Sampling enhancement for the quantum mechanical potential based molecular dynamics simulations: a general algorithm and its extension for free energy calculation on rugged energy surface

A Comparison of QM/MM Simulations with and without the Drude Oscillator Model Based on Hydration Free Energies of Simple Solutes

Maintaining a proper balance between specific intermolecular interactions and non-specific solvent interactions is of critical importance in molecular simulations, especially when predicting binding affinities or reaction rates in the condensed phase. The most rigorous metric for characterizing solvent affinity are solvation free energies, which correspond to a transfer from the gas phase into solution. Due to the drastic change of the electrostatic environment during this process, it is also a stringent test of polarization response in the model. Here, we employ both the CHARMM fixed charge and polarizable force fields to predict hydration free energies of twelve simple solutes. The resulting classical ensembles are then reweighted to obtain QM/MM hydration free energies using a variety of QM methods, including MP2, Hartree⁻Fock, density functional methods (BLYP, B3LYP, M06-2X) and semi-empirical methods (OM2 and AM1 ). Our simulations test the compatibility of quantum-mechanical methods with molecular-mechanical water models and solute Lennard⁻Jones parameters. In all cases, the resulting QM/MM hydration free energies were inferior to purely classical results, with the QM/MM Drude force field predictions being only marginally better than the QM/MM fixed charge results. In addition, the QM/MM results for different quantum methods are highly divergent, with almost inverted trends for polarizable and fixed charge water models. While this does not necessarily imply deficiencies in the QM models themselves, it underscores the need to develop consistent and balanced QM/MM interactions. Both the QM and the MM component of a QM/MM simulation have to match, in order to avoid artifacts due to biased solute⁻solvent interactions. Finally, we discuss strategies to improve the convergence and efficiency of multi-scale free energy simulations by automatically adapting the molecular-mechanics force field to the target quantum method.

On the convergence of multi-scale free energy simulations

In this work we employ simple model systems to evaluate the relative performance of two of the most important free energy methods: The Zwanzig equation (also known as "Free energy perturbation") and Bennett's acceptance ratio method (BAR). Although our examples should be transferable to other kinds of free energy simulations, we focus on applications of multi-scale free energy simulations. Such calculations are especially complex, since they connect two different levels of theory with very different requirements in terms of speed, accuracy, sampling and parallelizability. We try to reconcile all those different factors by developing some simple criteria to guide the early stages of the development of a free energy protocol. This is accomplished by quantifying how many λ intermediate steps and how many potential energy evaluations are necessary in order to reach a certain level of convergence.

Computing converged free energy differences between levels of theory via nonequilibrium work methods: Challenges and opportunities

We demonstrate that Jarzynski's equation can be used to reliably compute free energy differences between low and high level representations of systems. The need for such a calculation arises when employing the so-called "indirect" approach to free energy simulations with mixed quantum mechanical/molecular mechanical (QM/MM) Hamiltonians; a popular technique for circumventing extensive simulations involving quantum chemical computations. We have applied this methodology to several small and medium sized organic molecules, both in the gas phase and explicit solvent. Test cases include several systems for which the standard approach; that is, free energy perturbation between low and high level description, fails to converge. Finally, we identify three major areas in which the difference between low and high level representations make the calculation of ΔAlow→high difficult: bond stretching and angle bending, different preferred conformations, and the response of the MM region to the charge distribution of the QM region. © 2016 Wiley Periodicals, Inc.

