Optimal Perturbation Control of General Topology Molecular Networks

In this paper, we develop a comprehensive framework for optimal perturbation control of dynamic networks. The aim of the perturbation is to drive the network away from an undesirable steady-state distribution and to force it to converge towards a desired steady-state distribution. The proposed framework does not make any assumptions about the topology of the initial network, and is thus applicable to general-topology networks. We define the optimal perturbation control as the minimum-energy perturbation measured in terms of the Frobenius-norm between the initial and perturbed probability transition matrices of the dynamic network. We subsequently demonstrate that there exists at most one optimal perturbation that forces the network into the desirable steady-state distribution. In the event where the optimal perturbation does not exist, we construct a family of suboptimal perturbations, and show that the suboptimal perturbation can be used to approximate the optimal limiting distribution arbitrarily closely. Moreover, we investigate the robustness of the optimal perturbation control to errors in the probability transition matrix, and demonstrate that the proposed optimal perturbation control is robust to data and inference errors in the probability transition matrix of the initial network. Finally, we apply the proposed optimal perturbation control method to the Human melanoma gene regulatory network in order to force the network from an initial steady-state distribution associated with melanoma and into a desirable steady-state distribution corresponding to a benign cell.