Free energy calculation methods: A theoretical and empirical comparison of numerical errors and a new method qualitative estimates of free energy changes

On the Statistical Mechanics of Mass Accommodation at Liquid-Vapor Interfaces

We propose a framework for describing the dynamics associated with the adsorption of small molecules to liquid-vapor interfaces using an intermediate resolution between traditional continuum theories that are bereft of molecular detail and molecular dynamics simulations that are replete with them. In particular, we develop an effective single particle equation of motion capable of describing the physical processes that determine thermal and mass accommodation probabilities. The effective equation is parametrized with quantities that vary through space away from the liquid-vapor interface. Of particular importance in describing the early time dynamics is the spatially dependent friction, for which we propose a numerical scheme to evaluate from molecular simulation. Taken together with potentials of mean force computable with importance sampling methods, we illustrate how to compute the mass accommodation coefficient and residence time distribution. Throughout, we highlight the case of ozone adsorption in aqueous solutions and its dependence on electrolyte composition.

The slow but steady rise of binding free energy calculations in drug discovery

Binding free energy calculations are increasingly used in drug discovery research to predict protein-ligand binding affinities and to prioritize candidate drug molecules accordingly. It has taken decades of collective effort to transform this academic concept into a technology adopted by the pharmaceutical and biotech industry. Having personally witnessed and taken part in this transformation, here I recount the (incomplete) list of problems that had to be solved to make this computational tool practical and suggest areas of future development.

An Open Source Graph-Based Weighted Cycle Closure Method for Relative Binding Free Energy Calculations

Free energy perturbation-relative binding free energy (FEP-RBFE) prediction has shown its reliability and accuracy in the prediction of protein-ligand binding affinities, which plays a fundamental role in structure-based drug design. In FEP-RBFE predictions, the calculation of each mutation path is associated with a statistical error, and cycle closure (cc) has proven to be an effective method in improving the calculation accuracy by correcting the hysteresis (summation of errors) of each closed cycle to the theoretical value 0. However, a primary hypothesis was made in the current cycle closure method that the hysteresis is evenly distributed to all paths, which is unlikely to be true in practice and may limit the further improvement of the calculation accuracy when better error estimation methods are available. Moreover, being a closed source software makes the current cycle closure method unachievable in many studies. In this paper, a newly implemented open source graph-based weighted cycle closure (wcc) algorithm was developed and introduced, not only including functions from the original cc method but also containing a new wcc method which can consider different error contributions from different paths and further improve the calculation accuracy. The wcc program also provides a new path-independent molecular error calculation method, which can be quite useful in many studies (like structure-activity relationship (SAR)) compared with the path-dependent method of the original cc program.

Molecular Dynamics and Free Energy Calculations of Dicyclohexano-18-crown-6 Diastereoisomers with Sm2+, Eu2+, Dy2+, Yb2+, Cf2+, and Three Halide Salts in Tetrahydrofuran and Acetonitrile Using the AMOEBA Force Field

With the continual development of lanthanides (Ln) in current technological devices, an efficient separation process is needed that can recover greater amounts of these rare elements. Dicyclohexano-18-crown-6 (DCH18C6) is a crown ether that may be a promising candidate for Ln separation, but additional research is required. As such, molecular dynamics (MD) simulations have been performed on four divalent lanthanide halide salts (Sm2+, Eu2+, Dy2+, and Yb2+) and one divalent actinide halide salt (Cf2+) bound to three diastereoisomers of DCH18C6. Dy2+, Yb2+, Cf2+, DCH18C6, and tetrahydrofuran (THF) solvent were parameterized for the AMOEBA polarizable force field for the first time, whereas existing parameters for Sm2+ and Eu2+ were utilized from our previous efforts. A coordination number (CN) of six for Ln2+/An2+-O solvated in THF indicated that the cations interacted almost entirely with the oxygens of the polyether ring. A CN of one for Ln2+/An2+-N solvated in acetonitrile for systems containing iodide suggested that the N atom of acetonitrile was competitive with I- for cation interactions. Fluctuation between five and six CNs for Dy2+ and Yb2+ suggested that although the cations remained in the polyether ring, the size of the ring may not be an ideal fit as these cations possess comparatively smaller ionic radii. Gibbs binding free energies of Sm2+ in all DCH18C6 diastereoisomers solvated in THF were calculated. The binding free energy of the cis-syn-cis diastereoisomer was the most favorable, followed by cis-anti-cis, and then trans-anti-trans. Finally, two major types of conformation were observed for each diastereoisomer that were related to the electrostatic interactions and charge density of the cations.

