Accurate and Rigorous Prediction of the Changes in Protein Free Energies in a Large‐Scale Mutation Scan

The prediction of mutation‐induced free‐energy changes in protein thermostability or protein–protein binding is of particular interest in the fields of protein design, biotechnology, and bioengineering. Herein, we achieve remarkable accuracy in a scan of 762 mutations estimating changes in protein thermostability based on the first principles of statistical mechanics. The remaining error in the free‐energy estimates appears to be due to three sources in approximately equal parts, namely sampling, force‐field inaccuracies, and experimental uncertainty. We propose a consensus force‐field approach, which, together with an increased sampling time, leads to a free‐energy prediction accuracy that matches those reached in experiments. This versatile approach enables accurate free‐energy estimates for diverse proteins, including the prediction of changes in the melting temperature of the membrane protein neurotensin receptor 1.     

Functional assessment of “in vivo” and “in silico” mutations in the guanine binding site of RNase T1: A DFT study

Experimentally, the functional assessment of amino acid side chains in proteins is carried out by comparing parameters such as binding constants for the wild‐type protein and a mutant protein in which the considered side chain is deleted. In the present study, we apply a density functional theory (DFT) methodology to obtain changes in binding energy upon mutations in the enzyme ribonuclease T₁. Mutant structures were either taken directly from crystallographic data (“in vivo”) allowing for conformational changes upon mutation, or derived from the wild‐type (“in silico”). Excluding entropic contributions, the computed interaction energy changes upon mutation in vivo correlate qualitatively well with experimental binding free energy changes. In contrast, the in silico approach does not perform as well, especially for residues that contribute largely to binding. Subsequently, we assessed the applicability of the in vivo approach by analyzing the functional cooperativity between pairs of side chains. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

An efficient simulation technique for electrostatic free energies with applications to azurin

We present an efficient technique for Monte Carlo simulation of electrostatic free energy changes in biomolecular systems. It is a development of a recent method for the study of the influence of electrostatic interactions on the ion binding properties and redox potentials of biomolecules. The electrolyte solution is described by the primitive model, in which ions are treated as hard charged spheres and the solvent is replaced by a structureless continuum. The protein is kept fixed in the center of a spherical simulation cell, and the dielectric constant has the solvent value throughout the cell. By a multiparticle perturbation approach, it is possible to obtain a number of free energy changes within one simulation only. The usefulness of the method is illustrated with a study of the copper binding electron-transport protein azurin (from Alcaligenes denitrificans). The change in acidity of the histidine residues upon changing the redox state of the copper ion is calculated. The theoretical predictions agree well with available experiments. © 1995 by John Wiley & Sons, Inc.     

Efficient calculation of relative binding free energies by umbrella sampling perturbation

An important task of biomolecular simulation is the calculation of relative binding free energies upon chemical modification of partner molecules in a biomolecular complex. The potential of mean force (PMF) along a reaction coordinate for association or dissociation of the complex can be used to estimate binding affinities. A free energy perturbation approach, termed umbrella sampling (US) perturbation, has been designed that allows an efficient calculation of the change of the PMF upon modification of a binding partner based on the trajectories obtained for the wild type reference complex. The approach was tested on the interaction of modified water molecules in aqueous solution and applied to in silico alanine scanning of a peptide‐protein complex. For the water interaction test case, excellent agreement with an explicit PMF calculation for each modification was obtained as long as no long range electrostatic perturbations were considered. For the alanine scanning, the experimentally determined ranking and binding affinity changes upon alanine substitutions could be reproduced within 0.1–2.0 kcal/mol. In addition, good agreement with explicitly calculated PMFs was obtained mostly within the sampling uncertainty. The combined US and perturbation approach yields, under the condition of sufficiently small system modifications, rigorously derived changes in free energy and is applicable to any PMF calculation. © 2014 Wiley Periodicals, Inc.     