Toward polarizable AMOEBA thermodynamics at fixed charge efficiency using a dual force field approach: application to organic crystals

First principles prediction of the structure, thermodynamics and solubility of organic molecular crystals, which play a central role in chemical, material, pharmaceutical and engineering sciences, challenges both potential energy functions and sampling methodologies. Here we calculate absolute crystal deposition thermodynamics using a novel dual force field approach whose goal is to maintain the accuracy of advanced multipole force fields (e.g. the polarizable AMOEBA model) while performing more than 95% of the sampling in an inexpensive fixed charge (FC) force field (e.g. OPLS-AA). Absolute crystal sublimation/deposition phase transition free energies were determined using an alchemical path that grows the crystalline state from a vapor reference state based on sampling with the OPLS-AA force field, followed by dual force field thermodynamic corrections to change between FC and AMOEBA resolutions at both end states (we denote the three step path as AMOEBA/FC). Importantly, whereas the phase transition requires on the order of 200 ns of sampling per compound, only 5 ns of sampling was needed for the dual force field thermodynamic corrections to reach a mean statistical uncertainty of 0.05 kcal mol-1. For five organic compounds, the mean unsigned error between direct use of AMOEBA and the AMOEBA/FC dual force field path was only 0.2 kcal mol-1 and not statistically significant. Compared to experimental deposition thermodynamics, the mean unsigned error for AMOEBA/FC (1.4 kcal mol-1) was more than a factor of two smaller than uncorrected OPLS-AA (3.2 kcal mol-1). Overall, the dual force field thermodynamic corrections reduced condensed phase sampling in the expensive force field by a factor of 40, and may prove useful for protein stability or binding thermodynamics in the future.

Calculating distribution coefficients based on multi-scale free energy simulations: an evaluation of MM and QM/MM explicit solvent simulations of water-cyclohexane transfer in the SAMPL5 challenge

One of the central aspects of biomolecular recognition is the hydrophobic effect, which is experimentally evaluated by measuring the distribution coefficients of compounds between polar and apolar phases. We use our predictions of the distribution coefficients between water and cyclohexane from the SAMPL5 challenge to estimate the hydrophobicity of different explicit solvent simulation techniques. Based on molecular dynamics trajectories with the CHARMM General Force Field, we compare pure molecular mechanics (MM) with quantum-mechanical (QM) calculations based on QM/MM schemes that treat the solvent at the MM level. We perform QM/MM with both density functional theory (BLYP) and semi-empirical methods (OM1, OM2, OM3, PM3). The calculations also serve to test the sensitivity of partition coefficients to solute polarizability as well as the interplay of the quantum-mechanical region with the fixed-charge molecular mechanics environment. Our results indicate that QM/MM with both BLYP and OM2 outperforms pure MM. However, this observation is limited to a subset of cases where convergence of the free energy can be achieved.

Methods for Efficiently and Accurately Computing Quantum Mechanical Free Energies for Enzyme Catalysis

Enzyme activity is inherently linked to free energies of transition states, ligand binding, protonation/deprotonation, etc.; these free energies, and thus enzyme function, can be affected by residue mutations, allosterically induced conformational changes, and much more. Therefore, being able to predict free energies associated with enzymatic processes is critical to understanding and predicting their function. Free energy simulation (FES) has historically been a computational challenge as it requires both the accurate description of inter- and intramolecular interactions and adequate sampling of all relevant conformational degrees of freedom. The hybrid quantum mechanical molecular mechanical (QM/MM) framework is the current tool of choice when accurate computations of macromolecular systems are essential. Unfortunately, robust and efficient approaches that employ the high levels of computational theory needed to accurately describe many reactive processes (ie, ab initio, DFT), while also including explicit solvation effects and accounting for extensive conformational sampling are essentially nonexistent. In this chapter, we will give a brief overview of two recently developed methods that mitigate several major challenges associated with QM/MM FES: the QM non-Boltzmann Bennett's acceptance ratio method and the QM nonequilibrium work method. We will also describe usage of these methods to calculate free energies associated with (1) relative properties and (2) along reaction paths, using simple test cases with relevance to enzymes examples.