Fast Prediction of Binding Affinities of the SARS-CoV-2 Spike Protein Mutant N501Y (UK Variant) with ACE2 and Miniprotein Drug Candidates

A recently identified variant of SARS-CoV-2 virus, known as the United Kingdom (UK) variant (lineage B.1.1.7), has an N501Y mutation on its spike protein. SARS-CoV-2 spike protein binds with angiotensin-converting enzyme 2 (ACE2), a key protein for the viral entry into the host cells. Here, we report an efficient computational approach, including the simple energy minimizations and binding free energy calculations, starting from an experimental structure of the binding complex along with experimental calibration of the calculated binding free energies, to rapidly and reliably predict the binding affinities of the N501Y mutant with human ACE2 (hACE2) and recently reported miniprotein and hACE2 decoy (CTC-445.2) drug candidates. It has been demonstrated that the N501Y mutation markedly increases the ACE2-spike protein binding affinity (Kd) from 22 to 0.44 nM, which could partially explain why the UK variant is more infectious. The miniproteins are predicted to have ∼10,000- to 100,000-fold diminished binding affinities with the N501Y mutant, creating a need for design of novel therapeutic candidates to overcome the N501Y mutation-induced drug resistance. The N501Y mutation is also predicted to decrease the binding affinity of a hACE2 decoy (CTC-445.2) binding with the spike protein by ∼200-fold. This convenient computational approach along with experimental calibration may be similarly used in the future to predict the binding affinities of potential new variants of the spike protein.

Skewed thermodynamic geometry and optimal free energy estimation

Free energy differences are a central quantity of interest in physics, chemistry, and biology. We develop design principles that improve the precision and accuracy of free energy estimators, which have potential applications to screening for targeted drug discovery. Specifically, by exploiting the connection between the work statistics of time-reversed protocol pairs, we develop near-equilibrium approximations for moments of the excess work and analyze the dominant contributions to the precision and accuracy of standard nonequilibrium free-energy estimators. Within linear response, minimum-dissipation protocols follow the geodesics of the Riemannian metric induced by the Stokes friction tensor. We find that the next-order contribution arises from the rank-3 supra-Stokes tensor that skews the geometric structure such that minimum-dissipation protocols follow the geodesics of a generalized cubic Finsler metric. Thus, near equilibrium, the supra-Stokes tensor determines the leading-order contribution to the bias of bidirectional free-energy estimators.

Identify potent SARS-CoV-2 main protease inhibitors via accelerated free energy perturbation-based virtual screening of existing drugs

The COVID-19 pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has become a global crisis. There is no therapeutic treatment specific for COVID-19. It is highly desirable to identify potential antiviral agents against SARS-CoV-2 from existing drugs available for other diseases and thus repurpose them for treatment of COVID-19. In general, a drug repurposing effort for treatment of a new disease, such as COVID-19, usually starts from a virtual screening of existing drugs, followed by experimental validation, but the actual hit rate is generally rather low with traditional computational methods. Here we report a virtual screening approach with accelerated free energy perturbation-based absolute binding free energy (FEP-ABFE) predictions and its use in identifying drugs targeting SARS-CoV-2 main protease (Mpro). The accurate FEP-ABFE predictions were based on the use of a restraint energy distribution (RED) function, making the practical FEP-ABFE-based virtual screening of the existing drug library possible. As a result, out of 25 drugs predicted, 15 were confirmed as potent inhibitors of SARS-CoV-2 Mpro The most potent one is dipyridamole (inhibitory constant Ki = 0.04 µM) which has shown promising therapeutic effects in subsequently conducted clinical studies for treatment of patients with COVID-19. Additionally, hydroxychloroquine (Ki = 0.36 µM) and chloroquine (Ki = 0.56 µM) were also found to potently inhibit SARS-CoV-2 Mpro We anticipate that the FEP-ABFE prediction-based virtual screening approach will be useful in many other drug repurposing or discovery efforts.

Atomistic insights into the separation mechanism of multilayer graphene membranes for water desalination

Graphene-based membranes have been extensively explored owing to their excellent separation properties. In this paper, multiple factors regarding desalination performance were investigated by molecular dynamics (MD) simulations. These factors include the interlayer spacing distance (H), the gap width (dG), offset (O), and the number of gaps and layers in a multilayer graphene membrane (MGM). It is found that salt rejection is influenced significantly by the interlayer spacing distance owing to the largest free energy between ions and graphene sheets as well as the relatively larger size of the hydration layer around the ions. The optimal desalting parameter (dG = 1 nm, H = 0.8 nm) was selected; MGM systems based on the optimized parameter exhibited excellent salt rejection for NaCl, MgCl2 and CaCl2 solutions. These results can provide some ideas for the future design of graphene-based membranes.

Optimal Measurement Network of Pairwise Differences

When both the difference between two quantities and their individual values can be measured or computationally predicted, multiple quantities can be determined from the measurements or predictions of select individual quantities and select pairwise differences. These measurements and predictions form a network connecting the quantities through their differences. Here, I analyze the optimization of such networks, where the trace (A-optimal), the largest eigenvalue (E-optimal), or the determinant (D-optimal) of the covariance matrix associated with the estimated quantities are minimized with respect to the allocation of the measurement (or computational) cost to different measurements (or predictions). My statistical analysis of the performance of such optimal measurement networks-based on large sets of simulated data-suggests that they substantially accelerate the determination of the quantities and that they may be useful in applications such as the computational prediction of binding free energies of candidate drug molecules.

