Convergence of single-step free energy perturbation

Convergence-Adaptive Roundtrip Method Enables Rapid and Accurate FEP Calculations

The free energy perturbation (FEP) method is a powerful technique for accurate binding free energy calculations, which is crucial for identifying potent ligands with a high affinity in drug discovery. However, the widespread application of FEP is limited by the high computational cost required to achieve equilibrium sampling and the challenges in obtaining converged predictions. In this study, we present the convergence-adaptive roundtrip (CAR) method, which is an enhanced adaptive sampling approach, to address the key challenges in FEP calculations, including the precision-efficiency tradeoff, sampling efficiency, and convergence assessment. By employing on-the-fly convergence analysis to automatically adjust simulation times, enabling efficient traversal of the important phase space through rapid propagation of conformations between different states and eliminating the need for multiple parallel simulations, the CAR method increases convergence and minimizes computational overhead while maintaining calculation accuracy. The performance of the CAR method was evaluated through relative binding free energy (RBFE) calculations on benchmarks comprising four diverse protein-ligand systems. The results demonstrated a significant speedup of over 8-fold compared to conventional FEP methods while maintaining high accuracy. The overall R2 values of 0.65 and 0.56 were obtained using the combined-structure FEP approach and the single-step FEP approach, respectively, in conjunction with the CAR method. In-depth case studies further highlighted the superior performance of the CAR method in terms of convergence acceleration, improved predicted correlations, and reduced computational costs. The advancement of the CAR method makes it a highly effective approach, enhancing the applicability of FEP in drug discovery.

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.

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.

Reweighting from Molecular Mechanics Force Fields to the ANI-2x Neural Network Potential

To achieve chemical accuracy in free energy calculations, it is necessary to accurately describe the system's potential energy surface and efficiently sample configurations from its Boltzmann distribution. While neural network potentials (NNPs) have shown significantly higher accuracy than classical molecular mechanics (MM) force fields, they have a limited range of applicability and are considerably slower than MM potentials, often by orders of magnitude. To address this challenge, Rufa et al. [Rufa et al. bioRxiv 2020, 10.1101/2020.07.29.227959.] suggested a two-stage approach that uses a fast and established MM alchemical energy protocol, followed by reweighting the results using NNPs, known as endstate correction or indirect free energy calculation. This study systematically investigates the accuracy and robustness of reweighting from an MM reference to a neural network target potential (ANI-2x) for an established data set in vacuum, using single-step free-energy perturbation (FEP) and nonequilibrium (NEQ) switching simulation. We assess the influence of longer switching lengths and the impact of slow degrees of freedom on outliers in the work distribution and compare the results to those of multistate equilibrium free energy simulations. Our results demonstrate that free energy calculations between NNPs and MM potentials should be preferably performed using NEQ switching simulations to obtain accurate free energy estimates. NEQ switching simulations between the MM potentials and NNPs are efficient, robust, and trivial to implement.

Modern Alchemical Free Energy Methods for Drug Discovery Explained

This Perspective provides a contextual explanation of the current state-of-the-art alchemical free energy methods and their role in drug discovery as well as highlights select emerging technologies. The narrative attempts to answer basic questions about what goes on "under the hood" in free energy simulations and provide general guidelines for how to run simulations and analyze the results. It is the hope that this work will provide a valuable introduction to students and scientists in the field.

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.

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.

Exploring Routes to Enhance the Calculation of Free Energy Differences via Non-Equilibrium Work SQM/MM Switching Simulations Using Hybrid Charge Intermediates between MM and SQM Levels of Theory or Non-Linear Switching Schemes

