(416h) Stability Analysis of Stochastic SchlöGl Model

The behavior of chemical reaction networks is influenced by the number of molecules and the size of the system. When the number of molecules is asymptotically reaching large values, the system is in the macroscopic (thermodynamic) limit. In such cases, a deterministic modeling formalism is more suitable to describe the system. Away from it, at the microscopic limit, a stochastic approach is more suitable since the system’s size is small and thermal molecular fluctuations become significant [1].

We have previously discussed the lack of knowledge of the behavior of networks in the mesoscopic area, where the system is sizable yet still under the influence of thermal noise [2]. This lack is largely due to numerical and computational difficulties in stochastically modeling mesoscopic system [2]. We have also provided insight into how these issues can be resolved with the help of ZI-closure scheme method [3]. This is a numerical method to solve chemical master equations by maximizing the entropy of the reacting system.

We employ the Schlögl model reaction network, as a general example of bistable systems. A wide variety of system sizes is studied with ZI-closure. In the cases that the system has one stable attractor, the stochastic system recovers the deterministic behavior for large system sizes. However, the stochastic mesoscopic behavior only partially matches the deterministic solution in the region of kinetic parameter values that result in two stable deterministic attractors. In such cases, the stochastic system recovers only one of the deterministic attractors in the mesoscopic limit.

Stability analysis of the stochastic system provides a better insight into the system’s behavior. Eigenvalues of the stochastic system are calculated through the Jacobian matrix of the linearized system around the steady state [4]. The correlation between the eigenvalues of the stochastic system at the mesoscopic limit and the ones of the deterministic formalism reveals the underlying stability dynamics of the system. The stochastic formalism can distinguish between attractors and thus retrieves the more stable one. In contrast, the deterministic formalism finds all the stable attractors indistinguishably.

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. Vlysidis M, Kaznessis YN "The Mesoscopic Behavior of Stochastic Schlögl Model", 2016 AIChE Annual Meeting, San Francisco, CA
3. 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
4. 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