Absolute Binding Free Energy Calculation and Design of a Subnanomolar Inhibitor of Phosphodiesterase-10

Accurate prediction of absolute protein-ligand binding free energy could considerably enhance the success rate of structure-based drug design but is extremely challenging and time-consuming. Free energy perturbation (FEP) has been proven reliable but is limited to prediction of relative binding free energies of similar ligands (with only minor structural differences) in binding with a same drug target in practical drug design applications. Herein, a Gaussian algorithm-enhanced FEP (GA-FEP) protocol has been developed to enhance the FEP simulation performance, enabling to efficiently carry out the FEP simulations on vanishing the whole ligand and, thus, predict the absolute binding free energies (ABFEs). Using the GA-FEP protocol, the FEP simulations for the ABFE calculation (denoted as GA-FEP/ABFE) can achieve a satisfactory accuracy for both structurally similar and diverse ligands in a dataset of more than 100 receptor-ligand systems. Further, our GA-FEP/ABFE-guided lead optimization against phosphodiesterase-10 led to the discovery of a subnanomolar inhibitor (IC₅₀ = 0.87 nM, ∼2000-fold improvement in potency) with cocrystal confirmation.

Looking "Under the Hood" of Cellular Mechanotransduction with Computational Tools: A Systems Biomechanics Approach across Multiple Scales

Signal modulation has been developed in living cells throughout evolution to promote utilizing the same machinery for multiple cellular functions. Chemical and mechanical modules of signal transmission and transduction are interconnected and necessary for organ development and growth. However, due to the high complexity of the intercommunication of physical intracellular connections with biochemical pathways, there are many missing details in our overall understanding of mechanotransduction processes, i.e., the process by which mechanical signals are converted to biochemical cascades. Cell-matrix adhesions are mechanically coupled to the nucleus through the cytoskeleton. This modulated and tightly integrated network mediates the transmission of mechanochemical signals from the extracellular matrix to the nucleus. Various experimental and computational techniques have been utilized to understand the basic mechanisms of mechanotransduction, yet many aspects have remained elusive. Recently, in silico experiments have made important contributions to the field of mechanobiology. Herein, computational modeling efforts devoted to understanding integrin-mediated mechanotransduction pathways are reviewed, and an outlook is presented for future directions toward using suitable computational approaches and developing novel techniques for addressing important questions in the field of mechanotransduction.

Quantitative Characterization of the Binding and Unbinding of Millimolar Drug Fragments with Molecular Dynamics Simulations

A quantitative characterization of the binding properties of drug fragments to a target protein is an important component of a fragment-based drug discovery program. Fragments typically have a weak binding affinity, however, making it challenging to experimentally characterize key binding properties, including binding sites, poses, and affinities. Direct simulation of the binding equilibrium by molecular dynamics (MD) simulations can provide a computational route to characterize fragment binding, but this approach is so computationally intensive that it has thus far remained relatively unexplored. Here, we perform MD simulations of sufficient length to observe several different fragments spontaneously and repeatedly bind to and unbind from the protein FKBP, allowing the binding affinities, on- and off-rates, and relative occupancies of alternative binding sites and alternative poses within each binding site to be estimated, thereby illustrating the potential of long time scale MD as a quantitative tool for fragment-based drug discovery. The data from the long time scale fragment binding simulations reported here also provide a useful benchmark for testing alternative computational methods aimed at characterizing fragment binding properties. As an example, we calculated binding affinities for the same fragments using a standard free energy perturbation approach and found that the values agreed with those obtained from the fragment binding simulations within statistical error.

On the loading mechanism of ssDNA into carbon nanotubes

The loading mechanism and dynamics of ssDNA oligomers into single-walled carbon nanotubes was investigated. The binding free energy is in the order A-zigzag SWNT (24,0) > T-zigzag SWNT (24,0) > A-armchair SWNT (14,14) > T-armchair SWNT (14,14).

Constant pH Molecular Dynamics in Explicit Solvent with Enveloping Distribution Sampling and Hamiltonian Exchange

We present a new computational approach for constant pH simulations in explicit solvent based on the combination of the enveloping distribution sampling (EDS) and Hamiltonian replica exchange (HREX) methods. Unlike constant pH methods based on variable and continuous charge models, our method is based on discrete protonation states. EDS generates a hybrid Hamiltonian of different protonation states. A smoothness parameter s is used to control the heights of energy barriers of the hybrid-state energy landscape. A small s value facilitates state transitions by lowering energy barriers. Replica exchange between EDS potentials with different s values allows us to readily obtain a thermodynamically accurate ensemble of multiple protonation states with frequent state transitions. The analysis is performed with an ensemble obtained from an EDS Hamiltonian without smoothing, s = ∞, which strictly follows the minimum energy surface of the end states. The accuracy and efficiency of this method is tested on aspartic acid, lysine, and glutamic acid, which have two protonation states, a histidine with three states, a four-residue peptide with four states, and snake cardiotoxin with eight states. The pKa values estimated with the EDS-HREX method agree well with the experimental pKa values. The mean absolute errors of small benchmark systems range from 0.03 to 0.17 pKa units, and those of three titratable groups of snake cardiotoxin range from 0.2 to 1.6 pKa units. This study demonstrates that EDS-HREX is a potent theoretical framework, which gives the correct description of multiple protonation states and good calculated pKa values.