Non-equilibrium work switching simulations and Jarzynski's equation are a reliable method for computing free energy differences, ΔAlow→high, between two levels of theory, such as a pure molecular mechanical (MM) and a quantum mechanical/molecular mechanical (QM/MM) description of a system of interest. Despite the inherent parallelism, the computational cost of this approach can quickly become very high. This is particularly true for systems where the core region, the part of the system to be described at different levels of theory, is embedded in an environment such as explicit solvent water. We find that even for relatively simple solute-water systems, switching lengths of at least 5 ps are necessary to compute ΔAlow→high reliably. In this study, we investigate two approaches towards an affordable protocol, with an emphasis on keeping the switching length well below 5 ps. Inserting a hybrid charge intermediate state with modified partial charges, which resembles the charge distribution of the desired high level, makes it possible to obtain reliable calculations with 2 ps switches. Attempts using step-wise linear switching paths, on the other hand, did not lead to improvement, i.e., a faster convergence for all systems. To understand these findings, we analyzed the solutes' properties as a function of the partial charges used and the number of water molecules in direct contact with the solute, and studied the time needed for water molecules to reorient themselves upon a change in the solute's charge distribution.

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.

The performance of ANI-ML potentials for ligand-n(H2 O) interaction energies and estimation of hydration free energies from end-point MD simulations

Here, we investigate the performance of "Accurate NeurAl networK engINe for Molecular Energies" (ANI), trained on small organic compounds, on bulk systems including non-covalent interactions and applicability to estimate solvation (hydration) free energies using the interaction between the ligand and explicit solvent (water) from single-step MD simulations. The method is adopted from ANI using the Atomic Simulation Environment (ASE) and predicts the non-covalent interaction energies at the accuracy of wb97x/6-31G(d) level by a simple linear scaling for the conformations sampled by molecular dynamics (MD) simulations of ligand-n(H₂ O) systems. For the first time, we test ANI potentials' abilities to reproduce solvation free energies using linear interaction energy (LIE) formulism by modifying the original LIE equation. Our results on ~250 different complexes show that the method can be accurate and have a correlation of R²  = 0.88-0.89 (MAE <1.0 kcal/mol) to the experimental solvation free energies, outperforming current end-state methods. Moreover, it is competitive to other conventional free energy methods such as FEP and BAR with 15-20 × fold reduced computational cost.

Accurate Binding Free Energy Method from End-State MD Simulations

Herein, we introduce a new strategy to estimate binding free energies using end-state molecular dynamics simulation trajectories. The method is adopted from linear interaction energy (LIE) and ANI-2x neural network potentials (machine learning) for the atomic simulation environment (ASE). It predicts the single-point interaction energies between ligand–protein and ligand–solvent pairs at the accuracy of the wb97x/6-31G* level for the conformational space that is sampled by molecular dynamics (MD) simulations. Our results on 54 protein–ligand complexes show that the method can be accurate and have a correlation of R = 0.87–0.88 to the experimental binding free energies, outperforming current end-state methods with reduced computational cost. The method also allows us to compare BFEs of ligands with different scaffolds. The code is available free of charge (documentation and test files) at https://github.com/otayfuroglu/deepQM.

Comparative assessment of QM-based and MM-based models for prediction of protein-ligand binding affinity trends

Methods which accurately predict protein-ligand binding strengths are critical for drug discovery. In the last two decades, advances in chemical modelling have enabled steadily accelerating progress in the discovery and optimization of structure-based drug design. Most computational methods currently used in this context are based on molecular mechanics force fields that often have deficiencies in describing the quantum mechanical (QM) aspects of molecular binding. In this study, we show the competitiveness of our QM-based Molecules-in-Molecules (MIM) fragmentation method for characterizing binding energy trends for seven different datasets of protein-ligand complexes. By using molecular fragmentation, the MIM method allows for accelerated QM calculations. We demonstrate that for classes of structurally similar ligands bound to a common receptor, MIM provides excellent correlation to experiment, surpassing the more popular Molecular Mechanics Poisson-Boltzmann Surface Area (MM/PBSA) and Molecular Mechanics Generalized Born Surface Area (MM/GBSA) methods. The MIM method offers a relatively simple, well-defined protocol by which binding trends can be ascertained at the QM level and is suggested as a promising option for lead optimization in structure-based drug design.

