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 ($β$) 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 $Φ(x,V)$, $k\propto\exp\{-VΔΦ^‡(V)\}$, can be understood from a decomposition $ΔΦ^‡(V) \approxΔϕ_0^‡ Δϕ_1^‡/V$. Nonlinear bistability means $Δϕ_0^‡>0$ while stochastic bistability has $Δϕ_0^‡<0$ but $Δϕ_1^‡>0$. Stochastic bistabilities can be viewed as remants (or "ghosts) of nonlinear bifurcations or extinction phenomenon, and $Δϕ_0^‡$ and $Δϕ_1^‡$ as "enthalpic" and "entropic" barriers to a transition.

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