Hydration free energies of cyanide and hydroxide ions from molecular dynamics simulations with accurate force fields

The evaluation of hydration free energies is a sensitive test to assess force fields used in atomistic simulations. We showed recently that the vibrational relaxation times, 1D- and 2D-infrared spectroscopies for CN(-) in water can be quantitatively described from molecular dynamics (MD) simulations with multipolar force fields and slightly enlarged van der Waals radii for the C- and N-atoms. To validate such an approach, the present work investigates the solvation free energy of cyanide in water using MD simulations with accurate multipolar electrostatics. It is found that larger van der Waals radii are indeed necessary to obtain results close to the experimental values when a multipolar force field is used. For CN(-), the van der Waals ranges refined in our previous work yield hydration free energy between -72.0 and -77.2 kcal mol(-1), which is in excellent agreement with the experimental data. In addition to the cyanide ion, we also study the hydroxide ion to show that the method used here is readily applicable to similar systems. Hydration free energies are found to sensitively depend on the intermolecular interactions, while bonded interactions are less important, as expected. We also investigate in the present work the possibility of applying the multipolar force field in scoring trajectories generated using computationally inexpensive methods, which should be useful in broader parametrization studies with reduced computational resources, as scoring is much faster than the generation of the trajectories.

Using Multistate Reweighting to Rapidly and Efficiently Explore Molecular Simulation Parameters Space for Nonbonded Interactions

Multistate reweighting methods such as the multistate Bennett acceptance ratio (MBAR) can predict free energies and expectation values of thermodynamic observables at poorly sampled or unsampled thermodynamic states using simulations performed at only a few sampled states combined with single point energy reevaluations of these samples at the unsampled states. In this study, we demonstrate the power of this general reweighting formalism by exploring the effect of simulation parameters controlling Coulomb and Lennard-Jones cutoffs on free energy calculations and other observables. Using multistate reweighting, we can quickly identify, with very high sensitivity, the computationally least expensive nonbonded parameters required to obtain a specified accuracy in observables compared to the answer obtained using an expensive "gold standard" set of parameters. We specifically examine free energy estimates of three molecular transformations in a benchmark molecular set as well as the enthalpy of vaporization of TIP3P. The results demonstrates the power of this multistate reweighting approach for measuring changes in free energy differences or other estimators with respect to simulation or model parameters with very high precision and/or very low computational effort. The results also help to identify which simulation parameters affect free energy calculations and provide guidance to determine which simulation parameters are both appropriate and computationally efficient in general.

A flexible-protein molecular docking study of the binding of ruthenium complex compounds to PIM1, GSK-3β, and CDK2/Cyclin A protein kinases

We employ ensemble docking simulations to characterize the interactions of two enantiomeric forms of a Ru-complex compound (1-R and 1-S) with three protein kinases, namely PIM1, GSK-3β, and CDK2/cyclin A. We show that our ensemble docking computational protocol adequately models the structural features of these interactions and discriminates between competing conformational clusters of ligand-bound protein structures. Using the determined X-ray crystal structure of PIM1 complexed to the compound 1-R as a control, we discuss the importance of including the protein flexibility inherent in the ensemble docking protocol, for the accuracy of the structure prediction of the bound state. A comparison of our ensemble docking results suggests that PIM1 and GSK-3β bind the two enantiomers in similar fashion, through two primary binding modes: conformation I, which is very similar to the conformation presented in the existing PIM1/compound 1-R crystal structure; conformation II, which represents a 180° flip about an axis through the NH group of the pyridocarbazole moiety, relative to conformation I. In contrast, the binding of the enantiomers to CDK2 is found to have a different structural profile including a suggested bound conformation, which lacks the conserved hydrogen bond between the kinase and the ligand (i.e., ATP, staurosporine, Ru-complex compound). The top scoring conformation of the inhibitor bound to CDK2 is not present among the top-scoring conformations of the inhibitor bound to either PIM1 or GSK-3β and vice-versa. Collectively, our results help provide atomic-level insights into inhibitor selectivity among the three kinases.

