The Abridgment and Relaxation Time for a Linear Multi-Scale Model Based on Multiple Site Phosphorylation

Random effect in cellular systems is an important topic in systems biology and often simulated with Gillespie's stochastic simulation algorithm (SSA). Abridgment refers to model reduction that approximates a group of reactions by a smaller group with fewer species and reactions. This paper presents a theoretical analysis, based on comparison of the first exit time, for the abridgment on a linear chain reaction model motivated by systems with multiple phosphorylation sites. The analysis shows that if the relaxation time of the fast subsystem is much smaller than the mean firing time of the slow reactions, the abridgment can be applied with little error. This analysis is further verified with numerical experiments for models of bistable switch and oscillations in which linear chain system plays a critical role.