System Decomposition Of Boolean Control Networks

With the rapid development of systems biology, Boolean dynamical models, as an effective tool for analyzing the interaction of genes, have become a hot research topic. Since Boolean control networks are suitable for describing genetic regulatory networks, they have received considerable attention from the areas of systems biology and control science.In the past few years, Cheng et. al. have proposed matrix semi-tensor product, which is a new theoretical tool for studying Boolean control networks. With ma-trix semi-tensor product, the logical dynamic equations of Boolean control networks can be converted into bi-linear equations, which greatly promotes the development of Boolean control networks.This dissertation investigates the system decomposition problem of Boolean con-trol networks using algebraic and graph theory methods. The problems of decom-position w.r.t. inputs, decomposition w.r.t outputs and Kalman decomposition are proposed and completely solved. In the existing related literature, authors obtain suf-ficient conditions for these decomposability problems under the condition that the largest uncontrollable subspace and the largest unobservable subspace are regular. Compared with known results, this dissertation uses algebraic and graphic methods instead of state-space and subspace analytical methods, and it obtains necessary and sufficient conditions for the three kinds of decomposition problems. This dissertation is a collection of the author’s work during the period of pursuing her PhD degree. The main contents of this dissertation are listed as follows:1. Chapter 2 defines state transition diagram in the frame of the algebraic forms of Boolean control networks. Based on the knowledge of the equivalence relations of set theory, the relationship of the vertices and the directed edges in the state transi-tion diagram is analyzed, which effectively combines the structure of Boolean control networks and the knowledge of vertex partitions.2. Chapter 3 discusses the problem of decomposition w.r.t. inputs for Boolean control networks. This section mainly discusses whether the states of the system can be decomposed into two parts-one is control-dependent and the other is control-independent. First, the definition of decomposition w.r.t. inputs is given based on the logical coordinate transformation. With the relationship of the vertices and the direct-ed edges in the digraph, a necessary and sufficient condition for the decomposition w.r.t. inputs is derived. Then a logical coordinate transformation realizing the decom-position w.r.t. inputs is obtained constructively. Finally, an algorithm is designed to find the decomposition form w.r.t. inputs for Boolean control networks.3. Chapter 4 studies the problem of decomposition w.r.t. outputs for Boolean control networks. This section mainly studies whether the states of the system can be decomposed into two parts-one can affect the outputs and the other can not. First, the definition of decomposition w.r.t. outputs is given. Based on the outputs of each vertex in the digraph, a necessary and sufficient condition for the decomposition w.r.t. outputs is derived. Then a logical coordinate transformation is obtained constructively, and an algorithm is designed to find the decomposition form w.r.t. outputs for Boolean control networks. Finally, the relationship of decomposability w.r.t. outputs and the unobservability is analyzed.4. Chapter 5 investigates the Kalman decomposition problem based on the de-composition w.r.t. inputs and the decomposition w.r.t. outputs, i.e. analyzing whether a Boolean control network can be decomposed into four sub-systems according to whether the states can be affected by the controls and whether the states can affect the outputs. With the known results about the decomposition w.r.t. inputs and the decomposition w.r.t. outputs, a necessary and sufficient condition for the Kalman decomposition is proposed. Then an algorithm is designed to compute the logical coordinate transformation that results in the Kalman decomposition. Based on the obtained necessary and sufficient conditions, an essential property of Boolean control networks is revealed, i.e. not all the Boolean control networks can be decomposed into a Kalman decomposition form.