Structure-Preserving Numerical Methods For Differential Equations In Biology

Ordinary differential equations (DEs) have proved to be one of the powerful tools for modeling the complex dynamics of biological systems, where the change rates over time of quantities representing the system species are expressed as mathematical functions of those quantities. As the analytical solutions to ODEs in biology are not acquirable, numerical simulation becomes important and indispensable in the exploring the complexity of biological systems. So far, differential equations for genetic regulation are mostly solved by the classical Runge-Kutta (RK) methods. However, general-purpose RK methods have not taken into account the special structure of the system to be simulated and fail to capture the dynamical features of the system effectively, especially in the long time simulation. The purpose of this thesis is to investigate some new structure-preserving numerical methods adapted to the physical structures of biological systems.The thesis is divided into four chapters.In Chapter 1, we review the idea for modeling of biological systems in terms of ordinary differential equations (ODEs). Three typical ODE models, the Lotka-Volterra prey-predator system in ecology, genetic regulatory systems and a typical p53-Mdm2 protein system cell biology that will be simulated in later chapters, are introduced. Dynamical structures as well as structure-preserving numerical methods are also discussed.In chapter 2, we transforms the Lotka-Volterra system into a Hamiltonian system and integrates it with two highly efficient symplectic methods. The results of the numerical simulation show the robustness of these methods.In chapter 3, a novel family of exponential Runge-Kutta (expRK) methods are designed which incorporate the stable steady-state structure of the genetic regulatory systems. A natural and convenient approach to constructing new expRK methods on the base of tra-ditional RK methods is provided. In the numerical integration of the one-gene, two-gene and p53-mdm2 regulatory systems, the new expRK methods are shown to be more accu-rate than their prototype RK methods. Moreover, for non-stiff two-gene regulatory systems, the expRK methods are more efficient than some famous exponential RK integrators in the scientific literature.In chapter 4, in order to simulate gene regulatory oscillators more effectively, Runge-Kutta (RK) integrators are adapted to the limit-cycle structure of the system. Taking into ac-count the oscillatory feature of the gene regulatory oscillators, phase-fitted and amplification-fitted Runge-Kutta (FRK) methods are designed. New FRK methods with phase-fitted and amplification-fitted updated are also considered. The error coefficients and the error constant for each of new FRK methods are obtained. In the numerical simulation of the two-gene reg-ulatory system, the new methods are shown to be more accurate and more efficient than their prototype RK methods in the long term integration. An important discovery is that the best fitting frequency for a FRK method not only depends on the problem to be solved, but also on the method itself.