van't Hoff-Arrhenius Analysis of Mesoscopic and Macroscopic Dynamics of Simple Biochemical Systems: Stochastic vs. Nonlinear Bistabilities

Abstract

Multistability of mesoscopic, driven biochemical reaction systems has implications to a wide range of cellular processes. Using several simple models, we show that one class of bistable chemical systems has a deterministic counterpart in the nonlinear dynamics based on the Law of Mass Action, while another class, widely known as noise-induced stochastic bistability, does not. Observing the system's volume ($V$) playing a similar role as the inverse temperature (${\hat{I}}^2$) in classical rate theory, an van't Hoff-Arrhenius like analysis is introduced. In one-dimensional systems, a transition rate between two states, represented in terms of a barrier in the landscape for the dynamics $\hat{I}¦(x,V)$, $k\propto \exp \{-V\hat{I}”\hat{I}{¦}^{\hat{a}}€¡(V)\}$, can be understood from a decomposition $\hat{I}”\hat{I}{¦}^{\hat{a}}€¡(V)\approx \hat{I}”\ddot{I}{•}_0^{\hat{a}}€¡\hat{I}”\ddot{I}{•}_1^{\hat{a}}€¡/V$. Nonlinear bistability means $\hat{I}”\ddot{I}{•}_0^{\hat{a}}€¡>0$ while stochastic bistability has $\hat{I}”\ddot{I}{•}_0^{\hat{a}}€¡<0$ but $\hat{I}”\ddot{I}{•}_1^{\hat{a}}€¡>0$. Stochastic bistabilities can be viewed as remants (or "ghosts) of nonlinear bifurcations or extinction phenomenon, and $\hat{I}”\ddot{I}{•}_0^{\hat{a}}€¡$ and $\hat{I}”\ddot{I}{•}_1^{\hat{a}}€¡$ as "enthalpic" and "entropic" barriers to a transition.