This thesis is concerned with methodologies for the accurate quantitative modelling of molecular biological systems. The first part is devoted to the chemical Langevin equation (CLE), a stochastic differential equation driven by a multidimensional Wiener process. The CLE is an approximation to the standard discrete Markov jump process model of chemical reaction kinetics. It is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. We observe that the CLE is not a single equation, but a family of equations with shared finite-dimensional distributions. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m_1 pairs of reversible reactions and m_2 irreversible reactions, there is another, simple formulation of the CLE with only m_1+m_2 Wiener processes, whereas the standard approach uses 2m_1+m_2. Considerable computational savings are achieved with this latter formulation. A flaw of the CLE model is identified: trajectories may leave the nonnegative orthant with positive probability. The second part addresses the challenge when alternative, structurally different ordinary differential equation models of similar complexity fit the available experimental data equally well. We review optimal experiment design methods for choosing the initial state and structural changes on the biological system to maximally discriminate between the outputs of rival models in terms of L_2-distance. We determine the optimal stimulus (input) profile for externally excitable systems. The numerical implementation relies on sum of squares decompositions and is demonstrated on two rival models of signal processing in starving Dictyostelium amoebae. Such experiments accelerate the perfection of our understanding of biochemical mechanisms.

In spite of remarkable progress in molecular biology, our understanding of the dynamics and functions of intra- and inter-cellular biological networks has been hampered by their complexity. Kinetics modelling, an important type of mathematical modelling, provides a rigorous and reliable way to reveal the complexity of biological networks. In this thesis, two genetic regulatory networks have been investigated via kinetic models. In the first part of the study, a model is developed to represent the transcriptional regulatory network essential for the circadian rhythms in Drosophila. The model incorporates the transcriptional feedback loops revealed so far in the network of the circadian clock (PER/TIM and VRI/PDP1 loops). Conventional Hill functions are not used to describe the regulation of genes, instead the explicit reactions of binding and unbinding processes of transcription factors to promoters are modelled. The model is described by a set of ordinary differential equations and the parameters are estimated from the in vitro experimental data of the clocks’ components. The simulation results show that the model reproduces sustained circadian oscillations in mRNA and protein concentrations that are in agreement with experimental observations. It also simulates the entrainment by light-dark cycles, the disappearance of the rhythmicity in constant light and the shape of phase response curves resembling that of experimental results. The model is robust over a wide range of parameter variations. In addition, the simulated E-box mutation, perS and perL mutants are similar to that observed in the experiments. The deficiency between the simulated mRNA levels and experimental observations in per01, tim01 and clkJrk mutants suggests some differences in the model from reality. Finally, a possible function of VRI/PDP1 loops is proposed to increase the robustness of the clock. In the second part of the study, the sources of intrinsic noise and the influence of extrinsic noise are investigated on an intracellular viral infection system. The contribution of the intrinsic noise from each reaction is measured by means of a special form of stochastic differential equation, the chemical Langevin equation. The intrinsic noise of the system is the linear sum of the noise in each of the reactions. The intrinsic noise arises mainly from the degradation of mRNA and the transcription processes. Then, the effects of extrinsic noise are studied by means of a general form of stochastic differential equation. It is found that the noise of the viral components grows logarithmically with increasing noise intensities. The system is most susceptible to noise in the virus assembly process. A high level of noise in this process can even inhibit the replication of the viruses. In summary, the success of this thesis demonstrates the usefulness of models for interpreting experimental data, developing hypotheses, as well as for understanding the design principles of genetic regulatory networks.

In spite of remarkable progress in molecular biology, our understanding of the dynamics and functions of intra- and inter-cellular biological networks has been hampered by their complexity. Kinetics modelling, an important type of mathematical modelling, provides a rigorous and reliable way to reveal the complexity of biological networks. In this thesis, two genetic regulatory networks have been investigated via kinetic models. In the first part of the study, a model is developed to represent the transcriptional regulatory network essential for the circadian rhythms in Drosophila. The model incorporates the transcriptional feedback loops revealed so far in the network of the circadian clock (PER/TIM and VRI/PDP1 loops). Conventional Hill functions are not used to describe the regulation of genes, instead the explicit reactions of binding and unbinding processes of transcription factors to promoters are modelled. The model is described by a set of ordinary differential equations and the parameters are estimated from the in vitro experimental data of the clocks' components. The simulation results show that the model reproduces sustained circadian oscillations in mRNA and protein concentrations that are in agreement with experimental observations. It also simulates the entrainment by light-dark cycles, the disappearance of the rhythmicity in constant light and the shape of phase response curves resembling that of experimental results. The model is robust over a wide range of parameter variations. In addition, the simulated E-box mutation, perS and perL mutants are similar to that observed in the experiments. The deficiency between the simulated mRNA levels and experimental observations in per01, tim01 and clkJrk mutants suggests some differences in the model from reality. Finally, a possible function of VRI/PDP1 loops is proposed to increase the robustness of the clock. In the second part of the study, the sources of intrinsic noise and the influence of extrinsic noise are investigated on an intracellular viral infection system. The contribution of the intrinsic noise from each reaction is measured by means of a special form of stochastic differential equation, the chemical Langevin equation. The intrinsic noise of the system is the linear sum of the noise in each of the reactions. The intrinsic noise arises mainly from the degradation of mRNA and the transcription processes. Then, the effects of extrinsic noise are studied by means of a general form of stochastic differential equation. It is found that the noise of the viral components grows logarithmically with increasing noise intensities. The system is most susceptible to noise in the virus assembly process. A high level of noise in this process can even inhibit the replication of the viruses. In summary, the success of this thesis demonstrates the usefulness of models for interpreting experimental data, developing hypotheses, as well as for understanding the design principles of genetic regulatory networks.