Calculation of relative free energy via indirect pathways

Water Harvesting by Thermoresponsive Ionic Liquids: A Molecular Dynamics Study of the Water Absorption Kinetics and of the Role of Nanostructuring

Ionic liquids (ILs) whose water solutions are thermoresponsive provide an appealing route to harvest water from the atmosphere at an energy cost that can be accessed by solar heating. IL/water solutions that present a lower critical solution temperature (LCST), i.e., demix upon increasing temperature, represent the most promising choice for this task since they could absorb vapor during the night when its saturation is highest and release liquid water during the day. The kinetics of water absorption at the surface and the role of nanostructuring in this process have been investigated by atomistic molecular dynamics simulations for the ionic liquid tetrabutyl phosphonium 2,4-dimethylbenzenesulfonate whose LCST in water occurs at T_c = 36 °C for solutions of 50-50 wt % composition. The simulation results show that water molecules are readily adsorbed on the IL and migrate along the surface to form thick three-dimensional islands. On a slightly longer time scale, ions crawl on these islands, covering water and recreating the original surface whose free energy is particularly low. At a high deposition rate, this mechanism allows the fast incorporation of large amounts of water, producing subsurface water pockets that eventually merge into the populations of water-rich and IL-rich domains in the nanostructured bulk. Simulation results suggest that strong nanostructuring could ease the separation of water and water-contaminated IL phases even before macroscopic demixing.

PathDetect-SOM: A Neural Network Approach for the Identification of Pathways in Ligand Binding Simulations

Understanding the process of ligand–protein recognition is important to unveil biological mechanisms and to guide drug discovery and design. Enhanced-sampling molecular dynamics is now routinely used to simulate the ligand binding process, resulting in the need for suitable tools for the analysis of large data sets of binding events. Here, we designed, implemented, and tested PathDetect-SOM, a tool based on self-organizing maps to build concise visual models of the ligand binding pathways sampled along single simulations or replicas. The tool performs a geometric clustering of the trajectories and traces the pathways over an easily interpretable 2D map and, using an approximate transition matrix, it can build a graph model of concurrent pathways. The tool was tested on three study cases representing different types of problems and simulation techniques. A clear reconstruction of the sampled pathways was derived in all cases, and useful information on the energetic features of the processes was recovered. The tool is available at https://github.com/MottaStefano/PathDetect-SOM.

Umbrella Sampling-Based Method to Compute Ligand-Binding Affinity

Many proteins have a solvent-exposed binding cleft, which permits their inhibitors to bind and unbind without significant protein conformation transforms. The binding/unbinding pathways of these protein-inhibitor complexes can be rather straightforwardly sampled by using umbrella sampling (US) simulation methods. During a US simulation, the C_α atoms of the protein are restrained via a harmonic force. The potential of mean force (PMF) along the binding pathway can be estimated by using the weighted histogram analysis method (WHAM). The binding affinity is then computed as the difference in PMF between the binding and unbinding states.

Thermodynamics and Kinetics of Drug-Target Binding by Molecular Simulation

Computational studies play an increasingly important role in chemistry and biophysics, mainly thanks to improvements in hardware and algorithms. In drug discovery and development, computational studies can reduce the costs and risks of bringing a new medicine to market. Computational simulations are mainly used to optimize promising new compounds by estimating their binding affinity to proteins. This is challenging due to the complexity of the simulated system. To assess the present and future value of simulation for drug discovery, we review key applications of advanced methods for sampling complex free-energy landscapes at near nonergodicity conditions and for estimating the rate coefficients of very slow processes of pharmacological interest. We outline the statistical mechanics and computational background behind this research, including methods such as steered molecular dynamics and metadynamics. We review recent applications to pharmacology and drug discovery and discuss possible guidelines for the practitioner. Recent trends in machine learning are also briefly discussed. Thanks to the rapid development of methods for characterizing and quantifying rare events, simulation's role in drug discovery is likely to expand, making it a valuable complement to experimental and clinical approaches.

