Mean first passage time solution of the Smoluchowski equation: Application to relaxation dynamics in myoglobin

A robust numerical approach to solving the Smoluchowski equation describing a diffusive process is presented for the case where standard procedures are not so useful, in particular, diffusion along a spatially rough potential. The approach developed here makes use of an analytical expression for the mean first passage time for a system to get from one point to another along an arbitrary rough potential, and reduces the solution of the Smoluchowski equation to the solution of a relatively small number of first-order coupled differential equations. The results of this approach are compared with a discrete approximation solution of the Smoluchowski equation as well as with the analytical solution for the special case of a smooth harmonic potential. A significant reduction of computational time is achieved over the discrete approximation method. A model of configurational diffusion along a one-dimensional harmonic potential with coordinate-dependent diffusion coefficient is used to fit the highly nonexponential relaxation dynamics observed in myoglobin following the photodissociation of the bound carbon-monoxide. The relaxation is well described with an effective diffusion coefficient that decreases exponentially along the reaction coordinate. This decrease can arise from either an increase in the roughness of the potential surface or an increase in the friction along the reaction coordinate as the system approaches equilibrium. (C) 2000 American Institute of Physics. [S0021-9606(00)51205-7].