Potential for regulatory genetic networks of gene expression near a stable point

A description for regulatory genetic network based on generalized potential energy is constructed. The potential energy is derived from the steady state solution of linearized Fokker-Plank equation, and the result is shown to be equivalent to the system of coupled oscillators. The correspondence between the quantities from the mechanical picture and the steady-state fluctuations is established. Explicit calculation is given for auto-regulatory networks in which, the force constant associated with the degree of protein is very weak. Negative feedback not only suppresses the fluctuations but also increases the steepness of the potential. The results for the fluctuations agree completely with those obtained from linear noise Fokker-Planck equation.