The Mesoscopic Behavior of Stochastic SchlöGl Model

An important concept for describing chemical reaction networks is the thermodynamic limit, where the number of molecules and the size of the system are asymptotically large. Systems in the thermodynamic (macroscopic) limit can be modeled with a deterministic modeling formalism; away from it, at the microscopic limit, a stochastic approach is more suitable [1]. However, due to numerical and computational difficulties, little is known for the behavior of networks in the mesoscopic area, where the system is sizable yet still under the influence of thermal noise.

We attempt to provide useful information about the mesoscopic behavior of chemical reaction networks with the help of ZI-closure scheme method [2]. This is a numerical solution of chemical master equations, which utilizes the maximization of the system entropy. For our analysis, we study the Schlögl model reaction network, a simple single component system that can exhibit bistability.

The closure scheme makes it possible to investigate the behavior of the system for a wide variety of volumes. Notably, the stochastic mesoscopic behavior only partially matches the deterministic solution; in the region of kinetic parameter values that the deterministic system exhibits two attractors, the stochastic system may retrieve only one of them in the mesoscopic limit. The stochastic formalism distinguishes between the attractors of the system and recovers the more stable one, whereas the deterministic system cannot distinguish between the two attractors. An investigation of the passage time of the system gives a better insight into this inability of the deterministic model.

A look at the dependence of the system entropy on the kinetic constants provides an important insight into the mesoscopic system behavior. The entropy of the system has a global maximum with respect to the kinetic constants of the system. The analysis also reveals a first order phase transition and a critical point of the system with respect the kinetic parameters. Results suggest the coexistence of two, appropriately normalized delta-like functions at the thermodynamic limit of this critical point. Such behavior is also supported by a basic stability analysis [3]. To our knowledge this is the first time such an analysis is presented for the transition between stochastic and deterministic chemical reaction models.

1. Constantino PH, Vlysidis M, Smadbeck P, Kaznessis YN, " Modeling stochasticity in biochemical reaction networks", J Phys D: Appl Phys 2016 Feb 1; 49(9): 093001 10.1088/0022-3727/49/9/093001
2. Smadbeck P, Kaznessis YN, "A Closure Scheme for Chemical Master Equations", Proc Natl Acad Sci U S A. 2013 Aug 27; 110(35): 14261-5 10.1073/pnas.1306481110
3. Smadbeck P, Kaznessis YN, "On a theory of stability for nonlinear stochastic chemical reaction networks", J Chem Phys 2015 May 8; 142: 184101 10.1063/1.4919834