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

Software Infrastructure for Next-Generation QM/MM-ΔMLP Force Fields

We present software infrastructure for the design and testing of new quantum mechanical/molecular mechanical and machine-learning potential (QM/MM-ΔMLP) force fields for a wide range of applications. The software integrates Amber's molecular dynamics simulation capabilities with fast, approximate quantum models in the xtb package and machine-learning potential corrections in DeePMD-kit. The xtb package implements the recently developed density-functional tight-binding QM models with multipolar electrostatics and density-dependent dispersion (GFN2-xTB), and the interface with Amber enables their use in periodic boundary QM/MM simulations with linear-scaling QM/MM particle-mesh Ewald electrostatics. The accuracy of the semiempirical models is enhanced by including machine-learning correction potentials (ΔMLPs) enabled through an interface with the DeePMD-kit software. The goal of this paper is to present and validate the implementation of this software infrastructure in molecular dynamics and free energy simulations. The utility of the new infrastructure is demonstrated in proof-of-concept example applications. The software elements presented here are open source and freely available. Their interface provides a powerful enabling technology for the design of new QM/MM-ΔMLP models for studying a wide range of problems, including biomolecular reactivity and protein-ligand binding.

Perspectives on Computational Enzyme Modeling: From Mechanisms to Design and Drug Development

Understanding enzyme mechanisms is essential for unraveling the complex molecular machinery of life. In this review, we survey the field of computational enzymology, highlighting key principles governing enzyme mechanisms and discussing ongoing challenges and promising advances. Over the years, computer simulations have become indispensable in the study of enzyme mechanisms, with the integration of experimental and computational exploration now established as a holistic approach to gain deep insights into enzymatic catalysis. Numerous studies have demonstrated the power of computer simulations in characterizing reaction pathways, transition states, substrate selectivity, product distribution, and dynamic conformational changes for various enzymes. Nevertheless, significant challenges remain in investigating the mechanisms of complex multistep reactions, large-scale conformational changes, and allosteric regulation. Beyond mechanistic studies, computational enzyme modeling has emerged as an essential tool for computer-aided enzyme design and the rational discovery of covalent drugs for targeted therapies. Overall, enzyme design/engineering and covalent drug development can greatly benefit from our understanding of the detailed mechanisms of enzymes, such as protein dynamics, entropy contributions, and allostery, as revealed by computational studies. Such a convergence of different research approaches is expected to continue, creating synergies in enzyme research. This review, by outlining the ever-expanding field of enzyme research, aims to provide guidance for future research directions and facilitate new developments in this important and evolving field.

Perspective: Reference-Potential Methods for the Study of Thermodynamic Properties in Chemical Processes: Theory, Applications, and Pitfalls

In silico investigations of enzymatic reactions and chemical reactions in condensed phases often suffer from formidable computational costs due to a large number of degrees of freedom and enormous important volume in phase space. Usually, accuracy must be compromised to trade for efficiency by lowering the reliability of the Hamiltonians employed or reducing the sampling time. Reference-potential methods (RPMs) offer an alternative approach to reaching high accuracy of simulation without much loss of efficiency. In this Perspective, we summarize the idea of RPMs and showcase some recent applications. Most importantly, the pitfalls of these methods are also discussed, and remedies to these pitfalls are presented.

Affordable Ab Initio Path Integral for Thermodynamic Properties via Molecular Dynamics Simulations Using Semiempirical Reference Potential

Path integral molecular dynamics (PIMD) is becoming a routinely applied method for incorporating the nuclear quantum effect in computer simulations. However, direct PIMD simulations at an ab initio level of theory are formidably expensive. Using the protonated 1,8-bis(dimethylamino)naphthalene molecule as an example, we show in this work that the computational expense for the intramolecular proton transfer between the two nitrogen atoms can be remarkably reduced by implementing the idea of reference-potential methods. The simulation time can be easily extended to a scale of nanoseconds while maintaining the accuracy on an ab initio level of theory for thermodynamic properties. In addition, postprocessing can be carried out in parallel on massive computer nodes. A 545-fold reduction in the total CPU time can be achieved in this way as compared to a direct PIMD simulation at the same ab initio level of theory.