A Benchmark Test Set for Alchemical Free Energy Transformations and Its Use to Quantify Error in Common Free Energy Methods

There is a significant need for improved tools to validate thermophysical quantities computed via molecular simulation. In this paper we present the initial version of a benchmark set of testing methods for calculating free energies of molecular transformation in solution. This set is based on molecular changes common to many molecular design problems, such as insertion and deletion of atomic sites and changing atomic partial charges. We use this benchmark set to compare the statistical efficiency, reliability, and quality of uncertainty estimates for a number of published free energy methods, including thermodynamic integration, free energy perturbation, the Bennett acceptance ratio (BAR) and its multistate equivalent MBAR. We identify MBAR as the consistently best performing method, though other methods are frequently comparable in reliability and accuracy in many cases. We demonstrate that assumptions of Gaussian distributed errors in free energies are usually valid for most methods studied. We demonstrate that bootstrap error estimation is a robust and useful technique for estimating statistical variance for all free energy methods studied. This benchmark set is provided in a number of different file formats with the hope of becoming a useful and general tool for method comparisons.

Insights into the complex formed by matrix metalloproteinase-2 and alloxan inhibitors: molecular dynamics simulations and free energy calculations

Matrix metalloproteinases (MMP) are well-known biological targets implicated in tumour progression, homeostatic regulation, innate immunity, impaired delivery of pro-apoptotic ligands, and the release and cleavage of cell-surface receptors. Hence, the development of potent and selective inhibitors targeting these enzymes continues to be eagerly sought. In this paper, a number of alloxan-based compounds, initially conceived to bias other therapeutically relevant enzymes, were rationally modified and successfully repurposed to inhibit MMP-2 (also named gelatinase A) in the nanomolar range. Importantly, the alloxan core makes its debut as zinc binding group since it ensures a stable tetrahedral coordination of the catalytic zinc ion in concert with the three histidines of the HExxHxxGxxH metzincin signature motif, further stabilized by a hydrogen bond with the glutamate residue belonging to the same motif. The molecular decoration of the alloxan core with a biphenyl privileged structure allowed to sample the deep S₁′ specificity pocket of MMP-2 and to relate the high affinity towards this enzyme with the chance of forming a hydrogen bond network with the backbone of Leu116 and Asn147 and the side chains of Tyr144, Thr145 and Arg149 at the bottom of the pocket. The effect of even slight structural changes in determining the interaction at the S₁′ subsite of MMP-2 as well as the nature and strength of the binding is elucidated via molecular dynamics simulations and free energy calculations. Among the herein presented compounds, the highest affinity (pIC₅₀ = 7.06) is found for BAM, a compound exhibiting also selectivity (>20) towards MMP-2, as compared to MMP-9, the other member of the gelatinases.

Understanding free-energy perturbation calculations through a model of harmonic oscillators: theory and implications to improve the sampling efficiency by molecular simulation

Free-energy perturbation calculation is frequently used to calculate free-energy differences because it is easy to implement and the computation is fast. However, the calculation is subject to large inaccuracies in some circumstances due to the insufficient sampling of the relevant tails of the energy-difference distributions. Here we expand this knowledge of insufficient sampling into a two-dimensional (2D) energy space using a model of harmonic oscillators. We show analytically the relation between the energies of the sampling system and those of the desired target energy spaces, which provide the basis to understand the difficulties in free-energy perturbation calculations. We clarify the reasons of the inaccurate calculation in the different harmonic cases that stem from the spatial separations of the reference and the target energy pairs located in the two-dimensional energy space. The potential-energy space introduced into this 2D energy-space model provides additional clues to improve the sampling efficiency. Based on this understanding, we propose two ways to calculate the free-energy differences using the two schemes of the distribution method. We show that the distribution method implemented in the appropriate energy space--the energy-difference space and the potential-energy space, respectively--can improve the calculation of free energies in different circumstances. This analysis implies that the sampling can be improved if it is directed toward the appropriate region in the potential-energy space, which is easily implemented in various types of free-energy calculations. To test this, we calculate the free-energy surface of alanine dipeptide in gas phase and in aqueous phase, respectively. We demonstrate that the free-energy surface calculation is improved when the biased sampling of the potential energy is integrated into the sampling scheme.

Advances in implicit models of water solvent to compute conformational free energy and molecular dynamics of proteins at constant pH

Modern implicit solvent models for macromolecular simulations in water-proton bath are considered. The fundamental quantity that implicit models approximate is the solute potential of mean force, which is obtained by averaging over solvent degrees of freedom. The implicit solvent models suggest practical ways to calculate free energies of macromolecular conformations taking into account equilibrium interactions with water solvent and proton bath, while the explicit solvent approach is unable to do that due to the need to account for a large number of solvent degrees of freedom. The most advanced realizations of the implicit continuum models by different research groups are discussed, their accuracy are examined, and some applications of the implicit solvent models to macromolecular modeling, such as free energy calculations, protein folding, and constant pH molecular dynamics are highlighted.

