Mathematical Modeling of Metabolic-Genetic Networks

Abstract

Systems biology deals with the computational and mathematical modeling of complex biological systems. The aim is to understand the big picture of the system’s dynamics rather than the individual parts by integrating different sciences, e.g., mathematics, physics, biology, computer science, and engineering. In biological systems, mathematical models of biochemical networks are necessary for predicting and optimizing the behavior of cells in culture. Different mathematical models have been discussed, such as discrete models, continuous models, and hybrid models. In a discrete model, the biological system assumes discrete values. A continuous model uses a system of differential equations to describe the change of concentrations of substances in a cell over time. A hybrid model combines both discrete and continuous models. The main challenge in continuous models is to find the kinetic parameter values. In this thesis, we build a kinetic model of a metabolic-genetic network introduced in Covert et al., 2001 that mimics a discrete model of regulatory flux balance analysis (rFBA) which is based on steady-state assumptions. The kinetic model we introduce has unknown parameters, so it is necessary to perform parameter estimation techniques. We perform a parameter estimation technique using data sets generated from a simulation of the rFBA model. In nature, many phenomena of interest are high-dimensional and complex. Thus, model reduction is considered a vital topic in systems biology. Model reduction methods are mathematical techniques that aim to represent a high-dimensional, dynamical system by a low-dimensional system that roughly preserves the main features and characteristics of the original system. The idea of model order reduction is to use the reduced-order model instead of the full-order model in the simulation or optimization of the system to reduce the computational effort and the runtime of the simulations. In this thesis, we discuss two different model reduction methods. The first method assumes a time scale separation, i.e., it assumes two time scales, a fast time scale and a slow time scale, where the fast time scale dynamics converge to a quasi-steady state. The second approach, proper orthogonal decomposition, aims at obtaining low-dimensional approximate descriptions of high-dimensional processes while retaining the most important features of the dynamics. We apply these approaches to different biological system models from the BioModels database.