Semiempirical calculations of binding enthalpy for protein–ligand complexes

A systematic semiempirical quantum mechanical study of the interactions between proteins and ligands has been performed to determine the ability of this approach for the accurate estimation of the enthalpic contribution to the binding free energy of the protein–ligand systems. This approach has been applied for eight test protein–ligand complexes with experimentally known binding enthalpies. The calculations were performed using the semiempirical PM3 approach incorporated in the MOPAC 97, ZAVA originally elaborated in Algodign, and MOPAC 2002 with MOZYME facility packages. Special attention was paid to take into account structural water molecules, which were located in the protein–ligand binding site. It was shown that the results of binding enthalpy calculations fit experimental data within ～2 kcal/mol in the presented approach. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

The Confine-and-Release Method: Obtaining Correct Binding Free Energies in the Presence of Protein Conformational Change

Free energy calculations are increasingly being used to estimate absolute and relative binding free energies of ligands to proteins. However, computed free energies often appear to depend on the initial protein conformation, indicating incomplete sampling. This is especially true when proteins can change conformation on ligand binding, as free energies associated with these conformational changes are either ignored or assumed to be included by virtue of the sampling performed in the calculation. Here, we show that, in a model protein system (a designed binding site in T4 Lysozyme), conformational changes can make a difference of several kcal/mol in computed binding free energies, and that they are neglected in computed binding free energies if the system remains kinetically trapped in a particular metastable state on simulation timescales. We introduce a general "confine-and-release" framework for free energy calculations that accounts for these free energies of conformational change. We illustrate its use in this model system by demonstrating that an umbrella sampling protocol can obtain converged binding free energies that are independent of the starting protein structure and include these conformational change free energies.     

Fragment‐based quantum mechanical calculation of protein–protein binding affinities

The electrostatically embedded generalized molecular fractionation with conjugate caps (EE‐GMFCC) method has been successfully utilized for efficient linear‐scaling quantum mechanical (QM) calculation of protein energies. In this work, we applied the EE‐GMFCC method for calculation of binding affinity of Endonuclease colicin–immunity protein complex. The binding free energy changes between the wild‐type and mutants of the complex calculated by EE‐GMFCC are in good agreement with experimental results. The correlation coefficient (R) between the predicted binding energy changes and experimental values is 0.906 at the B3LYP/6‐31G*‐D level, based on the snapshot whose binding affinity is closest to the average result from the molecular mechanics/Poisson–Boltzmann surface area (MM/PBSA) calculation. The inclusion of the QM effects is important for accurate prediction of protein–protein binding affinities. Moreover, the self‐consistent calculation of PB solvation energy is required for accurate calculations of protein–protein binding free energies. This study demonstrates that the EE‐GMFCC method is capable of providing reliable prediction of relative binding affinities for protein–protein complexes. © 2018 Wiley Periodicals, Inc.     

Modeling Protein-Ligand Binding by Mining Minima

We present the first application of the mining minima algorithm to protein-small molecule binding. This end-point approach use an empirical force field and implicit solvent models, treats the protein binding-site as fully flexible and estimates free energies as sums over local energy wells. The calculations are found to yield encouraging agreement with experiment for three sets of HIV-1protease inhibitors and a set of phosphodiesterase 10a inhibitors. The contributions of various aspects of the model to its accuracy are examined, and the Poisson-Boltzmann correction is found to be the most critical. Interestingly, the computed changes in configurational entropy upon binding fall roughly along the same entropy-energy correlation previously observed for smaller host-guest systems. Strengths and weaknesses of the method are discussed, as are the prospects for enhancing accuracy and speed.     

