Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Abstract

Schlögl's model is the canonical example of a chemical reaction system that exhibits bistability. Because the biological examples of bistability and switching behaviour are increasingly numerous, this paper presents an integrated deterministic, stochastic and thermodynamic analysis of the model. After a brief review of the deterministic and stochastic modelling frameworks, the concepts of chemical and mathematical detailed balances are discussed and non-equilibrium conditions are shown to be necessary for bistability. Thermodynamic quantities such as the flux, chemical potential and entropy production rate are defined and compared across the two models. In the bistable region, the stochastic model exhibits an exchange of the global stability between the two stable states under changes in the pump parameters and volume size. The stochastic entropy production rate shows a sharp transition that mirrors this exchange. A new hybrid model that includes continuous diffusion and discrete jumps is suggested to deal with the multiscale dynamics of the bistable system. Accurate approximations of the exponentially small eigenvalue associated with the time scale of this switching and the full time-dependent solution are calculated using MATLAB. A breakdown of previously known asymptotic approximations on small volume scales is observed through comparison with these and Monte Carlo results. Finally, in the appendix section is an illustration of how the diffusion approximation of the chemical master equation can fail to represent correctly the mesoscopically interesting steady-state behaviour of the system.

Probability landscape of heritable and robust epigenetic state of lysogeny in phage lambda

“…Deterministic models based on the principles of mass action are often incapable of capturing the multistable nature of the network when copy numbers are small (11). Although the theory of the chemical master equation (CME) provides a general framework for studying stochastic networks (12,13), there are no analytical solutions to the CME except for simple toy problems (14).…”

Stochastic Phenotype Transition of a Single Cell in an Intermediate Region of Gene State Switching

“…Furthermore, the variance of local fluctuations around each stable steady state x * (phenotypic state) can be approximated by 1/ d 2 Φi(x) dx 2 | x=x * [10,20]. One can clearly see from It is indispensable to emphasize that although the diffusion approximation of CME that was always applied [12,15] can also give rise to a landscape function, in the worst-case scenario, it might even reverse the relative stability of the coexisting phenotypic states and give incorrect saddle-crossing rates [10,24].…”

Stochastic transitions in a bistable reaction system on the membrane

“…Simulations of Newtonian hard sphere dynamics provided evidence [21] that in the bistable perfectly stirred system the global attractor is correctly defined by the (stochastic) master equation, while using the Fokker-Planck equation with either linear (additive) or nonlinear (multiplicative) noise may lead to incorrect predictions [9]. Baras et al [21] used the Bird's direct simulation Monte Carlo method [22] to study the chemical kinetics in a homogeneous Boltzmann gas by associating the entire system volume with a single collisional cell.…”

“…In well-mixed reactors, however, the expected time to switch t depends exponentially on the system size, t / expðaVÞ, a . 0, assuming a constant concentration of molecules N/V [9]. The number of reacting molecules in the plasma membrane is of order N ¼ 10 3 to 10 5 [10,11], implying an infinitesimal rate of switching between macroscopic states of activity and inactivity in the well-mixed approximation.…”

“…-The process in the well-mixed reactor can be considered in the deterministic approximation when MFPTs of macroscopic state-to-state transitions are longer than the duration of other processes modifying the system; for instance, the duration of the cell cycle T. The characteristic MFPT t grows exponentially with the size of the well-mixed reactor [9], t ¼ 1 c 0 expðr P VÞ: ð3:1Þ…”