Revealing the Unbinding Kinetics and Mechanism of Type I and Type II Protein Kinase Inhibitors by Local-Scaled Molecular Dynamics Simulations

Protein kinase inhibitors disrupt phosphorylation of the target kinases, which are an important class of drug for treating cancer and other diseases. Conventional structure-based design methods (such as molecular docking) focus on the static binding mode of the kinase inhibitor with its target. However, dissociation kinetic properties of a drug molecule are found to correlate with its residence time in vivo and thus have drawn the attention of drug designers in recent years. In this study, we have applied the local-scaled molecular dynamics (MD) simulation enabled in GROMACS software to explore the unbinding mechanism of a total of 41 type I and type II kinase inhibitors. Our simulation considered multiple starting configurations as well as possible protonation states of kinase inhibitors. Based on our local-scaled MD results, we discovered that the integrals of the favorable binding energy during dissociation correlated well (R² = 0.64) with the experimental dissociation rate constants of those kinase inhibitors on the entire data set. Given its accuracy and technical advantage, this method may serve as a practical option for estimating this important property in reality. Our simulation also provided a reasonable explanation of the dynamic properties of kinase and its inhibitor as well as the role of relevant water molecules in dissociation.

Accelerating Rare Dissociative Processes in Biomolecules Using Selectively Scaled MD Simulations

Molecular dynamics (MD) simulations can be a powerful tool for modeling complex dissociative processes such as ligand unbinding. However, many biologically relevant dissociative processes occur on timescales that far exceed the timescales of typical MD simulations. Here, we implement and apply an enhanced sampling method in which specific energy terms in the potential energy function are selectively "scaled" to accelerate dissociative events during simulations. Using ligand unbinding as an example of a complex dissociative process, we selectively scaled up ligand-water interactions in an attempt to increase the rate of ligand unbinding. After applying our selectively scaled MD (ssMD) approach to several cyclin-dependent kinase-inhibitor complexes, we discovered that we could accelerate ligand unbinding, thereby allowing, in some cases, unbinding events to occur within as little as 2 ns. Moreover, we found that we could make realistic estimates of the initial unbinding times (τ_unbind^sim) as well as the accompanying change in free energy (ΔG^sim) of the inhibitors from our ssMD simulation data. To accomplish this, we employed a previously described Kramers'-based rate extrapolation method and a newly described free energy extrapolation method. Because our ssMD approach is general, it should find utility as an easy-to-deploy, enhanced sampling method for modeling other dissociative processes.

Kinetics of Drug Binding and Residence Time

The kinetics of drug binding and unbinding is assuming an increasingly crucial role in the long, costly process of bringing a new medicine to patients. For example, the time a drug spends in contact with its biological target is known as residence time (the inverse of the kinetic constant of the drug-target unbinding, 1/ koff). Recent reports suggest that residence time could predict drug efficacy in vivo, perhaps even more effectively than conventional thermodynamic parameters (free energy, enthalpy, entropy). There are many experimental and computational methods for predicting drug-target residence time at an early stage of drug discovery programs. Here, we review and discuss the methodological approaches to estimating drug binding kinetics and residence time. We first introduce the theoretical background of drug binding kinetics from a physicochemical standpoint. We then analyze the recent literature in the field, starting from the experimental methodologies and applications thereof and moving to theoretical and computational approaches to the kinetics of drug binding and unbinding. We acknowledge the central role of molecular dynamics and related methods, which comprise a great number of the computational methods and applications reviewed here. However, we also consider kinetic Monte Carlo. We conclude with the outlook that drug (un)binding kinetics may soon become a go/no go step in the discovery and development of new medicines.

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.

A generalized linear response framework for expanded ensemble and replica exchange simulations

Expanded ensemble simulation is a powerful technique for enhancing sampling over a range of thermodynamic parameters. However, although the premise is relatively simple, running successful simulations in practice still presents something of an ad hoc challenge. Three main difficulties exist: (1) the selection of the thermodynamic states, (2) the selection of the sampling weights, and (3) efficient sampling of the expanded parameter space. Here we consider these problems in the context of a pairwise linear response approach to the work fluctuation theorem. The approach offers comprehensive tactics for addressing the three difficulties and can be used as either an alternative or a complement to replica exchange simulations. Importantly, the results are trivially implemented for multi-dimensional parameter spaces and they recover results from the literature aimed at the special cases of simulated/parallel tempering and replica exchange umbrella sampling. Illustrative examples are shown using the NAMD simulation engine.

