An Efficient, Path-Independent Method for Free-Energy Calculations

PME Switching in Confinement Simulations of Charged Solutes

The confinement method is a reaction coordinate-free enhanced sampling method for the calculation of conformational free energy differences. We show that in explicit solvent, artifacts occur when treating charged solutes. These artifacts are resolved by switching off the particle mesh Ewald (PME) method at the start of confinement. Calculations of the free energy cost of this switching converge rapidly, with small statistical error. The effectiveness and accuracy of confinement with PME switching is demonstrated by its application to a series of solutes of different charge; its ability to treat complex systems is illustrated by evaluating the free energy difference between B and Z-DNA.

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.

Smooth orientation-dependent scoring function for coarse-grained protein quality assessment

MOTIVATION

Protein quality assessment (QA) is a crucial element of protein structure prediction, a fundamental and yet open problem in structural bioinformatics. QA aims at ranking predicted protein models to select the best candidates. The assessment can be performed based either on a single model or on a consensus derived from an ensemble of models. The latter strategy can yield very high performance but substantially depends on the pool of available candidate models, which limits its applicability. Hence, single-model QA methods remain an important research target, also because they can assist the sampling of candidate models.

RESULTS

We present a novel single-model QA method called SBROD. The SBROD (Smooth Backbone-Reliant Orientation-Dependent) method uses only the backbone protein conformation, and hence it can be applied to scoring coarse-grained protein models. The proposed method deduces its scoring function from a training set of protein models. The SBROD scoring function is composed of four terms related to different structural features: residue-residue orientations, contacts between backbone atoms, hydrogen bonding and solvent-solute interactions. It is smooth with respect to atomic coordinates and thus is potentially applicable to continuous gradient-based optimization of protein conformations. Furthermore, it can also be used for coarse-grained protein modeling and computational protein design. SBROD proved to achieve similar performance to state-of-the-art single-model QA methods on diverse datasets (CASP11, CASP12 and MOULDER).

AVAILABILITY AND IMPLEMENTATION

The standalone application implemented in C++ and Python is freely available at https://gitlab.inria.fr/grudinin/sbrod and supported on Linux, MacOS and Windows.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Calculating the absolute binding free energy of the insulin dimer in an explicit solvent

Insulin is a significant hormone in the regulation of glucose level in the blood. Its monomers bind to each other to form dimers or hexamers through a complex process. To study the binding of the insulin dimer, we first calculate its absolute binding free energy by the steered molecular dynamics method and the confinement method based on a fictitious thermodynamic cycle. After considering some special correction terms, the final calculated binding free energy at 298 K is −8.97 ± 1.41 kcal mol⁻¹, which is close to the experimental value of −7.2 ± 0.8 kcal mol⁻¹. Furthermore, we discuss the important residue–residue interactions between the insulin monomers, including hydrophobic interactions, π–π interactions and hydrogen bond interactions. The analysis reveals five key residues, Vla^B12, Tyr^B16, Phe^B24, Phe^B25, and Tyr^B26, for the dimerization of the insulin. We also perform MM-PBSA calculations for the wild-type dimer and some mutants and study the roles of the key residues by the change of the binding energy of the insulin dimer.

In this paper, we calculate the absolute binding free energy of an insulin dimer by steered MD method. The result of −8.97 kcal mol⁻¹ is close to the experimental value −7.2 kcal mol⁻¹. We also analyze the residue–residue interactions.

Differential Local Stability Governs the Metamorphic Fold Switch of Bacterial Virulence Factor RfaH

RfaH, a two-domain protein from a universally conserved NusG/Spt5 family of regulators, is required for the transcription and translation of long virulence and conjugation operons in many Gram-negative bacterial pathogens. Escherichia coli RfaH action is controlled by a unique large-scale structural rearrangement triggered by recruitment to transcription elongation complexes through a specific DNA element. Upon recruitment, the C-terminal domain of RfaH refolds from an α-hairpin, which is bound to RNA polymerase binding site within the N-terminal domain, into an unbound β-barrel that interacts with the ribosome. Although structures of the autoinhibited (α-hairpin) and active (β-barrel) states and plausible refolding pathways have been reported, how this reversible switch is encoded within RfaH sequence and structure is poorly understood. Here, we combined hydrogen-deuterium exchange measurements by mass spectrometry and nuclear magnetic resonance with molecular dynamics to evaluate the differential local stability between both RfaH folds. Deuteron incorporation reveals that the tip of the C-terminal hairpin (residues 125-145) is stably folded in the autoinhibited state (∼20% deuteron incorporation), whereas the rest of this domain is highly flexible (>40% deuteron incorporation), and its flexibility only decreases in the β-folded state. Computationally predicted ΔG agree with these results by displaying similar anisotropic stability within the tip of the α-hairpin and on neighboring N-terminal domain residues. Remarkably, the β-folded state shows comparable structural flexibility than nonmetamorphic homologs. Our findings provide information critical for understanding the metamorphic behavior of RfaH and other chameleon proteins and for devising targeted strategies to combat bacterial infections.

