Diffusion of a protein in configuration space

¹Theoretical Biology and Biophysics Group, T10, MS K710 ²Center for Nonlinear Studies and ³CIC-3, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, U.S.A.

Proceedings of the Ninth Conversation in Biomolecular Stereodynamics, edited by R. H. Sarma and M. H. Sarma (Adenine Press, Schenectady, NY, 1996) pp. 267-280. [LA-UR 95-2162]

Simulations of biomolecular dynamics are commonly interpreted in terms of harmonic or quasi-harmonic models for the dynamics of the system. These models assume that biomolecules exhibit oscillations around a single energy minimum. However, spectroscopic data on myoglobin suggest that proteins sample multiple minima. Transitions between minima reveal a broad distribution of energy barriers. This behavior has been observed in other biomolecular systems.

To elucidate the nature of protein dynamics we have studied a 1.2 ns molecular dynamics trajectory of crambin in aqueous solution. This trajectory samples multiple local energy minima. Transitions among minima involve collective motions of amino acids over long distances. We show that nonlinear motions are responsible for most of the atomic fluctuations of the protein. These atomic fluctuations are not well described by large motions of individual atoms or a small group of atoms, but rather by concerted motions of many atoms. These nonlinear motions describe transitions between different basins of attraction. The signature of these motions manifests in local and global structural variables.

A method for extracting Molecule Optimal Dynamic Coordinates (MODC) is presented. A generalization of this method is used to identify small (1-3) dimensional subspaces of the configuration space that describe the dynamics of the protein within the context of nonlinear, multi-basin dynamics.

We present a model for describing the dynamics of biomolecules in terms of an open Newtonian system (the protein) coupled to a stochastic system (solvent). Autocorrelation functions of the displacements along relevant MODC directions show that the protein loses memory of its configuration within a few picoseconds. The diffusion of the protein in configuration space is anomalous, namely, the time dependence of the mean square displacement is not proportional to time, but to t^2HD where 2HD is a nontrivial fractional exponent. Therefore, transitions among energy minima far apart in configuration space exhibit a stretched-exponential time dependence, scaling as t^-2HDexp(-t^-2HD), with HD < 0.5. This picture is consistent with a model suggested by Frauenfelder and collaborators to explain multiple timescale relaxation processes observed in myoglobin.