Fully Flexible Docking via Reaction-Coordinate-Independent Molecular Dynamics Simulations

Predicting the geometry of protein-ligand binding complexes is of primary importance for structure-based drug discovery. Molecular dynamics (MD) is emerging as a reliable computational tool for use in conjunction with, or an alternative to, docking methods. However, simulating the protein-ligand binding process often requires very expensive simulations. This drastically limits the practical application of MD-based approaches. Here, we propose a general framework to accelerate the generation of putative protein-ligand binding modes using potential-scaled MD simulations. The proposed dynamical protocol has been applied to two pharmaceutically relevant systems (GSK-3β and the N-terminal domain of HSP90α). Our approach is fully independent of any predefined reaction coordinate (or collective variable). It identified the correct binding mode of several ligands and can thus save valuable computational time in dynamic docking simulations.

Dynamic Docking: A Paradigm Shift in Computational Drug Discovery

Molecular docking is the methodology of choice for studying in silico protein-ligand binding and for prioritizing compounds to discover new lead candidates. Traditional docking simulations suffer from major limitations, mostly related to the static or semi-flexible treatment of ligands and targets. They also neglect solvation and entropic effects, which strongly limits their predictive power. During the last decade, methods based on full atomistic molecular dynamics (MD) have emerged as a valid alternative for simulating macromolecular complexes. In principle, compared to traditional docking, MD allows the full exploration of drug-target recognition and binding from both the mechanistic and energetic points of view (dynamic docking). Binding and unbinding kinetic constants can also be determined. While dynamic docking is still too computationally expensive to be routinely used in fast-paced drug discovery programs, the advent of faster computing architectures and advanced simulation methodologies are changing this scenario. It is feasible that dynamic docking will replace static docking approaches in the near future, leading to a major paradigm shift in in silico drug discovery. Against this background, we review the key achievements that have paved the way for this progress.

Protein-ligand (un)binding kinetics as a new paradigm for drug discovery at the crossroad between experiments and modelling

In the last three decades, protein and nucleic acid structure determination and comprehension of the mechanisms, leading to their physiological and pathological functions, have become a cornerstone of biomedical sciences. A deep understanding of the principles governing the fates of cells and tissue at the molecular level has been gained over the years, offering a solid basis for the rational design of drugs aimed at the pharmacological treatment of numerous diseases. Historically, affinity indicators (i.e. Kd and IC50/EC50) have been assumed to be valid indicators of the in vivo efficacy of a drug. However, recent studies pointed out that the kinetics of the drug-receptor binding process could be as important or even more important than affinity in determining the drug efficacy. This eventually led to a growing interest in the characterisation and prediction of the rate constants of protein-ligand association and dissociation. For instance, a drug with a longer residence time can kinetically select a given receptor over another, even if the affinity for both receptors is comparable, thus increasing its therapeutic index. Therefore, understanding the molecular features underlying binding and unbinding processes is of central interest towards the rational control of drug binding kinetics. In this review, we report the theoretical framework behind protein-ligand association and highlight the latest advances in the experimental and computational approaches exploited to investigate the binding kinetics.

Kinetics of protein-ligand unbinding via smoothed potential molecular dynamics simulations

Drug discovery is expensive and high-risk. Its main reasons of failure are lack of efficacy and toxicity of a drug candidate. Binding affinity for the biological target has been usually considered one of the most relevant figures of merit to judge a drug candidate along with bioavailability, selectivity and metabolic properties, which could depend on off-target interactions. Nevertheless, affinity does not always satisfactorily correlate with in vivo drug efficacy. It is indeed becoming increasingly evident that the time a drug spends in contact with its target (aka residence time) can be a more reliable figure of merit. Experimental kinetic measurements are operatively limited by the cost and the time needed to synthesize compounds to be tested, to express and purify the target, and to setup the assays. We present here a simple and efficient molecular-dynamics-based computational approach to prioritize compounds according to their residence time. We devised a multiple-replica scaled molecular dynamics protocol with suitably defined harmonic restraints to accelerate the unbinding events while preserving the native fold. Ligands are ranked according to the mean observed scaled unbinding time. The approach, trivially parallel and easily implementable, was validated against experimental information available on biological systems of pharmacological relevance.

