Free Energy, Enthalpy and Entropy from Implicit Solvent End-Point Simulations

Free energy is the key quantity to describe the thermodynamics of biological systems. In this perspective we consider the calculation of free energy, enthalpy and entropy from end-point molecular dynamics simulations. Since the enthalpy may be calculated as the ensemble average over equilibrated simulation snapshots the difficulties related to free energy calculation are ultimately related to the calculation of the entropy of the system and in particular of the solvent entropy. In the last two decades implicit solvent models have been used to circumvent the problem and to take into account solvent entropy implicitly in the solvation terms. More recently outstanding advancement in both implicit solvent models and in entropy calculations are making the goal of free energy estimation from end-point simulations more feasible than ever before. We review briefly the basic theory and discuss the advancements in light of practical applications. \ua9 2018 Fogolari, Corazza and Esposito

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

Testing the mutual information expansion of entropy with multivariate Gaussian distributions

The mutual information expansion (MIE) represents an approximation of the configurational entropy in terms of low-dimensional integrals. It is frequently employed to compute entropies from simulation data of large systems, such as macromolecules, for which brute-force evaluation of the full configurational integral is intractable. Here, we test the validity of MIE for systems consisting of more than m = 100 degrees of freedom (dofs). The dofs are distributed according to multivariate Gaussian distributions which were generated from protein structures using a variant of the anisotropic network model. For the Gaussian distributions, we have semi-analytical access to the configurational entropy as well as to all contributions of MIE. This allows us to accurately assess the validity of MIE for different situations. We find that MIE diverges for systems containing long-range correlations which means that the error of consecutive MIE approximations grows with the truncation order n for all tractable n ≪ m. This fact implies severe limitations on the applicability of MIE, which are discussed in the article. For systems with correlations that decay exponentially with distance, MIE represents an asymptotic expansion of entropy, where the first successive MIE approximations approach the exact entropy, while MIE also diverges for larger orders. In this case, MIE serves as a useful entropy expansion when truncated up to a specific truncation order which depends on the correlation length of the system

Efficient Calculation of Molecular Configurational Entropies Using an Information Theoretic Approximation

Accurate computation of free energy changes upon molecular binding remains a challenging problem, and changes in configurational entropy are especially difficult due to the potentially large numbers of local minima, anharmonicity, and high-order coupling among degrees of freedom. Here we propose a new method to compute molecular entropies based on the maximum information spanning tree (MIST) approximation that we have previously developed. Estimates of high-order couplings using only low-order terms provide excellent convergence properties, and the theory is also guaranteed to bound the entropy. The theory is presented together with applications to the calculation of the entropies of a variety of small molecules and the binding entropy change for a series of HIV protease inhibitors. The MIST framework developed here is demonstrated to compare favorably with results computed using the related mutual information expansion (MIE) approach, and an analysis of similarities between the methods is presented.National Institutes of Health (U.S.) (GM065418)National Institutes of Health (U.S.) (GM082209)National Science Foundation (U.S.) (0821391

Quantifying the entropy of binding for water molecules in protein cavities by computing correlations.

Protein structural analysis demonstrates that water molecules are commonly found in the internal cavities of proteins. Analysis of experimental data on the entropies of inorganic crystals suggests that the entropic cost of transferring such a water molecule to a protein cavity will not typically be greater than 7.0 cal/mol/K per water molecule, corresponding to a contribution of approximately +2.0 kcal/mol to the free energy. In this study, we employ the statistical mechanical method of inhomogeneous fluid solvation theory to quantify the enthalpic and entropic contributions of individual water molecules in 19 protein cavities across five different proteins. We utilize information theory to develop a rigorous estimate of the total two-particle entropy, yielding a complete framework to calculate hydration free energies. We show that predictions from inhomogeneous fluid solvation theory are in excellent agreement with predictions from free energy perturbation (FEP) and that these predictions are consistent with experimental estimates. However, the results suggest that water molecules in protein cavities containing charged residues may be subject to entropy changes that contribute more than +2.0 kcal/mol to the free energy. In all cases, these unfavorable entropy changes are predicted to be dominated by highly favorable enthalpy changes. These findings are relevant to the study of bridging water molecules at protein-protein interfaces as well as in complexes with cognate ligands and small-molecule inhibitors.Acknowledgments go to the Cambridge High Performance Computing Service for technical assistance, Professor Mike Gilson for helpful discussions, and Professor Bruce Tidor for support. This work was supported by the MRC under grant ML/L007266/1. All calculations were performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http://www.hpc.cam.ac.uk/) provided by Dell, Inc., using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England and were funded by the EPSRC under grants EP/F032773/1 and EP/J017639/1.This is the final published version. It first appeared in Biophysics Journal, Volume 108 February 2015, 928–936 DOI: http://dx.doi.org/10.1016/j.bpj.2014.12.035

Extending fragment-based free energy calculations with library Monte Carlo simulation: Annealing in interaction space

Pre-calculated libraries of molecular fragment configurations have previously been used as a basis for both equilibrium sampling (via "library-based Monte Carlo") and for obtaining absolute free energies using a polymer-growth formalism. Here, we combine the two approaches to extend the size of systems for which free energies can be calculated. We study a series of all-atom poly-alanine systems in a simple dielectric "solvent" and find that precise free energies can be obtained rapidly. For instance, for 12 residues, less than an hour of single-processor is required. The combined approach is formally equivalent to the "annealed importance sampling" algorithm; instead of annealing by decreasing temperature, however, interactions among fragments are gradually added as the molecule is "grown." We discuss implications for future binding affinity calculations in which a ligand is grown into a binding site

The influence of chain connectivity on thermodynamic and structural properties of lattice polymers : integral equation and partition function theories

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