Good practices in free-energy calculations

As access to computational resources continues to increase, free-energy calculations have emerged as a powerful tool that can play a predictive role in a wide range of research areas. Yet, the reliability of these calculations can often be improved significantly if a number of precepts, or good practices, are followed. Although the theory upon which these good practices rely has largely been known for many years, it is often overlooked or simply ignored. In other cases, the theoretical developments are too recent for their potential to be fully grasped and merged into popular platforms for the computation of free-energy differences. In this contribution, the current best practices for carrying out free-energy calculations using free energy perturbation and nonequilibrium work methods are discussed, demonstrating that at little to no additional cost, free-energy estimates could be markedly improved and bounded by meaningful error estimates. Monitoring the probability distributions that underlie the transformation between the states of interest, performing the calculation bidirectionally, stratifying the reaction pathway, and choosing the most appropriate paradigms and algorithms for transforming between states offer significant gains in both accuracy and precision.

Minimizing thermodynamic length to select intermediate states for free-energy calculations and replica-exchange simulations

In computational thermodynamics, a sequence of intermediate states is commonly introduced to connect two equilibrium states. We consider two cases where the choice of intermediate states is particularly important: minimizing statistical error in free-energy difference calculations and maximizing average acceptance probabilities in replica-exchange simulations. We derive bounds for these quantities in terms of the thermodynamic distance between the intermediates, and show that in both cases the intermediates should be chosen as equidistant points along a geodesic connecting the end states.

QM/MM linear response method distinguishes ligand affinities for closely related metalloproteins

Design of selective ligands for closely related targets is becoming one of the most important tasks in the drug development. New tools, more precise than fast scoring functions and less demanding than sophisticated Free Energy Perturbation methods, are necessary to help accomplish this goal. The methods of intermediate complexity, characterizing individual contributions to the binding energy, have been an area of intense research in the past few years. Our recently developed quantum mechanical/molecular mechanical (QM/MM) modification of the Linear Response (LR) method describes the binding free energies as the sum of empirically weighted contributions of the QM/MM interaction energies and solvent-accessible surface areas for the time-averaged structures of hydrated complexes, obtained by molecular dynamics (MD) simulations. The method was applied to published data on 27 inhibitors of matrix metalloproteinase-3 (MMP-3). The two descriptors explained 90% of variance in the inhibition constants with RMSE of 0.245 log units. The QM/MM treatment is indispensable for characterization of the systems lacking suitable force-field expressions. In this case, it provided characteristics of H-bonds of the inhibitors to Glu202, charges of binding site atoms, and accurate coordination geometries of the ligands to catalytic zinc. The geometries were constrained during the MD simulations, which characterized conformational flexibility of the complexes and helped in the elucidation of the binding differences for related compounds. A comparison of the presented QM/MM LR results with those previously published for inhibition of MMP-9 by the same set of ligands showed that the QM/MM LR approach was able to distinguish subtle differences in binding affinities for MMP-3 and MMP-9, which did not exceed one order of magnitude. This precision level makes the approach a useful tool for design of selective ligands to similar targets, because the results can be safely extrapolated to maximize selectivity.

Absolute FKBP binding affinities obtained via nonequilibrium unbinding simulations

We compute the absolute binding affinities for two ligands bound to the FKBP protein using nonequilibrium unbinding simulations. The methodology is straightforward requiring little or no modification to many modern molecular simulation packages. The approach makes use of a physical pathway, eliminating the need for complicated alchemical decoupling schemes. We compare our nonequilibrium results to those obtained via a fully equilibrium approach and to experiment. The results of this study suggest that to obtain accurate results using nonequilibrium approaches one should use the stiff-spring approximation with the second cumulant expansion. From this study we conclude that nonequilibrium simulation could provide a simple means to estimate protein-ligand binding affinities.

Molecular modeling of nucleic acid structure: setup and analysis

The last in a set of units by these authors, this unit addresses some important remaining questions about molecular modeling of nucleic acids. It describes how to choose an appropriate molecular mechanics force field; how to set up and equilibrate the system for accurate simulation of a nucleic acid in an explicit solvent by molecular dynamics or Monte Carlo simulation; and how to analyze molecular dynamics trajectories.

Comparison of free energy methods for molecular systems

We present a detailed comparison of computational efficiency and precision for several free energy difference (DeltaF) methods. The analysis includes both equilibrium and nonequilibrium approaches, and distinguishes between unidirectional and bidirectional methodologies. We are primarily interested in comparing two recently proposed approaches, adaptive integration, and single-ensemble path sampling to more established methodologies. As test cases, we study relative solvation free energies of large changes to the size or charge of a Lennard-Jones particle in explicit water. The results show that, for the systems used in this study, both adaptive integration and path sampling offer unique advantages over the more traditional approaches. Specifically, adaptive integration is found to provide very precise long-simulation DeltaF estimates as compared to other methods used in this report, while also offering rapid estimation of DeltaF. The results demonstrate that the adaptive integration approach is the best overall method for the systems studied here. The single-ensemble path sampling approach is found to be superior to ordinary Jarzynski averaging for the unidirectional, "fast-growth" nonequilibrium case. Closer examination of the path sampling approach on a two-dimensional system suggests it may be the overall method of choice when conformational sampling barriers are high. However, it appears that the free energy landscapes for the systems used in this study have rather modest configurational sampling barriers.