Accelerated Computation of Free Energy Profile at Ab Initio Quantum Mechanical/Molecular Mechanics Accuracy via a Semiempirical Reference Potential. 4. Adaptive QM/MM

Although quantum mechanical/molecular mechanics (QM/MM) methods are now routinely applied to the studies of chemical reactions in condensed phases and enzymatic reactions, they may experience technical difficulties when the reactive region is varying over time. For instance, when the solvent molecules are directly participating in the reaction, the exchange of water molecules between the QM and MM regions may occur on a time scale comparable to the reaction time. To cope with this situation, several adaptive QM/MM schemes have been proposed. However, these methods either add significantly to the computational cost or introduce artificial restraints to the system. In this work, we developed a novel adaptive QM/MM scheme and applied it to the study of a nucleophilic addition reaction. In this scheme, the configuration sampling was performed with a small QM region (without solvent molecules), and the thermodynamic properties under another potential energy function with a larger QM region (with a certain number of solvent molecules and/or different levels of QM theory) are computed via extrapolation using the reference-potential method. Our simulation results show that this adaptive QM/MM scheme is numerically stable, at least for the case studied in this work. Furthermore, this method also offers an inexpensive way to examine the convergence of the QM/MM calculation with respect to the size of the QM region.

Accelerated Computation of Free Energy Profile at Ab Initio Quantum Mechanical/Molecular Mechanics Accuracy via a Semiempirical Reference Potential. 3. Gaussian Smoothing on Density-of-States

Calculations of the free energy profile, also known as potential of mean force (PMF), along a chosen collective variable (CV) are now routinely applied in the studies of chemical processes, such as enzymatic reactions and chemical reactions in condensed phases. However, if the ab initio quantum mechanical/molecular mechanics (QM/MM) level of accuracy is required for the PMF, it can be formidably demanding even with the most advanced enhanced sampling methods, such as umbrella sampling. To ameliorate this difficulty, we developed a novel method for the computation of the free energy profile based on the reference-potential method recently, in which a low-level reference Hamiltonian is employed for phase space sampling and the free energy profile can be corrected to the level of interest (the target Hamiltonian) by energy reweighting in a nonparametric way. However, when the reference Hamiltonian is very different from the target Hamiltonian, the calculated ensemble averages, including the PMF, often suffer from numerical instability, which mainly comes from the overestimation of the density-of-states (DoS) in the low-energy region. Stochastic samplings of these low-energy configurations are rare events, and some low-energy conformations may get oversampled in simulations of a finite length. In this work, an assumption of Gaussian distribution is applied to the DoS in each CV bin, and the weight of each configuration is rescaled according to the accumulated DoS. The results show that this smoothing process can remarkably reduce the ruggedness of the PMF and increase the reliability of the reference-potential method.

Multi-phase Boltzmann weighting: accounting for local inhomogeneity in molecular simulations of water-octanol partition coefficients in the SAMPL6 challenge

Accurately computing partition coefficients is a pivotal part of drug discovery. Specifically, octanol-water partition coefficients can provide information into hydrophobicity of drug-like molecules, as well as a de facto representation of membrane permeability. However, one challenge facing the computation of partition coefficients is the need to encapsulate various microscopic environments. These include areas of largely bulk solvent (i.e., either water or octanol) or regions where octanol is saturated with water or areas of higher salt concentration. Also, tautomeric effects require consideration. Thus, we present a Boltzmann weighting approach that incorporates transfer free energies across varying microscopic media, as well as varying tautomeric state, to compute partition coefficients in the SAMPL6 challenge.

Hierarchical Markov State Model Building to Describe Molecular Processes

Markov state models can describe ensembles of pathways via kinetic networks but are difficult to create when large free-energy barriers limit unbiased sampling. Chain-of-states simulations allow sampling over large free-energy barriers but are often constructed using a single pathway that is unlikely to thermodynamically average over orthogonal degrees of freedom in complex systems. Here, we combine the advantages of these two approaches in the form of a Markov state model of Markov state models, which we call a Hierarchical Markov state model. In this approach, independent Markov models are constructed in regions of configuration space that are locally well sampled but are separated by large free-energy barriers from other regions. A string method is used to construct an ensemble of pathways connecting the states of these different local Markov models, and the rate through each pathway is then estimated. These rates are then combined with the rate information from the local Markov models in a master equation to predict global rates, fluxes, and populations. By applying this hierarchical approach to tractable systems, a toy potential and dipeptides, we demonstrate that it is more accurate than the conventional single-pathway description. The advantages of this approach are that it (i) is more realistic than the conventional chain-of-states approach, as an ensemble of pathways rather than a single pathway is used to describe processes in high-dimensional systems, and (ii) it resolves the issue of poor sampling in Markov State model building when large free-energy barriers are present. The divide-and-conquer strategy inherent to this approach should make this procedure straightforward to apply to more complex systems.

