Harmonicity and anharmonicity in protein dynamics: a normal mode analysis and principal component analysis

A comparison of a normal mode analysis and principal component analysis of a 200-ps molecular dynamics trajectory of bovine pancreatic trypsin inhibitor in vacuum has been made in order to further elucidate the harmonic and anharmonic aspects in the dynamics of proteins. An anharmonicity factor is defined which measures the degree of anharmonicity in the modes, be they principal modes or normal modes, and it is shown that the principal mode system naturally divides into anharmonic modes with peak frequencies below 80 cm-1, and harmonic modes with frequencies above this value. In general the larger the mean-square fluctuation of a principal mode, the greater the degree of anharmonicity in its motion. The anharmonic modes represent only 12% of the total number of variables, but account for 98% of the total mean-square fluctuation. The transitional nature of the anharmonic motion is demonstrated. The results strongly suggest that in a large subspace, the free energy surface, as probed by the simulation, is approximated by a multi-dimensional parabola which is just a rescaled version of the parabola corresponding to the harmonic approximation to the conformational energy surface at a single minimum. After 200 ps, the rescaling factor, termed the "normal mode rescaling factor," has apparently converged to a value whereby the mean-square fluctuation within the subspace is about twice that predicted by the normal mode analysis.