Steady-state probabilities for attractors in probabilistic Boolean networks

Boolean networks form a class of disordered dynamical systems that have been studied in physics owing to their relationships with disordered systems in statistical mechanics and in biology as models of genetic regulatory networks. Recently they have been generalized to probabilistic Boolean networks (PBNs) to facilitate the incorporation of uncertainty in the model and to represent cellular context changes in biological modeling. In essence, a PBN is composed of a family of Boolean networks between which the PBN switches in a stochastic fashion. In whatever framework Boolean networks are studied, their most important attribute is their attractors. Left to run, a Boolean network will settle into one of a collection of state cycles called attractors. The set of states from which the network will transition into a specific attractor forms the basin of the attractor. The attractors represent the essential long-run behavior of the network. In a classical Boolean network, the network remains in an attractor once there; in a Boolean network with perturbation, the states form an ergodic Markov chain and the network can escape an attractor, but it will return to it or a different attractor unless interrupted by another perturbation; in a probabilistic Boolean network, so long as the PBN remains in one of its constituent Boolean networks it will behave as a Boolean network with perturbation, but upon a switch it will move to an attractor of the new constituent Boolean network. Given the ergodic nature of the model, the steady-state probabilities of the attractors are critical to network understanding. Heretofore they have been found by simulation; in this paper we derive analytic expressions for these probabilities, first for Boolean networks with perturbation and then for PBNs.