Robust control of uncertain context-sensitive probabilistic Boolean networks

Uncertainty is an intrinsic phenomenon in control of gene regulatory networks (GRNs). The presence of uncertainty is related to impreciseness of GRN models due to: (1) Errors caused by imperfection of measurement devices and (2) Models´ inability to fully capture a complex structure of the GRN. Consequently, there is a discrepancy between actual behaviour of the GRN and what is predicted by its mathematical model. This can result in false control signals, which can drive a cell to an undesirable state. To address the problem of control under uncertainties, a risk-sensitive control paradigm is proposed. Robustness is accomplished by minimisation of the mean exponential cost as opposed to, for instance, minimisation of the mean square cost by risk-neutral controllers. The authors derive an optimal risk-sensitive controller when a GRN is modelled by a context-sensitive probabilistic Boolean network (CSPBN). By using a relation between the relative entropy and free-energy, a relative stability of the cost achieved by the risk-sensitive controller is demonstrated when the distribution of the CSPBN attractors is perturbed, as opposed to the cost of the risk-neutral controller that exhibits increase. The use of the relation between the relative entropy and free-energy to analyse the influence of a particular attractor on the robustness of the controller is studied. The efficiency of the risk-sensitive controller is tested for the CSPBN obtained from the study of malignant melanoma.