Convergence Of Free Energy Research Articles

Local convexity of the TAP free energy and AMP convergence for Z2-synchronization

We study mean-field variational Bayesian inference using the TAP approach, for Z2-synchronization as a prototypical example of a high-dimensional Bayesian model. We show that for any signal strength λ>1 (the weak-recovery threshold), there exists a unique local minimizer of the TAP free energy functional near the mean of the Bayes posterior law. Furthermore, the TAP free energy in a local neighborhood of this minimizer is strongly convex. Consequently, a natural-gradient/mirror-descent algorithm achieves linear convergence to this minimizer from a local initialization, which may be obtained by a constant number of iterations of Approximate Message Passing (AMP). This provides a rigorous foundation for variational inference in high dimensions via minimization of the TAP free energy. We also analyze the finite-sample convergence of AMP, showing that AMP is asymptotically stable at the TAP minimizer for any λ>1, and is linearly convergent to this minimizer from a spectral initialization for sufficiently large λ. Such a guarantee is stronger than results obtainable by state evolution analyses, which only describe a fixed number of AMP iterations in the infinite-sample limit. Our proofs combine the Kac–Rice formula and Sudakov–Fernique Gaussian comparison inequality to analyze the complexity of critical points that satisfy strong convexity and stability conditions within their local neighborhoods.

Free energy convergence in short- and long-length hydrophobic hydration

We show, through molecular dynamics, that the hydration free energy of model hydrocarbons, exhibits a convergence temperature in supercooled water, TG∗ ~ 255 K ± 10 K, around which, hydration entropy and enthalpy minima are observed. Further, although entropy convergence at TS∗ ~ 400 K ± 10 K is found to be violated by large non-spherical hydrocarbons, due to “folding”, a free energy maximum is still observed, opposite to hard spheres. Evidence is provided that the entropy and enthalpy minima are associated with shell-bulk water structural differences, also observed between low-density liquid and high-density liquid water, around TG∗, specifically, the occurrence of an orientational order maximum.

Improvement in predicting drug sensitivity changes associated with protein mutations using a molecular dynamics based alchemical mutation method

While molecular-targeted drugs have demonstrated strong therapeutic efficacy against diverse diseases such as cancer and infection, the appearance of drug resistance associated with genetic variations in individual patients or pathogens has severely limited their clinical efficacy. Therefore, precision medicine approaches based on the personal genomic background provide promising strategies to enhance the effectiveness of molecular-targeted therapies. However, identifying drug resistance mutations in individuals by combining DNA sequencing and in vitro analyses is generally time consuming and costly. In contrast, in silico computation of protein-drug binding free energies allows for the rapid prediction of drug sensitivity changes associated with specific genetic mutations. Although conventional alchemical free energy computation methods have been used to quantify mutation-induced drug sensitivity changes in some protein targets, these methods are often adversely affected by free energy convergence. In this paper, we demonstrate significant improvements in prediction performance and free energy convergence by employing an alchemical mutation protocol, MutationFEP, which directly estimates binding free energy differences associated with protein mutations in three types of a protein and drug system. The superior performance of MutationFEP appears to be attributable to its more-moderate perturbation scheme. Therefore, this study provides a deeper level of insight into computer-assisted precision medicine.

An Improved Free Energy Perturbation FEP+ Sampling Protocol for Flexible Ligand-Binding Domains

Recent improvements to the free energy perturbation (FEP) calculations, especially FEP+ , established their utility for pharmaceutical lead optimization. Herein, we propose a modified version of the FEP/REST (i.e., replica exchange with solute tempering) sampling protocol, based on detail studies on several targets by probing a large number of perturbations with different sampling schemes. Improved FEP+ binding affinity predictions for regular flexible-loop motions and considerable structural changes can be obtained by extending the prior to REST (pre-REST) sampling time from 0.24 ns/λ to 5 ns/λ and 2 × 10 ns/λ, respectively. With this new protocol, much more precise ∆∆G values of the individual perturbations, including the sign of the transformations and decreased error were obtained. We extended the REST simulations from 5 ns to 8 ns to achieve reasonable free energy convergence. Implementing REST to the entire ligand as opposed to solely the perturbed region, and also some important flexible protein residues (pREST region) in the ligand binding domain (LBD) has considerably improved the FEP+ results in most of the studied cases. Preliminary molecular dynamics (MD) runs were useful for establishing the correct binding mode of the compounds and thus precise alignment for FEP+ . Our improved protocol may further increase the FEP+ accuracy.

