How Many Conformations Need To Be Sampled To Obtain Converged QM/MM Energies? The Curse of Exponential Averaging

Elucidation of factors shaping reactivity of 5'-deoxyadenosyl - a prominent organic radical in biology

This study investigates the factors modulating the reactivity of 5'-deoxyadenosyl (5'dAdo˙) radical, a potent hydrogen atom abstractor that forms in the active sites of radical SAM enzymes and that otherwise undergoes a rapid self-decay in aqueous solution. Here, we compare hydrogen atom abstraction (HAA) reactions between native substrates of radical SAM enzymes and 5'dAdo˙ in aqueous solution and in two enzymatic microenvironments. With that we reveal that HAA efficiency of 5'dAdo˙ is due to (i) the in situ formation of 5'dAdo˙ in a pre-ordered complex with a substrate, which attenuates the unfavorable effect of substrate:5'dAdo˙ complex formation, and (ii) the prevention of the conformational changes associated with self-decay by a tight active-site cavity. The enzymatic cavity, however, does not have a strong effect on the HAA activity of 5'dAdo˙. Thus, we performed an analysis of in-water HAA performed by 5'dAdo˙ based on a three-component thermodynamic model incorporating the diagonal effect of the free energy of reaction, and the off-diagonal effect of asynchronicity and frustration. To this aim, we took advantage of the straightforward relationship between the off-diagonal thermodynamic effects and the electronic-structure descriptor - the redistribution of charge between the reactants during the reaction. It allows to access HAA-competent redox and acidobasic properties of 5'dAdo˙ that are otherwise unavailable due to its instability upon one-electron reduction and protonation. The results show that all reactions feature a favourable thermodynamic driving force and tunneling, the latter of which lowers systematically barriers by ∼2 kcal mol-1. In addition, most of the reactions experience a favourable off-diagonal thermodynamic contribution. In HAA reactions, 5'dAdo˙ acts as a weak oxidant as well as a base, also 5'dAdo˙-promoted HAA reactions proceed with a quite low degree of asynchronicity of proton and electron transfer. Finally, the study elucidates the crucial and dual role of asynchronicity. It directly lowers the barrier as a part of the off-diagonal thermodynamic contribution, but also indirectly increases the non-thermodynamic part of the barrier by presumably controlling the adiabatic coupling between proton and electron transfer. The latter signals that the reaction proceeds as a hydrogen atom transfer rather than a proton-coupled electron transfer.

Insights and Challenges in Correcting Force Field Based Solvation Free Energies Using a Neural Network Potential

We present a comprehensive study investigating the potential gain in accuracy for calculating absolute solvation free energies (ASFE) using a neural network potential to describe the intramolecular energy of the solute. We calculated the ASFE for most compounds from the FreeSolv database using the Open Force Field (OpenFF) and compared them to earlier results obtained with the CHARMM General Force Field (CGenFF). By applying a nonequilibrium (NEQ) switching approach between the molecular mechanics (MM) description (either OpenFF or CGenFF) and the neural net potential (NNP)/MM level of theory (using ANI-2x as the NNP potential), we attempted to improve the accuracy of the calculated ASFEs. The predictive performance of the results did not change when this approach was applied to all 589 small molecules in the FreeSolv database that ANI-2x can describe. When selecting a subset of 156 molecules, focusing on compounds where the force fields performed poorly, we saw a slight improvement in the root-mean-square error (RMSE) and mean absolute error (MAE). The majority of our calculations utilized unidirectional NEQ protocols based on Jarzynski's equation. Additionally, we conducted bidirectional NEQ switching for a subset of 156 solutes. Notably, only a small fraction (10 out of 156) exhibited statistically significant discrepancies between unidirectional and bidirectional NEQ switching free energy estimates.

Convergence criteria for single-step free-energy calculations: the relation between the Π bias measure and the sample variance

Free energy calculations play a crucial role in simulating chemical processes, enzymatic reactions, and drug design. However, assessing the reliability and convergence of these calculations remains a challenge. This study focuses on single-step free-energy calculations using thermodynamic perturbation. It explores how the sample distributions influence the estimated results and evaluates the reliability of various convergence criteria, including Kofke's bias measure Π and the standard deviation of the energy difference ΔU, σ ΔU . The findings reveal that for Gaussian distributions, there is a straightforward relationship between Π and σ ΔU , free energies can be accurately approximated using a second-order cumulant expansion, and reliable results are attainable for σ ΔU up to 25 kcal mol-1. However, interpreting non-Gaussian distributions is more complex. If the distribution is skewed towards more positive values than a Gaussian, converging the free energy becomes easier, rendering standard convergence criteria overly stringent. Conversely, distributions that are skewed towards more negative values than a Gaussian present greater challenges in achieving convergence, making standard criteria unreliable. We propose a practical approach to assess the convergence of estimated free energies.

The influence of model building schemes and molecular dynamics sampling on QM-cluster models: the chorismate mutase case study

Most QM-cluster models of enzymes are constructed based on X-ray crystal structures, which limits comparison to in vivo structure and mechanism. The active site of chorismate mutase from Bacillus subtilis and the enzymatic transformation of chorismate to prephenate is used as a case study to guide construction of QM-cluster models built first from the X-ray crystal structure, then from molecular dynamics (MD) simulation snapshots. The Residue Interaction Network ResidUe Selector (RINRUS) software toolkit, developed by our group to simplify and automate the construction of QM-cluster models, is expanded to handle MD to QM-cluster model workflows. Several options, some employing novel topological clustering from residue interaction network (RIN) information, are evaluated for generating conformational clustering from MD simulation. RINRUS then generates a statistical thermodynamic framework for QM-cluster modeling of the chorismate mutase mechanism via refining 250 MD frames with density functional theory (DFT). The 250 QM-cluster models sampled provide a mean ΔG‡ of 10.3 ± 2.6 kcal mol-1 compared to the experimental value of 15.4 kcal mol-1 at 25 °C. While the difference between theory and experiment is consequential, the level of theory used is modest and therefore "chemical" accuracy is unexpected. More important are the comparisons made between QM-cluster models designed from the X-ray crystal structure versus those from MD frames. The large variations in kinetic and thermodynamic properties arise from geometric changes in the ensemble of QM-cluster models, rather from the composition of the QM-cluster models or from the active site-solvent interface. The findings open the way for further quantitative and reproducible calibration in the field of computational enzymology using the model construction framework afforded with the RINRUS software toolkit.

