitable and accurate representation of the full-dimensional landscape. In the second step, the dPCA energy landscape can be employed (e.g., in a Langevin simulation) to facilitate a detailed investigation of biomolecular dynamics in low dimensions. Finally, several ways to visualize the multidimensional energy landscape are discussed.     

Combination of molecular dynamics method and 3D-RISM theory for conformational sampling of large flexible molecules in solution

We have developed an algorithm for sampling the conformational space of large flexible molecules in solution, which combines the molecular dynamics (MD) method and the three-dimensional reference interaction site model (3D-RISM) theory. The solvent-induced force acting on solute atoms was evaluated as the gradient of the solvation free energy with respect to the solute-atom coordinates. To enhance the computation speed, we have applied a multiple timestep algorithm based on the RESPA (Reversible System Propagator Algorithm) to the combined MD/3D-RISM method. By virtue of the algorithm, one can choose a longer timestep for renewing the solvent-induced force compared with that of the conformational update. To illustrate the present MD/3D-RISM simulation, we applied the method to a model of acetylacetone in aqueous solution. The multiple timestep algorithm succeeded in enhancing the computation speed by 3.4 times for this model case. Acetylacetone possesses an intramolecular hydrogen-bonding capability between the hydroxyl group and the carbonyl oxygen atom, and the molecule is significantly stabilized due to this hydrogen bond, especially in gas phase. The intramolecular hydrogen bond was kept intact during almost entire course of the MD simulation in gas phase, while in the aqueous solutions the bond is disrupted in a significant number of conformations. This result qualitatively agrees with the behavior on a free energy barrier lying upon the process for rotating a torsional degree of freedom of the hydroxyl group, where it is significantly reduced in aqueous solution by a cancellation between the electrostatic interaction and the solvation free energy.     

Computing Alchemical Free Energy Differences with Hamiltonian Replica Exchange Molecular Dynamics (H-REMD) Simulations

Alchemical free energy calculations play a very important role in the field of molecular modeling. Efforts have been made to improve the accuracy and precision of those calculations. One of the efforts is to employ a Hamiltonian replica exchange molecular dynamics (H-REMD) method to enhance conformational sampling. In this paper, we demonstrated that HREMD method not only improves convergence in alchemical free energy calculations but also can be used to compute free energy differences directly via the Free Energy Perturbation (FEP)algorithm. We show a direct mapping between the H-REMD and the usual FEP equations, which are then used directly to compute free energies. The H-REMD alchemical free energy calculation (Replica exchange Free Energy Perturbation, REFEP) was tested on predicting the pK(a) value of the buried Asp26 in thioredoxin. We compare the results of REFEP with TI and regular FEP simulations. REFEP calculations converged faster than those from TI and regular FEP simulations. The final predicted pK(a) value from the H-REMD simulation was also very accurate, only 0.4 pK(a) unit above the experimental value. Utilizing the REFEP algorithm significantly improves conformational sampling, and this in turn improves the convergence of alchemical free energy simulations.     

Tork: Conformational analysis method for molecules and complexes

A conformational search method for organic molecules and bimolecular complexes is presented. The method, termed Tork, uses normal-mode analysis in bond-angle-torsion coordinates and focuses on a key subset of torsional coordinates to identify natural molecular motions that lead the initial conformation to new energy minima. New conformations are generated via distortion along these modes and their pairwise combinations, followed by energy minimization. For complexes, special treatment is accorded to the six coordinates that specify the position and orientation of one molecule relative to the other. Tests described here show that Tork is highly efficient for cyclic, acyclic, and mixed single molecules, as well as for host-guest complexes.     