Population based reweighting of scaled molecular dynamics

Molecular dynamics simulation using enhanced sampling methods is one of the powerful computational tools used to explore protein conformations and free energy landscapes. Enhanced sampling methods often employ either an increase in temperature or a flattening of the potential energy surface to rapidly sample phase space, and a corresponding reweighting algorithm is used to recover the Boltzmann statistics. However, potential energies of complex biomolecules usually involve large fluctuations on a magnitude of hundreds of kcal/mol despite minimal structural changes during simulation. This leads to noisy reweighting statistics and complicates the obtainment of accurate final results. To overcome this common issue in enhanced conformational sampling, we propose a scaled molecular dynamics method, which modifies the biomolecular potential energy surface and employs a reweighting scheme based on configurational populations. Statistical mechanical theory is applied to derive the reweighting formula, and the canonical ensemble of simulated structures is recovered accordingly. Test simulations on alanine dipeptide and the fast folding polypeptide Chignolin exhibit sufficiently enhanced conformational sampling and accurate recovery of free energy surfaces and thermodynamic properties. The results are comparable to long conventional molecular dynamics simulations and exhibit better recovery of canonical statistics over methods which employ a potential energy term in reweighting.

Improved Binding Free Energy Predictions from Single-Reference Thermodynamic Integration Augmented with Hamiltonian Replica Exchange

Reliable predictions of relative binding free energies are essential in drug discovery, where chemists modify promising compounds with the aim of increasing binding affinity. Conventional Thermodynamic Integration (TI) approaches can estimate corresponding changes in binding free energies, but suffer from inadequate sampling due to ruggedness of the molecular energy surfaces. Here, we present an improved TI strategy for computing relative binding free energies of congeneric ligands. This strategy employs a specific, unphysical single-reference (SR) state and Hamiltonian replica exchange (HREX) to locally enhance sampling. We then apply this strategy to compute relative binding free energies of twelve ligands in the L99A mutant of T4 Lysozyme. Besides the ligands, our approach enhances hindered rotations of the important V111, as well as V87 and L118 sidechains. Concurrently, we devise practical strategies to monitor and improve HREX-SRTI efficiency. Overall, the HREX-SRTI results agree well (R(2) = 0.76, RMSE = 0.3 kcal/mol) with available experimental data. When optimized for efficiency, the HREX-SRTI precision matches that of experimental measurements.

Alchemical free energy simulations for biological complexes: powerful but temperamental

Free energy simulations compare multiple ligand:receptor complexes by "alchemically" transforming one into another, yielding binding free energy differences. Since their introduction in the 1980s, many technical and theoretical obstacles were surmounted, and the method ("MDFE," since molecular dynamics are often used) has matured into a powerful tool. We describe its current status, its effectiveness, and the challenges it faces. MDFE has provided chemical accuracy for many systems but remains expensive, with significant human overhead costs. The bottlenecks have shifted, partly due to increased computer power. To study diverse sets of ligands, force field availability and accuracy can be a major difficulty. Another difficulty is the frequent need to consider multiple states, related to sidechain protonation or buried waters, for example. Sophisticated, automated methods to sample these states are maturing, such as constant pH simulations. Meanwhile, combinations of MDFE and simpler approaches, like continuum dielectric models, can be very effective. As illustrations, we show how, with careful force field parameterization, MDFE accurately predicts binding specificities between complex tetracycline ligands and their targets. We describe substrate binding to the aspartyl-tRNA synthetase enzyme, where many distinct electrostatic states play a role, and a histidine and a Mg(2+) ion act as coupled switches that help enforce a strict preference for the aspartate substrate, relative to several analogs. Overall, MDFE has achieved a predictive status, where novel ligands can be studied and molecular recognition elucidated in depth. It should play an increasing role in the analysis of complex cellular processes and biomolecular engineering.