Tinker‐OpenMM: Absolute and relative alchemical free energies using AMOEBA on GPUs

The capabilities of the polarizable force fields for alchemical free energy calculations have been limited by the high computational cost and complexity of the underlying potential energy functions. In this work, we present a GPU‐based general alchemical free energy simulation platform for polarizable potential AMOEBA. Tinker‐OpenMM, the OpenMM implementation of the AMOEBA simulation engine has been modified to enable both absolute and relative alchemical simulations on GPUs, which leads to a ∼200‐fold improvement in simulation speed over a single CPU core. We show that free energy values calculated using this platform agree with the results of Tinker simulations for the hydration of organic compounds and binding of host–guest systems within the statistical errors. In addition to absolute binding, we designed a relative alchemical approach for computing relative binding affinities of ligands to the same host, where a special path was applied to avoid numerical instability due to polarization between the different ligands that bind to the same site. This scheme is general and does not require ligands to have similar scaffolds. We show that relative hydration and binding free energy calculated using this approach match those computed from the absolute free energy approach. © 2017 Wiley Periodicals, Inc.     

Absolute Protein Binding Free Energy Simulations for Ligands with Multiple Poses,a Thermodynamic Path That Avoids Exhaustive Enumeration of the Poses

We propose a free energy calculation method for receptor–ligand binding, which have multiple binding poses that avoids exhaustive enumeration of the poses. For systems with multiple binding poses, the standard procedure is to enumerate orientations of the binding poses, restrain the ligand to each orientation, and then, calculate the binding free energies for each binding pose. In this study, we modify a part of the thermodynamic cycle in order to sample a broader conformational space of the ligand in the binding site. This modification leads to more accurate free energy calculation without performing separate free energy simulations for each binding pose. We applied our modification to simple model host–guest systems as a test, which have only two binding poses, by using a single decoupling method (SDM) in implicit solvent. The results showed that the binding free energies obtained from our method without knowing the two binding poses were in good agreement with the benchmark results obtained by explicit enumeration of the binding poses. Our method is applicable to other alchemical binding free energy calculation methods such as the double decoupling method (DDM) in explicit solvent. We performed a calculation for a protein–ligand system with explicit solvent using our modified thermodynamic path. The results of the free energy simulation along our modified path were in good agreement with the results of conventional DDM, which requires a separate binding free energy calculation for each of the binding poses of the example of phenol binding to T4 lysozyme in explicit solvent. © 2019 Wiley Periodicals, Inc.     

Converging free energy estimates: MM-PB(GB)SA studies on the protein-protein complex Ras-Raf

Estimating protein-protein interaction energies is a very challenging task for current simulation protocols. Here, absolute binding free energies are reported for the complex H-Ras/C-Raf1 using the MM-PB(GB)SA approach, testing the internal consistency and model dependence of the results. Averaging gas-phase energies (MM), solvation free energies as determined by Generalized Born models (GB/SA), and entropic contributions calculated by normal mode analysis for snapshots obtained from 10 ns explicit-solvent molecular dynamics in general results in an overestimation of the binding affinity when a solvent-accessible surface area-dependent model is used to estimate the nonpolar solvation contribution. Applying the sum of a cavity solvation free energy and explicitly modeled solute-solvent van der Waals interaction energies instead provides less negative estimates for the nonpolar solvation contribution. When the polar contribution to the solvation free energy is determined by solving the Poisson-Boltzmann equation (PB) instead, the calculated binding affinity strongly depends on the atomic radii set chosen. For three GB models investigated, different absolute deviations from PB energies were found for the unbound proteins and the complex. As an alternative to normal-mode calculations, quasiharmonic analyses have been performed to estimate entropic contributions due to changes of solute flexibility upon binding. However, such entropy estimates do not converge after 10 ns of simulation time, indicating that sampling issues may limit the applicability of this approach. Finally, binding free energies estimated from snapshots of the unbound proteins extracted from the complex trajectory result in an underestimate of binding affinity. This points to the need to exercise caution in applying the computationally cheaper "one-trajectory-alternative" to systems where there may be significant changes in flexibility and structure due to binding. The best estimate for the binding free energy of Ras-Raf obtained in this study of -8.3 kcal mol(-1) is in good agreement with the experimental result of -9.6 kcal mol(-1), however, further probing the transferability of the applied protocol that led to this result is necessary.     

