Stability analysis and control of piecewise affine models for genetic regulatory networks with applications in systems and synthetic biology

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This project sits at the intersection between the areas of Control Engineering, Systems Biology and Synthetic Biology. The main object of our research are genetic regulatory networks, which describe the interaction between genes and their products. A mathematical model of such networks is needed and among the different models available [1] to describe them, the piecewise affine model [2] has been chosen for the present work, due to its capability to merge together the advantages of both qualitative and quantitative approaches.The goal of this project is to develop theoretical and practical instruments for the stability analysis and control of these biological systems. This means that, in first instance, we want to understand the long term behaviour of the system, the existence of equilibria and their stable or unstable nature and secondly we aim to design external devices or other genetic regulatory networks that are able to influence such behaviour.The intended approach for the stability analysis of the system is the construction, and the study, of a Lyapunov function, which is a sort of energy measure for the network. Studying the evolution in time of such function gives information on the original system, but its construction is non-trivial due to the non-linearities and the possible presence of multiple equilibria in the system. An optimization problem needs to be set up in order to find a description of such function, if one exists.In regard to the control of biological systems, an extensive literature review on the topic will be made and ideally the results found regarding the stability of piecewise affine model of genetic regulatory networks will be applied to the control problem.Synthetic biology, which is one of the main area in which the foundation of this work lay, is a promising and ever growing field. Many applications include cell-based therapy in cancer treatment, therapies against diabetes and the production of drugs, fuels and other materials using biological entities. Moreover this research finds application also in Systems biology, studying already existing, but not well characterised, biological systems, and in the field of Control engineering for the interesting challenges related to the non-linearities in the system and the presence of multiple equilibria.The project is aligned with the EPSRC research areas of Control engineering and Synthetic Biology.[1] Saadatpour, A., & Albert, R. (2016). A comparative study of qualitative and quantitative dynamic models of biological regulatory networks. EPJ Nonlinear Biomedical Physics, 4(1), 5. [2] Casey, R., De Jong, H., & Gouzé, J. L. (2006). Piecewise-linear models of genetic regulatory networks: Equilibria and their stability. Journal of Mathematical Biology, 52(1), 27-56.