Accelerating Convergence of Free Energy Computations with Hamiltonian Simulated Annealing of Solvent (HSAS).

Coupling between binding of a ligand to a receptor and the displacement of a number of bound water molecules is a common event in molecular recognition processes. When the binding site is deeply buried and the exchange of water molecules with the bulk region is difficult to sample, the convergence and accuracy in free energy calculations can be severely compromised. Traditionally, Grand Canonical Monte Carlo (GCMC) based methods have been used to accelerate equilibration of water-at the expense, however, of lengthy trials before a molecular dynamics (MD) simulation. In this paper, a user-friendly and cost-efficient method, Hamiltonian simulated annealing of solvent in combination with λ-exchange of free energy perturbation (FEP) is proposed to accelerate the sampling of water molecules in free energy calculations. As an illustrative example with reliable data from previous GCMC simulations, absolute binding affinity of camphor to cytochrome P450 was calculated. The simulated hydration state change in the buried binding pocket quantitatively agrees with GCMC simulations. It is shown that the new protocol significantly accelerates sampling of water in a buried binding pocket and the convergence of free energy, with negligible setup and computing costs compared to GCMC methods.

Correction to Intermediate Thermodynamic States Contribute Equally to Free Energy Convergence: A Demonstration with Replica Exchange.

ADVERTISEMENT RETURN TO ISSUEPREVErratumNEXTORIGINAL ARTICLEThis notice is a correctionCorrection to Intermediate Thermodynamic States Contribute Equally to Free Energy Convergence: A Demonstration with Replica ExchangeTrung Hai NguyenTrung Hai NguyenMore by Trung Hai Nguyenhttp://orcid.org/0000-0003-1848-3963 and David D. L. Minh*David D. L. MinhMore by David D. L. Minhhttp://orcid.org/0000-0002-4802-2618Cite this: J. Chem. Theory Comput. 2018, 14, 6, 3364Publication Date (Web):May 31, 2018Publication History Published online31 May 2018Published inissue 12 June 2018https://doi.org/10.1021/acs.jctc.8b00459Copyright © 2018 American Chemical SocietyRIGHTS & PERMISSIONSArticle Views478Altmetric-Citations-LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (216 KB) Get e-Alerts Get e-Alerts

Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation

We study a phase-field variational model for the solvaiton of charged molecules with an implicit solvent. The solvation free-energy functional of all phase fields consists of the surface energy, solute excluded volume and solute-solvent van der Waals dispersion energy, and electrostatic free energy. The surface energy is defined by the van der Waals--Cahn--Hilliard functional with squared gradient and a double-well potential. The electrostatic part of free energy is defined through the electrostatic potential governed by the Poisson--Boltzmann equation in which the dielectric coefficient is defined through the underlying phase field. We prove the continuity of the electrostatics---its potential, free energy, and dielectric boundary force---with respect to the perturbation of dielectric boundary. We also prove the $\Gamma$-convergence of the phase-field free-energy functionals to their sharp-interface limit, and the equivalence of the convergence of total free energies to that of all individual parts of free energy. We finally prove the convergence of phase-field forces to their sharp-interface limit. Such forces are defined as the negative first variations of the free-energy functional; and arise from stress tensors. In particular, we obtain the force convergence for the van der Waals--Cahn--Hilliard functionals with minimal assumptions.

Exploring high dimensional free energy landscapes: Temperature accelerated sliced sampling

Biased sampling of collective variables is widely used to accelerate rare events in molecular simulations and to explore free energy surfaces. However, computational efficiency of these methods decreases with increasing number of collective variables, which severely limits the predictive power of the enhanced sampling approaches. Here we propose a method called Temperature Accelerated Sliced Sampling (TASS) that combines temperature accelerated molecular dynamics with umbrella sampling and metadynamics to sample the collective variable space in an efficient manner. The presented method can sample a large number of collective variables and is advantageous for controlled exploration of broad and unbound free energy basins. TASS is also shown to achieve quick free energy convergence and is practically usable with ab initio molecular dynamics techniques.