Obtaining QM/MM binding free energies in the SAMPL8 drugs of abuse challenge: indirect approaches

Accurately predicting free energy differences is essential in realizing the full potential of rational drug design. Unfortunately, high levels of accuracy often require computationally expensive QM/MM Hamiltonians. Fortuitously, the cost of employing QM/MM approaches in rigorous free energy simulation can be reduced through the use of the so-called “indirect” approach to QM/MM free energies, in which the need for QM/MM simulations is avoided via a QM/MM “correction” at the classical endpoints of interest. Herein, we focus on the computation of QM/MM binding free energies in the context of the SAMPL8 Drugs of Abuse host–guest challenge. Of the 5 QM/MM correction coupled with force-matching submissions, PM6-D3H4/MM ranked submission proved the best overall QM/MM entry, with an RMSE from experimental results of 2.43 kcal/mol (best in ranked submissions), a Pearson’s correlation of 0.78 (second-best in ranked submissions), and a Kendall \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document}τ correlation of 0.52 (best in ranked submissions).

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.

Optimizing the Calculation of Free Energy Differences in Nonequilibrium Work SQM/MM Switching Simulations

A key step during indirect alchemical free energy simulations using quantum mechanical/molecular mechanical (QM/MM) hybrid potential energy functions is the calculation of the free energy difference ΔAlow→high between the low level (e.g., pure MM) and the high level of theory (QM/MM). A reliable approach uses nonequilibrium work (NEW) switching simulations in combination with Jarzynski's equation; however, it is computationally expensive. In this study, we investigate whether it is more efficient to use more shorter switches or fewer but longer switches. We compare results obtained with various protocols to reference free energy differences calculated with Crooks' equation. The central finding is that fewer longer switches give better converged results. As few as 200 sufficiently long switches lead to ΔAlow→high values in good agreement with the reference results. This optimized protocol reduces the computational cost by a factor of 40 compared to earlier work. We also describe two tools/ways of analyzing the raw data to detect sources of poor convergence. Specifically, we find it helpful to analyze the raw data (work values from the NEW switching simulations) in a quasi-time series-like manner. Principal component analysis helps to detect cases where one or more conformational degrees of freedom are different at the low and high level of theory.

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.

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 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.

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.

Alchemical Binding Free Energy Calculations in AMBER20: Advances and Best Practices for Drug Discovery

Predicting protein-ligand binding affinities and the associated thermodynamics of biomolecular recognition is a primary objective of structure-based drug design. Alchemical free energy simulations offer a highly accurate and computationally efficient route to achieving this goal. While the AMBER molecular dynamics package has successfully been used for alchemical free energy simulations in academic research groups for decades, widespread impact in industrial drug discovery settings has been minimal because of the previous limitations within the AMBER alchemical code, coupled with challenges in system setup and postprocessing workflows. Through a close academia-industry collaboration we have addressed many of the previous limitations with an aim to improve accuracy, efficiency, and robustness of alchemical binding free energy simulations in industrial drug discovery applications. Here, we highlight some of the recent advances in AMBER20 with a focus on alchemical binding free energy (BFE) calculations, which are less computationally intensive than alternative binding free energy methods where full binding/unbinding paths are explored. In addition to scientific and technical advances in AMBER20, we also describe the essential practical aspects associated with running relative alchemical BFE calculations, along with recommendations for best practices, highlighting the importance not only of the alchemical simulation code but also the auxiliary functionalities and expertise required to obtain accurate and reliable results. This work is intended to provide a contemporary overview of the scientific, technical, and practical issues associated with running relative BFE simulations in AMBER20, with a focus on real-world drug discovery applications.

