External control in a special class of probabilistic Boolean networks

Instantaneously Random Probabilistic Boolean Networks (PBN's) have been recently introduced as a rule-based paradigm for modeling gene regulatory networks. Furthermore, it has been shown how ideas from optimal control of Markov decision processes can be used to desirably affect the dynamic evolution of the state of such a network. This paper considers the problem of optimal intervention in context sensitive PBNs, i.e. PBNs in which the state evolves over one or more time steps as a Boolean network with a fixed set of predictor functions until a random event such as an external stimulus (or a novel context) causes the network to switch to a new Boolean one. In addition, the paper seeks to accomodate random gene perturbations such as one or more gene flippings provided, at a given time step, the state either evolves according to the predictor functions or undergoes random perturbations but both do not occur simultaneously. Another novelty of the results reported in this paper is that the example PBN used for control is derived from steady-state (long run) considerations and the concept of influence is used to choose the intervention gene. For a PBN with n genes and k possible predictor sets, two possible solutionHomiletics to the control problem are presented. In the first, the dimension of the state space is artificially increased to 2ⁿk while in the second, it is shrunk back to 2ⁿ, the usual state dimension encountered in earlier work with instantaneously random PBNs.