Reducing the bias and uncertainty of free energy estimates by using regression to fit thermodynamic integration data

This report presents the application of polynomial regression for estimating free energy differences using thermodynamic integration data, i.e., slope of free energy with respect to the switching variable lambda. We employ linear regression to construct a polynomial that optimally fits the thermodynamic integration data, and thus reduces the bias and uncertainty of the resulting free energy estimate. Two test systems with analytical solutions were used to verify the accuracy and precision of the approach. Our results suggest that use of regression with high degree of polynomials provides the most accurate free energy difference estimates, but often with slightly larger uncertainty, compared to commonly used quadrature techniques. High degree polynomials possess the flexibility to closely fit the thermodynamic integration data but are often sensitive to small changes in the data points. Thus, we also used Chebyshev nodes to guide in the selection of nonequidistant lambda values for use in thermodynamic integration. We conclude that polynomial regression with nonequidistant lambda values delivers the most accurate and precise free energy estimates for thermodynamic integration data for the systems considered here. Software and documentation is available at http://www.phys.uidaho.edu/ytreberg/software.

Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages

Existing optimal estimators of nonequilibrium path-ensemble averages are shown to fall within the framework of extended bridge sampling. Using this framework, we derive a general minimal-variance estimator that can combine nonequilibrium trajectory data sampled from multiple path-ensembles to estimate arbitrary functions of nonequilibrium expectations. The framework is also applied to obtain asymptotic variance estimates, which are a useful measure of statistical uncertainty. In particular, we develop asymptotic variance estimates pertaining to Jarzynski's equality for free energies and the Hummer-Szabo expressions for the potential of mean force, calculated from uni- or bidirectional path samples. These estimators are demonstrated on a model single-molecule pulling experiment. In these simulations, the asymptotic variance expression is found to accurately characterize the confidence intervals around estimators when the bias is small. Hence, the confidence intervals are inaccurately described for unidirectional estimates with large bias, but for this model it largely reflects the true error in a bidirectional estimator derived by Minh and Adib.

Density-dependent analysis of nonequilibrium paths improves free energy estimates

When a system is driven out of equilibrium by a time-dependent protocol that modifies the Hamiltonian, it follows a nonequilibrium path. Samples of these paths can be used in nonequilibrium work theorems to estimate equilibrium quantities such as free energy differences. Here, we consider analyzing paths generated with one protocol using another one. It is posited that analysis protocols which minimize the lag, the difference between the nonequilibrium and the instantaneous equilibrium densities, will reduce the dissipation of reprocessed trajectories and lead to better free energy estimates. Indeed, when minimal lag analysis protocols based on exactly soluble propagators or relative entropies are applied to several test cases, substantial gains in the accuracy and precision of estimated free energy differences are observed.

Efficient free energy calculations on small molecule host-guest systems - a combined linear interaction energy/one-step perturbation approach

Two efficient methods to calculate binding affinities of ligands with proteins have been critically evaluated by using sixteen small ligand host-guest complexes. It is shown that both the one-step (OS) perturbation method and the linear interaction energy (LIE) method have complementing strengths and weaknesses and can be optimally combined in a new manner. The OS method has a sound theoretical basis to address the free energy of cavity formation, whereas the LIE approach is more versatile and efficient to calculate the free energy of adding charges to such cavities. The off-term, which is neglected in the original LIE equation, can be calculated without additional costs from the OS, offering a powerful synergy between the two methods. The LIE/OS approach presented here combines the best of two worlds and for the model systems studied here, is more accurate than and as efficient as the original methods. It has a sound theoretical background and no longer requires any empirical parameters. The method appears very well suited for application in lead-optimization programmes in drug research, where the structure and dynamics of a series of molecules is of interest, together with an accurate calculation of the binding free energy.

