Abstract
We propose a mathematical method to analyze the transient change in the probability of gene regulation protein particle ( a repressor protein ) for the first time arrival at the target operator gene region of the DNA chain. The differential equation for the probabilistic behavior of the repressor molecular particle was diffusion type. The probability was expressed by the modified Bessel functions. By applying the Allen's approximation for the modified Bessel function, we obtained an approximation form of the first time arrival probability of the gene regulation protein particle immediately after the onset of the reaction. The impulse response of the probability showed considerable oscillation. The present result indicates the genetic expression particularly at its initial phase is unstable. The present method will be available for evaluating the temporal change in the probability of the gene regulation.