Catalytic Effects of Active Site Conformational Change in the Allosteric Activation of Imidazole Glycerol Phosphate Synthase

Imidazole glycerol phosphate synthase (IGPS) is a class-I glutamine amidotransferase (GAT) that hydrolyzes glutamine. Ammonia is produced and transferred to a second active site, where it reacts with N1-(5'-phosphoribosyl)-formimino-5-aminoimidazole-4-carboxamide ribonucleotide (PrFAR) to form precursors to purine and histidine biosynthesis. Binding of PrFAR over 25 Å away from the active site increases glutaminase efficiency by ∼4500-fold, primarily altering the glutamine turnover number. IGPS has been the focus of many studies on allosteric communication; however, atomic details for how the glutamine hydrolysis rate increases in the presence of PrFAR are lacking. We present a density functional theory study on 237-atom active site cluster models of IGPS based on crystallized structures representing the inactive and allosterically active conformations and investigate the multistep reaction leading to thioester formation and ammonia production. The proposed mechanism is supported by similar, well-studied enzyme mechanisms, and the corresponding energy profile is consistent with steady-state kinetic studies of PrFAR + IGPS. Additional active site models are constructed to examine the relationship between active site structural change and transition-state stabilization via energy decomposition schemes. The results reveal that the inactive IGPS conformation does not provide an adequately formed oxyanion hole structure and that repositioning of the oxyanion strand relative to the substrate is vital for a catalysis-competent oxyanion hole, with or without the hVal51 dihedral flip. These findings are valuable for future endeavors in modeling the IGPS allosteric mechanism by providing insight into the atomistic changes required for rate enhancement that can inform suitable reaction coordinates for subsequent investigations.

Bridging semiempirical and ab initio QM/MM potentials by Gaussian process regression and its sparse variants for free energy simulation

Free energy simulations that employ combined quantum mechanical and molecular mechanical (QM/MM) potentials at ab initio QM (AI) levels are computationally highly demanding. Here, we present a machine-learning-facilitated approach for obtaining AI/MM-quality free energy profiles at the cost of efficient semiempirical QM/MM (SE/MM) methods. Specifically, we use Gaussian process regression (GPR) to learn the potential energy corrections needed for an SE/MM level to match an AI/MM target along the minimum free energy path (MFEP). Force modification using gradients of the GPR potential allows us to improve configurational sampling and update the MFEP. To adaptively train our model, we further employ the sparse variational GP (SVGP) and streaming sparse GPR (SSGPR) methods, which efficiently incorporate previous sample information without significantly increasing the training data size. We applied the QM-(SS)GPR/MM method to the solution-phase SN2 Menshutkin reaction, NH3+CH3Cl→CH3NH3++Cl-, using AM1/MM and B3LYP/6-31+G(d,p)/MM as the base and target levels, respectively. For 4000 configurations sampled along the MFEP, the iteratively optimized AM1-SSGPR-4/MM model reduces the energy error in AM1/MM from 18.2 to 4.4 kcal/mol. Although not explicitly fitting forces, our method also reduces the key internal force errors from 25.5 to 11.1 kcal/mol/Å and from 30.2 to 10.3 kcal/mol/Å for the N-C and C-Cl bonds, respectively. Compared to the uncorrected simulations, the AM1-SSGPR-4/MM method lowers the predicted free energy barrier from 28.7 to 11.7 kcal/mol and decreases the reaction free energy from -12.4 to -41.9 kcal/mol, bringing these results into closer agreement with their AI/MM and experimental benchmarks.

Accelerated Quantum Mechanics/Molecular Mechanics Simulations via Neural Networks Incorporated with Mechanical Embedding Scheme

A powerful tool to study the mechanism of reactions in solutions or enzymes is to perform the ab initio quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) simulations. However, the computational cost is too high due to the explicit electronic structure calculations at every time step of the simulation. A neural network (NN) method can accelerate the QM/MM-MD simulations, but it has long been a problem to accurately describe the QM/MM electrostatic coupling by NN in the electrostatic embedding (EE) scheme. In this work, we developed a new method to accelerate QM/MM calculations in the mechanic embedding (ME) scheme. The potentials and partial point charges of QM atoms are first learned in vacuo by the embedded atom neural networks (EANN) approach. MD simulations are then performed on this EANN/MM potential energy surface (PES) to obtain free energy (FE) profiles for reactions, in which the QM/MM electrostatic coupling is treated in the mechanic embedding (ME) scheme. Finally, a weighted thermodynamic perturbation (wTP) corrects the FE profiles in the ME scheme to the EE scheme. For two reactions in water and one in methanol, our simulations reproduced the B3LYP/MM free energy profiles within 0.5 kcal/mol with a speed-up of 30-60-fold. The results show that the strategy of combining EANN potential in the ME scheme with the wTP correction is efficient and reliable for chemical reaction simulations in liquid. Another advantage of our method is that the QM PES is independent of the MM subsystem, so it can be applied to various MM environments as demonstrated by an S_N2 reaction studied in water and methanol individually, which used the same EANN PES. The free energy profiles are in excellent accordance with the results obtained from B3LYP/MM-MD simulations. In future, this method will be applied to the reactions of enzymes and their variants.