BAR-based multi-dimensional nonequilibrium pulling for indirect construction of QM/MM free energy landscapes: from semi-empirical to ab initio

The indirect method for the construction of quantum mechanics (QM)/molecular mechanics (MM) free energy landscapes provides a cheaper alternative for free energy simulations at the QM level. The indirect method features a direct calculation of the free energy profile with a computationally efficient but less accurate Hamiltonian (i.e. low-level Hamiltonian) and a low-level-to-high-level correction. In the thermodynamic cycle, the direct low-level calculation along the physically meaningful reaction coordinate is corrected via the alchemical method, which is often achieved with perturbation-based techniques. In our previous work, a multi-dimensional nonequilibrium pulling framework is proposed for the indirect construction of QM/MM free energy landscapes. Previously, we focused on obtaining semi-empirical QM (SQM) results indirectly from direct MM simulations and MM to SQM corrections. In this work, we apply this method to obtain results under ab initio QM Hamiltonians by combining direct SQM results and SQM to QM corrections. A series of SQM and QM Hamiltonians are benchmarked. It is observed that PM6 achieves the best performance among the low-level Hamiltonians. Therefore, we recommend using PM6 as the low-level theory in the indirect free energy simulation. Considering its higher similarity to the high-level Hamiltonians, PM6 corrected with the bond charge correction could be more accurate than the existing AM1-BCC model. Another central result in the current work is a basic protocol of choosing the strength of restraints and an appropriate time step in nonequilibrium free energy simulation at the stiff spring limit. We provide theoretical derivations to emphasize the importance of using a sufficiently large force constant and choosing an appropriate time step. It is worth noting that a general rule of thumb for choosing the time step, according to our derivation, is that a time step of 1 fs or smaller should be used, as long as the stiff spring approximation is employed, even in simulations with constraints on bonds involving hydrogen atoms.

Development of a Robust Indirect Approach for MM → QM Free Energy Calculations That Combines Force-Matched Reference Potential and Bennett's Acceptance Ratio Methods

We use the PBE0/6-31G* density functional method to perform ab initio quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) simulations under periodic boundary conditions with rigorous electrostatics using the ambient potential composite Ewald method in order to test the convergence of MM → QM/MM free energy corrections for the prediction of 17 small-molecule solvation free energies and eight ligand binding free energies to T4 lysozyme. The "indirect" thermodynamic cycle for calculating free energies is used to explore whether a series of reference potentials improve the statistical quality of the predictions. Specifically, we construct a series of reference potentials that optimize a molecular mechanical (MM) force field's parameters to reproduce the ab initio QM/MM forces from a QM/MM simulation. The optimizations form a systematic progression of successively expanded parameters that include bond, angle, dihedral, and charge parameters. For each reference potential, we calculate benchmark quality reference values for the MM → QM/MM correction by performing the mixed MM and QM/MM Hamiltonians at 11 intermediate states, each for 200 ps. We then compare forward and reverse application of Zwanzig's relation, thermodynamic integration (TI), and Bennett's acceptance ratio (BAR) methods as a function of reference potential, simulation time, and the number of simulated intermediate states. We find that Zwanzig's equation is inadequate unless a large number of intermediate states are explicitly simulated. The TI and BAR mean signed errors are very small even when only the end-state simulations are considered, and the standard deviations of the TI and BAR errors are decreased by choosing a reference potential that optimizes the bond and angle parameters. We find a robust approach for the data sets of fairly rigid molecules considered here is to use bond + angle reference potential together with the end-state-only BAR analysis. This requires QM/MM simulations to be performed in order to generate reference data to parametrize the bond + angle reference potential, and then this same simulation serves a dual purpose as the full QM/MM end state. The convergence of the results with respect to time suggests that computational resources may be used more efficiently by running multiple simulations for no more than 50 ps, rather than running one long simulation.