An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence

We extend the $L^{p}$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of $W$-random graphs.

Intermediate Thermodynamic States Contribute Equally to Free Energy Convergence: A Demonstration with Replica Exchange.

We investigate the relationship between the number of intermediate thermodynamic states along a pathway and the precision of free energy estimates. With a sufficient number of states, the asymptotic variance collapses as a function of the total sample size. Our analytical result is corroborated by replica exchange molecular dynamics simulations of model systems in which the neighbor exchange rate exceeds 35%. Precision collapse is also observed in heat capacity estimates based on the multistate Bennett acceptance ratio. In contrast to the relaxation and mean first-passage times, the autocorrelation time of state indices is found to be relevant to free energy convergence.

Replica State Exchange Metadynamics for Improving the Convergence of Free Energy Estimates

Metadynamics (MTD) is a powerful enhanced sampling method for systems with rugged energy landscapes. It constructs a bias potential in a collective variable (CV) space to overcome barriers between metastable states. In the bias-exchange MTD (BE-MTD), multiple replicas approximate the CV space by exchanging bias potentials (replica conditions). A replica state exchange metadynamics (RSE-MTD) is proposed, using a more sophisticated replica-exchange scheme to improve the convergence of free energy for a given simulation time. Published in Journal of Computational Chemistry 2015, 36, 1446–1455

Thermodynamics of the 2-D Ising Model From a Random Path Sampling Method

We explore the thermodynamics of the 2-D Ising model from a random path sampling method. We show how convergence of free energy and magnetization is obtained in a number of sweeps orders of magnitude below the number of total configurations. The effects of external parameters, such as magnetic field (H) and temperature (T) are accounted in a computationally simple way, allowing very detailed (H, T)-dependent calculations. The effects of more complex interactions can also be considered, particularly magnetovolume coupling, which has proven to be difficult to tackle using traditional approaches. We show its effect on magnetic entropy change (magnetocaloric effect) in this microscopic scenario. As magnetic refrigeration technology matures, the microscopic optimization of magnetovolume effects to enhance the magnetocaloric properties may become a valuable tool in the search of new magnetic refrigerants.

Free-energy molecular simulations of the inclusion complex of Ne with fullerene C60 in water

Binding free energy of Ne in the endohedral fullerene Ne@C60 in water was calculated using closed thermodynamic cycle that involves annihilation in silico of Ne in the free and bound states. The underlying microscopic force field molecular dynamics simulations were performed at 300 K in the constant-volume 27.9 Å periodic box of the TIP3P water molecules. The reference binding free energy of a dummy atom to C60 was calculated from the ideal gas theory equations. The free energies of atom annihilation were computed by means of the free-energy perturbation and Bennett acceptance ratio methods. Introduction of intermediate particle of small size in the mutation scheme in C60 substantially improved free-energy convergence. The predicted binding free energies of Ne and Ne cavity to C60 at the standard state concentration are about − 5 kJ mol− 1 and +22 kJ mol− 1, respectively. The free energy of reversible annihilation of Ne is controlled by the insertion step for simulations both in water and in C60.

Accelerated convergence of molecular free energy via superposition approximation-based reference states.

The free energy of a molecular system can, at least in principle, be computed by thermodynamic perturbation from a reference system whose free energy is known. The convergence of such a calculation depends critically on the conformational overlap between the reference and the physical systems. One approach to defining a suitable reference system is to construct it from the one-dimensional marginal probability distribution functions (PDFs) of internal coordinates observed in a molecular simulation. However, the conformational overlap of this reference system tends to decline steeply with increasing dimensionality, due to the neglect of correlations among the coordinates. Here, we test a reference system that can account for pairwise correlations among the internal coordinates, as captured by their two-dimensional marginal PDFs derived from a molecular simulation. Incorporating pairwise correlations in the reference system is found to dramatically improve the convergence of the free energy estimates relative to the first-order reference system, due to increased conformational overlap with the physical distribution.

On the simulated scaling based free energy simulations: Adaptive optimization of the scaling parameter intervals

Recently, we developed a generalized ensemble based free energy simulation technique, the simulated scaling (SS) method [Li et al., J. Chem. Phys. 126, 024106 (2007)]. In the SS simulations, random walks in the scaling parameter space are realized and free energy values can be conveniently estimated based on trial biasing weights. To improve free energy convergence in the SS simulations, we adopt a recent adaptive algorithm to systematically optimize the scaling parameter intervals; here, the optimization target is the round-trip rate between two end chemical states. As demonstrated in our model studies on the solvation of chloride ion and methane, free energy convergence can be greatly improved when the round-trip rates are accelerated.

