An optimal algorithm for enumerating state space of stochastic molecular networks with small copy numbers of molecules

Stochasticity plays central role in molecular networks of small copy numbers, including those important in protein synthesis and gene regulation. The combination of copy numbers of molecular species defines the microscopic state of molecular interactions. With this formulation, nonlinear reactions can be effectively modeled through chemical master equations. However, currently little is known about the state space associated with stochastic networks, other than the defeatist admission that it is exponentially large. There is neither closed-form solution nor computational algorithm that can effectively characterize the state space of molecular networks. Such a characterization is a prerequisite for directly solving the chemical master equation. In this study, we describe an algorithm that can exhaustively characterize all possible states of a molecular networks with small copy numbers of species for a given initial condition. Our algorithm works for networks of arbitrary stoichiometry, and is optimal in both storage and time complexity. It allows the approach of solving chemical master equation to be applicable to a larger class of stochastic molecular networks. We show an example of application of our method to the MAPK cascade network.