Abstract
Modeling gene regulation is one of the most significant issues in genetic regulatory networks, the logical behaviors of which can be modeled by Boolean networks. As a conventional logical system, Boolean network was first proposed by Kauffman to disclose the inherent logical behaviors of genes, DNAs, gene products, etc. The logical dynamics of a Boolean network are determined by both the network structure describing the neighbor relationships between each node and also by a series of value update logical functions. Thus, systematic analysis on Boolean networks will be an important issue in genomic research. In this thesis, systematic analysis on Boolean networks using semi-tensor product method will be studied to better understand and model genetic regulatory networks.Semi-tensor product of matrices is a new matrix product, which breaks the traditional dimension matching condition of conventional matrix product. It was firstly proposed by Daizhan Cheng and his colleagues, as an efficient tool to analyze logical functions, Boolean networks, game theory, Petri networks and other finite-valued systems. Based on a bijective equivalence between logical variables and vectors, any logical function can be expressed by a standard discrete-time algebraic representation. Thus, based on the semi-tensor product approach, a Boolean network will be converted into an equivalent algebraic linear system. Under the framework of algebraic system, systematic analysis of Boolean networks can be addressed including attractors, basins, controllability, stability and oscillations.
In genetic regulatory networks, some structures of networks appear with certain different models. One typical example is the switching phenomenon in the lysis/lysogeny gene network of bacteriophage λ switching from lysogenic mode to lysis mode. Another representative example is that different cells can be switched among growth, differentiation, apoptosis, quiescence, motility. This motivates us to investigate the evolutionary behaviors of Boolean networks with switching structure. Based on the semi-tensor product method, the output tracking problems with respect to a constant and a periodic output signal for switched Boolean networks are studied. Further, a new design of switching-signal-triggered pinning controllers is proposed to achieve output tracking. Finally, the discussions of an apoptosis network show that the theoretic results are effective in designing the switching-signal-triggered pinning controllers to achieve output tracking.
In consequence of some external conditions or genes mutations, perturbation related issues have been widely studied to address some biomedical problems. The consequences of function perturbations especially the one-bit perturbation may have a significant impact on state transitions, outputs and steady-state properties in Boolean networks. In this thesis, an output feedback pinning control design for output tracking issues on the number of active output signals are studied, which is called activation output tracking. By resorting to semi-tensor product method, the activation output tracking issues with respect to certain numbers of active output signals are studied, and some criteria are also obtained. In addition, the impact of one-bit perturbation on the activation output tracking issues is also addressed. If a one-bit perturbation is a valid perturbation on the activation output tracking problem, two algorithms and an output feedback pinning control is proposed to recover activation output tracking. Finally, a D. melanogaster network and a reduced signal transduction network have been given to illustrate the effectiveness of the obtained results.
As is known, external disturbances are ubiquitous in many real genetic regulatory networks, and some disturbances may lead coupled networks to some unexpected behaviors. Thus, when modeling genetic regulatory networks, internal/external disturbances should be taken into account, as the states of genes may be subject to abrupt disturbances. In this thesis, global robust stability and stabilization of Boolean networks with disturbances are investigated using the semi-tensor product method. By using the algebraic state space representation of disturbed Boolean networks, some necessary and sufficient criteria are obtained to ensure global robust stability with respect to a steady state or a limit cycle. In contrast, if a given disturbed Boolean network is not globally robust stable, a matrix transformation technique is proposed to guarantee stability. However, it is a difficult task to find such a suitable matrix transformation. Then, a state feedback pinning control design is proposed to find a suitable transformation. Based on the proposed state feedback pinning control design, global robust stabilization is achieved. Some numerical examples are given to demonstrate the effectiveness of proposed results.
Since the dynamical behavior of Boolean networks is determined by both the network structure and also by logical functions, it will be an interesting topic to disclose the relationships between network structure of Boolean networks and its dynamics such as the number of steady states and cycles. In this thesis, a new framework on pinning control design for global stabilization (with a unique steady state as attractor) of Boolean networks is studied based on Boolean networks' network structure. By deleting the minimum number of edges to lead an acyclic structure, global stability is guaranteed. Then, a state feedback pinning control based on the neighbors of controlled nodes are designed to achieve global stabilization. Compared with existing literatures, the design of pinning control is based on network structure describing coupling connections among nodes, but not on state transition matrix of Boolean networks. In addition, without using state transition matrix, global state information is no longer needed, and the design of pinning control is just based on neighbors' local information, which is easier to be implemented. The proposed method is well demonstrated by two T-LGL survival signaling biological networks. In the simulation, a network with eighteen nodes is verified by the proposed method. The results are shown to be simple and concise, while it is impossible to use traditional pinning control for such a high dimensional state transition matrix.
Oscillations play a significant role in many dynamic cellular processes and cell biology. In this thesis, pinning control design for oscillations of Boolean networks is studied based on positive feedback loops. Firstly, using model reduction technique, the size of a Boolean network can be decreased to a small-scale network. The reduction method leads to network structure with minimal in-degree at least one. In order to further let network structure without positive feedback loops, two methods are presented. One is transforming positive feedback loops into negative feedback loops, and another one is destroying all the positive feedback loops. Based on the proposed two methods, a state feedback pinning control is designed to achieve oscillations. Compared with recent references on pinning control design, the design of pinning control in this thesis is based on interaction digraph characterization network structure of Boolean networks, without using state transition matrix. In addition, compared with traditional pinning control design, the proposed pinning control design in this thesis is effective for some large-scale Boolean networks. Finally, a T-LGL survival signaling biological network and an apoptosis network are presented to show the potential applications of theoretical results.
| Date of Award | 6 Jul 2018 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Wing Cheong Daniel HO (Supervisor) |
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