In this work, the mathematical models describing the dynamics of the gene regulatory network of the lac operon are considered. The lac operon is one of the simplest biological systems which involves the regulation network of three genes. The mathematical models of the regulatory mechanisms of the lac system, developed in the literature are based on deterministic or fully stochastic approach to the problem. The aim of the thesis is the development of two stochastic models (reduced and full) based on extension of existing deterministic models with noise terms. The two models reflect different level of complexity of the regulatory processes. The advantage of this approach is based on the realistic description of protein concentrations, protein kinetics and time delays. The research considers first order stochastic delayed differential equations (SDDEs) and their solutions. Stability properties of the stochastic models are investigated by linearization of the systems of SDDEs. New sufficient conditions of mean square stability are obtained analytically for these models using Lyapunov function. Additionally, the threshold values for SDDEs are derived. These conditions and threshold values are applied to nd analytical solutions of the two models of nonlinear SDDE. Further, numerical solutions of these equations are obtained using Euler Maruyama method. A detailed analysis of the stability regions of the models is performed, analytically and numerically. A specific attention is given to the bistable region as it reflects important biological features of the system linked to the positive regulatory mechanism. It is concluded that the stochasticity can change the boundaries of the bistable region which cannot be obtained in the case of the deterministic model of the lac operon. This thesis provides a thorough investigation of the stochastic stability of two lac operon models and demonstrates that the system behaviour is very sensitive to protein concentrations. It also provides a novel way for estimating such concentrations.
|Publication status||Accepted/In press - Dec 2013|