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Gilson MK, Kurtzman T. Free Energy Density of a Fluid and Its Role in Solvation and Binding. J Chem Theory Comput 2024; 20:2871-2887. [PMID: 38536144 PMCID: PMC11197885 DOI: 10.1021/acs.jctc.3c01173] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/10/2024]
Abstract
The concept that a fluid has a position-dependent free energy density appears in the literature but has not been fully developed or accepted. We set this concept on an unambiguous theoretical footing via the following strategy. First, we set forth four desiderata that should be satisfied by any definition of the position-dependent free energy density, f(R), in a system comprising only a fluid and a rigid solute: its volume integral, plus the fixed internal energy of the solute, should be the system free energy; it deviates from its bulk value, fbulk, near a solute but should asymptotically approach fbulk with increasing distance from the solute; it should go to zero where the solvent density goes to zero; and it should be well-defined in the most general case of a fluid made up of flexible molecules with an arbitrary interaction potential. Second, we use statistical thermodynamics to formulate a definition of the free energy density that satisfies these desiderata. Third, we show how any free energy density satisfying the desiderata may be used to analyze molecular processes in solution. In particular, because the spatial integral of f(R) equals the free energy of the system, it can be used to compute free energy changes that result from the rearrangement of solutes as well as the forces exerted on the solutes by the solvent. This enables the use of a thermodynamic analysis of water in protein binding sites to inform ligand design. Finally, we discuss related literature and address published concerns regarding the thermodynamic plausibility of a position-dependent free energy density. The theory presented here has applications in theoretical and computational chemistry and may be further generalizable beyond fluids, such as to solids and macromolecules.
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Affiliation(s)
- Michael K Gilson
- Skaggs School of Pharmacy and Pharmaceutical Sciences, and Department of Chemistry and Biochemistry, UC San Diego, La Jolla, CA, 92093, USA
| | - Tom Kurtzman
- PhD Programs in Chemistry, Biochemistry, and Biology, The Graduate Center of the City University of New York, New York, 10016, USA; Department of Chemistry, Lehman College, The City University of New York, Bronx, New York, 10468, USA
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Heinz L, Grubmüller H. Why Solvent Response Contributions to Solvation Free Energies Are Compatible with Ben-Naim's Theorem. J Chem Theory Comput 2023; 19:8013-8019. [PMID: 37948626 PMCID: PMC10687852 DOI: 10.1021/acs.jctc.3c00655] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2023] [Revised: 10/19/2023] [Accepted: 10/23/2023] [Indexed: 11/12/2023]
Abstract
We resolve a seeming paradox arising from a common misinterpretation of Ben-Naim's theorem, which rests on the decomposition of the Hamiltonian of a molecular solute/solvent system into solute-solvent and solvent-solvent interactions. According to this theorem, the solvation entropy can also be decomposed into a solute-solvent term and a remaining solvent-solvent term that is commonly referred to as the solvent reorganization term. Crucially, the latter equals the average solvent-solvent interaction energy such that these two solvent-solvent terms cancel and thus do not change the total solvation free energy. This analytical result implies that changes in the solvent-solvent interactions cannot contribute to any thermodynamic driving force. The solvent reorganization term is often identified with the contribution of many-body solvent correlations to the solvation entropy, which seems to imply that these correlations, too, cannot contribute to solvation. However, recent calculations based on atomistic simulations of a solvated globular protein and spatially resolved mutual information expansions revealed substantial contributions of many-body solvent correlations to the solvation free energy, which are not canceled by the enthalpy change of the solvent. Here, we resolved this seeming contradiction and illustrate by two examples─a simple Ising model and a solvated Lennard-Jones particle─that the solvent reorganization entropy and the actual entropy contribution arising from many-body solvent correlations differ both conceptually and numerically. Whereas the solvent reorganization entropy in fact arises from both solvent-solvent as well as solute-solvent interactions and thus has no straightforward intuitive interpretation, the mutual information expansion permits an interpretation in terms of the entropy contribution of solvent-solvent correlations to the solvation free energy.
