Distance-Based Configurational Entropy of Proteins from Molecular Dynamics Simulations

Abstract

Estimation of configurational entropy from molecular dynamics trajectories is a difficult task which is often performed using quasi-harmonic or histogram analysis. An entirely different approach, proposed recently, estimates local density distribution around each conformational sample by measuring the distance from its nearest neighbors. In this work we show this theoretically well grounded the method can be easily applied to estimate the entropy from conformational sampling. We consider a set of systems that are representative of important biomolecular processes. In particular: reference entropies for amino acids in unfolded proteins are obtained from a database of residues not participating in secondary structure elements;the conformational entropy of folding of β2-microglobulin is computed from molecular dynamics simulations using reference entropies for the unfolded state;backbone conformational entropy is computed from molecular dynamics simulations of four different states of the EPAC protein and compared with order parameters (often used as a measure of entropy);the conformational and rototranslational entropy of binding is computed from simulations of 20 tripeptides bound to the peptide binding protein OppA and of β2-microglobulin bound to a citrate coated gold surface. This work shows the potential of the method in the most representative biological processes involving proteins, and provides a valuable alternative, principally in the shown cases, where other approaches are problematic.

Fig 1. Entropy for residues in unfolded proteins calculated according to the k ^th nearest neighbor vs. the average distance of the k ^th nearest neighbor.

Distances are increasing with k. The curves are relative to alanine (continuous line), valine (dotted line), isoleucine (short dashed line), tryptophane (long dashed line) and lysine (dot dashed line).

Fig 2. Entropy versus residue number from 30 ns β2-microglobulin molecular dynamics simulation.

Residue computed entropy (dashed line), summed entropy corresponding to residue neighborhood (continuous line). In the low part of the figure residues with defined secondary structure are indicated by black boxes. Black dots with error bars represent experimentally determined entropy from Bluu-Tramp hydrogen-deuterium exchange NMR experiments [3]. The latter experiments measure the enthalpy and entropy of the process (partial or global unfolding) which exposes protein amide hydrogens to solvent.

Fig 3. Entropy versus residue number from 30 ns β2-microglobulin molecular dynamics simulation (thick line) and from the first and second half of the simulation (thin lines).

Fig 4. HN vector order parameter versus the computed backbone entropy on all four EPAC MD trajectories.

The single simlations show some differences, but deviations from a sigmoidal behaviour is found for all four simulations.

Fig 5. Residue entropy computed for all four EPAC MD simulations.

On the x-axis the entropy is computed using all residue torsion angles, on the y-axis the entropy is fitted as a linear combination of backbone entropy, the number of sidechain degrees of freedom and the solvent accessible surface area of the residue. Glycine, Alanine and Proline residues that do not possess sidechain torsion angles, are not reported in the figure.

Fig 6. Graphical representation of how the rotation matrix and translation vector describing the rototranslational state of the tripeptide ligand with respect to OppA is obtained.

First, overall rototranslation of the complex is removed by superposition of frame residues. Second, the central residue N, CA and C atoms are superimposed on the reference structure. The resulting rotation matrix and translation vector describe the rototranslational state of the tripeptide ligand.