Multiscale Modeling of Phosphate···π Contacts in RNA U-Turns Exposes Differences between Quantum-Chemical and AMBER Force Field Descriptions

Phosphate···π, also called anion···π, contacts occur between nucleobases and anionic phosphate oxygens (OP2) in r(GNRA) and r(UNNN) U-turn motifs (N = A,G,C,U; R = A,G). These contacts were investigated using state-of-the-art quantum-chemical methods (QM) to characterize their physicochemical properties and to serve as a reference to evaluate AMBER force field (AFF) performance. We found that phosphate···π interaction energies calculated with the AFF for dimethyl phosphate···nucleobase model systems are less stabilizing in comparison with double-hybrid DFT and that minimum contact distances are larger for all nucleobases. These distance stretches are also observed in large-scale AFF vs QM/MM computations and classical molecular dynamics (MD) simulations on several r(gcGNRAgc) tetraloop hairpins when compared to experimental data extracted from X-ray/cryo-EM structures (res. ≤ 2.5 Å) using the WebFR3D bioinformatic tool. MD simulations further revealed shifted OP2/nucleobase positions. We propose that discrepancies between the QM and AFF result from a combination of missing polarization in the AFF combined with too large AFF Lennard-Jones (LJ) radii of nucleobase carbon atoms in addition to an exaggerated short-range repulsion of the r^-12 LJ repulsive term. We compared these results with earlier data gathered on lone pair···π contacts in CpG Z-steps occurring in r(UNCG) tetraloops. In both instances, charge transfer calculations do not support any significant n → π* donation effects. We also investigated thiophosphate···π contacts that showed reduced stabilizing interaction energies when compared to phosphate···π contacts. Thus, we challenge suggestions that the experimentally observed enhanced thermodynamic stability of phosphorothioated r(GNRA) tetraloops can be explained by larger London dispersion.

Influence of the Greater Protein Environment on the Electrostatic Potential in Metalloenzyme Active Sites: The Case of Formate Dehydrogenase

The Mo/W-containing metalloenzyme formate dehydrogenase (FDH) is an efficient and selective natural catalyst that reversibly converts CO₂ to formate under ambient conditions. In this study, we investigate the impact of the greater protein environment on the electrostatic potential (ESP) of the active site. To model the enzyme environment, we used a combination of classical molecular dynamics and multiscale quantum-mechanical (QM)/molecular-mechanical (MM) simulations. We leverage charge shift analysis to systematically construct QM regions and analyze the electronic environment of the active site by evaluating the degree of charge transfer between the core active site and the protein environment. The contribution of the terminal chalcogen ligand to the ESP of the metal center is substantial and dependent on the chalcogen identity, with similar, less negative ESPs for Se and S terminal chalcogens in comparison to O regardless of whether the metal is Mo or W. The orientation of the side chains and conformations of the cofactor also affect the ESP, highlighting the importance of sampling dynamic fluctuations in the protein. Overall, our observations suggest that the terminal chalcogen ligand identity plays an important role in the enzymatic activity of FDH, suggesting opportunities for a rational bioinspired catalyst design.

Mechanistic Insights into Enzyme Catalysis from Explaining Machine-Learned Quantum Mechanical and Molecular Mechanical Minimum Energy Pathways

With the increasing popularity of machine learning (ML) applications, the demand for explainable artificial intelligence techniques to explain ML models developed for computational chemistry has also emerged. In this study, we present the development of the Boltzmann-weighted cumulative integrated gradients (BCIG) approach for effective explanation of mechanistic insights into ML models trained on high-level quantum mechanical and molecular mechanical (QM/MM) minimum energy pathways. Using the acylation reactions of the Toho-1 β-lactamase and two antibiotics (ampicillin and cefalexin) as the model systems, we show that the BCIG approach could quantitatively attribute the energetic contribution in one system and the relative reactivity of individual steps across different systems to specific chemical processes such as the bond making/breaking and proton transfers. The proposed BCIG contribution attribution method quantifies chemistry-interpretable insights in terms of contributions from each elementary chemical process, which is in agreement with the validating QM/MM calculations and our intuitive mechanistic understandings of the model reactions.

Computational Modeling of Conformer Stability in Benenodin-1, a Thermally Actuated Lasso Peptide Switch

Benenodin-1 is a thermally actuated lasso peptide rotaxane switch with two primary translational isomers that differ in the relative position of the residue Gln15. The conversion from one conformer to the other involves substantial enthalpy-entropy compensation: one conformer is energetically favored and the other is entropically favored. Here, we take a multi-scale quantum mechanical (QM) and classical molecular dynamic (MD) approach to reveal residue-specific sources of these differences in stability. QM reveals that the two benenodin-1 conformers involve distinct hydrogen bonding networks, with the enthalpically favored conformer having more intra-peptide hydrogen bonds between the Gln15 side chain and nearby residues. The evaluation of configurational entropy over the MD-sampled geometries reveals that the entropically favored conformer has enhanced conformational flexibility. By computing the by-residue-sum entropies, we identify the role of Gln15 and neighboring Glu14 in mediating the entropic variation during the switching process. These computational insights help explain the effects of Glu14Ala and Gln15Ala mutations on the conformational population of benenodin-1 observed experimentally.

