Macromolecular Entropy Can Be Accurately Computed from Force

Abstract

A method is presented to evaluate a molecule's entropy from the atomic forces calculated in a molecular dynamics simulation. Specifically, diagonalization of the mass-weighted force covariance matrix produces eigenvalues which in the harmonic approximation can be related to vibrational frequencies. The harmonic oscillator entropies of each vibrational mode may be summed to give the total entropy. The results for a series of hydrocarbons, dialanine and a β hairpin are found to agree much better with values derived from thermodynamic integration than results calculated using quasiharmonic analysis. Forces are found to follow a harmonic distribution more closely than coordinate displacements and better capture the underlying potential energy surface. The method's accuracy, simplicity, and computational similarity to quasiharmonic analysis, requiring as input force trajectories instead of coordinate trajectories, makes it readily applicable to a wide range of problems.

Evaluation of water displacement energetics in protein binding sites with grid cell theory

“…This was used here to determine the spatial extent of the energetics perturbations in a water network upon modification of the chemical structure of a ligand. Further developments of the GCT formalism would be desirable to account for associated changes in protein and ligand entropy, 45 and to automatically assess the conformational dependence of hydration free energies. Overall the current GCT implementation appears well suited for clarifying the role of water in protein-ligand binding, and applications in combination with for instance alchemical free energy methods, 46 should be envisioned.…”

Entropy of flexible liquids from hierarchical force–torque covariance and coordination

“…The associated harmonic potential may be parametrized using either the Hessian at the energy minimum by normal mode analysis, 24 coordinate covariance by quasiharmonic analysis, 25 or forces. [26][27][28][29][30] An alternative implementation of the harmonic approximation derives entropy from the vibrational density of states from the velocity autocorrelation function. 31,32 It has been shown 33 that the matrix of Fourier transforms of the velocity autocorrelation function is equivalent to the force covariance matrix 29,30 as well as the ensemble average of the Hessian.…”

“…[26][27][28][29][30] An alternative implementation of the harmonic approximation derives entropy from the vibrational density of states from the velocity autocorrelation function. 31,32 It has been shown 33 that the matrix of Fourier transforms of the velocity autocorrelation function is equivalent to the force covariance matrix 29,30 as well as the ensemble average of the Hessian. A notable advantage of harmonic approaches is that quantum effects can be incorporated easily, [34][35][36][37][38] especially important for covalent bonds because of their high frequency.…”

“…The method builds on our earlier work for liquid water, 27,37 organic liquids 71 and single flexible molecules of united atoms. 29 Previously for water, vibrational frequencies had been extracted from force and torque magnitudes 27 or squared forces and squared torques 37 and the number of energy wells had been calculated using a generalized Pauling model [52][53][54][55]72 originally developed for the residual entropy of ice. 73 For single, flexible, united-atom molecules, vibrational frequencies had been extracted from the eigenvalues of the force covariance matrix so as to account for correlations between covalently bonded atoms.…”

“…73 For single, flexible, united-atom molecules, vibrational frequencies had been extracted from the eigenvalues of the force covariance matrix so as to account for correlations between covalently bonded atoms. 29 Here, for liquids of flexible molecules, we treat each molecule as a covalently bonded collection of rigid-body united atoms in the mean-field of its neighbouring molecules. We derive vibrational frequencies from the eigenvalues of a force-torque covariance matrix at the united-atom level and from another force-torque covariance matrix at the molecule level so as to capture flexibility at different length-scales.…”