In Silico Prediction of Drug Solubility. 3. Free Energy of Solvation in Pure Amorphous Matter

The solubility of drugs in water is investigated in a series of papers. In this work, we address the process of bringing a drug molecule from the vapor into a pure drug amorphous phase. This step enables us to actually calculate the solubility of amorphous drugs in water. In our general approach, we, on one hand, perform rigorous free energy simulations using a combination of the free energy perturbation and thermodynamic integration methods. On the other hand, we develop an approximate theory containing parameters that are easily accessible from conventional Monte Carlo simulations, thereby reducing the computation time significantly. In the theory for solvation, we assume that DeltaG* = DeltaGcav + ELJ + EC/2, where the free energy of cavity formation, DeltaGcav, in pure drug systems is obtained using a theory for hard-oblate spheroids, and ELJ and EC are the Lennard-Jones and Coulomb interaction energies between the chosen molecule and the others in the fluid. The theoretical predictions for the free energy of solvation in pure amorphous matter are in good agreement with free energy simulation data for 46 different drug molecules. These results together with our previous studies support our theoretical approach. By using our previous data for the free energy of hydration, we compute the total free energy change of bringing a molecule from the amorphous phase into water. We obtain good agreement between the theory and simulations. It should be noted that to obtain accurate results for the total process, high precision data are needed for the individual subprocesses. Finally, for eight different substances, we compare the experimental amorphous and crystalline solubility in water with the results obtained by the proposed theory with reasonable success.

The Common Chemical Motifs Within Anesthetic Binding Sites

BACKGROUND

It is not yet possible to obtain crystal structures of anesthetic molecules bound to proteins that are plausible neuronal targets; for example, ligand-gated ion channels. However, there are x-ray crystal structures in which anesthetics are complexed with proteins that are not directly related to anesthetic action. Much useful information about anesthetic-protein interactions can be derived from the x-ray crystal structures of halothane-cholesterol oxidase, bromoform-luciferase, halothane-albumin, and dichloroethane-dehalogenase. These structures show anesthetic-protein interactions at the atomic level.

METHODS

We obtained the known coordinate files for bromoform-luciferase, halothane- albumin, dichloroethane-dehalogenase, and halothane-cholesterol oxidase. These were then modified by adding hydrogens, edited into subsets, and underwent a series of restrained molecular mechanics optimizations. Final analysis of anesthetic polarization within the anesthetic binding site occurred via combined molecular mechanics-quantum mechanics calculations.

RESULTS

The anesthetic binding sites within these well-characterized anesthetic-protein complexes possess a set of common characteristics that we refer to as "binding motifs." The common features of these motifs are polar and nonpolar interactions within an amphiphilic binding cavity, including the presence of weak hydrogen bond interactions with amino acids and water molecules. Calculations also demonstrated the polarizing effect of the amphipathic binding sites on what are otherwise considered quite hydrophobic anesthetics. This polarization appears energetically favorable.

CONCLUSIONS

Anesthetic binding to proteins involves amphipathic interactions.

The transition between the B and Z conformations of DNA investigated by targeted molecular dynamics simulations with explicit solvation

The transition between the B and Z conformations of double-helical deoxyribonucleic acid (DNA) belongs to the most complex and elusive conformational changes occurring in biomolecules. Since the accidental discovery of the left-handed Z-DNA form in the late 1970s, research on this DNA morphology has been engaged in resolving questions relative to its stability, occurrence, and function in biological processes. While the occurrence of Z-DNA in vivo is now widely recognized and the major factors influencing its thermodynamical stability are largely understood, the intricate conformational changes that take place during the B-to-Z transition are still unknown at the atomic level. In this article, we report simulations of this transition for the 3'-(CGCGCG)-5' hexamer duplex using targeted molecular dynamics with the GROMOS96 force field in explicit water under different ionic-strength conditions. The results suggest that for this oligomer length and sequence, the transition mechanism involves: 1), a stretched intermediate conformation, which provides a simple solution to the important sterical constraints involved in this transition; 2), the transient disruption of Watson-Crick hydrogen-bond pairing, partly compensated energetically by an increase in the number of solute-solvent hydrogen bonds; and 3), an asynchronous flipping of the bases compatible with a zipperlike progression mechanism.

Hybrid integral equation/simulation model for enhancing free energy computations

Integral equation theory is used for extrapolating free energy data from molecular simulations of a reference state with respect to a modification of the interaction potential. The methodology is applied to the correction of artefacts arising from potential shifting and truncation. Corrective contributions for the hydration free energy with respect to the full potential are analysed for the case that both the solute-solvent as well as the solvent-solvent potentials are truncated and modified by a shifted-force term, reaching beyond the range of the dielectric continuum approximation and simple long-range correction expressions. The model systems argon in water and pure water are used as examples for apolar and polar solutes, revealing significant correction contributions even for the short-ranged dispersive interactions and the magnitude of solute-solvent and solvent-solvent components. In comparison with simulation-based extrapolation techniques the integral equation method is shown to be capable of quantitatively predicting truncation artefacts at negligible computational overhead.

