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.

<p>This dissertation focuses on Boolean networks with a view to their applications in Systems Biology. We study two notions of stability, based on Hamming distance and on maintenance of a stable period length. Algorithms are given for the determination of Boolean networks from both complete and partial dynamics. The dynamics of ring networks are systematically studied. An algebraic structure is developed for derivation of adjacency matrices for the dynamics of Boolean networks from simple building blocks, both by edge-swapping and by gluing simple building blocks. Some results are implemented in Python and conclusions drawn for theta networks, a class of networks only slightly more complex than rings. A short section on applications to a known biological system closes the dissertation.</p> / Dissertation

New tools are available for reconstructing the attractor basins of discrete dynamical networks where state-space is linked according the network's dynamics. In this thesis the computer software "Discrete Dynamics Lab" is applied to examine simple networks ranging from cellular automata (CA) to random Boolean networks (RBN), that have been widely applied as idealised models of physical and biological systems, to search for general principles underlying their dynamics. The algorithms and methods for generating pre-images for both CA and RBN, and reconstructing and representing attractor basins are described, and also considered in the mathematical context of random directed graphs. RBN and CA provide contrasting notions of self-organisation. RBN provide models of hierarchical categorisation in biology, for example memory in neural and genomic networks. CA provide models at the lower level of emergent complex pattern. New measures and results are presented on CA attractor basins and how they relate to measures on local dynamics and the Z parameter, characterising ordered to "complex" to chaotic behaviour. A method is described for classifying CA rules by an entropy-variance measure which allows glider rules and related complex rules to be found automatically giving a virtually unlimited sample for further study. The dynamics of RBN and intermediate network architectures are examined in the context of memory, where categorisation occurs at the roots of subtrees as well as at attractors. Learning algorithms are proposed for "sculpting" the basin of attraction field. RBN are proposed as a possible neural network model, and also discussed as a model of genomic regulatory networks, where cell types have been explained as attractors

The development of cDNA microarray technology has made it possible to simultaneously monitor the expression status of thousands of genes. A natural use for this vast amount of information would be to try and figure out inter-gene relationships by studying the gene expression patterns across different experimental conditions and to build Gene Regulatory Networks from these data. In this thesis, we study some of the issues involved in Genetic Regulatory Networks. One of them is to discover and elucidate multivariate logical predictive relations among gene expressions and to demonstrate how these logical relations based on coarse quantization closely reflect corresponding relations in the continuous data. The other issue involves construction of synthetic Probabilistic Boolean Networks with particular attractor structures. These synthetic networks help in testing of various algorithms like Bayesian Connectivity based approach for design of Probabilistic Boolean Networks.

<p>Switching networks are a common model for biological systems, especially</p><p>for genetic transcription networks. Stuart Kaufman originally proposed</p><p>the usefulness of the Boolean framework, but much of the dynamical </p><p>features there are not realizable in a continuous analogue. We introduce the notion</p><p>of braid-like dynamics as a bridge between Boolean and continuous dynamics and</p><p>study its importance in the local dynamics of ring and ring-like networks. We discuss</p><p>a near-theorem on the global dynamics of general feedback networks, and in the final</p><p>chapter study the main ideas of this thesis in models of a yeast cell transcription network.</p> / Dissertation

The development of cDNA microarray technology has made it possible to simultaneously monitor the expression status of thousands of genes. A natural use for this vast amount of information would be to try and &#64257;gure out inter-gene relationships by studying the gene expression patterns across different experimental conditions and to build Gene Regulatory Networks from these data. In this thesis, we study some of the issues involved in Genetic Regulatory Networks. One of them is to discover and elucidate multivariate logical predictive relations among gene expressions and to demonstrate how these logical relations based on coarse quantization closely re&#64258;ect corresponding relations in the continuous data. The other issue involves construction of synthetic Probabilistic Boolean Networks with particular attractor structures. These synthetic networks help in testing of various algorithms like Bayesian Connectivity based approach for design of Probabilistic Boolean Networks.

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.

Cells behave as complex systems with regulatory processes that make use of many elements such as switches based on thresholds, memory, feedback, error-checking, and other components commonly encountered in electrical engineering. It is therefore not surprising that these complex systems are amenable to study by engineering methods. A great deal of effort has been spent on observing how cells store, modify, and use information. Still, an understanding of how one uses this knowledge to exert control over cells within a living organism is unavailable. Our prime objective is "Personalized Cancer Therapy" which is based on characterizing the treatment for every individual cancer patient. Knowing how one can systematically alter the behavior of an abnormal cancerous cell will lead towards personalized cancer therapy. Towards this objective, it is required to construct a model for the regulation of the cell and utilize this model to devise effective treatment strategies. The proposed treatments will have to be validated experimentally, but selecting good treatment candidates is a monumental task by itself. It is also a process where an analytic approach to systems biology can provide significant breakthrough. In this dissertation, theoretical frameworks towards effective treatment strategies in the context of probabilistic Boolean networks, a class of gene regulatory networks, are addressed. These proposed analytical tools provide insight into the design of effective therapeutic interventions.

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Boolean networks are coarse-grained models of the regulatory dynamics that controls the survival and proliferation of a living cell. The dynamics is time- and state-discrete. This Boolean abstraction assumes that small differences in concentration levels are irrelevant; and the binary distinction of a low or a high concentration of each biomolecule is sufficient to capture the dynamics. In this work, we briefly introduce the gene regulatory models, where with the advent of system-specific Boolean models, new conceptual questions and analytical and numerical challenges arise. In particular, the response of the system to external intervention presents a novel area of research. Thus first we investigate how to quantify a node\\\'s individual impact on dynamics in a more detailed manner than an averaging against all eligible perturbations. Since each node now represents a specific biochemical entity, it is the subject of our interest. The prediction of nodes\\\' dynamical impacts from the model may be compared to the empirical data from biological experiments. Then we develop a hybrid model that incorporates both continuous and discrete random Boolean networks to compare the reaction of the dynamics against small as well as flip perturbations in different regimes. We show that the chaotic behaviour disappears in high sensitive Boolean ensembles with respect to continuous small fluctuations in contrast to the flipping. Finally, we discuss the role of distributing delays in stabilizing of the Boolean dynamics against noise. These studies are expected to trigger additional experiments and lead to improvement of models in gene regulatory dynamics.