Practically Efficient QM/MM Alchemical Free Energy Simulations: The Orthogonal Space Random Walk Strategy

The difference between free energy changes occurring at two chemical states can be rigorously estimated via alchemical free energy (AFE) simulations. Traditionally, most AFE simulations are carried out under the classical energy potential treatment; then, accuracy and applicability of AFE simulations are limited. In the present work, we integrate a recent second-order generalized ensemble strategy, the orthogonal space random walk (OSRW) method, into the combined quantum mechanical/molecular mechanical (QM/MM) potential based AFE simulation scheme. Thereby, within a commonly affordable simulation length, accurate QM/MM alchemical free energy simulations can be achieved. As revealed by the model study on the equilibrium of a tautomerization process of hydrated 3-hydroxypyrazole and by the model calculations of the redox potentials of two flavin derivatives, lumichrome (LC) and riboflavin (RF) in aqueous solution, the present OSRW-based scheme could be a viable path toward the realization of practically efficient QM/MM AFE simulations.

Efficiently computing pathway free energies: New approaches based on chain-of-replica and Non-Boltzmann Bennett reweighting schemes

BACKGROUND

Accurately modeling condensed phase processes is one of computation's most difficult challenges. Include the possibility that conformational dynamics may be coupled to chemical reactions, where multiscale (i.e., QM/MM) methods are needed, and this task becomes even more daunting.

METHODS

Free energy simulations (i.e., molecular dynamics), multiscale modeling, and reweighting schemes.

RESULTS

Herein, we present two new approaches for mitigating the aforementioned challenges. The first is a new chain-of-replica method (off-path simulations, OPS) for computing potentials of mean force (PMFs) along an easily defined reaction coordinate. This development is coupled with a new distributed, highly-parallel replica framework (REPDstr) within the CHARMM package. Validation of these new schemes is carried out on two processes that undergo conformational changes. First is the simple torsional rotation of butane, while a much more challenging glycosidic rotation (in vacuo and solvated) is the second. Additionally, a new approach that greatly improves (i.e., possibly an order of magnitude) the efficiency of computing QM/MM PMFs is introduced and compared to standard schemes. Our efforts are grounded in the recently developed method for efficiently computing QM-based free energies (i.e., QM-Non-Boltzmann Bennett, QM-NBB). Again, we validate this new technique by computing the QM/MM PMF of butane's torsional rotation.

CONCLUSIONS

The OPS-REPDstr method is a promising new approach that overcomes many limitations of standard pathway simulations in CHARMM. The combination of QM-NBB with pathway techniques is very promising as it offers significant advantages over current procedures.

GENERAL SIGNIFICANCE

Efficiently computing potentials of mean force is a major, unresolved, area of interest. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Multiscale Free Energy Simulations: An Efficient Method for Connecting Classical MD Simulations to QM or QM/MM Free Energies Using Non-Boltzmann Bennett Reweighting Schemes

The reliability of free energy simulations (FES) is limited by two factors: (a) the need for correct sampling and (b) the accuracy of the computational method employed. Classical methods (e.g., force fields) are typically used for FES and present a myriad of challenges, with parametrization being a principle one. On the other hand, parameter-free quantum mechanical (QM) methods tend to be too computationally expensive for adequate sampling. One widely used approach is a combination of methods, where the free energy difference between the two end states is computed by, e.g., molecular mechanics (MM), and the end states are corrected by more accurate methods, such as QM or hybrid QM/MM techniques. Here we report two new approaches that significantly improve the aforementioned scheme; with a focus on how to compute corrections between, e.g., the MM and the more accurate QM calculations. First, a molecular dynamics trajectory that properly samples relevant conformational degrees of freedom is generated. Next, potential energies of each trajectory frame are generated with a QM or QM/MM Hamiltonian. Free energy differences are then calculated based on the QM or QM/MM energies using either a non-Boltzmann Bennett approach (QM-NBB) or non-Boltzmann free energy perturbation (NB-FEP). Both approaches are applied to calculate relative and absolute solvation free energies in explicit and implicit solvent environments. Solvation free energy differences (relative and absolute) between ethane and methanol in explicit solvent are used as the initial test case for QM-NBB. Next, implicit solvent methods are employed in conjunction with both QM-NBB and NB-FEP to compute absolute solvation free energies for 21 compounds. These compounds range from small molecules such as ethane and methanol to fairly large, flexible solutes, such as triacetyl glycerol. Several technical aspects were investigated. Ultimately some best practices are suggested for improving methods that seek to connect MM to QM (or QM/MM) levels of theory in FES.