Allosterism and signal transfer in DNA

We analysed the basic mechanisms of signal transmission in DNA and the origins of the allostery exhibited by systems such as the ternary complex BAMHI-DNA-GRDBD. We found that perturbation information generated by a primary protein binding event travels as a wave to distant regions of DNA following a hopping mechanism. However, such a structural perturbation is transient and does not lead to permanent changes in the DNA geometry and interaction properties at the secondary binding site. The BAMHI-DNA-GRDBD allosteric mechanism does not occur through any traditional models: direct (protein-protein), indirect (reorganization of the secondary site) readout or solvent-release. On the contrary, it is generated by a subtle and less common entropy-mediated mechanism, which might have an important role to explain other DNA-mediated cooperative effects.

Modeling the adsorption equilibrium of small-molecule gases on graphene: effect of the volume to surface ratio

The adsorption probability of molecules at surfaces depends on the available volume-to-surface ratio with important implications for 2D self-assembly.

Computation of Hemagglutinin Free Energy Difference by the Confinement Method

Hemagglutinin (HA) mediates membrane fusion, a crucial step during influenza virus cell entry. How many HAs are needed for this process is still subject to debate. To aid in this discussion, the confinement free energy method was used to calculate the conformational free energy difference between the extended intermediate and postfusion state of HA. Special care was taken to comply with the general guidelines for free energy calculations, thereby obtaining convergence and demonstrating reliability of the results. The energy that one HA trimer contributes to fusion was found to be 34.2 ± 3.4kBT, similar to the known contributions from other fusion proteins. Although computationally expensive, the technique used is a promising tool for the further energetic characterization of fusion protein mechanisms. Knowledge of the energetic contributions per protein, and of conserved residues that are crucial for fusion, aids in the development of fusion inhibitors for antiviral drugs.

Fast exploration of an optimal path on the multidimensional free energy surface

In a reaction, determination of an optimal path with a high reaction rate (or a low free energy barrier) is important for the study of the reaction mechanism. This is a complicated problem that involves lots of degrees of freedom. For simple models, one can build an initial path in the collective variable space by the interpolation method first and then update the whole path constantly in the optimization. However, such interpolation method could be risky in the high dimensional space for large molecules. On the path, steric clashes between neighboring atoms could cause extremely high energy barriers and thus fail the optimization. Moreover, performing simulations for all the snapshots on the path is also time-consuming. In this paper, we build and optimize the path by a growing method on the free energy surface. The method grows a path from the reactant and extends its length in the collective variable space step by step. The growing direction is determined by both the free energy gradient at the end of the path and the direction vector pointing at the product. With fewer snapshots on the path, this strategy can let the path avoid the high energy states in the growing process and save the precious simulation time at each iteration step. Applications show that the presented method is efficient enough to produce optimal paths on either the two-dimensional or the twelve-dimensional free energy surfaces of different small molecules.

Balancing Accuracy and Cost of Confinement Simulations by Interpolation and Extrapolation of Confinement Energies

Improvements to the confinement method for the calculation of conformational free energy differences are presented. By taking advantage of phase space overlap between simulations at different frequencies, significant gains in accuracy and speed are reached. The optimal frequency spacing for the simulations is obtained from extrapolations of the confinement energy, and relaxation time analysis is used to determine time steps, simulation lengths, and friction coefficients. At postprocessing, interpolation of confinement energies is used to significantly reduce discretization errors in the calculation of conformational free energies. The efficiency of this protocol is illustrated by applications to alanine n-peptides and lactoferricin. For the alanine-n-peptide, errors were reduced between 2- and 10-fold and sampling times between 8- and 67-fold, while for lactoferricin the long sampling times at low frequencies were reduced 10-100-fold.

Assessment of Mutational Effects on Peptide Stability through Confinement Simulations

The evaluation of free energy differences between specific states of a system is of fundamental interest in the study of (bio)chemical systems. Herein, we examine the use of the recently introduced confinement method (CM) to evaluate relative free energy changes upon protein/peptide mutations. CM is a path-independent technique that involves the transformation of a configurational state of the system into an ideal crystal permitting the direct computation of free energy differences. We illustrate the method by evaluating the differential stabilities between native and mutant sequences of a model peptide that has been extensively characterized by experimental approaches, the GB1 hairpin. We show a good correlation between calculated and experimental relative stabilities and discuss other possible applications of this method in the context of complex molecular conversions.

