Error bound for hybrid models of two-scaled stochastic reaction systems

Abstract

Biochemical reaction systems are often modeled by a Markov jump process in order to account for the discreteness of the populations and the stochastic nature of their evolution. The associated time-dependent probability distribution is the solution of the Chemical Master Equation (CME), but solving the CME numerically is a considerable challenge due to the high dimension of the state space. In many applications, however, species with rather small population numbers interact with abundant species, and only the former group exhibits stochastic behavior. This has motivated the derivation of hybrid models where a low-dimensional CME is coupled to a set of ordinary differential equations representing the abundant species. Using such a hybrid model decreases the number of unknowns significantly but –in addition to the numerical error – causes a modeling error. We investigate the accuracy of the MRCE (= model reduction based on conditional expectations) approach with respect to a particular scaling of the reaction system and prove that the error is proportional to the scaling parameter.