Use of Interaction Energies in QM/MM Free Energy Simulations

The use of the most accurate (i.e., QM or QM/MM) levels of theory for free energy simulations (FES) is typically not possible. Primarily, this is because the computational cost associated with the extensive configurational sampling needed for converging FES is prohibitive. To ensure the feasibility of QM-based FES, the "indirect" approach is generally taken, necessitating a free energy calculation between the MM and QM/MM potential energy surfaces. Ideally, this step is performed with standard free energy perturbation (Zwanzig's equation) as it only requires simulations be carried out at the low level of theory; however, work from several groups over the past few years has conclusively shown that Zwanzig's equation is ill-suited to this task. As such, many approximations have arisen to mitigate difficulties with Zwanzig's equation. One particularly popular notion is that the convergence of Zwanzig's equation can be improved by using interaction energy differences instead of total energy differences. Although problematic numerical fluctuations (a major problem when using Zwanzig's equation) are indeed reduced, our results and analysis demonstrate that this "interaction energy approximation" (IEA) is theoretically incorrect, and the implicit approximation invoked is spurious at best. Herein, we demonstrate this via solvation free energy calculations using IEA from two different low levels of theory to the same target high level. Results from this proof-of-concept consistently yield the wrong results, deviating by ∼1.5 kcal/mol from the rigorously obtained value.

The Good, the Bad, and the Ugly: "HiPen", a New Dataset for Validating (S)QM/MM Free Energy Simulations

Indirect (S)QM/MM free energy simulations (FES) are vital to efficiently incorporating sufficient sampling and accurate (QM) energetic evaluations when estimating free energies of practical/experimental interest. Connecting between levels of theory, i.e., calculating Δ A l o w → h i g h , remains to be the most challenging step within an indirect FES protocol. To improve calculations of Δ A l o w → h i g h , we must: (1) compare the performance of all FES methods currently available; and (2) compile and maintain datasets of Δ A l o w → h i g h calculated for a wide-variety of molecules so that future practitioners may replicate or improve upon the current state-of-the-art. Towards these two aims, we introduce a new dataset, "HiPen", which tabulates Δ A g a s M M → 3 o b (the free energy associated with switching from an M M to an S C C - D F T B molecular description using the 3ob parameter set in gas phase), calculated for 22 drug-like small molecules. We compare the calculation of this value using free energy perturbation, Bennett's acceptance ratio, Jarzynski's equation, and Crooks' equation. We also predict the reliability of each calculated Δ A g a s M M → 3 o b by evaluating several convergence criteria including sample size hysteresis, overlap statistics, and bias metric ( Π ). Within the total dataset, three distinct categories of molecules emerge: the "good" molecules, for which we can obtain converged Δ A g a s M M → 3 o b using Jarzynski's equation; "bad" molecules which require Crooks' equation to obtain a converged Δ A g a s M M → 3 o b ; and "ugly" molecules for which we cannot obtain reliably converged Δ A g a s M M → 3 o b with either Jarzynski's or Crooks' equations. We discuss, in depth, results from several example molecules in each of these categories and describe how dihedral discrepancies between levels of theory cause convergence failures even for these gas phase free energy simulations.

Accelerating QM/MM Free Energy Computations via Intramolecular Force Matching

The calculation of free energy differences between levels of theory has numerous potential pitfalls. Chief among them is the lack of overlap, i.e., ensembles generated at one level of theory (e.g., "low") not being good approximations of ensembles at the other (e.g., "high"). Numerous strategies have been devised to mitigate this issue. However, the most straightforward approach is to ensure that the "low" level ensemble more closely resembles that of the "high". Ideally, this is done without increasing computational cost. Herein, we demonstrate that by reparametrizing classical intramolecular potentials to reproduce high level forces (i.e., force matching) configurational overlap between a "low" (i.e., classical) and "high" (i.e., quantum) level can be significantly improved. This procedure is validated on two test cases and results in vastly improved convergence of free energy simulations.

Force matching as a stepping stone to QM/MM CB[8] host/guest binding free energies: a SAMPL6 cautionary tale

Use of quantum mechanical/molecular mechanical (QM/MM) methods in binding free energy calculations, particularly in the SAMPL challenge, often fail to achieve improvement over standard additive (MM) force fields. Frequently, the implementation is through use of reference potentials, or the so-called "indirect approach", and inherently relies on sufficient overlap existing between MM and QM/MM configurational spaces. This overlap is generally poor, particularly for the use of free energy perturbation to perform the MM to QM/MM free energy correction at the end states of interest (e.g., bound and unbound states). However, by utilizing MM parameters that best reproduce forces obtained at the desired QM level of theory, it is possible to lessen the configurational disparity between MM and QM/MM. To this end, we sought to use force matching to generate MM parameters for the SAMPL6 CB[8] host-guest binding challenge, classically compute binding free energies, and apply energetic end state corrections to obtain QM/MM binding free energy differences. For the standard set of 11 molecules and the bonus set (including three additional challenge molecules), error statistics, such as the root mean square deviation (RMSE) were moderately poor (5.5 and 5.4 kcal/mol). Correlation statistics, however, were in the top two for both standard and bonus set submissions ([Formula: see text] of 0.42 and 0.26, [Formula: see text] of 0.64 and 0.47 respectively). High RMSE and moderate correlation strongly indicated the presence of systematic error. Identifiable issues were ameliorated for two of the guest molecules, resulting in a reduction of error and pointing to strong prospects for the future use of this methodology.