Free-energy calculations for semi-flexible macromolecules: applications to DNA knotting and looping

We present a method to obtain numerically accurate values of configurational free energies of semiflexible macromolecular systems, based on the technique of thermodynamic integration combined with normal-mode analysis of a reference system subject to harmonic constraints. Compared with previous free-energy calculations that depend on a reference state, our approach introduces two innovations, namely, the use of internal coordinates to constrain the reference states and the ability to freely select these reference states. As a consequence, it is possible to explore systems that undergo substantially larger fluctuations than those considered in previous calculations, including semiflexible biopolymers having arbitrary ratios of contour length L to persistence length P. To validate the method, high accuracy is demonstrated for free energies of prime DNA knots with L/P = 20 and L/P = 40, corresponding to DNA lengths of 3000 and 6000 base pairs, respectively. We then apply the method to study the free-energy landscape for a model of a synaptic nucleoprotein complex containing a pair of looped domains, revealing a bifurcation in the location of optimal synapse (crossover) sites. This transition is relevant to target-site selection by DNA-binding proteins that occupy multiple DNA sites separated by large linear distances along the genome, a problem that arises naturally in gene regulation, DNA recombination, and the action of type-II topoisomerases.

Computing the relative stabilities and the per-residue components in protein conformational changes

Protein molecules often undergo conformational changes. In order to gain insights into the forces that drive such changes, it would be useful to have a method that computes the per-residue contributions to the conversion free energy. Here, we describe the "confine-convert-release" (CCR) method, which is applicable to large conformational changes. We show that CCR correctly predicts the stable states of several "chameleon" sequences that have previously been challenging for molecular simulations. CCR can often discriminate better from worse predictions of native protein models in critical assessment of protein structure prediction (CASP). We show how the total conversion free energies can be parsed into per-residue free-energy components. Such parsing gives insights into which amino acids are most responsible for given transformations. For example, here we are able to "reverse-engineer" the known design principles of the chameleon proteins. This opens up the possibility for systematic improvements in structure-prediction scoring functions, in the design of protein conformational switches, and in interpreting protein mechanisms at the amino-acid level.

Absolute free energies of biomolecules from unperturbed ensembles

Computing the absolute free energy of a macromolecule's structural state, F, is a challenging problem of high relevance. This study presents a method that computes F using only information from an unperturbed simulation of the macromolecule in the relevant conformational state, ensemble, and environment. Absolute free energies produced by this method, dubbed Valuation of Local Configuration Integral with Dynamics (VALOCIDY), enable comparison of alternative states. For example, comparing explicitly solvated and vaporous states of amino acid side-chain analogs produces solvation free energies in good agreement with experiments. Also, comparisons between alternative conformational states of model heptapeptides (including the unfolded state) produce free energy differences in agreement with data from μs molecular-dynamics simulations and experimental propensities. The potential of using VALOCIDY in computational protein design is explored via a small design problem of stabilizing a β-turn structure. When VALOCIDY-based estimation of folding free energy is used as the design metric, the resulting sequence folds into the desired structure within the atomistic force field used in design. The VALOCIDY-based approach also recognizes the distinct status of the native sequence regardless of minor details of the starting template structure, in stark contrast with a traditional fixed-backbone approach.

A simplified confinement method for calculating absolute free energies and free energy and entropy differences

A simple and robust formulation of the path-independent confinement method for the calculation of free energies is presented. The simplified confinement method (SCM) does not require matrix diagonalization or switching off the molecular force field, and has a simple convergence criterion. The method can be readily implemented in molecular dynamics programs with minimal or no code modifications. Because the confinement method is a special case of thermodynamic integration, it is trivially parallel over the integration variable. The accuracy of the method is demonstrated using a model diatomic molecule, for which exact results can be computed analytically. The method is then applied to the alanine dipeptide in vacuum, and to the α-helix ↔ β-sheet transition in a 16-residue peptide modeled in implicit solvent. The SCM requires less effort for the calculation of free energy differences than previous formulations because it does not require computing normal modes. The SCM has a diminished advantage for determining absolute free energy values, because it requires decreasing the MD integration step to obtain accurate results. An approximate confinement procedure is introduced, which can be used to estimate directly the configurational entropy difference between two macrostates, without the need for additional computation of the difference in the free energy or enthalpy. The approximation has convergence properties similar to those of the standard confinement method for the calculation of free energies. The use of the approximation requires about 5 times less wall-clock simulation time than that needed to compute enthalpy differences to similar precision from an MD trajectory. For the biomolecular systems considered in this study, the errors in the entropy approximation are under 10%. Practical applications of the methods to proteins are currently limited to implicit solvent simulations.

