Stochastic diffusion control for gene-regulating protein particles

Abstract

We introduce a precise analytical method for computing the temporal changes in concentration and fluxes of gene-regulating repressor protein particles. The temporal changes in repressor particle concentration is described by an integral-differential diffusion equation in cylindrical coordinates. The equation consists of the memory-less first-time-arrival probability and the integration of the return probability of the particles to the operator region of the DNA. By using the Laplace transformation, we could derive analytical forms of the temporal changes in concentration and flux in a radial direction, the total flux, and the first-time-arrival probability. We also computed the impulse responses of the first-time-arrival probability of the repressor to the sink. The computed diffusion of the repressor particles decreased rapidly from the onset of the reaction. As the diffusion constant in the medium around the DNA increased, the first-time-arrival probability, the diffusion, and the flux of the particles decreased, while the total flux into the target sink increased. As the chemical factor became predominantly a diffusion factor, the first-time-arrival probability, the diffusion, and the flux of the particles decreased. As the dissociation rate of the particles increased, the flux into the sink increased. The number of dissociated particles was significantly more influenced by the chemical factor than by the diffusion. The first-time-arrival probability oscillated significantly at the onest of the reaction. When this method has been extended, it will be available for predicing genetic expression and creating artificial life.