Finding Acceptable Parameter Regions of Stochastic Hill functions for Multisite Phosphorylation Mechanism

Multisite phosphorylation plays an important role in regulating switchlike protein activity and has been used widely in mathematical models. With the development of new experimental techniques and more molecular data, molecular phosphorylation processes emerge in many systems with increasing complexity and sizes. These developments call for simple yet valid stochastic models to describe various multisite phosphorylation processes, especially in large and complex biochemical networks. To reduce model complexity, this work aims to simplify the multisite phosphorylation mechanism by a stochastic Hill function model. Further, this work optimizes regions of parameter space to match simulation results from the stochastic Hill function with the distributive multisite phosphorylation process. While traditional parameter optimization methods have been focusing on finding the best parameter vector, in most circumstances modelers would like to find a set of parameter vectors that generate similar system dynamics and results. This paper proposes a general $\alpha$-$\beta$-$\gamma$ rule to return an acceptable parameter region of the stochastic Hill function based on a quasi-Newton stochastic optimization (QNSTOP) algorithm. Different objective functions are investigated characterizing different features of the simulation-based empirical data, among which the approximate maximum log-likelihood method is recommended for general applications. Numerical results demonstrate that with an appropriate parameter vector value, the stochastic Hill function model depicts the multisite phosphorylation process well except the initial (transient) period.