Approximation of a multi-scale model based on multiple site phosphorylation

The increasing complexity of the modeling of the budding yeast cell cycle control system demands development of efficient stochastic simulation algorithms. Specifically, models with multi-scale and multi-state features present great difficulties in simulation efficiency. The coexistence of the very fast and very slow reactions in one model usually leads to great challenges to the Stochastic Simulation Algorithm (SSA) in terms of both accuracy and efficiency. The Stochastic Quasi-Steady State Assumption (SQSSA) and the hybrid simulation method, which combines the ODE solver and SSA, are two commonly used approximation methods for stochastic simulation. In this paper we consider biological models based on multiple site phosphorylation. We focus on the criterion of how to select the best method to simulate systems with multi-scale and multi-site features. We propose an efficient method to approximate the relaxation time of the subsystem that is supposed to be on the fast time scale. Then we discuss the best scenario for each method to be selected using the proposed approximated relaxation time. At last, we apply the SQSSA and the hybrid method to a bistable switch model based on multiple site phosphorylation, and compare the accuracy and efficiency of the two methods with the exact SSA via numerical experiments.