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Affiliation(s)
| | - Helmut Grubmüller
- Department of Theoretical
and Computational Biophysics, Max-Planck
Institute for Multidisciplinary Sciences, 37077 Göttingen, Germany
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Jwo DJ, Cho TS, Biswal A. Geometric Insights into the Multivariate Gaussian Distribution and Its Entropy and Mutual Information. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1177. [PMID: 37628206 PMCID: PMC10453792 DOI: 10.3390/e25081177] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2023] [Revised: 07/31/2023] [Accepted: 08/04/2023] [Indexed: 08/27/2023]
Abstract
In this paper, we provide geometric insights with visualization into the multivariate Gaussian distribution and its entropy and mutual information. In order to develop the multivariate Gaussian distribution with entropy and mutual information, several significant methodologies are presented through the discussion, supported by illustrations, both technically and statistically. The paper examines broad measurements of structure for the Gaussian distributions, which show that they can be described in terms of the information theory between the given covariance matrix and correlated random variables (in terms of relative entropy). The content obtained allows readers to better perceive concepts, comprehend techniques, and properly execute software programs for future study on the topic's science and implementations. It also helps readers grasp the themes' fundamental concepts to study the application of multivariate sets of data in Gaussian distribution. The simulation results also convey the behavior of different elliptical interpretations based on the multivariate Gaussian distribution with entropy for real-world applications in our daily lives, including information coding, nonlinear signal detection, etc. Involving the relative entropy and mutual information as well as the potential correlated covariance analysis, a wide range of information is addressed, including basic application concerns as well as clinical diagnostics to detect the multi-disease effects.
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Affiliation(s)
- Dah-Jing Jwo
- Department of Communications, Navigation and Control Engineering, National Taiwan Ocean University, 2 Peining Rd., Keelung 202301, Taiwan;
| | - Ta-Shun Cho
- Department of Business Administration, Asia University, 500 Liufeng Road, Wufeng, Taichung 41354, Taiwan;
| | - Amita Biswal
- Department of Communications, Navigation and Control Engineering, National Taiwan Ocean University, 2 Peining Rd., Keelung 202301, Taiwan;
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Khade P, Jernigan RL. Entropies Derived from the Packing Geometries within a Single Protein Structure. ACS OMEGA 2022; 7:20719-20730. [PMID: 35755337 PMCID: PMC9219053 DOI: 10.1021/acsomega.2c00999] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/18/2022] [Accepted: 05/17/2022] [Indexed: 05/17/2023]
Abstract
A fast, simple, yet robust method to calculate protein entropy from a single protein structure is presented here. The focus is on the atomic packing details, which are calculated by combining Voronoi diagrams and Delaunay tessellations. Even though the method is simple, the entropies computed exhibit an extremely high correlation with the entropies previously derived by other methods based on quasi-harmonic motions, quantum mechanics, and molecular dynamics simulations. These packing-based entropies account directly for the local freedom and provide entropy for any individual protein structure that could be used to compute free energies directly during simulations for the generation of more reliable trajectories and also for better evaluations of modeled protein structures. Physico-chemical properties of amino acids are compared with these packing entropies to uncover the relationships with the entropies of different residue types. A public packing entropy web server is provided at packing-entropy.bb.iastate.edu, and the application programing interface is available within the PACKMAN (https://github.com/Pranavkhade/PACKMAN) package.