Exponential averaging versus umbrella sampling for computing the QM/MM free energy barrier of the initial step of the desuccinylation reaction catalyzed by sirtuin 5

The computational characterization of enzymatic reactions poses a great challenge which arises from the high dimensional and often rough potential energy surfaces commonly explored by static QM/MM methods such as adiabatic mapping (AM). The present study highlights the difficulties in estimating free energy barriers via exponential averaging over AM pathways. Based on our previous study [von der Esch et al., J. Chem. Theory Comput., 2019, 15, 6660-6667], where we analyzed the first reaction step of the desuccinylation reaction catalyzed by human sirtuin 5 (SIRT5) by means of QM/MM adiabatic mapping and machine learning, we use, here, umbrella sampling to compute the free energy profile of the initial reaction step. The computational investigations show that the initial step of the desuccinylation reaction proceeds via an S_N2-type reaction mechanism in SIRT5, suggesting that the first step of the deacylation reactions catalyzed by sirtuins is highly conserved. In addition, the direct comparison of the extrapolated free energy barrier from minimal energy paths and the computed free energy path from umbrella sampling further underlines the importance of extensive sampling.

Factors That Determine the Variation of Equilibrium and Kinetic Properties of QM/MM Enzyme Simulations: QM Region, Conformation, and Boundary Condition

To analyze the impact of various technical details on the results of quantum mechanical (QM)/molecular mechanical (MM) enzyme simulations, including the QM region size, catechol-O-methyltransferase (COMT) is studied as a model system using an approximate QM/MM method (DFTB3/CHARMM). The results show that key equilibrium and kinetic properties for methyl transfer in COMT exhibit limited variations with respect to the size of the QM region, which ranges from ∼100 to ∼500 atoms in this study. With extensive sampling, local and global structural characteristics of the enzyme are largely conserved across the studied QM regions, while the nature of the transition state (e.g., secondary kinetic isotope effect) and reaction exergonicity are largely maintained. Deviations in the free energy profile with different QM region sizes are similar in magnitude to those observed with changes in other simulation protocols, such as different initial enzyme conformations and boundary conditions. Electronic structural properties, such as the covariance matrix of residual charge fluctuations, appear to exhibit rather long-range correlations, especially when the peptide backbone is included in the QM region; this observation holds when a range-separated DFT approach is used as the QM region, suggesting that delocalization error is unlikely the origin. Overall, the analyses suggest that multiple simulation details determine the results of QM/MM enzyme simulations with comparable contributions.

Machine-Learning-Assisted Free Energy Simulation of Solution-Phase and Enzyme Reactions

Despite recent advances in the development of machine learning potentials (MLPs) for biomolecular simulations, there has been limited effort on developing stable and accurate MLPs for enzymatic reactions. Here we report a protocol for performing machine-learning-assisted free energy simulation of solution-phase and enzyme reactions at the ab initio quantum-mechanical/molecular-mechanical (ai-QM/MM) level of accuracy. Within our protocol, the MLP is built to reproduce the ai-QM/MM energy and forces on both QM (reactive) and MM (solvent/enzyme) atoms. As an alternative strategy, a delta machine learning potential (ΔMLP) is trained to reproduce the differences between the ai-QM/MM and semiempirical (se) QM/MM energies and forces. To account for the effect of the condensed-phase environment in both MLP and ΔMLP, the DeePMD representation of a molecular system is extended to incorporate the external electrostatic potential and field on each QM atom. Using the Menshutkin and chorismate mutase reactions as examples, we show that the developed MLP and ΔMLP reproduce the ai-QM/MM energy and forces with errors that on average are less than 1.0 kcal/mol and 1.0 kcal mol-1 Å-1, respectively, for representative configurations along the reaction pathway. For both reactions, MLP/ΔMLP-based simulations yielded free energy profiles that differed by less than 1.0 kcal/mol from the reference ai-QM/MM results at only a fraction of the computational cost.

On the Accuracy of QM/MM Models: A Systematic Study of Intramolecular Proton Transfer Reactions of Amino Acids in Water

This work presents a systematic assessment of QM/QM' and QM/MM models with respect to direct QM calculations for the tautomerization (neutral to zwitterion) reactions of amino acids (glycine, alanine, valine, aspartate, and neutral and protonated histidine) solvated in a 160 water cluster. The effect of varying QM region size and choice of embedding potentials, including fixed-charge and polarizable molecular mechanics force fields (TIP3P and EFP) and various semiempirical QM methods (PM7, GFN2-xTB, DFTBA, DFTB3, HF-3c, and PBEh-3c), on the accuracy of the models was examined. A surprising finding was that molecular mechanics force fields outperformed many of the semiempirical methods. Generally, the errors in the QM/QM' and QM/MM models converge slowly with respect to the QM region size, requiring 50 or more waters to be included in the QM region before the error in the model falls below 1 kcal mol^-1 of its pure QM result. Different QM region selection schemes were also compared, and it was found that selection based on Natural Population Analysis (NPA) atomic charges significantly reduced the error in the QM/QM' and QM/MM models particularly if a low-quality embedding potential was used. It is envisaged that these results will be useful for the development of future hybrid QM models.

Harder, better, faster, stronger: Large-scale QM and QM/MM for predictive modeling in enzymes and proteins

Computational prediction of enzyme mechanism and protein function requires accurate physics-based models and suitable sampling. We discuss recent advances in large-scale quantum mechanical (QM) modeling of biochemical systems that have reduced the cost of high-accuracy models. Tradeoffs between sampling and accuracy have motivated modeling with molecular mechanics (MM) in a multiscale QM/MM or iterative approach. Limitations to both conventional density-functional theory and classical MM force fields remain for describing noncovalent interactions in comparison to experiment or wavefunction theory. Because predictions of enzyme action (i.e. electrostatics), free energy barriers, and mechanisms are sensitive to the protocol and embedding method in QM/MM, convergence tests and systematic methods for quantifying QM-level interactions are a needed, active area of development.

