Steady-state solution of probabilistic gene regulatory networks.

We introduce a probability model for gene regulatory networks, based on a system of Chapman-Kolmogorov equations that represent the dynamics of the concentration levels of each agent in the network. This unifying approach includes the representation of excitatory and inhibitory interactions between agents, second-order interactions which allow any two agents to jointly act on other agents, and Boolean dependencies between agents. The probability model represents the concentration or quantity of each agent, and we obtain the equilibrium solution for the joint probability distribution of each of the concentrations. The result is an exact solution in "product form," where the joint equilibrium probability distribution of the concentration for each gene is the product of the marginal distribution for each of the concentrations. The analysis we present yields the probability distribution of the concentration or quantity of all of the agents in a network that includes both logical dependencies and excitatory-inhibitory relationships between agents.