Predicting hydration free energies with a hybrid QM/MM approach: an evaluation of implicit and explicit solvation models in SAMPL4

The correct representation of solute-water interactions is essential for the accurate simulation of most biological phenomena. Several highly accurate quantum methods are available to deal with solvation by using both implicit and explicit solvents. So far, however, most evaluations of those methods were based on a single conformation, which neglects solute entropy. Here, we present the first test of a novel approach to determine hydration free energies that uses molecular mechanics (MM) to sample phase space and quantum mechanics (QM) to evaluate the potential energies. Free energies are determined by using re-weighting with the Non-Boltzmann Bennett (NBB) method. In this context, the method is referred to as QM-NBB. Based on snapshots from MM sampling and accounting for their correct Boltzmann weight, it is possible to obtain hydration free energies that incorporate the effect of solute entropy. We evaluate the performance of several QM implicit solvent models, as well as explicit solvent QM/MM for the blind subset of the SAMPL4 hydration free energy challenge. While classical free energy simulations with molecular dynamics give root mean square deviations (RMSD) of 2.8 and 2.3 kcal/mol, the hybrid approach yields an improved RMSD of 1.6 kcal/mol. By selecting an appropriate functional and basis set, the RMSD can be reduced to 1 kcal/mol for calculations based on a single conformation. Results for a selected set of challenging molecules imply that this RMSD can be further reduced by using NBB to reweight MM trajectories with the SMD implicit solvent model.

Replica exchanging self-guided Langevin dynamics for efficient and accurate conformational sampling

This work presents a replica exchanging self-guided Langevin dynamics (RXSGLD) simulation method for efficient conformational searching and sampling. Unlike temperature-based replica exchanging simulations, which use high temperatures to accelerate conformational motion, this method uses self-guided Langevin dynamics (SGLD) to enhance conformational searching without the need to elevate temperatures. A RXSGLD simulation includes a series of SGLD simulations, with simulation conditions differing in the guiding effect and/or temperature. These simulation conditions are called stages and the base stage is one with no guiding effect. Replicas of a simulation system are simulated at the stages and are exchanged according to the replica exchanging probability derived from the SGLD partition function. Because SGLD causes less perturbation on conformational distribution than high temperatures, exchanges between SGLD stages have much higher probabilities than those between different temperatures. Therefore, RXSGLD simulations have higher conformational searching ability than temperature based replica exchange simulations. Through three example systems, we demonstrate that RXSGLD can generate target canonical ensemble distribution at the base stage and achieve accelerated conformational searching. Especially for large systems, RXSGLD has remarkable advantages in terms of replica exchange efficiency, conformational searching ability, and system size extensiveness.

Tunable, mixed-resolution modeling using library-based Monte Carlo and graphics processing units

Building on our recently introduced library-based Monte Carlo (LBMC) approach, we describe a flexible protocol for mixed coarse-grained (CG)/all-atom (AA) simulation of proteins and ligands. In the present implementation of LBMC, protein side chain configurations are pre-calculated and stored in libraries, while bonded interactions along the backbone are treated explicitly. Because the AA side chain coordinates are maintained at minimal run-time cost, arbitrary sites and interaction terms can be turned on to create mixed-resolution models. For example, an AA region of interest such as a binding site can be coupled to a CG model for the rest of the protein. We have additionally developed a hybrid implementation of the generalized Born/surface area (GBSA) implicit solvent model suitable for mixed-resolution models, which in turn was ported to a graphics processing unit (GPU) for faster calculation. The new software was applied to study two systems: (i) the behavior of spin labels on the B1 domain of protein G (GB1) and (ii) docking of randomly initialized estradiol configurations to the ligand binding domain of the estrogen receptor (ERα). The performance of the GPU version of the code was also benchmarked in a number of additional systems.