On the Use of Interaction Entropy and Related Methods to Estimate Binding Entropies

Molecular mechanics combined with Poisson-Boltzmann or generalized Born and solvent-accessible area solvation energies (MM/PBSA and MM/GBSA) are popular methods to estimate the free energy for the binding of small molecules to biomacromolecules. However, the estimation of the entropy has been problematic and time-consuming. Traditionally, normal-mode analysis has been used to estimate the entropy, but more recently, alternative approaches have been suggested. In particular, it has been suggested that exponential averaging of the electrostatic and Lennard-Jones interaction energies may provide much faster and more accurate entropies, the interaction entropy (IE) approach. In this study, we show that this exponential averaging is extremely poorly conditioned. Using stochastic simulations, assuming that the interaction energies follow a Gaussian distribution, we show that if the standard deviation of the interaction energies (σIE) is larger than 15 kJ/mol, it becomes practically impossible to converge the interaction entropies (more than 10 million energies are needed, and the number increases exponentially). A cumulant approximation to the second order of the exponential average shows a better convergence, but for σIE > 25 kJ/mol, it gives entropies that are unrealistically large. Moreover, in practical applications, both methods show a steady increase in the entropy with the number of energies considered.

Free energy change estimation: The Divide and Conquer MBAR method

In the present study, the Divide and Conquer MBAR (DC-MBAR) method is proposed to predict the free energies based on the data sampled by multi-states simulations. For DC-MBAR method, the overlap between any two alchemical states is calculated first and those with sufficient overlap are defined as the adjacent states. Unlike the traditional MBAR method, which calculates the free energy of each state using all the data at once, DC-MBAR focuses on predicting the free energy changes between adjacent states. To estimate the free energy changes accurately, the other states with overlaps with the two adjacent states bigger than the defined threshold are included in the MBAR equation. At a specific threshold, the free energies predicted by DC-MBAR are very close to those calculated by the traditional MABR method. Furthermore, DC-MBAR scheme can reduce both the computation and memory cost. One important characteristic of DC-MBAR method is linear scaling, which means the CPU time with the change of the number of states is a straight-line relation. As the pair-based calculations are mutually independent and parallelizable, all accessible CPU cores on the HPC cluster could be utilized, which makes DC-MBAR strategy more efficient.

Biomolecular QM/MM Simulations: What Are Some of the "Burning Issues"?

QM/MM simulations have become an indispensable tool in many chemical and biochemical investigations. Considering the tremendous degree of success, including recognition by a 2013 Nobel Prize in Chemistry, are there still "burning challenges" in QM/MM methods, especially for biomolecular systems? In this short Perspective, we discuss several issues that we believe greatly impact the robustness and quantitative applicability of QM/MM simulations to many, if not all, biomolecules. We highlight these issues with observations and relevant advances from recent studies in our group and others in the field. Despite such limited scope, we hope the discussions are of general interest and will stimulate additional developments that help push the field forward in meaningful directions.

A simplified charge projection scheme for long-range electrostatics in ab initio QM/MM calculations

In a previous work [Pan et al., Molecules 23, 2500 (2018)], a charge projection scheme was reported, where outer molecular mechanical (MM) charges [>10 Å from the quantum mechanical (QM) region] were projected onto the electrostatic potential (ESP) grid of the QM region to accurately and efficiently capture long-range electrostatics in ab initio QM/MM calculations. Here, a further simplification to the model is proposed, where the outer MM charges are projected onto inner MM atom positions (instead of ESP grid positions). This enables a representation of the long-range MM electrostatic potential via augmentary charges (AC) on inner MM atoms. Combined with the long-range electrostatic correction function from Cisneros et al. [J. Chem. Phys. 143, 044103 (2015)] to smoothly switch between inner and outer MM regions, this new QM/MM-AC electrostatic model yields accurate and continuous ab initio QM/MM electrostatic energies with a 10 Å cutoff between inner and outer MM regions. This model enables efficient QM/MM cluster calculations with a large number of MM atoms as well as QM/MM calculations with periodic boundary conditions.

Dependency of solvation effects on metal identity in surface reactions

Solvent interactions with adsorbed moieties involved in surface reactions are often believed to be similar for different metal surfaces. However, solvents alter the electronic structures of surface atoms, which in turn affects their interaction with adsorbed moieties. To reveal the importance of metal identity on aqueous solvent effects in heterogeneous catalysis, we studied solvent effects on the activation free energies of the O-H and C-H bond cleavages of ethylene glycol over the (111) facet of six transition metals (Ni, Pd, Pt, Cu, Ag, Au) using an explicit solvation approach based on a hybrid quantum mechanical/molecular mechanical (QM/MM) description of the potential energy surface. A significant metal dependence on aqueous solvation effects was observed that suggests solvation effects must be studied in detail for every reaction system. The main reason for this dependence could be traced back to a different amount of charge-transfer between the adsorbed moieties and metals in the reactant and transition states for the different metal surfaces.