Standard free energy of releasing a localized water molecule from the binding pockets of proteins: double-decoupling method

Localized water molecules in the binding pockets of proteins play an important role in noncovalent association of proteins and small drug compounds. At times, the dominant contribution to the binding free energy comes from the release of localized water molecules in the binding pockets of biomolecules. Therefore, to quantify the energetic importance of these water molecules for drug design purposes, we have used the double-decoupling approach to calculate the standard free energy of tying up a water molecule in the binding pockets of two protein complexes. The double-decoupling approach is based on the underlying principle of statistical thermodynamics. We have calculated the standard free energies of tying up the water molecule in the binding pockets of these complexes to be favorable. These water molecules stabilize the protein-drug complexes by interacting with the ligands and binding pockets. Our results offer ideas that could be used in optimizing protein-drug interactions, by designing ligands that are capable of targeting localized water molecules in protein binding sites. The resulting free energy of ligand binding could benefit from the potential free energy gain accompanying the release of these water molecules. Furthermore, we have examined the theoretical background of the double-decoupling method and its connection to the molecular dynamics thermodynamic integration techniques.     

Free energy of binding of a small molecule to an amorphous polymer in a solvent

Crystallization is a commonly used purification process in industrial practice. It usually begins with heterogeneous nucleation on a foreign surface. The complicated mechanism of heterogeneous nucleation is not well understood, but we hypothesize that a possible correlation between binding affinity to a surface and nucleation enhancement might exist. Amorphous polymers have been used in controlling crystallization. However, to our knowledge, no attempt has been made to calculate the free energy of binding of a small molecule to an amorphous polymer in a solvent, and to characterize the binding sites/conformations of this system at a molecular level. We developed a two-step approach, first using Adsorption Locator to identify probable binding sites and molecular dynamics to screen for the best binding sites and then using the Blue-Moon Ensemble method to compute the free energy of binding. A system of ethylene glycol, polyvinyl alcohol (PVA), and heavy water (D(2)O) was used for validation, since experimental data exists on a related system. Looking at four independently constructed surfaces, we found that ethylene glycol binds to an indentation on the surface or in a hole beneath the surface. We focused on the indentation binding sites because they are easily accessible and do not have large free energy barriers. The closest system for which experimental data on binding energetics exists is ethylene glycol on PVA in aqueous solutions/gels, and the magnitudes of the free energy of binding to the three best indentation binding sites are close to the experimental value, 0.4-3.7 kcal/mol higher. Our approach offers a way to compute the free energy of binding and characterize the binding sites/conformations, and is general enough to apply to other small molecule/amorphous polymer/solvent systems.     

Effects of Biomolecular Flexibility on Alchemical Calculations of Absolute Binding Free Energies

The independent trajectory thermodynamic integration (IT-TI) approach (Lawrenz et. al J. Chem. Theory. Comput. 2009, 5:1106-1116(1)) for free energy calculations with distributed computing is employed to study two distinct cases of protein-ligand binding: first, the influenza surface protein N1 neuraminidase bound to the inhibitor oseltamivir, and second, the M. tuberculosis enzyme RmlC complexed with the molecule CID 77074. For both systems, finite molecular dynamics (MD) sampling and varied molecular flexibility give rise to IT-TI free energy distributions that are remarkably centered on the target experimental values, with a spread directly related to protein, ligand, and solvent dynamics. Using over 2 μs of total MD simulation, alternative protocols for the practical, general implementation of IT-TI are investigated, including the optimal use of distributed computing, the total number of alchemical intermediates, and the procedure to perturb electrostatics and van der Waals interactions. A protocol that maximizes predictive power and computational efficiency is proposed. IT-TI outperforms traditional TI predictions and allows a straightforward evaluation of the reliability of free energy estimates. Our study has broad implications for the use of distributed computing in free energy calculations of macromolecular systems.     