Optimization of replica exchange molecular dynamics by fast mimicking

We present an approach to mimic replica exchange molecular dynamics simulations (REMD) on a microsecond time scale within a few minutes rather than the years, which would be required for real REMD. The speed of mimicked REMD makes it a useful tool for "testing" the efficiency of different settings for REMD and then to select those settings, that give the highest efficiency. We present an optimization approach with the example of Hamiltonian REMD using soft-core interactions on two model systems, GTP and 8-Br-GTP. The optimization process using REMD mimicking is very fast. Optimization of Hamiltonian-REMD settings of GTP in explicit water took us less than one week. In our study we focus not only on finding the optimal distances between neighboring replicas, but also on finding the proper placement of the highest level of softness. In addition we suggest different REMD simulation settings at this softness level. We allow several replicas to be simulated at the same Hamiltonian simultaneously and reduce the frequency of switching attempts between them. This approach allows for more efficient conversions from one stable conformation to the other.

Synergistic approach to improve "alchemical" free energy calculation in rugged energy surface

The authors present an integrated approach to "alchemical" free energy simulation, which permits efficient calculation of the free energy difference on rugged energy surface. The method is designed to obtain efficient canonical sampling for rapid free energy convergence. The proposal is motivated by the insight that both the exchange efficiency in the presently designed dual-topology alchemical Hamiltonian replica exchange method (HREM), and the confidence of the free energy determination using the overlap histogramming method, depend on the same criterion, viz., the overlaps of the energy difference histograms between all pairs of neighboring states. Hence, integrating these two techniques can produce a joint solution to the problems of the free energy convergence and conformational sampling in the free energy simulations, in which lambda parameter plays two roles to simultaneously facilitate the conformational sampling and improve the phase space overlap for the free energy determination. Specifically, in contrast with other alchemical HREM based free energy simulation methods, the dual-topology approach can ensure robust conformational sampling. Due to these features (a synergistic solution to the free energy convergence and canonical sampling, and the improvement of the sampling efficiency with the dual-topology treatment), the present approach, as demonstrated in the model studies of the authors, is highly efficient in obtaining accurate free energy differences, especially for the systems with rough energy landscapes.

Sampling enhancement for the quantum mechanical potential based molecular dynamics simulations: a general algorithm and its extension for free energy calculation on rugged energy surface

An approach is developed in the replica exchange framework to enhance conformational sampling for the quantum mechanical (QM) potential based molecular dynamics simulations. Importantly, with our enhanced sampling treatment, a decent convergence for electronic structure self-consistent-field calculation is robustly guaranteed, which is made possible in our replica exchange design by avoiding direct structure exchanges between the QM-related replicas and the activated (scaled by low scaling parameters or treated with high "effective temperatures") molecular mechanical (MM) replicas. Although the present approach represents one of the early efforts in the enhanced sampling developments specifically for quantum mechanical potentials, the QM-based simulations treated with the present technique can possess the similar sampling efficiency to the MM based simulations treated with the Hamiltonian replica exchange method (HREM). In the present paper, by combining this sampling method with one of our recent developments (the dual-topology alchemical HREM approach), we also introduce a method for the sampling enhanced QM-based free energy calculations.

Free energies of binding of R- and S-propranolol to wild-type and F483A mutant cytochrome P450 2D6 from molecular dynamics simulations

Detailed molecular dynamics (MD) simulations have been performed to reproduce and rationalize the experimental finding that the F483A mutant of CYP2D6 has lower affinity for R-propranolol than for S-propranolol. Wild-type (WT) CYP2D6 does not show this stereospecificity. Four different approaches to calculate the free energy differences have been investigated and were compared to the experimental binding data. From the differences between calculations based on forward and backward processes and the closure of thermodynamic cycles, it was clear that not all simulations converged sufficiently. The approach that calculates the free energies of exchanging R-propranolol with S-propranolol in the F483A mutant relative to the exchange free energy in WT CYP2D6 accurately reproduced the experimental binding data. Careful inspection of the end-points of the MD simulations involved in this approach, allowed for a molecular interpretation of the observed differences.

