Mathematical and computational analysis of biochemical reaction networks

Cell biology is based in large part on the operation of modules consisting of intricate biochemical reaction networks whose dynamics is usually governed by highly nonlinear systems of differential equations. This thesis considers three different aspects of biochemical dynamical systems: identification of reaction networks and parameters given experimental data; features and limitations of model reduction strategies; and dynamical properties of biochemical network models.;Inference about the network structure often involves collecting time-dependent chemical concentration data. Many mathematical and computational methods have been developed to identify chemical reaction systems and their parameters from measurements of chemical species concentrations. An inherent difficulty in such identification methodologies is the "fundamental dogma of chemical kinetics": different reaction networks can generate the same ODE model. We address this important limitation by describing conditions for two different reaction networks to produce identical mass-action models. Also, we present a novel method for identifying a biochemical reaction network, using its intrinsic stochasticity.;Complex biochemical network models can sometimes be simplified by eliminating variables using slow/fast scale analysis. A standard way to accomplish this model reduction is quasi-steady state approximation (QSSA). We use algebraic tools to describe an algorithm that call determine whether the QSSA reduction can be achieved. In particular, we conclude that the 100-year-old approach of classic QSSA model reduction cannot be achieved for relatively simple networks. We also show that common alternatives to solving the QSSA equations do not achieve the requirements of model reduction.;Two main open questions about the limiting behavior of biochemical dynamical systems are the Global Attractor Conjecture (GAC) and the Persistence Conjecture. GAC asserts that under certain conditions, a mass-action system has a unique positive equilibrium in each stoichiometric class and this equilibrium is globally asymptotically stable. The Persistence Conjecture states that, under certain hypotheses, a trajectory starting in the interior of the positive orthant does not approach its boundary. GAC is known to be true when the stoichiometric subspace is two-dimensional; the Persistence Conjecture was open in any dimension. I prove the Persistence Conjecture for two-dimensional stoichiometry, and GAC for three-dimensional stoichiometry.