On the faithfulness of molecular mechanics representations of proteins towards quantum-mechanical energy surfaces

Force fields based on molecular mechanics (MM) are the main computational tool to study the relationship between protein structure and function at the molecular level. To validate the quality of such force fields, high-level quantum-mechanical (QM) data are employed to test their capability to reproduce the features of all major conformational substates of a series of blocked amino acids. The phase-space overlap between MM and QM is quantified in terms of the average structural reorganization energies over all energy minima. Here, the structural reorganization energy is the MM potential-energy difference between the structure of the respective QM energy minimum and the structure of the closest MM energy minimum. Thus, it serves as a measure for the relative probability of visiting the QM minimum during an MM simulation. We evaluate variants of the AMBER, CHARMM, GROMOS and OPLS biomolecular force fields. In addition, the two blocked amino acids alanine and serine are used to demonstrate the dependence of the measured agreement on the QM method, the phase, and the conformational preferences. Blocked serine serves as an example to discuss possible improvements of the force fields, such as including polarization with Drude particles, or using tailored force fields. The results show that none of the evaluated force fields satisfactorily reproduces all energy minima. By decomposing the average structural reorganization energies in terms of individual energy terms, we can further assess the individual weaknesses of the parametrization strategies of each force field. The dominant problem for most force fields appears to be the van der Waals parameters, followed to a lesser degree by dihedral and bonded terms. Our results show that performing a simple QM energy optimization from an MM-optimized structure can be a first test of the validity of a force field for a particular target molecule.

Improved Alchemical Free Energy Calculations with Optimized Smoothstep Softcore Potentials

Progress in the development of GPU-accelerated free energy simulation software has enabled practical applications on complex biological systems and fueled efforts to develop more accurate and robust predictive methods. In particular, this work re-examines concerted (a.k.a., one-step or unified) alchemical transformations commonly used in the prediction of hydration and relative binding free energies (RBFEs). We first classify several known challenges in these calculations into three categories: endpoint catastrophes, particle collapse, and large gradient-jumps. While endpoint catastrophes have long been addressed using softcore potentials, the remaining two problems occur much more sporadically and can result in either numerical instability (i.e., complete failure of a simulation) or inconsistent estimation (i.e., stochastic convergence to an incorrect result). The particle collapse problem stems from an imbalance in short-range electrostatic and repulsive interactions and can, in principle, be solved by appropriately balancing the respective softcore parameters. However, the large gradient-jump problem itself arises from the sensitivity of the free energy to large values of the softcore parameters, as might be used in trying to solve the particle collapse issue. Often, no satisfactory compromise exists with the existing softcore potential form. As a framework for solving these problems, we developed a new family of smoothstep softcore (SSC) potentials motivated by an analysis of the derivatives along the alchemical path. The smoothstep polynomials generalize the monomial functions that are used in most implementations and provide an additional path-dependent smoothing parameter. The effectiveness of this approach is demonstrated on simple yet pathological cases that illustrate the three problems outlined. With appropriate parameter selection, we find that a second-order SSC(2) potential does at least as well as the conventional approach and provides vast improvement in terms of consistency across all cases. Last, we compare the concerted SSC(2) approach against the gold-standard stepwise (a.k.a., decoupled or multistep) scheme over a large set of RBFE calculations as might be encountered in drug discovery.

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.

Advances in the calculation of binding free energies

In recent years, calculations of binding affinities from molecular simulations seem to have matured significantly. While the number of applications of such methods in drug design and biotechnology increases, the number of truly new methodological developments decreases. This review provides an overview of the current status of the field as reflected in recent publications. The focus is on the challenges that remain when using endstate, alchemical and pathway methods. For endstate methods this is the calculation of entropic contributions. For alchemical methods there are unsolved problems associated with the solvation of the active site, sampling slow degrees of freedom and when modifying the net charge. For pathway methods achieving sufficient sampling remains challenging. New trends are also highlighted, including the use of pathway methods for the quantification of protein-protein interactions.

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.

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.