UUCG RNA Tetraloop as a Formidable Force-Field Challenge for MD Simulations

Explicit solvent atomistic molecular dynamics (MD) simulations represent an established technique to study structural dynamics of RNA molecules and an important complement for diverse experimental methods. However, performance of molecular mechanical (MM) force fields (ff's) remains far from satisfactory even after decades of development, as apparent from a problematic structural description of some important RNA motifs. Actually, some of the smallest RNA molecules belong to the most challenging systems for MD simulations and, among them, the UUCG tetraloop is saliently difficult. We report a detailed analysis of UUCG MD simulations, depicting the sequence of events leading to the loss of the UUCG native state during MD simulations. The total amount of MD simulation data analyzed in this work is close to 1.3 ms. We identify molecular interactions, backbone conformations, and substates that are involved in the process. Then, we unravel specific ff deficiencies using diverse quantum mechanical/molecular mechanical (QM/MM) and QM calculations. Comparison between the MM and QM methods shows discrepancies in the description of the 5'-flanking phosphate moiety and both signature sugar-base interactions. Our work indicates that poor behavior of the UUCG tetraloop in simulations is a complex issue that cannot be attributed to one dominant and straightforwardly correctable factor. Instead, there is a concerted effect of multiple ff inaccuracies that are coupled and amplifying each other. We attempted to improve the simulation behavior by some carefully tailored interventions, but the results were still far from satisfactory, underlying the difficulties in development of accurate nucleic acid ff's.

Multi-level free energy simulation with a staged transformation approach

Combining multiple levels of theory in free energy simulations to balance computational accuracy and efficiency is a promising approach for studying processes in the condensed phase. While the basic idea has been proposed and explored for quite some time, it remains challenging to achieve convergence for such multi-level free energy simulations as it requires a favorable distribution overlap between different levels of theory. Previous efforts focused on improving the distribution overlap by either altering the low-level of theory for the specific system of interest or ignoring certain degrees of freedom. Here, we propose an alternative strategy that first identifies the degrees of freedom that lead to gaps in the distributions of different levels of theory and then treats them separately with either constraints or restraints or by introducing an intermediate model that better connects the low and high levels of theory. As a result, the conversion from the low level to the high level model is done in a staged fashion that ensures a favorable distribution overlap along the way. Free energy components associated with different steps are mostly evaluated explicitly, and thus, the final result can be meaningfully compared to the rigorous free energy difference between the two levels of theory with limited and well-defined approximations. The additional free energy component calculations involve simulations at the low level of theory and therefore do not incur high computational costs. The approach is illustrated with two simple but non-trivial solution examples, and factors that dictate the reliability of the result are discussed.

Both Configuration and QM Region Size Matter: Zinc Stability in QM/MM Models of DNA Methyltransferase

Quantum-mechanical/molecular-mechanical (QM/MM) methods are essential to the study of metalloproteins, but the relative importance of sampling and degree of QM treatment in achieving quantitative predictions is poorly understood. We study the relative magnitude of configurational and QM-region sensitivity of energetic and electronic properties in a representative Zn²⁺ metal binding site of a DNA methyltransferase. To quantify property variations, we analyze snapshots extracted from 250 ns of molecular dynamics simulation. To understand the degree of QM-region sensitivity, we perform analysis using QM regions ranging from a minimal 49-atom region consisting only of the Zn²⁺ metal and its four coordinating Cys residues up to a 628-atom QM region that includes residues within 12 Å of the metal center. Over the configurations sampled, we observe that illustrative properties (e.g., rigid Zn²⁺ removal energy) exhibit large fluctuations that are well captured with even minimal QM regions. Nevertheless, for both energetic and electronic properties, we observe a slow approach to asymptotic limits with similarly large changes in absolute values that converge only with larger (ca. 300-atom) QM region sizes. For the smaller QM regions, the electronic description of Zn²⁺ binding is incomplete: the metal binds too tightly and is too stabilized by the strong electrostatic potential of MM point charges, and the Zn-S bond covalency is overestimated. Overall, this work suggests that efficient sampling with QM/MM in small QM regions is an effective method to explore the influence of enzyme structure on target properties. At the same time, accurate descriptions of electronic and energetic properties require a larger QM region than the minimal metal-coordinating residues in order to converge treatment of both metal-local bonding and the overall electrostatic environment.

Quantum Chemical and QM/MM Models in Biochemistry

Quantum chemical (QC) calculations provide a basis for deriving a microscopic understanding of enzymes and photobiological systems. Here we describe how QC models can be used to explore the electronic structure, dynamics, and energetics of biomolecules. We introduce the hybrid quantum mechanics/classical mechanics (QM/MM) approach, where a quantum mechanically described system of interest is embedded in a classically described force field representation of the biochemical surroundings. We also discuss the QM cluster model approach, as well as embedding theories, that provide complementary methodologies to model quantum mechanical effects in biomolecules. The chapter also provides some practical guides for building quantum biochemical models using the quinone reduction catalysis in respiratory complex I and a model reaction in solution as examples.

Evidence of Skewness and Sub-Gaussian Character in Temperature-Dependent Distributions of One Million Electronic Excitation Energies in PbS Quantum Dots

Obtaining statistical distributions by sampling a large number of conformations is vital for an accurate description of temperature-dependent properties of chemical systems. However, constructing distributions with 10⁵-10⁶ samples is computationally challenging because of the prohibitively high computational cost of performing first-principles quantum mechanical calculations. In this work, we present a new technique called the effective stochastic potential configuration interaction singles (ESP-CIS) method to obtain excitation energies. The ESP-CIS method uses random matrix theory for the construction of an effective stochastic representation of the Fock operator and combines it with the CIS method. Excited-state energies of PbS quantum dots (0.75-1.75 nm) at temperatures of 10-400 K were calculated using the ESP-CIS method. Results from a total of 27 million excitation energy calculations revealed the distributions to be sub-Gaussian in nature with negative skewness, which progressively became red-shifted with increasing temperature. This study demonstrates the efficacy of the ESP-CIS method as a general-purpose method for efficient excited-state calculations.