Extraction of configurational entropy from molecular simulations via an expansion approximation

A method is presented for extracting the configurational entropy of solute molecules from molecular dynamics simulations, in which the entropy is computed as an expansion of multidimensional mutual information terms, which account for correlated motions among the various internal degrees of freedom of the molecule. The mutual information expansion is demonstrated to be equivalent to estimating the full-dimensional configurational probability density function (PDF) using the generalized Kirkwood superposition approximation (GKSA). While the mutual information expansion is derived to the full dimensionality of the molecule, the current application uses a truncated form of the expansion in which all fourth- and higher-order mutual information terms are neglected. Truncation of the mutual information expansion at the nth order is shown to be equivalent to approximating the full-dimensional PDF using joint PDFs with dimensionality of n or smaller by successive application of the GKSA. The expansion method is used to compute the absolute (classical) configurational entropy in a basis of bond-angle-torsion internal coordinates for several small molecules as well as the change in entropy upon binding for a small host-guest system. Convergence properties of the computed entropy values as a function of simulation time are investigated and comparisons are made with entropy values from the second generation Mining Minima software. These comparisons demonstrate a deviation in -TS of no more than about 2 kcal/mol for all cases in which convergence has been obtained.     

Conformational sensitivity of polyether macrocycles to electrostatic potential: Partial atomic charges,molecular mechanics,and conformational prediction

Molecular recognition (whether by enzymes, the immune system, or chelating ligands) depends critically on molecular conformation. Molecular mechanics predicts energetically favorable molecular conformations by locating low energy conformations using an empirical fit of molecular potential energy as a function of internal coordinates. Molecular mechanics analysis of 18-crown-6 demonstrates that the nonbonded term (primarily the electrostatic part) is the largest contributor to the conformational energy. Nevertheless, common methods of treating the electrostatic interaction for 18-crown-6 yield inconsistent values for conformational energies partly because partial charges assigned to each atom can change with conformation due to through-space inductive effects which are not considered in most molecular mechanics programs. Similar findings from several other groups are reviewed to support our conclusions. We argue for care and caution in predicting conformational preferences of molecules with two or more highly polar atoms. We also discuss the desirability of using an empirical method of partial charge determination such as the charge equilibration algorithm of Rappé and Goddard (or a suitable generalization which includes polarization) as a method of including these effects in molecular mechanics and molecular dynamics calculations.     

Potential energy constrained molecular dynamics simulations

A method for carrying out molecular dynamics simulations in which the potential energy U of the molecular system is constrained at its initial value is developed and thoroughly tested. The constraint is not introduced within the framework of the Lagrange multipliers technique, rather it is fulfilled in a natural way by carrying out the simulations in terms of suitable sets of delocalized coordinates. Such coordinates are defined by an appropriate tuning of the Baker, Kessi, and Delley internal delocalized nonredundant coordinates technique [J. Chem. Phys. 105, 192 (1996)]. The proposed method requires multiple evaluations of energy and gradients in each step of the molecular dynamics simulation, so that constant U simulations suffer some overhead compared to ordinary simulations. But the particular formulation of the delocalized coordinates and of the equations of motion greatly simplifies all the various steps required by the Baker's technique, thus allowing for the efficient implementation of the method itself. The technique is reliable and allows for very high accuracy in the potential energy conservation during the whole simulation. Moreover, it proved to be free of drift troubles which can occur when standard constraint methods are straightforwardly implemented without the application of appropriate correcting techniques.     

Enveloping distribution sampling: a method to calculate free energy differences from a single simulation

The authors present a method to calculate free energy differences between two states A and B "on the fly" from a single molecular dynamics simulation of a reference state R. No computer time has to be spent on the simulation of intermediate states. Only one state is sampled, i.e., the reference state R which is designed such that the subset of phase space important to it is the union of the parts of phase space important to A and B. Therefore, an accurate estimate of the relative free energy can be obtained by construction. The authors applied the method to four test systems (dipole inversion, van der Waals interaction perturbation, charge inversion, and water to methanol conversion) and compared the results to thermodynamic integration estimates. In two cases, the enveloping distribution sampling calculation was straightforward. However, in the charge inversion and the water to methanol conversion, Hamiltonian replica-exchange molecular dynamics of the reference state was necessary to observe transitions in the reference state simulation between the parts of phase space important to A and B, respectively. This can be explained by the total absence of phase space overlap of A and B in these two cases.     

