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

摘要

Stochasticity plays important roles in molecular networks when molecularconcentrations are in the range of $0.1 \mu$M to $10 n$M (about 100 to 10copies in a cell). The chemical master equation provides a fundamentalframework for studying these networks, and the time-varying landscapeprobability distribution over the full microstates provide a fullcharacterization of the network dynamics. A complete characterization of thespace of the microstates is a prerequisite for obtaining the full landscapeprobability distribution of a network. However, there are neither closed-formsolutions nor algorithms fully describing all microstates for a given molecularnetwork. We have developed an algorithm that can exhaustively enumerate themicrostates of a molecular network of small copy numbers under the finitebuffer condition that the net gain in newly synthesized molecules is smallerthan a predefined limit. We also describe a simple method for computing theexact mean or steady state landscape probability distribution over microstates.We show how the full landscape probability for the gene networks of theself-regulating gene and the toggle-switch in the steady state can be fullycharacterized. We also give an example using the MAPK cascade network. Our algorithm works for networks of small copy numbers buffered with a finitecopy number of net molecules that can be synthesized, regardless of thereaction stoichiometry, and is optimal in both storage and time complexity. Thebuffer size is limited by the available memory or disk storage. Our algorithmis applicable to a class of biological networks when the copy numbers ofmolecules are small and the network is closed, or the network is open but thenet gain in newly synthesized molecules does not exceed a predefined buffercapacity.