Free-energy calculations along a high-dimensional fragmented path with constrained dynamics

Free-energy calculations for high-dimensional systems, such as peptides or proteins, always suffer from a serious sampling problem in a huge conformational space. For such systems, path-based free-energy methods, such as thermodynamic integration or free-energy perturbation, are good choices. However, both of them need sufficient sampling along a predefined transition path, which can only be controlled using restrained or constrained dynamics. Constrained simulations produce more reasonable free-energy profiles than restrained simulations. But calculations of standard constrained dynamics require an explicit expression of reaction coordinates as a function of Cartesian coordinates of all related atoms, which may be difficult to find for the complex transition of biomolecules. In this paper, we propose a practical solution: (1) We use restrained dynamics to define an optimized transition path, divide it into small fragments, and define a virtual reaction coordinate to denote a position along the path. (2) We use constrained dynamics to perform a formal free-energy calculation for each fragment and collect the values together to provide the entire free-energy profile. This method avoids the requirement to explicitly define reaction coordinates in Cartesian coordinates and provides a novel strategy to perform free-energy calculations for biomolecules along any complex transition path.

A conformational transition in the myosin VI converter contributes to the variable step size

Myosin VI (MVI) is a dimeric molecular motor that translocates backwards on actin filaments with a surprisingly large and variable step size, given its short lever arm. A recent x-ray structure of MVI indicates that the large step size can be explained in part by a novel conformation of the converter subdomain in the prepowerstroke state, in which a 53-residue insert, unique to MVI, reorients the lever arm nearly parallel to the actin filament. To determine whether the existence of the novel converter conformation could contribute to the step-size variability, we used a path-based free-energy simulation tool, the string method, to show that there is a small free-energy difference between the novel converter conformation and the conventional conformation found in other myosins. This result suggests that MVI can bind to actin with the converter in either conformation. Models of MVI/MV chimeric dimers show that the variability in the tilting angle of the lever arm that results from the two converter conformations can lead to step-size variations of ∼12 nm. These variations, in combination with other proposed mechanisms, could explain the experimentally determined step-size variability of ∼25 nm for wild-type MVI. Mutations to test the findings by experiment are suggested.

Accounting for ligand conformational restriction in calculations of protein-ligand binding affinities

The conformation adopted by a ligand on binding to a receptor may differ from its lowest-energy conformation in solution. In addition, the bound ligand is more conformationally restricted, which is associated with a configurational entropy loss. The free energy change due to these effects is often neglected or treated crudely in current models for predicting binding affinity. We present a method for estimating this contribution, based on perturbation theory using the quasi-harmonic model of Karplus and Kushick as a reference system. The consistency of the method is checked for small model systems. Subsequently we use the method, along with an estimate for the enthalpic contribution due to ligand-receptor interactions, to calculate relative binding affinities. The AMBER force field and generalized Born implicit solvent model is used. Binding affinities were estimated for a test set of 233 protein-ligand complexes for which crystal structures and measured binding affinities are available. In most cases, the ligand conformation in the bound state was significantly different from the most favorable conformation in solution. In general, the correlation between measured and calculated ligand binding affinities including the free energy change due to ligand conformational change is comparable to or slightly better than that obtained by using an empirically-trained docking score. Both entropic and enthalpic contributions to this free energy change are significant.

Methods for calculating the absolute entropy and free energy of biological systems based on ideas from polymer physics

The commonly used simulation techniques, Metropolis Monte Carlo (MC) and molecular dynamics (MD) are of a dynamical type which enables one to sample system configurations i correctly with the Boltzmann probability, P(i)(B), while the value of P(i)(B) is not provided directly; therefore, it is difficult to obtain the absolute entropy, S approximately -ln P(i)(B), and the Helmholtz free energy, F. With a different simulation approach developed in polymer physics, a chain is grown step-by-step with transition probabilities (TPs), and thus their product is the value of the construction probability; therefore, the entropy is known. Because all exact simulation methods are equivalent, i.e. they lead to the same averages and fluctuations of physical properties, one can treat an MC or MD sample as if its members have rather been generated step-by-step. Thus, each configuration i of the sample can be reconstructed (from nothing) by calculating the TPs with which it could have been constructed. This idea applies also to bulk systems such as fluids or magnets. This approach has led earlier to the "local states" (LS) and the "hypothetical scanning" (HS) methods, which are approximate in nature. A recent development is the hypothetical scanning Monte Carlo (HSMC) (or molecular dynamics, HSMD) method which is based on stochastic TPs where all interactions are taken into account. In this respect, HSMC(D) can be viewed as exact and the only approximation involved is due to insufficient MC(MD) sampling for calculating the TPs. The validity of HSMC has been established by applying it first to liquid argon, TIP3P water, self-avoiding walks (SAW), and polyglycine models, where the results for F were found to agree with those obtained by other methods. Subsequently, HSMD was applied to mobile loops of the enzymes porcine pancreatic alpha-amylase and acetylcholinesterase in explicit water, where the difference in F between the bound and free states of the loop was calculated. Currently, HSMD is being extended for calculating the absolute and relative free energies of ligand-enzyme binding. We describe the whole approach and discuss future directions.