Molecular dynamics coupled with a virtual system for effective conformational sampling

An enhanced conformational sampling method is proposed: virtual‐system coupled canonical molecular dynamics (VcMD). Although VcMD enhances sampling along a reaction coordinate, this method is free from estimation of a canonical distribution function along the reaction coordinate. This method introduces a virtual system that does not necessarily obey a physical law. To enhance sampling the virtual system couples with a molecular system to be studied. Resultant snapshots produce a canonical ensemble. This method was applied to a system consisting of two short peptides in an explicit solvent. Conventional molecular dynamics simulation, which is ten times longer than VcMD, was performed along with adaptive umbrella sampling. Free‐energy landscapes computed from the three simulations mutually converged well. The VcMD provided quicker association/dissociation motions of peptides than the conventional molecular dynamics did. The VcMD method is applicable to various complicated systems because of its methodological simplicity. © 2018 Wiley Periodicals, Inc.     

Using Selectively Applied Accelerated Molecular Dynamics to Enhance Free Energy Calculations

Accelerated molecular dynamics (aMD) has been shown to enhance conformational space sampling relative to classical molecular dynamics; however, the exponential reweighting of aMD trajectories, which is necessary for the calculation of free energies relating to the classical system, is oftentimes problematic, especially for systems larger than small poly peptides. Here, we propose a method of accelerating only the degrees of freedom most pertinent to sampling, thereby reducing the total acceleration added to the system and improving the convergence of calculated ensemble averages, which we term selective aMD. Its application is highlighted in two biomolecular cases. First, the model system alanine dipeptide is simulated with classical MD, all-dihedral aMD, and selective aMD, and these results are compared to the infinite sampling limit as calculated with metadynamics. We show that both forms of aMD enhance the convergence of the underlying free energy landscape by 5-fold relative to classical MD; however, selective aMD can produce improved statistics over all-dihedral aMD due to the improved reweighting. Then we focus on the pharmaceutically relevant case of computing the free energy of the decoupling of oseltamivir in the active site of neuraminidase. Results show that selective aMD greatly reduces the cost of this alchemical free energy transformation, whereas all-dihedral aMD produces unreliable free energy estimates.     

On the convergence improvement in the metadynamics simulations: a Wang-Landau recursion approach

As a popular tool in exploring free energy landscapes, the metadynamics method has been widely applied to elucidate various chemical or biochemical processes. As deeply discussed by Laio et al. [J. Phys. Chem. B 109, 6714 (2005)], the size of the updating Gaussian function is pivotal to the free energy convergence toward the target free energy surface. For instance, a greater Gaussian height can facilitate the quick visit of a conformation region of interest; however, it may lead to a larger error of the calculated free energy surface. In contrast, a lower Gaussian height can guarantee a better resolution of the calculated free energy surface; however, it will take longer time for such a simulation to navigate through the defined conformational region. In order to reconcile such confliction, the authors present a method by implementing the Wang-Landau recursion scheme in the metadynamics simulations to adaptively update the height of the unit Gaussian function. As demonstrated in their model studies on both a toy system, and a realistic molecular system treated with the hybrid quantum mechanical and molecular mechanical (QMMM) potential, the present approach can quickly result in more decently converged free energy surfaces, compared with the classical metadynamics simulations employing the fixed Gaussian heights.     

Self-consistent molecular theory of polymers in melts and solutions

We propose a self-consistent molecular theory of conformational properties of flexible polymers in melts and solutions. The method employs the polymer reference interaction site model for the intermolecular correlations and the Green function technique for the intramolecular correlations. We demonstrate this method on n-alkane molecules in different environments: water, hexane, and in melt, corresponding to poor, good, and theta condition, respectively. The numerical results of the intramolecular correlation function, the radius of gyration, and the characteristic ratio of a polymer chain are indicative of conformational changes from one environment to another and are in agreement with other findings in the literature. Scaling laws for the chain size with respect to the number of monomers are discussed. We show results for the intra- and intermolecular correlation functions and the medium-induced potential. We also extract the Kuhn length and the characteristic ratio for the infinite chain limit for melts. The latter is compared to the experimental results and computer simulation. The conformational free energy per monomer in different solvents is calculated. Our treatment can be generalized readily to other polymer-solvent systems, for example, those containing branched copolymers and polar solvents.