Finite reservoir replica exchange to enhance canonical sampling in rugged energy surfaces

A "finite reservoir" replica exchange method is presented to further enhance sampling upon the regular replica exchange method (REM) in a rugged energy surface. The present method can facilitate important sampling more efficiently by exchanging structures with configurations randomly selected from a finite-sized reservoir; this finite reservoir is pregenerated and updated by a mechanism of replica exchange with neighboring "temperature" simulations. In practice, this proposal revises exchange schedule in REM simulations in order to make productive exchange for conformational "tunneling" more frequent.

Conformational sampling via a self-regulating effective energy surface

The difficulty of efficiently sampling the phase space of complex systems with rough energy surfaces is well known. Typical solutions to the problem involve accelerating the crossing of barriers, but such methods often have the secondary problem that the low-energy states of interest are inadequately sampled, unless the parameters of the search algorithm are modified as the system evolves. A method is presented to improve the sampling with particular emphasis on the low-energy conformations, which make the most important contributions to the thermodynamics of the system. The algorithm proposed here samples the details of the minima, while easily surmounting barriers. This is achieved by introducing a self-regulating sampling variable which depends on the current state of the system. Two replicas of the system are introduced and the sampling variable is treated as a particle coupled to the physical system. The method is illustrated with a simple model system and is applied to the realistic example of barrier crossing in a protein-ligand complex.

Calculation of the Redox Potential of the Protein Azurin and Some Mutants

Azurin from Pseudomonas aeruginosa is a small 128-residue, copper-containing protein. Its redox potential can be modified by mutating the protein. Free-energy calculations based on classical molecular-dynamics simulations of the protein and from mutants in aqueous solution at different pH values were used to compute relative redox potentials. The precision of the free-energy calculations with the lambda coupling-parameter approach is evaluated as function of the number and sequence of lambda values, the sampling time and initial conditions. It is found that the precision is critically dependent on the relaxation of hydrogen-bonding networks when changing the atomic-charge distribution due to a change of redox state or pH value. The errors in the free energies range from 1 to 10 k(B)T, depending on the type of process. Only qualitative estimates of the change in redox potential by protein mutation can be obtained.

Efficient use of nonequilibrium measurement to estimate free energy differences for molecular systems

A promising method for calculating free energy differences DeltaF is to generate nonequilibrium data via "fast-growth" simulations or by experiments--and then use Jarzynski's equality. However, a difficulty with using Jarzynski's equality is that DeltaF estimates converge very slowly and unreliably due to the nonlinear nature of the calculation--thus requiring large, costly data sets. The purpose of the work presented here is to determine the best estimate for DeltaF given a (finite) set of work values previously generated by simulation or experiment. Exploiting statistical properties of Jarzynski's equality, we present two fully automated analyses of nonequilibrium data from a toy model, and various simulated molecular systems. Both schemes remove at least several k(B)T of bias from DeltaF estimates, compared to direct application of Jarzynski's equality, for modest sized data sets (100 work values), in all tested systems. Results from one of the new methods suggest that good estimates of DeltaF can be obtained using 5-40-fold less data than was previously possible. Extending previous work, the new results exploit the systematic behavior of bias due to finite sample size. A key innovation is better use of the more statistically reliable information available from the raw data.

Improved precision and efficiency of free energy calculations for small systems using lambda-scaled atomic masses and separating conformational and transformational sampling

We present results showing the importance of appropriate treatment of atomic masses in molecular dynamics (MD)-based single topology free-energy perturbations (FEPs) on small molecule systems. The reversibility of gas phase simulations is significantly improved by scaling the atomic mass of mutated atoms with the lambda variable normally used for the scaling of energy terms. Because this effect is less pronounced for solvated systems, it will not cancel in estimates of the relative hydration free energy difference. The advantage of mass scaling is demonstrated by a null mutation of ethane to ethane and the calculation of the relative hydration free energy difference between ethane and n-propane. Furthermore, it is found that the simulation time necessary for converged MD/FEPs is prohibitively large for relative hydration free energy calculations on cyclic alkanes. Therefore, we explore an alternative free energy pathway including strongly constrained conformations to improve convergence in FEP simulations of flexible molecules.

