Dynamics and asymptotic behaviors of biochemical networks

Abstract

The purpose of this dissertation is to study the dynamics and asymptotic behaviors of biochemical networks using a modular'' approach. New mathematics is motivated and developed to analyze modules in terms of the number of steady states and the stability of the steady states. One of the main contributions of the thesis is to extend Hirsch's generic convergence result from monotone systems to systems close'' to monotone using geometric singular perturbation theory. A monotone system is a dynamical system for which the comparison principle holds, that is, bigger'' initial states lead to bigger'' future states. In monotone systems, every net feedback loop is positive. On the other hand, negative feedback loops are important features of many systems, since they are required for adaptation and precision. We show that, provided that these negative loops act at a comparatively fast time scale, the generic convergence property still holds. This is an appealing result, which suggests that monotonicity has broader implications than previously thought. One particular application of great interest is that of double phosphorylation dephosphorylation cycles. Other recurring modules in biochemical networks are also analyzed in detail. For systems without time scale separation, we study the global stability of one special class of systems, called monotone tridiagonal systems with negative feedback. The key technique is to rule out periodic orbits using the theory of second compound matrices. We also investigate the effect of diffusion on the stability of a constant steady state for systems with more general structures represented by graphs. This work extends the passivity-based stability criterion developed by Arcak and Sontag to reaction diffusion equations. Upon assembling different modules, the dynamics of individual modules might be affected. One particular effect is called retroactivity'' in the systems biology literature. We propose designs and conditions under which the retroactivity can be attenuated.%%%%