BAR-based optimum adaptive sampling regime for variance minimization in alchemical transformation: the nonequilibrium stratification

Following the previously proposed equilibrate-state sampling based adaptive sampling regime Optimum Bennett Acceptance Ratio (OBAR), we introduce its nonequilibrium extension, the Optimum Crooks' Equation (OCE) in the current work. The efficiency of the NonEquilibrium Work (NEW) stratification is improved by adaptively manipulating the significance of each nonequilibrium realization followed by importance sampling. As is exhibited in the equilibrium case, the nonequilibrium extension outperforms the simple equal time rule used in nonequilibrium stratification in the sense of minimizing the total variance of the free energy estimate. The speedup of this non-equal time rule is more than 1-fold. The Time Derivative of total Variance (TDV) proposed for the OBAR criterion is extended to determine the importance of each nonequilibrium transformation, which is linearly dependent on the variance. The TDV in the nonequilibrium case gives a totally different importance rank from the standard errors of the free energy differences and OBAR TDV due to the duration of nonequilibrium pulling being added into the OCE equation. The performance of the OCE workflow is demonstrated in the solvation of several small molecules with a series of lambda increments and relaxation times between successive perturbations. To the best of our knowledge, such a nonequilibrium adaptive sampling regime in alchemical transformation is unprecedented.

Dynamics Sampling in Transition Pathway Space

The minimum energy pathway contains important information describing the transition between two states on a potential energy surface (PES). Chain-of-states methods were developed to efficiently calculate minimum energy pathways connecting two stable states. In the chain-of-states framework, a series of structures are generated and optimized to represent the minimum energy pathway connecting two states. However, multiple pathways may exist connecting two existing states and should be identified to obtain a full view of the transitions. Therefore, we developed an enhanced sampling method, named as the direct pathway dynamics sampling (DPDS) method, to facilitate exploration of a PES for multiple pathways connecting two stable states as well as addition minima and their associated transition pathways. In the DPDS method, molecular dynamics simulations are carried out on the targeting PES within a chain-of-states framework to directly sample the transition pathway space. The simulations of DPDS could be regulated by two parameters controlling distance among states along the pathway and smoothness of the pathway. One advantage of the chain-of-states framework is that no specific reaction coordinates are necessary to generate the reaction pathway, because such information is implicitly represented by the structures along the pathway. The chain-of-states setup in a DPDS method greatly enhances the sufficient sampling in high-energy space between two end states, such as transition states. By removing the constraint on the end states of the pathway, DPDS will also sample pathways connecting minima on a PES in addition to the end points of the starting pathway. This feature makes DPDS an ideal method to directly explore transition pathway space. Three examples demonstrate the efficiency of DPDS methods in sampling the high-energy area important for reactions on the PES.

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.

BAR-based optimum adaptive sampling regime for variance minimization in alchemical transformation

The efficiency of alchemical free energy simulations with a staging strategy is improved by adaptively manipulating the significance of each ensemble followed by importance sampling.

Blind prediction of distribution in the SAMPL5 challenge with QM based protomer and pK a corrections

The computation of distribution coefficients between polar and apolar phases requires both an accurate characterization of transfer free energies between phases and proper accounting of ionization and protomerization. We present a protocol for accurately predicting partition coefficients between two immiscible phases, and then apply it to 53 drug-like molecules in the SAMPL5 blind prediction challenge. Our results combine implicit solvent QM calculations with classical MD simulations using the non-Boltzmann Bennett free energy estimator. The OLYP/DZP/SMD method yields predictions that have a small deviation from experiment (RMSD = 2.3 [Formula: see text] D units), relative to other participants in the challenge. Our free energy corrections based on QM protomer and [Formula: see text] calculations increase the correlation between predicted and experimental distribution coefficients, for all methods used. Unfortunately, these corrections are overly hydrophilic, and fail to account for additional effects such as aggregation, water dragging and the presence of polar impurities in the apolar phase. We show that, although expensive, QM-NBB free energy calculations offer an accurate and robust method that is superior to standard MM and QM techniques alone.

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.