Estimating absolute configurational entropies of macromolecules: the minimally coupled subspace approach

We develop a general minimally coupled subspace approach (MCSA) to compute absolute entropies of macromolecules, such as proteins, from computer generated canonical ensembles. Our approach overcomes limitations of current estimates such as the quasi-harmonic approximation which neglects non-linear and higher-order correlations as well as multi-minima characteristics of protein energy landscapes. Here, Full Correlation Analysis, adaptive kernel density estimation, and mutual information expansions are combined and high accuracy is demonstrated for a number of test systems ranging from alkanes to a 14 residue peptide. We further computed the configurational entropy for the full 67-residue cofactor of the TATA box binding protein illustrating that MCSA yields improved results also for large macromolecular systems.

Calculation of free-energy differences by confinement simulations. Application to peptide conformers

Conformational free-energy differences are key quantities for understanding important phenomena in molecular biology that involve large structural changes of macromolecules. In this paper, an improved version of the confinement approach, which is based on earlier developments, is used to determine the free energy of individual molecular states by progressively restraining the corresponding molecular structures to pure harmonic basins, whose absolute free energy can be computed by normal-mode analysis. The method is used to calculate the free-energy difference between two conformational states of the alanine dipeptide in vacuo, and of the beta-hairpin from protein G with an implicit solvation model. In all cases, the confinement results are in excellent agreement with the ones obtained from converged equilibrium molecular dynamics simulations, which have a much larger computational cost. The systematic and statistical errors of the results are evaluated and the origin of the errors is identified. The sensitivity of the calculated free-energy differences to structure-based definitions of the molecular states is discussed. A variant of the method, which closes the thermodynamic cycle by a quasi-harmonic rather than harmonic analysis, is introduced. The latter is proposed for possible use with explicit solvent simulations.

Adaptive anisotropic kernels for nonparametric estimation of absolute configurational entropies in high-dimensional configuration spaces

The quasiharmonic approximation is the most widely used estimate for the configurational entropy of macromolecules from configurational ensembles generated from atomistic simulations. This method, however, rests on two assumptions that severely limit its applicability, (i) that a principal component analysis yields sufficiently uncorrelated modes and (ii) that configurational densities can be well approximated by Gaussian functions. In this paper we introduce a nonparametric density estimation method which rests on adaptive anisotropic kernels. It is shown that this method provides accurate configurational entropies for up to 45 dimensions thus improving on the quasiharmonic approximation. When embedded in the minimally coupled subspace framework, large macromolecules of biological interest become accessible, as demonstrated for the 67-residue coldshock protein.

Computing conformational free energy by deactivated morphing

Despite the significant advances in free-energy computations for biomolecules, there exists no general method to evaluate the free-energy difference between two conformations of a macromolecule that differ significantly from each other. A crucial ingredient of such a method is the ability to find a path between different conformations that allows an efficient computation of the free energy. In this paper, we introduce a method called "deactivated morphing," in which one conformation is morphed into another after the internal interactions are completely turned off. An important feature of this method is the (shameless) use of nonphysical paths, which makes the method robustly applicable to conformational changes of arbitrary complexity.

Recent developments in methodologies for calculating the entropy and free energy of biological systems by computer simulation

The Helmholtz free energy, F, plays an important role in proteins because of their rugged potential energy surface, which is 'decorated' with a tremendous number of local wells (denoted microstates, m). F governs protein folding, whereas differences DeltaF(mn) determine the relative populations of microstates that are visited by a flexible cyclic peptide or a flexible protein segment (e.g. a surface loop). Recently developed methodologies for calculating DeltaF(mn) (and entropy differences, DeltaS(mn)) mainly use thermodynamic integration and calculation of the absolute F; interesting new approaches in these categories are the adaptive integration method and the hypothetical scanning molecular dynamics method, respectively.