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

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 ($\beta$) 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 $\Phi(x,V)$, $k\propto\exp\{-V\Delta\Phi^{\ddag}(V)\}$, can be understood from a decomposition $\Delta\Phi^{\ddag}(V) \approx\Delta\phi_0^{\ddag} \Delta\phi_1^{\ddag}/V$. Nonlinear bistability means $\Delta\phi_0^{\ddag}>0$ while stochastic bistability has $\Delta\phi_0^{\ddag}<0$ but $\Delta\phi_1^{\ddag}>0$. Stochastic bistabilities can be viewed as remants (or "ghosts) of nonlinear bifurcations or extinction phenomenon, and $\Delta\phi_0^{\ddag}$ and $\Delta\phi_1^{\ddag}$ as "enthalpic" and "entropic" barriers to a transition.