This dissertation attempts to answer some of the vital questions involved in the genetic regulatory networks: inference, optimization and robustness of the mathe- matical models. Network inference constitutes one of the central goals of genomic signal processing. When inferring rule-based Boolean models of genetic regulations, the same values of predictor genes can correspond to di®erent values of the target gene because of inconsistencies in the data set. To resolve this issue, a consistency-based inference method is developed to model a probabilistic genetic regulatory network, which consists of a family of Boolean networks, each governed by a set of regulatory functions. The existence of alternative function outputs can be interpreted as the result of random switches between the constituent networks. This model focuses on the global behavior of genetic networks and re°ects the biological determinism and stochasticity. When inferring a network from microarray data, it is often the case that the sample size is not su±ciently large to infer the network fully, such that it is neces- sary to perform model selection through an optimization procedure. To this end, the network connectivity and the physical realization of the regulatory rules should be taken into consideration. Two algorithms are developed for the purpose. One algo- rithm ¯nds the minimal realization of the network constrained by the connectivity, and the other algorithm is mathematically proven to provide the minimally connected network constrained by the minimal realization. Genetic regulatory networks are subject to modeling uncertainties and perturba- tions, which brings the issue of robustness. From the perspective of network stability, robustness is desirable; however, from the perspective of intervention to exert in- °uence on network behavior, it is undesirable. A theory is developed to study the impact of function perturbations in Boolean networks: It ¯nds the exact number of a®ected state transitions and attractors, and predicts the new state transitions and robust/fragile attractors given a speci¯c perturbation. Based on the theory, one algorithm is proposed to structurally alter the network to achieve a more favorable steady-state distribution, and the other is designed to identify function perturbations that have caused changes in the network behavior, respectively.

In recent years Boolean Networks (BN) and Probabilistic Boolean Networks (PBN) have become popular paradigms for modeling gene regulation. A PBN is a collection of BNs in which the gene state vector transitions according to the rules of one of the constituent BNs, and the network choice is governed by a selection distribution. Intervention in the context of PBNs was first proposed with an objective of avoid- ing undesirable states, such as those associated with a disease. The early methods of intervention were ad hoc, using concepts like mean first passage time and alteration of rule based structure. Since then, the problem has been recognized and posed as one of optimal control of a Markov Network, where the objective is to find optimal strategies for manipulating external control variables to guide the network away from the set of undesirable states towards the set of desirable states. This development made it possible to use the elegant theory of Markov decision processes (MDP) to solve an array of problems in the area of control in gene regulatory networks, the main theme of this work. We first introduce the optimal control problem in the context of PBN models and review our solution using the dynamic programming approach. We next discuss a case in which the network state is not observable but for which measurements that are probabilistically related to the underlying state are available. We then address the issue of terminal penalty assignment, considering long term prospective behavior and the special attractor structure of these networks. We finally discuss our recent work on optimal intervention for the case of a family of BNs. Here we consider simultaneously controlling a set of Boolean Models that satisfy the constraints imposed by the underlying biology and the data. This situation arises in a case where the data is assumed to arise by sampling the steady state of the real biological network.

A basic issue for translational genomics is to model gene interactions via gene regulatory networks (GRNs) and thereby provide an informatics environment to derive and study effective interventions eradicating the tumor. In this dissertation, we present two different approaches to intervention methods in cancer-related GRNs. Decisions regarding possible interventions are assumed to be made at every state transition of the network. To account for dosing constraints, a model for the sequence of treatment windows is considered, where treatments are allowed only at the beginning of each treatment cycle followed by a recovery phase. Due to biological variabilities within tumor cells, the action period of an antitumor drug can vary among a population of patients. That is, a treatment typically has a random duration of action. We propose a unified approach to such intervention models for any Markovian GRN governing the tumor. To accomplish this, we place the problem in the general framework of partially controlled decision intervals with infinite horizon discounting cost. We present a methodology to devise optimal intervention policies for synthetically generated gene regulatory networks as well as a mutated mammalian cell-cycle network. As a different approach, we view the phenotype as a characterization of the long- run behavior of the Markovian GRN and desire interventions that optimally move the probability mass from undesirable to desirable states. We employ a linear programming approach to formulate the maximal shift problem, that is, optimization is directly based on the amount of shift. Moreover, the same basic linear programming structure is used for a constrained optimization, where there is a limit on the amount of mass that may be shifted to states that are not directly undesirable relative to the pathology of interest, but which bear some perceived risk. We demonstrate the performance of optimal policies on synthetic networks as well as two real GRNs derived from the metastatic melanoma and mammalian cell cycle. These methods, as any effective cancer treatment must, aim to carry out their actions rapidly and with high efficiency such that a very large percentage of tumor cells die or shift into a state where they stop proliferating.