Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of steady state landscape probability

Stochasticity plays important roles in molecular networks when molecular concentrations are in the range of $0.1 μ$M to $10 n$M (about 100 to 10 copies in a cell). The chemical master equation provides a fundamental framework for studying these networks, and the time-varying landscape probability distribution over the full microstates provide a full characterization of the network dynamics. A complete characterization of the space of the microstates is a prerequisite for obtaining the full landscape probability distribution of a network. However, there are neither closed-form solutions nor algorithms fully describing all microstates for a given molecular network. We have developed an algorithm that can exhaustively enumerate the microstates of a molecular network of small copy numbers under the finite buffer condition that the net gain in newly synthesized molecules is smaller than a predefined limit. We also describe a simple method for computing the exact mean or steady state landscape probability distribution over microstates. We show how the full landscape probability for the gene networks of the self-regulating gene and the toggle-switch in the steady state can be fully characterized. We also give an example using the MAPK cascade network. Our algorithm works for networks of small copy numbers buffered with a finite copy number of net molecules that can be synthesized, regardless of the reaction stoichiometry, and is optimal in both storage and time complexity. The buffer size is limited by the available memory or disk storage. Our algorithm is applicable to a class of biological networks when the copy numbers of molecules are small and the network is closed, or the network is open but the net gain in newly synthesized molecules does not exceed a predefined buffer capacity.