Optimal control policy for probabilistic Boolean networks with hard constraints

It is well known that the control/intervention of some genes in a genetic regulatory network is useful for avoiding undesirable states associated with some diseases like cancer. For this purpose, both optimal finite-horizon control and infinite-horizon control policies have been proposed. Boolean networks (BNs) and its extension probabilistic Boolean networks (PBNs) as useful and effective tools for modelling gene regulatory systems have received much attention in the biophysics community. The control problem for these models has been studied widely. The optimal control problem in a PBN can be formulated as a probabilistic dynamic programming problem. In the previous studies, the optimal control problems did not take into account the hard constraints, i.e. to include an upper bound for the number of controls that can be applied to the captured PBN. This is important as more treatments may bring more side effects and the patients may not bear too many treatments. A formulation for the optimal finite-horizon control problem with hard constraints introduced by the authors. This model is state independent and the objective function is only dependent on the distance between the desirable states and the terminal states. An approximation method is also given to reduce the computational cost in solving the problem. Experimental results are given to demonstrate the efficiency of our proposed formulations and methods.