Dependency of control parameters on a rate coefficient giving the potential curvature in a reaction cascade model

Many chemical reactions in vivo are self-controlled by fluxes of chemical energy and matter through biological systems, so the induction of such reactions can be governed by changes in the control parameters of the rate equation. A potential of a system is assumed to be given by Gibbs' functionG(T, P, x), which is continuously differentiable, and the rate equation can be derived from the differential (–G/x) of Taylor's expansion ofG_(T,P)(x) for the order parameterx, which corresponds to the product number, at around the critical pointC(T_C, P_C). The equation is described bydx/dt=(x)–k₁x–k₂x³, andk₂>0. In this equation,k₁ andk₂ are functions of the control parameters, temperatureT and pressureP, andk₁ is allowed to have a positive or negative values as (T, P). Thenk₁ is an important factor that decides the induction conditions of the reactions with a phase transition in the steady statex=0. Because bothk₁ (the transition parameter) andG are the quantity of state, they are given by the total differential, and functions that decideG andk₁ are related to a mutual inverse function. From the above relation, the rate of change ink₁ by G, which corresponds to the reaction energy of the system, is uniquely determined by a function ofk₁, [f(k₁^±)] andf(k₁^±) is described approximately by ±₁k₁^± in the transient process thatk₁ approaches zero, where ₁ implies 1/RT. These results indicate that internal driving forces caused by a stimulus in a system are proportional tok₁^± and that the system is regulated by competition of the forces. an approximate function fork₁ in the transient process is described by tanh (G/RT) and Arrhenius' law is elucidated from this theory.Decreased January 19, 1992