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Chakravorty A, Higham J, Henchman RH. Entropy of Proteins Using Multiscale Cell Correlation. J Chem Inf Model 2020; 60:5540-5551. [PMID: 32955869 DOI: 10.1021/acs.jcim.0c00611] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
A new multiscale method is presented to calculate the entropy of proteins from molecular dynamics simulations. Termed Multiscale Cell Correlation (MCC), the method decomposes the protein into sets of rigid-body units based on their covalent-bond connectivity at three levels of hierarchy: molecule, residue, and united atom. It evaluates the vibrational and topographical entropy from forces, torques, and dihedrals at each level, taking into account correlations between sets of constituent units that together make up a larger unit at the coarser length scale. MCC gives entropies in close agreement with normal-mode analysis and smaller than those using quasiharmonic analysis as well as providing much faster convergence. Moreover, MCC provides an insightful decomposition of entropy at each length scale and for each type of amino acid according to their solvent exposure and whether they are terminal residues. While the residue entropy depends weakly on solvent exposure, there is greater variation in entropy components for larger, more polar amino acids, which have increased conformational entropy but reduced vibrational entropy with greater solvent exposure.
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Affiliation(s)
- Arghya Chakravorty
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Jonathan Higham
- MRC Human Genetics Unit, Institute of Genetics & Molecular Medicine, The University of Edinburgh, Western General Hospital, Crewe Road South, Edinburgh EH4 2XU, United Kingdom
| | - Richard H Henchman
- Manchester Institute of Biotechnology, The University of Manchester, 131 Princess Street, Manchester M1 7DN, United Kingdom.,Department of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
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Heinz LP, Grubmüller H. Computing Spatially Resolved Rotational Hydration Entropies from Atomistic Simulations. J Chem Theory Comput 2019; 16:108-118. [DOI: 10.1021/acs.jctc.9b00926] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Affiliation(s)
- Leonard P. Heinz
- Department of Theoretical and Computational Biophysics, Max-Planck Institute for Biophysical Chemistry, 37077 Göttingen, Germany
| | - Helmut Grubmüller
- Department of Theoretical and Computational Biophysics, Max-Planck Institute for Biophysical Chemistry, 37077 Göttingen, Germany
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Ali HS, Higham J, Henchman RH. Entropy of Simulated Liquids Using Multiscale Cell Correlation. ENTROPY 2019; 21:e21080750. [PMID: 33267464 PMCID: PMC7515279 DOI: 10.3390/e21080750] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/29/2019] [Revised: 07/22/2019] [Accepted: 07/28/2019] [Indexed: 12/16/2022]
Abstract
Accurately calculating the entropy of liquids is an important goal, given that many processes take place in the liquid phase. Of almost equal importance is understanding the values obtained. However, there are few methods that can calculate the entropy of such systems, and fewer still to make sense of the values obtained. We present our multiscale cell correlation (MCC) method to calculate the entropy of liquids from molecular dynamics simulations. The method uses forces and torques at the molecule and united-atom levels and probability distributions of molecular coordinations and conformations. The main differences with previous work are the consistent treatment of the mean-field cell approximation to the approriate degrees of freedom, the separation of the force and torque covariance matrices, and the inclusion of conformation correlation for molecules with multiple dihedrals. MCC is applied to a broader set of 56 important industrial liquids modeled using the Generalized AMBER Force Field (GAFF) and Optimized Potentials for Liquid Simulations (OPLS) force fields with 1.14*CM1A charges. Unsigned errors versus experimental entropies are 8.7 J K - 1 mol - 1 for GAFF and 9.8 J K - 1 mol - 1 for OPLS. This is significantly better than the 2-Phase Thermodynamics method for the subset of molecules in common, which is the only other method that has been applied to such systems. MCC makes clear why the entropy has the value it does by providing a decomposition in terms of translational and rotational vibrational entropy and topographical entropy at the molecular and united-atom levels.
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Affiliation(s)
- Hafiz Saqib Ali
- Manchester Institute of Biotechnology, The University of Manchester, 131 Princess Street, Manchester M1 7DN, UK
- School of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
| | - Jonathan Higham
- Manchester Institute of Biotechnology, The University of Manchester, 131 Princess Street, Manchester M1 7DN, UK
- School of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
| | - Richard H. Henchman
- Manchester Institute of Biotechnology, The University of Manchester, 131 Princess Street, Manchester M1 7DN, UK
- School of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
- Correspondence: ; Tel.: +44-161-306-5194
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