pmx: Automated protein structure and topology generation for alchemical perturbations

Computational protein design requires methods to accurately estimate free energy changes in protein stability or binding upon an amino acid mutation. From the different approaches available, molecular dynamics‐based alchemical free energy calculations are unique in their accuracy and solid theoretical basis. The challenge in using these methods lies in the need to generate hybrid structures and topologies representing two physical states of a system. A custom made hybrid topology may prove useful for a particular mutation of interest, however, a high throughput mutation analysis calls for a more general approach. In this work, we present an automated procedure to generate hybrid structures and topologies for the amino acid mutations in all commonly used force fields. The described software is compatible with the Gromacs simulation package. The mutation libraries are readily supported for five force fields, namely Amber99SB, Amber99SB*‐ILDN, OPLS‐AA/L, Charmm22*, and Charmm36. © 2014 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

(Thermo)dynamic Role of Receptor Flexibility,Entropy, and Motional Correlation in Protein–Ligand Binding

The binding of 2‐amino‐5‐methylthiazole to the W191G cavity mutant of cytochrome c peroxidase is an ideal test case to investigate the entropic contribution to the binding free energy due to changes in receptor flexibility. The dynamic and thermodynamic role of receptor flexibility are studied by 50 ns‐long explicit‐solvent molecular dynamics simulations of three separate receptor ensembles: W191G binding a K⁺ ion, W191G–2a5mt complex with a closed 190–195 gating loop, and apo with an open loop. We employ a method recently proposed to estimate accurate absolute single‐molecule configurational entropies and their differences for systems undergoing conformational transitions. We find that receptor flexibility plays a generally underestimated role in protein–ligand binding (thermo)dynamics and that changes of receptor motional correlation determine such large entropy contributions.     

Multiple protonation equilibria in electrostatics of protein-protein binding

All proteins contain groups capable of exchanging protons with their environment. We present here an approach, based on a rigorous thermodynamic cycle and the partition functions for energy levels characterizing protonation states of the associating proteins and their complex, to compute the electrostatic pH-dependent contribution to the free energy of protein-protein binding. The computed electrostatic binding free energies include the pH of the solution as the variable of state, mutual "polarization" of associating proteins reflected as changes in the distribution of their protonation states upon binding and fluctuations between available protonation states. The only fixed property of both proteins is the conformation; the structure of the monomers is kept in the same conformation as they have in the complex structure. As a reference, we use the electrostatic binding free energies obtained from the traditional Poisson-Boltzmann model, computed for a single macromolecular conformation fixed in a given protonation state, appropriate for given solution conditions. The new approach was tested for 12 protein-protein complexes. It is shown that explicit inclusion of protonation degrees of freedom might lead to a substantially different estimation of the electrostatic contribution to the binding free energy than that based on the traditional Poisson-Boltzmann model. This has important implications for the balancing of different contributions to the energetics of protein-protein binding and other related problems, for example, the choice of protein models for Brownian dynamics simulations of their association. Our procedure can be generalized to include conformational degrees of freedom by combining it with molecular dynamics simulations at constant pH. Unfortunately, in practice, a prohibitive factor is an enormous requirement for computer time and power. However, there may be some hope for solving this problem by combining existing constant pH molecular dynamics algorithms with so-called accelerated molecular dynamics algorithms.     