Enhancing QM/MM molecular dynamics sampling in explicit environments via an orthogonal-space-random-walk-based strategy

Accurate prediction of molecular conformations in explicit environments, such as aqueous solution and protein interiors, can facilitate our understanding of various molecular recognition processes. Most computational approaches are limited as a result of their compromised choices between the underlying energy model and the sampling length. Taking advantage of a recent second-order generalized ensemble scheme [e.g., the orthogonal space random walk (OSRW) strategy], which can synergistically accelerate the motion of a focused region and its coupled environmental response, we are presenting a QM/MM (combined quantum mechanical/molecular mechanical)-based molecular dynamics sampling technique to explore molecular conformational landscapes in explicit environments. The present QM/MM potential scaling-based OSRW sampling scheme is employed to study the binding of DMSO to the FKBP12 protein, the conformation distribution of a novel mercaptosulfonamide inhibitor in aqueous solution, and its binding poses in zinc-containing matrix metalloproteinase-9 (MMP-9). As demonstrated, the present QM/MM second-order generalized ensemble sampling technique enables feasible usage of the QM/MM model to sample molecular conformations in condensed environments.

Balancing simulation accuracy and efficiency with the Amber united atom force field

We have analyzed the quality of a recently proposed Amber united-atom model and its overall efficiency in ab initio folding and thermodynamic sampling of two stable beta-hairpins. It is found that the mean backbone structures are quite consistent between the simulations in the united-atom and its corresponding all-atom models in Amber. More importantly, the simulated beta turns are also consistent between the two models. Finally, the chemical shifts on H alpha are highly consistent between simulations in the two models, although the simulated chemical shifts are lower than experiment, indicating less structured peptides, probably due to the omission of the hydrophobic term in the simulations. More interestingly, the stabilities of both beta-hairpins at room temperature are similar to those derived from the NMR measurement, whether the united-atom or the all-atom model is used. Detailed analysis shows high percentages of backbone torsion angles within the beta region and high percentages of native contacts. Given the reasonable quality of the united-atom model with respect to experimental data, we have further studied the simulation efficiency of the united-atom model over the all-atom model. Our data shows that the united-atom model is a factor of 6-8 faster than the all-atom model as measured with the ab initio first pass folding time for the two tested beta-hairpins. Detailed structural analysis shows that all ab initio folded trajectories enter the native basin, whether the united-atom model or the all-atom model is used. Finally, we have also studied the simulation efficiency of the united-atom model as measured in terms of how fast thermodynamic convergence can be achieved. It is apparent that the united-atom simulations reach convergence faster than the all-atom simulations with respect to both mean potential energies and mean native contacts. These findings show that the efficiency of the united-atom model is clearly beyond the per-step dynamics simulation of about 2 over the all-atom model. Thus, reasonable reduction of a protein model can be achieved with improved sampling efficiency while still preserving a high level of accuracy for applications in both ab initio folding and thermodynamic sampling. This study motivates us to develop more simplified protein models with sufficient consistency with the all-atom models for enhanced conformational sampling.

CHARMM: the biomolecular simulation program

CHARMM (Chemistry at HARvard Molecular Mechanics) is a highly versatile and widely used molecular simulation program. It has been developed over the last three decades with a primary focus on molecules of biological interest, including proteins, peptides, lipids, nucleic acids, carbohydrates, and small molecule ligands, as they occur in solution, crystals, and membrane environments. For the study of such systems, the program provides a large suite of computational tools that include numerous conformational and path sampling methods, free energy estimators, molecular minimization, dynamics, and analysis techniques, and model-building capabilities. The CHARMM program is applicable to problems involving a much broader class of many-particle systems. Calculations with CHARMM can be performed using a number of different energy functions and models, from mixed quantum mechanical-molecular mechanical force fields, to all-atom classical potential energy functions with explicit solvent and various boundary conditions, to implicit solvent and membrane models. The program has been ported to numerous platforms in both serial and parallel architectures. This article provides an overview of the program as it exists today with an emphasis on developments since the publication of the original CHARMM article in 1983.