Design and SAR Analysis of Covalent Inhibitors Driven by Hybrid QM/MM Simulations

Quantum mechanics/molecular mechanics (QM/MM) hybrid technique is emerging as a reliable computational method to investigate and characterize chemical reactions occurring in enzymes. From a drug discovery perspective, a thorough understanding of enzyme catalysis appears pivotal to assist the design of inhibitors able to covalently bind one of the residues belonging to the enzyme catalytic machinery. Thanks to the current advances in computer power, and the availability of more efficient algorithms for QM-based simulations, the use of QM/MM methodology is becoming a viable option in the field of covalent inhibitor design. In the present review, we summarized our experience in the field of QM/MM simulations applied to drug design problems which involved the optimization of agents working on two well-known drug targets, namely fatty acid amide hydrolase (FAAH) and epidermal growth factor receptor (EGFR). In this context, QM/MM simulations gave valuable information in terms of geometry (i.e., of transition states and metastable intermediates) and reaction energetics that allowed to correctly predict inhibitor binding orientation and substituent effect on enzyme inhibition. What is more, enzyme reaction modelling with QM/MM provided insights that were translated into the synthesis of new covalent inhibitor featured by a unique combination of intrinsic reactivity, on-target activity, and selectivity.

Finding Reactive Configurations: A Machine Learning Approach for Estimating Energy Barriers Applied to Sirtuin 5

Sirtuin 5 is a class III histone deacetylase that, unlike its classification, mainly catalyzes desuccinylation and demanoylation reactions. It is an interesting drug target that we use here to test new ideas for calculating reaction pathways of large molecular systems such as enzymes. A major issue with most schemes (e.g., adiabatic mapping) is that the resulting activation barrier height heavily depends on the chosen educt conformation. This makes the selection of the initial structure decisive for the success of the characterization. Here, we apply machine learning to a large number of molecular dynamics frames and potential energy barriers obtained by quantum mechanics/molecular mechanics calculations in order to identify (1) suitable start-conformations for reaction path calculations and (2) structural features relevant for the first step of the desuccinylation reaction catalyzed by Sirtuin 5. The latter generally aids the understanding of reaction mechanisms and important interactions in active centers. Using our novel approach, we found eleven key features that govern the reactivity. We were able to estimate reaction barriers with a mean absolute error of 3.6 kcal/mol and identified reactive configurations.

Accelerated computation of free energy profile at ab initio quantum mechanical/molecular mechanical accuracy via a semi-empirical reference potential. II. Recalibrating semi-empirical parameters with force matching

An efficient and accurate reference potential simulation protocol is proposed for producing ab initio quantum mechanical/molecular mechanical (AI-QM/MM) quality free energy profiles for chemical reactions in a solvent or macromolecular environment. This protocol involves three stages: (a) using force matching to recalibrate a semi-empirical quantum mechanical (SE-QM) Hamiltonian for the specific reaction under study; (b) employing the recalibrated SE-QM Hamiltonian (in combination with molecular mechanical force fields) as the reference potential to drive umbrella samplings along the reaction pathway; and (c) computing AI-QM/MM energy values for collected configurations from the sampling and performing weighted thermodynamic perturbation to acquire an AI-QM/MM corrected reaction free energy profile. For three model reactions (identity SN2 reaction, Menshutkin reaction, and glycine proton transfer reaction) in aqueous solution and one enzyme reaction (Claisen arrangement in chorismate mutase), our simulations using recalibrated PM3 SE-QM Hamiltonians well reproduced QM/MM free energy profiles at the B3LYP/6-31G* level of theory all within 1 kcal mol-1 with a 20 to 45 fold reduction in the computer time.

Harnessing fluoroacetate dehalogenase for defluorination of fluorocarboxylic acids: in silico and in vitro approach

Widely distributed fluorocarboxylic acids have aroused worldwide environmental concerns due to its toxicity, persistence, and bioaccumulation. Enzyme-based eco-friendly biodegradation techniques have become increasingly important in treating fluorocarboxylic acids. Here we utilized in silico and in vitro approaches to investigate the defluorination mechanism of fluoroacetate dehalogenase (FAcD) toward monofluoropropionic acids at atomic-level. The experimentally determined k_cat and k_M for defluorination of 2-fluoropropionic acid are 330 ± 60 min^-1 and 6.12 ± 0.13 mM. The in silico results demonstrated positive/negative correlations between activation barriers and structural parameters (e.g. distance and angle) under different enzymatic conformations. We also screened computationally and tested in vitro (enzyme assay and kinetic study) the catalytic proficiency of FAcD toward polyfluoropropionic acids and perfluoropropionic acids which are known to be challenging for enzymatic degradation. The results revealed potential degradation activity of FAcD enzyme toward 2,3,3,3-tetrafluoropropionic acids. Our work will initiate the development of a new "integrated approach" for enzyme engineering to degrade environmentally persistent fluorocarboxylic acids.

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.

Quantum Mechanical Description of Electrostatics Provides a Unified Picture of Catalytic Action Across Methyltransferases

Methyl transferases (MTases) are a well-studied class of enzymes for which competing enzymatic enhancement mechanisms have been suggested, ranging from structural methyl group CH···X hydrogen bonds (HBs) to electrostatic- and charge-transfer-driven stabilization of the transition state (TS). We identified all Class I MTases for which reasonable resolution (<2.0 Å) crystal structures could be used to form catalytically competent ternary complexes for multiscale (i.e., quantum-mechanical/molecular-mechanical or QM/MM) simulation of the S_N2 methyl transfer reaction coordinate. The four Class I MTases studied have both distinct functions (e.g., protein repair or biosynthesis) and substrate nucleophiles (i.e., C, N, or O). While CH···X HBs stabilize all reactant complexes, no universal TS stabilization role is found for these interactions in MTases. A consistent picture is instead obtained through analysis of charge transfer and electrostatics, wherein much of cofactor-substrate charge separation is maintained in the TS region, and electrostatic potential is correlated with substrate nucleophilicity (i.e., intrinsic reactivity).

