External control in Markovian Genetic Regulatory Networks

Probabilistic Boolean Networks (PBN's) have been recently introduced as a rule-based paradigm for modeling gene regulatory networks. Such networks, which form a subclass of Markovian Genetic Regulatory Networks, provide a convenient tool for studying interactions between different genes while allowing for uncertainty in the knowledge of these relationships. This paper deals with the issue of control in probabilistic Boolean networks. More precisely, given a general Markovian Genetic Regulatory Network whose state transition probabilities depend on an external ( control) variable, the paper develops a procedure by which one can choose the sequence of control actions that minimize a given performance index over a finite number of steps. The procedure is based on the theory of controlled Markov chains and makes use of the classical technique of Dynamic Programming. The choice of the finite horizon performance index is motivated by cancer treatment applications where one would ideally like to intervene only over a finite time horizon, then suspend treatment and observe the effects over some additional time before deciding if further intervention is necessary. The undiscounted finite horizon cost minimization problem considered here is the simplest one to formulate and solve, and is selected mainly for clarity of exposition, although more complicated costs could be used, provided appropriate technical conditions are satisfied.