Optimal sampling efficiency in Monte Carlo simulation with an approximate potential

Building on the work of Iftimie et al. [J. Chem. Phys. 113, 4852 (2000)] and Gelb [J. Chem. Phys. 118, 7747 (2003)], Boltzmann sampling of an approximate potential (the "reference" system) is used to build a Markov chain in the isothermal-isobaric ensemble. At the end points of the chain, the energy is evaluated at a more accurate level (the "full" system) and a composite move encompassing all of the intervening steps is accepted on the basis of a modified Metropolis criterion. For reference system chains of sufficient length, consecutive full energies are statistically decorrelated and thus far fewer are required to build ensemble averages with a given variance. Without modifying the original algorithm, however, the maximum reference chain length is too short to decorrelate full configurations without dramatically lowering the acceptance probability of the composite move. This difficulty stems from the fact that the reference and full potentials sample different statistical distributions. By manipulating the thermodynamic variables characterizing the reference system (pressure and temperature, in this case), we maximize the average acceptance probability of composite moves, lengthening significantly the random walk between consecutive full energy evaluations. In this manner, the number of full energy evaluations needed to precisely characterize equilibrium properties is dramatically reduced. The method is applied to a model fluid, but implications for sampling high-dimensional systems with ab initio or density functional theory potentials are discussed.

pKa of residue 66 in Staphylococal nuclease. I. Insights from QM/MM simulations with conventional sampling

A combined quantum mechanical/molecular mechanical (QM/MM) potential function is used in a thermodynamic integration approach to calculate the pK(a) of residue 66 in two mutants (V66E, V66D) of Staphylococal nuclease relative to solution. Despite the similarity in chemical nature and experimentally measured pK(a) of the two buried titritable residues, the behaviors of the two mutants and the computed pK(a) values vary greatly in the simulations. For Glu66, the side chain is consistently observed to spontaneously flip out from the protein interior during titration, and the overall protein structure remains stable throughout the simulations. The computed pK(a) shifts using conventional sampling techniques with multiple nanoseconds per lambda window (Set A and B) are generally close to the experimental value, therefore indicating that large-scale conformational rearrangements are not as important for V66E as suggested by the recent study of Warshel and co-worker. For Asp66, by contrast, flipping of the shorter side chain is not sufficient for getting adequate solvent stabilization of the ionized state. As a result, more complex behaviors such as partial unfolding of a nearby beta-sheet region is observed, and the computed pK(a) shift is substantially higher than the experimental value unless Asp66 is biased to adopt the similar configurations as Glu66 in the V66E simulations. Collectively, these studies suggest that the lack of electronic polarization is not expected to be the dominant source of error in microscopic pK(a) shift calculations, while the need of enhanced sampling is more compelling for predicting the pK(a) of buried residues. Furthermore, the comparison between V66E and V66D also highlights that the microscopic interpretation of similar apparent pK(a) values and effective "dielectric constants" of proteins can vary greatly in terms of the residues that make key contributions and the scale of structural/hydration response to titration, the latter of which is difficult to predict a priori. Perturbative analyses of interactions that contribute to the titration free energy point to mutants that can be used to verify the microscopic mechanisms of titration in V66E/D SNase proteins.

Quantitative computer simulations of biomolecules: A snapshot

A recent workshop titled "Quantitative Computational Biophysics" at Florida State University provided an overview of the state of the art in quantitative modeling of biomolecular systems. The presentations covered a wide range of interrelated topics, including the development and validation of force fields, the modeling of protein-protein interactions, the sampling of conformational space, and the assessment of equilibration and statistical errors. Substantial progress in all these areas was reported.