Proline cis-trans isomerization in staphylococcal nuclease: multi-substrate free energy perturbation calculations

Staphylococcal nuclease A exists in two folded forms that differ in the isomerization state of the Lys 116-Pro 117 peptide bond. The dominant form (90% occupancy) adopts a cis peptide bond, which is observed in the crystal structure. NMR studies show that the relatively small difference in free energy between the cis and trans forms (delta Gcis-->trans approximately 1.2 kcal/mol) results from large and nearly compensating differences in enthalpy and entropy (delta Hcis-->trans approximately delta TScis-->trans approximately 10 kcal/mol). There is evidence from X-ray crystal structures that the structural differences between the cis and the trans forms of nuclease are confined to the conformation of residues 112-117, a solvated protein loop. Here, we obtain a thermodynamic and structural description of the conformational equilibrium of this protein loop through an exhaustive conformational search that identified several substates followed by free energy simulations between the substrates. By partitioning the search into conformational substates, we overcame the multiple minima problem in this particular case and obtained precise and reproducible free energy values. The protein and water environment was implicitly modeled by appropriately chosen nonbonded terms between the explicitly treated loop and the rest of the protein. These simulations correctly predicted a small free energy difference between the cis and trans forms composed of larger, compensating differences in enthalpy and entropy. The structural predictions of these simulations were qualitatively consistent with known X-ray structures of nuclease variants and yield a model of the unknown minor trans conformation.

Protein structure prediction: recognition of primary, secondary, and tertiary structural features from amino acid sequence

This review attempts a critical stock-taking of the current state of the science aimed at predicting structural features of proteins from their amino acid sequences. At the primary structure level, methods are considered for detection of remotely related sequences and for recognizing amino acid patterns to predict posttranslational modifications and binding sites. The techniques involving secondary structural features include prediction of secondary structure, membrane-spanning regions, and secondary structural class. At the tertiary structural level, methods for threading a sequence into a mainchain fold, homology modeling and assigning sequences to protein families with similar folds are discussed. A literature analysis suggests that, to date, threading techniques are not able to show their superiority over sequence pattern recognition methods. Recent progress in the state of ab initio structure calculation is reviewed in detail. The analysis shows that many structural features can be predicted from the amino acid sequence much better than just a few years ago and with attendant utility in experimental research. Best prediction can be achieved for new protein sequences that can be assigned to well-studied protein families. For single sequences without homologues, the folding problem has not yet been solved.

Fundamentals of drug design from a biophysical viewpoint

Drug design means many things to many people. Commercially the aim is the development of compounds that can be patented and meet a variety of regulatory standards. In drug design, for medical purposes, toxicity and bio-availability are major considerations.

Site specific point mutation changes specificity: a molecular modeling study by free energy simulations and enzyme kinetics of the thermodynamics in ribonuclease T1 substrate interactions

We have theoretically and experimentally studied the binding of two different ligands to wild-type ribonuclease T1 (RNT1) and to a mutant of RNT1 with Glu-46 replaced by Gln. The binding of the natural substrate 3'-GMP has been compared with the binding of a fluorescent probe, 2-aminopurine 3'-monophosphate (2AP), and relative free energies of binding of these ligands to the mutant and the wild-type (wt) enzyme have been calculated by free energy perturbation methods. The free energy perturbations predict that the mutant RNT1-Gln-46 binds 2AP better than 3'GMP, in agreement with experiments on dinucleotides. Four free energy perturbations, forming a closed loop, have been performed to allow the detection of systematic errors in the simulation procedure. Because of the larger number of atoms involved, it was necessary to use a much longer simulation time for the change in the protein, i,e., the perturbation from Glu to Gln, than in the perturbation from 3'-GMP to 2AP. Finally the structure of the binding site is analyzed for understanding differences in catalytic speed and binding strength.