Conformational Transitions and Convergence of Absolute Binding Free Energy Calculations

The Binding Energy Distribution Analysis Method (BEDAM) is employed to compute the standard binding free energies of a series of ligands to a FK506 binding protein (FKBP12) with implicit solvation. Binding free energy estimates are in reasonably good agreement with experimental affinities. The conformations of the complexes identified by the simulations are in good agreement with crystallographic data, which was not used to restrain ligand orientations. The BEDAM method is based on λ -hopping Hamiltonian parallel Replica Exchange (HREM) molecular dynamics conformational sampling, the OPLS-AA/AGBNP2 effective potential, and multi-state free energy estimators (MBAR). Achieving converged and accurate results depends on all of these elements of the calculation. Convergence of the binding free energy is tied to the level of convergence of binding energy distributions at critical intermediate states where bound and unbound states are at equilibrium, and where the rate of binding/unbinding conformational transitions is maximal. This finding mirrors similar observations in the context of order/disorder transitions as for example in protein folding. Insights concerning the physical mechanism of ligand binding and unbinding are obtained. Convergence for the largest FK506 ligand is achieved only after imposing strict conformational restraints, which however require accurate prior structural knowledge of the structure of the complex. The analytical AGBNP2 model is found to underestimate the magnitude of the hydrophobic driving force towards binding in these systems characterized by loosely packed protein-ligand binding interfaces. Rescoring of the binding energies using a numerical surface area model corrects this deficiency. This study illustrates the complex interplay between energy models, exploration of conformational space, and free energy estimators needed to obtain robust estimates from binding free energy calculations.     

An estimation method of binding free energy in terms of ABEEMσπ/MM and continuum electrostatics fused into LIE method

A method is proposed for the estimation of absolute binding free energy of interaction between proteins and ligands. The linear interaction energy method is combined with atom‐bond electronegativity equalization method at σπ level Force field (fused into molecular mechanics) and generalized Born continuum model calculation of electrostatic solvation for the estimation of the absolute free energy of binding. The parameters of this method are calibrated by using a training set of 24 HIV‐1 protease–inhibitor complexes (PDB entry 1AAQ). A correlation coefficient of 0.93 was obtained with a root mean square deviation of 0.70 kcal mol^?1. This approach is further tested on seven inhibitor and protease complexes, and it provides small root mean square deviation between the calculated binding free energy and experimental binding free energy without reparametrization. By comparing the radii of gyration and the hydrogen bond distances between ligand and protein of three training model molecules, the consistent comparison result of binding free energy is obtained. It proves that this method of calculating the binding free energy with appropriate structural analysis can be applied to quickly assess new inhibitors of HIV‐1 proteases. To test whether the parameters of this method can apply to other drug targets, we have validated this method for the drug target cyclooxygenase‐2. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Solvated interaction energy (SIE) for scoring protein-ligand binding affinities. 1. Exploring the parameter space

We present a binding free energy function that consists of force field terms supplemented by solvation terms. We used this function to calibrate the solvation model along with the binding interaction terms in a self-consistent manner. The motivation for this approach was that the solute dielectric-constant dependence of calculated hydration gas-to-water transfer free energies is markedly different from that of binding free energies (J. Comput. Chem. 2003, 24, 954). Hence, we sought to calibrate directly the solvation terms in the context of a binding calculation. The five parameters of the model were systematically scanned to best reproduce the absolute binding free energies for a set of 99 protein-ligand complexes. We obtained a mean unsigned error of 1.29 kcal/mol for the predicted absolute binding affinity in a parameter space that was fairly shallow near the optimum. The lowest errors were obtained with solute dielectric values of Din = 20 or higher and scaling of the intermolecular van der Waals interaction energy by factors ranging from 0.03 to 0.15. The high apparent Din and strong van der Waals scaling may reflect the anticorrelation of the change in solvated potential energy and configurational entropy, that is, enthalpy-entropy compensation in ligand binding (Biophys. J. 2004, 87, 3035-3049). Five variations of preparing the protein-ligand data set were explored in order to examine the effect of energy refinement and the presence of bound water on the calculated results. We find that retaining water in the final protein structure used for calculating the binding free energy is not necessary to obtain good results; that is the continuum solvation model is sufficient. Virtual screening enrichment studies on estrogen receptor and thymidine kinase showed a good ability of the binding free energy function to recover true hits in a collection of decoys.