Computational Fluorine Scanning Using Free-Energy Perturbation

We present perturbative fluorine scanning, a computational fluorine scanning approach using free-energy perturbation. This method can be applied to molecular dynamics simulations of a single compound and make predictions for the best binders out of numerous fluorinated analogues. We tested the method on nine test systems: renin, DPP4, menin, P38, factor Xa, CDK2, AKT, JAK2, and androgen receptor. The predictions were in excellent agreement with more rigorous alchemical free-energy calculations and in good agreement with experimental data for most of the test systems. However, the agreement with experiment was very poor in some of the test systems, and this highlights the need for improved force fields in addition to accurate treatment of tautomeric and protonation states. The method is of particular interest due to the wide use of fluorine in medicinal chemistry to improve binding affinity and ADME properties. The promising results on this test case suggest that perturbative fluorine scanning will be a useful addition to the available arsenal of free-energy methods.

Alchemical Free-Energy Calculations by Multiple-Replica λ-Dynamics: The Conveyor Belt Thermodynamic Integration Scheme

A new method is proposed to calculate alchemical free-energy differences based on molecular dynamics (MD) simulations, called the conveyor belt thermodynamic integration (CBTI) scheme. As in thermodynamic integration (TI), K replicas of the system are simulated at different values of the alchemical coupling parameter λ. The number K is taken to be even, and the replicas are equally spaced on a forward-turn-backward-turn path, akin to a conveyor belt (CB) between the two physical end-states; and as in λ-dynamics (λD), the λ-values associated with the individual systems evolve in time along the simulation. However, they do so in a concerted fashion, determined by the evolution of a single dynamical variable Λ of period 2π controlling the advance of the entire CB. Thus, a change of Λ is always associated with K/2 equispaced replicas moving forward and K/2 equispaced replicas moving backward along λ. As a result, the effective free-energy profile of the replica system along Λ is periodic of period 2 πK^-1, and the magnitude of its variations decreases rapidly upon increasing K, at least as K^-1 in the limit of large K. When a sufficient number of replicas is used, these variations become small, which enables a complete and quasi-homogeneous coverage of the λ-range by the replica system, without application of any biasing potential. If desired, a memory-based biasing potential can still be added to further homogenize the sampling, the preoptimization of which is computationally inexpensive. The final free-energy profile along λ is calculated similarly to TI, by binning of the Hamiltonian λ-derivative as a function of λ considering all replicas simultaneously, followed by quadrature integration. The associated quadrature error can be kept very low owing to the continuous and quasi-homogeneous λ-sampling. The CBTI scheme can be viewed as a continuous/deterministic/dynamical analog of the Hamiltonian replica-exchange/permutation (HRE/HRP) schemes or as a correlated multiple-replica analog of the λD or λ-local elevation umbrella sampling (λ-LEUS) schemes. Compared to TI, it shares the advantage of the latter schemes in terms of enhanced orthogonal sampling, i.e. the availability of variable-λ paths to circumvent conformational barriers present at specific λ-values. Compared to HRE/HRP, it permits a deterministic and continuous sampling of the λ-range, is expected to be less sensitive to possible artifacts of the thermo- and barostating schemes, and bypasses the need to carefully preselect a λ-ladder and a swapping-attempt frequency. Compared to λ-LEUS, it eliminates (or drastically reduces) the dead time associated with the preoptimization of a biasing potential. The goal of this article is to provide the mathematical/physical formulation of the proposed CBTI scheme, along with an initial application of the method to the calculation of the hydration free energy of methanol.

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.

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.

Analysis of Phosphoryl-Transfer Enzymes with QM/MM Free Energy Simulations

We discuss the application of quantum mechanics/molecular mechanics (QM/MM) free energy simulations to the analysis of phosphoryl transfers catalyzed by two enzymes: alkaline phosphatase and myosin. We focus on the nature of the transition state and the issue of mechanochemical coupling, respectively, in the two enzymes. The results illustrate unique insights that emerged from the QM/MM simulations, especially concerning the interpretation of experimental data regarding the nature of enzymatic transition states and coupling between global structural transition and catalysis in the active site. We also highlight a number of technical issues worthy of attention when applying QM/MM free energy simulations, and comment on a number of technical and mechanistic issues that require further studies.

On the convergence of multi-scale free energy simulations

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