2D Entropy of Discrete Molecular Ensembles

A method is presented for the estimation of the conformational entropy of discrete macromolecular ensembles associated with multiple rotameric dihedral angle states. A covariance matrix is constructed of all mobile dihedral angles, which are represented as complex numbers on the unit circle, and subjected to a principal component analysis. The total entropy is decomposed into additive contributions from each eigenmode, for which a 2D entropy is computed after convolution of the projection coefficients of the conformer ensemble for that mode with a 2D Gaussian function. The method is tested for ensembles of linear polymer chains for which the exact conformational entropies are known. These include chains with up to 15 dihedral angles exhibiting two or three rotamers per dihedral angle. The performance of the method is tested for molecular ensembles that exhibit various forms of correlation effects, such as ensembles with mutually exclusive combinations of rotamers, ensembles with conformer populations biased toward compact conformers, ensembles with Gaussian distributed pairwise rotamer energies, and ensembles with electrostatic intramolecular interactions. For all these ensembles, the method generally provides good estimates for the exact conformational entropy. The method is applied to a protein molecular dynamics simulation to assess the effect of side-chain-backbone and side-chain-side-chain correlations on the conformational entropy.