Rotational Probability Flux from Discrete Differential Forms of Stochastic Reaction Kinetics in High Dimensional Reaction Networks

Abstract

Biochemical reaction networks often display different behaviors and exhibit different phenotypes. In non-equilibrium networks, switching between different phenotypes can occur spontaneously under certain conditions. While there have been significant efforts towards understanding the mechanism of dynamic phenotype switching, a full understanding in many cases remains lacking. Calculation of the rotational probability flux can help elucidating the mechanism of dynamic switching. However, when the copy numbers of molecular species are small, the assumption of a system with continuous state is invalid, and the calculation of rotational probability flux becomes difficult. To the best of our knowledge, rotational probability flux has not been formulated in discrete state space. Here, we first develop a theoretical framework of discrete differential forms for stochastic reaction kinetics. We then introduce the concept of discrete rotational probability flux in high dimensional reaction networks and show how it can be computed. Results of discrete rotational probability flux in the toggle switch network are presented and compared with the amount of entropy production to facilitate understanding of dynamic phenotype switching.