Dynamics preserving size reduction mappings for probabilistic Boolean networks

A probabilistic Boolean network (PBN) is a discrete network composed of a family of Boolean networks such that at each time instant the transition of the PBN is governed by one of its constituent networks. A random variable determines when the governing Boolean network switches and, when a switch occurs, a new constituent network is chosen according to a selection distribution to govern the PBN transitions until another switch is called for. As nonlinear models of genetic regulatory networks, PBNs incorporate the indeterminacy owing to latent variables external to the model that have biological interaction with genes in the model. Besides being used to model biologically phenomena, such as cellular state dynamics and the switch-like behavior of certain genes, PBNs have served as the main model for the application of intervention methods, including optimal control strategies, to favorably effect system dynamics. An obstacle in applying PBNs to large-scale networks is the computational complexity of the model. It is sometimes necessary to construct computationally tractable subnetworks while still carrying sufficient structure for the application at hand. Hence, there is a need for size reducing mappings. This paper proposes a reduction strategy that preserves the dynamical structure of the network, a crucial requirement for the development of intervention strategies based on control theory. In particular, we focus on the following two issues when deleting a gene from the network: 1) maintaining the same number of constituent Boolean networks, and 2) preserving, in an optimal sense, the attractor structure, the relative sizes of the basins of attraction, and the level structures of the state transition diagrams of the constituent Boolean networks. Preservation of the attractor structure is critical because the attractors of a PBN determine its steady-state behavior.