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.

Solvation free energies via alchemical simulations: let's get honest about sampling, once more

Free energy perturbation (FEP) approaches with stratification have seen widespread and increasing use in computational studies of biologically relevant molecules. However, when the molecular systems are characterized by a complex conformational free energy landscape, the assessment of convergence remains a concern for many practitioners. The sampling problem in FEP has been authoritatively addressed in a recent perspective paper [D. Mobley, J. Comput.-Aided Mol. Des., 2012, 26, 93], incisively entitled "Let's get honest about sampling". Here, I return to the issue of sampling in the determination of the octanol-water partition coefficient for a synthetic precursor of kinase inhibitors that has been included in the recent extension of the SAMPL6 blind challenge of log P coefficients. I will show that even for this simple compound, whose conformational space is essentially dictated by two sp₃ rotable bonds connecting rigid planar units, canonical sampling using standard techniques can be surprisingly hard to achieve. I will also show how the conformational sampling problem can be effectively bypassed using unidirectional and bidirectional nonequilibrium work methods, reliably recovering the solvation energy with minimal methodological uncertainty.

Projector-Based Embedding Eliminates Density Functional Dependence for QM/MM Calculations of Reactions in Enzymes and Solution

Combined quantum mechanics/molecular mechanics (QM/MM) methods are increasingly widely utilized in studies of reactions in enzymes and other large systems. Here, we apply a range of QM/MM methods to investigate the Claisen rearrangement of chorismate to prephenate, in solution, and in the enzyme chorismate mutase. Using projector-based embedding in a QM/MM framework, we apply treatments up to the CCSD(T) level. We test a range of density functional QM/MM methods and QM region sizes. The results show that the calculated reaction energetics are significantly more sensitive to the choice of density functional than they are to the size of the QM region in these systems. Projector-based embedding of a wave function method in DFT reduced the 13 kcal/mol spread in barrier heights calculated at the DFT/MM level to a spread of just 0.3 kcal/mol, essentially eliminating dependence on the functional. Projector-based embedding of correlated ab initio methods provides a practical method for achieving high accuracy for energy profiles derived from DFT and DFT/MM calculations for reactions in condensed phases.

Host-Guest Relative Binding Affinities at Density-Functional Theory Level from Semiempirical Molecular Dynamics Simulations

Relative free energies for the binding of nine cyclic carboxylate ligands to the octa-acid deep-cavity host were calculated at the combined density-functional theory and molecular mechanics (DFT/MM) level of theory. The DFT calculations employed the BLYP functional and the 6-31G* basis set for the ligand. We employed free-energy perturbations (FEP) with the reference-potential approach and used molecular dynamics (MD) simulations with the semiempirical quantum mechanical (SQM) PM6-DH+ method for the ligand as an intermediate level between MM and DFT/MM to improve the convergence. Thus, the relative binding free energy of two ligands was first calculated at the MM level by an alchemical transformation from one ligand to another in both the bound and unbound states. Then, for each ligand the free-energy correction for going from the MM to the SQM/MM potentials was calculated using explicit SQM/MM MD simulations. Finally, the free-energy correction for going from the SQM/MM to the DFT/MM potentials was estimated with FEP without running any DFT/MM simulations. Instead, the free energy was calculated by single-step exponential averaging (ssEA) or employing the cumulant approximation to the second order (CA). The results show that CA converges much better than ssEA, and with 500-4500 DFT/MM single-point energy calculations, converged free energies with a precision of 0.3 kJ/mol can be obtained. These free energies reproduce the experimental binding free energy differences with a mean absolute deviation of 3.4 kJ/mol, a correlation ( R²) of 0.97, and correct signs for all of the eight free-energy differences. This is appreciably better than the results obtained at the SQM/MM level of theory and also slightly better than those obtained with MM. We show that the convergence of the SQM/MM → DFT/MM perturbations can be monitored by the use of Wu and Kofke's bias metric Π and by the standard deviation of the difference between the SQM/MM and DFT/MM energies. Finally, we show that the use of the intermediate SQM/MM MD simulations improves the convergence of the free energies by a factor of at least two, compared to doing direct MM → DFT/MM perturbations.

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.

Solvation Free Energy Calculation Using a Fixed-Charge Model: Implicit and Explicit Treatments of the Polarization Effect

In this work, IPolQ-Mod charges and the reference potential scheme are used to calculate the solvation free energies of a set of organic molecules. Both methods could capture the phase transfer of a solute with accompanying polarization cost utilizing a fixed-charge model. The IPolQ-Mod charges, which are the average of two charge sets fitted in a vacuum state and a condensed phase, take account of the polarization effect implicitly. For the reference potential method, the quantum mechanics polarization corrections are calculated explicitly by thermodynamic perturbation. The polarization effect captured by the IPolQ-Mod charges is an approximation to that of the reference potential method theoretically. In the present study, the reference potential method shows a slight improvement over the classical restrained electrostatic potential (RESP) charges, which perform pretty well in predicting the solvation free energy. However, IPolQ-Mod(MP2) shows a poor agreement with the experimental data. Compared with IPolQ-Mod(MP2), IPolQ-Mod(M06-2X) or IPolQ-Mod(ωB97X) is found to give more appropriate prediction of the molecule's dipole and the solvation free energies calculated by IPolQ-Mod(M06-2X) or IPolQ-Mod(ωB97X) are more compatible with those of the RESP charges. If the other force field parameters remain unchanged, M06-2X or ωB97X is recommended to derive the IPolQ-Mod charges.

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.