THEORETICAL STUDIES ON THE DYNAMIC ASPECTS OF MACROMOLECULAR FUNCTION

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In this reporting period significant progress that lead to seven publications was made in the following broad areas: (1) diffusion influenced reactions (2) the calculation of free energy differences from computer simulations, and (3) the analysis of NMR relaxation data. Two projects are briefly described below. The kinetics of the irreversible diffusion-influenced reactions between a protein and a ligand is studied when the reactivity is stochasitically gated due to conformations fluctuations of one of the species. If gating is due to the ligand, we show that the Smoluchowski rate equations can be generalized by simply using a stochastically-gated time-dependent rate coefficient,. However, if gating is due to the protein, this is no longer true, except when the gating dynamics is sufficiently fast or the ligand concentration is very low. The dynamics of all the ligands around a protein become correlated even when they diffuse independently. An approximate theory for the kinetics of protein-gated reactions that is exact in both the fast and slow gating limits is developed. In order to test this theory, a Brownian dynamics simulation algorithm based on a path- integral formulation is introduced to calculate the time dependence of the protein concentration. Illustrative simulations using a simple model are carried out for a variety of gating rates. The results are in good agreement with the approximate theory. A class of simple expressions of increasing accuracy for the free- energy difference between two states is derived based on numerical thermodynamic integration. The implementation of these formulas requires simulations of the initial and final (and possibly a few intermediate) states. They involve higher free-energy derivatives at these states which are related to the moments of the probability distribution of the perturbations. Given a specified number of such derivatives, these integrations formulas are optimal in the sense that they are exact to the highest possible order of free-energy perturbation theory. The utility of this approach is illustrated for the hydration free energy of water. This problem provides a quire stringent test because the free energy is a highly nonlinear function of the charge so that even fourth order perturbation theory gives a very poor estimate of the free-energy change. Our results should prove most useful for complex, computationally demanding problems where free-energy differences arise primarily from changes in the electrostatic interactions (e.g., electron transfer, charging of ions, protonation of amino acids in proteins).