An efficient protocol for obtaining accurate hydration free energies using quantum chemistry and reweighting from molecular dynamics simulations

The non-Boltzmann Bennett (NBB) free energy estimator method is applied to 21 molecules from the blind subset of the SAMPL4 challenge. When NBB is applied with the SMD implicit solvent model, and the OLYP/DZP level of quantum chemistry, highly accurate hydration free energy calculations are obtained with respect to experiment (RMSD=0.89kcal·mol^-1). Other quantum chemical methods are also tested, and the effects of solvent model, density functional, basis set are explored in this benchmarking study, providing a framework for improvements in calculating hydration free energies. We provide a practical guide for using the best QM-NBB protocols that are consistently more accurate than either pure QM or pure MM alone. In situations where high accuracy hydration free energy predictions are needed, the QM-NBB method with SMD implicit solvent should be the first choice of quantum chemists.

Role of magnesium ions in the reaction mechanism at the interface between Tm1631 protein and its DNA ligand

A protein, Tm1631 from the hyperthermophilic organism Thermotoga maritima belongs to a domain of unknown function protein family. It was predicted that Tm1631 binds with the DNA and that the Tm1631–DNA complex is an endonuclease repair system with a DNA repair function (Konc et al. PLoS Comput Biol 9(11): e1003341, 2013). We observed that the severely bent, strained DNA binds to the protein for the entire 90 ns of classical molecular dynamics (MD) performed; we could observe no significant changes in the most distorted region of the DNA, where the cleavage of phosphodiester bond occurs. In this article, we modeled the reaction mechanism at the interface between Tm1631 and its proposed ligand, the DNA molecule, focusing on cleavage of the phosphodiester bond. After addition of two Mg²⁺ ions to the reaction center and extension of classical MD by 50 ns (totaling 140 ns), the DNA ligand stayed bolted to the protein. Results from density functional theory quantum mechanics/molecular mechanics (QM/MM) calculations suggest that the reaction is analogous to known endonuclease mechanisms: an enzyme reaction mechanism with two Mg²⁺ ions in the reaction center and a pentacovalent intermediate. The minimum energy pathway profile shows that the phosphodiester bond cleavage step of the reaction is kinetically controlled and not thermodynamically because of a lack of any energy barrier above the accuracy of the energy profile calculation. The role of ions is shown by comparing the results with the reaction mechanisms in the absence of the Mg²⁺ ions where there is a significantly higher reaction barrier than in the presence of the Mg²⁺ ions.Graphical abstract

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.

Use of Nonequilibrium Work Methods to Compute Free Energy Differences Between Molecular Mechanical and Quantum Mechanical Representations of Molecular Systems

Carrying out free energy simulations (FES) using quantum mechanical (QM) Hamiltonians remains an attractive, albeit elusive goal. Renewed efforts in this area have focused on using "indirect" thermodynamic cycles to connect "low level" simulation results to "high level" free energies. The main obstacle to computing converged free energy results between molecular mechanical (MM) and QM (ΔA(MM→QM)), as recently demonstrated by us and others, is differences in the so-called "stiff" degrees of freedom (e.g., bond stretching) between the respective energy surfaces. Herein, we demonstrate that this problem can be efficiently circumvented using nonequilibrium work (NEW) techniques, i.e., Jarzynski's and Crooks' equations. Initial applications of computing ΔA(NEW)(MM→QM), for blocked amino acids alanine and serine as well as to generate butane's potentials of mean force via the indirect QM/MM FES method, showed marked improvement over traditional FES approaches.

Free energy and hidden barriers of the β-sheet structure of prion protein

On-the-fly free-energy parametrization is a new collective variable biasing approach akin to metadynamics with one important distinction: rather than acquiring an accelerated distribution via a history-dependent bias potential, sampling on this distribution is achieved from the beginning of the simulation using temperature-accelerated molecular dynamics. In the present work, we compare the performance of both approaches to compute the free-energy profile along a scalar collective variable measuring the H-bond registry of the β-sheet structure of the mouse Prion protein. Both methods agree on the location of the free-energy minimum, but free-energy profiles from well-tempered metadynamics are subject to a much higher degree of statistical noise due to hidden barriers. The sensitivity of metadynamics to hidden barriers is shown to be a consequence of the history dependence of the bias potential, and we detail the nature of these barriers for the prion β-sheet. In contrast, on-the-fly parametrization is much less sensitive to these barriers and thus displays improved convergence behavior relative to that of metadynamics. While hidden barriers are a frequent and central issue in free-energy methods, on-the-fly free-energy parametrization appears to be a robust and preferable method to confront this issue.