On the convergence improvement in the metadynamics simulations: A Wang-Landau recursion approach

As a popular tool in exploring free energy landscapes, the metadynamics method has been widely applied to elucidate various chemical or biochemical processes. As deeply discussed by Laio et al. [J. Phys. Chem. B 109, 6714 (2005)], the size of the updating Gaussian function is pivotal to the free energy convergence toward the target free energy surface. For instance, a greater Gaussian height can facilitate the quick visit of a conformation region of interest; however, it may lead to a larger error of the calculated free energy surface. In contrast, a lower Gaussian height can guarantee a better resolution of the calculated free energy surface; however, it will take longer time for such a simulation to navigate through the defined conformational region. In order to reconcile such confliction, the authors present a method by implementing the Wang-Landau recursion scheme in the metadynamics simulations to adaptively update the height of the unit Gaussian function. As demonstrated in their model studies on both a toy system, and a realistic molecular system treated with the hybrid quantum mechanical and molecular mechanical (QMMM) potential, the present approach can quickly result in more decently converged free energy surfaces, compared with the classical metadynamics simulations employing the fixed Gaussian heights.

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.

Improved convergence of binding affinities with free energy perturbation: application to nonpeptide ligands with pp60src SH2 domain.

Free Energy Perturbations (FEP) in the context of Monte Carlo (MC) simulations were conducted to predict the relative free energies of binding for a series of human Src SH2 domain ligands. Two procedures for disappearing atoms during a single-topology FEP are investigated and dramatic differences in free energy convergence behavior are seen. Comparison of these two protocols suggests that the coupling of the removal of angular constraints with the disappearance of an atom may significantly slow free energy convergence. The series of ligands under investigation here cover a range of modifications at the 3-position of 4-([[4-(cyclohexyl methoxy)benzyl]amino]carbonyl) phenyl phosphate. Unlike any other compound in this study, the 3-amide analog can form two hydrogen bonds within the region of the perturbation, one to a backbone amide hydrogen and one to a highly coordinated water molecule. Agreement with experimental trends in binding affinity is seen, although the computed relative free energy of binding of the amido compound is underestimated. These results are reconciled by examination of the hydration energies of model systems, which predict primary amides as too hydrophilic.

Ab initio protein folding simulations with genetic algorithms: Simulations on the complete sequence of small proteins

Ab-initio folding simulations have been performed on three small proteins using a genetic algorithm- (GA-) based search method which operates on an all atom representation. Simulations were also performed on a number of small peptides expected to be independent folding units. The present genetic algorithm incorporates the results of developments made to the method first tested in CASP1. Additional operators have been introduced into the search in order to allow the simulation of longer sequences and to avoid the simulation of longer sequences and to avoid premature free energy convergence. Secondary structure information derived from a consensus of eight methods and Monte Carlo simulations on sets of homologous sequences has been used to bias the starting populations used in the GA simulations. For the fragment simulations, the results generally have approximately correct local structure, but tend to be too compact, leading to poor RMS error values. One of the three small protein structures has the topology and most of the general organization correct, although many of the details are incorrect.

Ab initio protein folding simulations with genetic algorithms: simulations on the complete sequence of small proteins.

Ab-initio folding simulations have been performed on three small proteins using a genetic algorithm- (GA-) based search method which operates on an all atom representation. Simulations were also performed on a number of small peptides expected to be independent folding units. The present genetic algorithm incorporates the results of developments made to the method first tested in CASP1. Additional operators have been introduced into the search in order to allow the simulation of longer sequences and to avoid the simulation of longer sequences and to avoid premature free energy convergence. Secondary structure information derived from a consensus of eight methods and Monte Carlo simulations on sets of homologous sequences has been used to bias the starting populations used in the GA simulations. For the fragment simulations, the results generally have approximately correct local structure, but tend to be too compact, leading to poor RMS error values. One of the three small protein structures has the topology and most of the general organization correct, although many of the details are incorrect.