Through the channel and around the channel: Validating and comparing microscopic approaches for the evaluation of free energy profiles for ion penetration through ion channels

Microscopic calculations of free energy profiles for ion transport through biological ion channels present a very serious challenge to modern simulation approaches. The main problem is due to the major convergence problems associated with the heterogeneous landscape of the electrostatic environment in ion channels and with the need to evaluate the profile associated with the transfer of the ion from bulk water to the channel environment. This problem is compounded by the lack of reliable and relevant benchmarks that can discriminate between alternative approaches. The present study is aimed at reducing the above problems by defining benchmarks that are directly relevant to ion channels and can also give converging results. This is done by constructing a series of models of a truncated gramicidin channel with different numbers of water molecules and by comparing the profiles for going around the channel and through the channel. These discriminating models are then used to validate and compare the adiabatic charging free energy perturbation (FEP) approach combined with an umbrella sampling approach (Warshel, A. J. Phys. Chem. 1982, 86, 2218) and the potential of mean force (PMF) approach used frequently in studies of ion channels. It is found that both approaches work quite well until one moves to the case of the fully solvated channel. In this limit, the PMF approach may give different results for the overall work of going through the channel and around the channel, while the FEP approach gives physically consistent results. The present benchmark also indicates that the weighted histogram analysis method (WHAM) approach does not offer a significant advantage over earlier approaches at least as much as studies of ion channels are concerned. Finally, it is concluded that the FEP approach may be more useful in evaluating the overall barrier for moving ions from water to ion channels and that in some cases it might be beneficial to use the FEP approach for selective points along the channel and then to connect these points by PMF calculations.

Determination of solvation free energies by adaptive expanded ensemble molecular dynamics

A new method of calculating absolute free energies is presented. It was developed as an extension to the expanded ensemble molecular dynamics scheme and uses probability density estimation to continuously optimize the expanded ensemble parameters. The new method is much faster as it removes the time-consuming and expertise-requiring step of determining balancing factors. Its efficiency and accuracy are demonstrated for the dissolution of three qualitatively very different chemical species in water: methane, ionic salts, and benzylamine. A recently suggested optimization scheme by Wang and Landau [Phys. Rev. Lett. 86, 2050 (2001)] was also implemented and found to be computationally less efficient than the proposed adaptive expanded ensemble method.

Systematic and statistical error in histogram-based free energy calculations

A common technique for the numerical calculation of free energies involves estimation of the probability density along a given coordinate from a set of configurations generated via simulation. The process requires discretization of one or more reaction coordinates to generate a histogram from which the continuous probability density is inferred. We show that the finite size of the intervals used to construct the histogram leads to quantifiable systematic error. The width of these intervals also determines the statistical error in the free energy, and the choice of the appropriate interval is therefore driven by the need to balance the two sources of error. We present a method for the construction of the optimal histogram for a given system, and show that the use of this technique requires little additional computational expense. We demonstrate the efficacy of the technique for a model system, and discuss how the principles governing the choice of discretization interval could be used to improve extended sampling techniques.

Computer simulation of the chemical catalysis of DNA polymerases: discriminating between alternative nucleotide insertion mechanisms for T7 DNA polymerase

Understanding the chemical step in the catalytic reaction of DNA polymerases is essential for elucidating the molecular basis of the fidelity of DNA replication. The present work evaluates the free energy surface for the nucleotide transfer reaction of T7 polymerase by free energy perturbation/empirical valence bond (FEP/EVB) calculations. A key aspect of the enzyme simulation is a comparison of enzymatic free energy profiles with the corresponding reference reactions in water using the same computational methodology, thereby enabling a quantitative estimate for the free energy of the nucleotide insertion reaction. The reaction is driven by the FEP/EVB methodology between valence bond structures representing the reactant, pentacovalent intermediate, and the product states. This pathway corresponds to three microscopic chemical steps, deprotonation of the attacking group, a nucleophilic attack on the P(alpha) atom of the dNTP substrate, and departure of the leaving group. Three different mechanisms for the first microscopic step, the generation of the RO(-) nucleophile from the 3'-OH hydroxyl of the primer, are examined: (i) proton transfer to the bulk solvent, (ii) proton transfer to one of the ionic oxygens of the P(alpha) phosphate group, and (iii) proton transfer to the ionized Asp654 residue. The most favorable reaction mechanism in T7 pol is predicted to involve the proton transfer to Asp654. This finding sheds light on the long standing issue of the actual role of conserved aspartates. The structural preorganization that helps to catalyze the reaction is also considered and analyzed. The overall calculated mechanism consists of three subsequent steps with a similar activation free energy of about 12 kcal/mol. The similarity of the activation barriers of the three microscopic chemical steps indicates that the T7 polymerase may select against the incorrect dNTP substrate by raising any of these barriers. The relative height of these barriers comparing right and wrong dNTP substrates should therefore be a primary focus of future computational studies of the fidelity of DNA polymerases.