Date of Award

1-1-2012

Document Type

Campus Access Dissertation

Department

Mathematics

First Advisor

Xinfeng Liu,

Second Advisor

Qi Wang

Abstract

The mitogen-activated protein kinase (MAPK) cascades that are evolutionally conserved from yeast to mammals play a pivotal role in many aspects of cellular functions. The mating decision in yeast is a switch-like or bistability response that allows cells to filter out weak pheromone signals or avoiding improper mating when a mate is sufficiently close. In many cases, scaffold proteins are thought to play a key role during this process. The molecular mechanisms that control the bistability decision is not yet fully understood. Here we show that bistability mechanism can arise from multisite phosphorylation system with substrate sequestration when phosphorylation and dephosphorylation occurs at different locations. This scaffold binding in a multisite phosphorylation system can robustly result in multiple steady states. By developing generic mathematical models, we argue that the scaffold protein plays an important role in creating bistability, and by treating parameters symbolically, we also thereby reduce the complexity of calculating steady states from simulating differential equations to finding the roots of polynomials, of which the degree depends on the number of phosphorylation sites N. In the next problem, we developed a generic mathematical model to incorporate MAPK cascade with two substrates in the presence of scaffold proteins, and for the simplification of presentation while keeping the key components of MAPK cascades, we assume that each substrate has only one phosphorylation site. Even for this simplified model, it consists of a total of 13 unknowns, which makes the mathematical analysis and computation much more complicated than previous models. By treating parameters systematically, the complicated model can be reduced to a system of solving four algebraic equations. Our computation reveals that multisite phosphorylation with scaffold binding in cascade leads to bistability even more robustly than the models without cascade. Finally, we present an efficient numerical technique for solving reaction-diffusion-advection equations. For reaction-diffusion-advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. For reaction-diffusion systems with both stiff reaction and diffusion terms, implicit integration factor (IIF) method and its higher dimensional analog compact IIF (cIIF) serve as an efficient class of time-stepping methods, and their second order version is linearly unconditionally stable. For nonlinear hyperbolic equations, we incorporated the backward error compensation method (BFECC) introduced in into semi-Lagrangian scheme, i.e., Courant-Issacson-Rees (CIR) scheme. In this thesis, we couple IIF/cIIF with BFECC methods using the operator splitting approach to solve reaction-diffusion-advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the BFECC method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties.

Rights

© 2012, Kanadpriya Basu

Download

Share

COinS