Cells are able to function and survive due to a delicate orchestration of the expression of genes and their downstream products at the genetic, transcriptomic, proteomic, and metabolic levels. Since metabolites are ultimately the causative agents for physiological responses and responsible for much of the functionality of the organism, a comprehensive understanding of cellular functioning mandates deep insights into how metabolism works. Gaining these insights is impeded by the fact that the regulation and dynamics of metabolic networks are often too complex to allow intuitive predictions, which thus renders mathematical modeling necessary as a means for assessing and understanding metabolic systems. The most difficult step of the modeling process is the extraction of information regarding the structure and regulation of the system from experimental data. The work presented here addresses this "inverse" task with three new methods that are applied to models within Biochemical Systems Theory (BST). Alternating Regression (AR) dissects the nonlinear estimation task into iterative steps of linear regression by utilizing the fact that power-law functions are linear in logarithmic space. Eigenvector Optimization (EO) is an extension of AR that is particularly well suited for the identification of model structure. Dynamic Flux Estimation (DFE) is a more general approach that can involve AR and EO and resolves open issues of model validity and quality beyond residual data fitting errors. The necessity of fast solutions to biological inverse problems is discussed in the context of concept map modeling, which allows the conversion of hypothetical network diagrams into mathematical models.

Lignin polymers provide natural rigidity to plant cell walls by forming complex molecular networks with polysaccharides such as cellulose and hemicellulose. This evolved strategy equips plants with recalcitrance to biological and chemical degradation. While naturally beneficial, recalcitrance complicates the use of inedible plant materials as feedstocks for biofuel production. Genetically modifying lignin biosynthesis is an effective way to generate varieties of bioenergy crops with reduced recalcitrance, but certain lignin-modified plants display undesirable phenotypes and/or unexplained effects on lignin composition, suggesting that the process and regulation of lignin biosynthesis is not fully understood. Given the intrinsic complexities of metabolic pathways in plants and the technical hurdles in understanding them purely with experimental methods, the objective of this dissertation is to develop novel computational tools combining static, constraint-based, and dynamic, kinetics-based modeling approaches for a systematic analysis of lignin biosynthesis in wild-type and genetically engineered plants. Pathway models are constructed and analyzed, yielding insights that are difficult to obtain with traditional molecular and biochemical approaches and allowing the formulation of new, testable hypotheses with respect to pathway regulation. These model-based insights, once they are verified experimentally, will form a solid foundation for the rational design of genetic modification strategies towards the generation of lignin-modified crops with reduced recalcitrance. More generically, the methods developed in this dissertation are likely to have wide applicability in similar studies of complex, ill-characterized pathways where regulation occurring at the metabolic level is not entirely known.

This dissertation addresses the development of integrative modeling strategies capable of combining deterministic and stochastic, discrete and continuous, as well as multi-scale features. The first set of studies combines the purely deterministic modeling methodology of Biochemical Systems Theory (BST) with a hybrid approach, using Functional Petri Nets, which permits the account of discrete features or events, stochasticity, and different types of delays. The efficiency and significance of this combination is demonstrated with several examples, including generic biochemical networks with feedback controls, gene regulatory modules, and dopamine based neuronal signal transduction. A study expanding the use of stochasticity toward systems with small numbers of molecules proposes a rather general strategy for converting a deterministic process model into a corresponding stochastic model. The strategy characterizes the mathematical connection between a stochastic framework and the deterministic analog. The deterministic framework is assumed to be a generalized mass action system and the stochastic analogue is in the format of the chemical master equation. The analysis identifies situations where internal noise affecting the system needs to be taken into account for a valid conversion from a deterministic to a stochastic model. The conversion procedure is illustrated with several representative examples, including elemental reactions, Michaelis-Menten enzyme kinetics, a genetic regulatory motif, and stochastic focusing. The last study establishes two novel, particle-based methods to simulate biochemical diffusion-reaction systems within crowded environments. These simulation methods effectively simulate and quantify crowding effects, including reduced reaction volumes, reduced diffusion rates, and reduced accessibility between potentially reacting particles. The proposed methods account for fractal-like kinetics, where the reaction rate depends on the local concentrations of the molecules undergoing the reaction. Rooted in an agent based modeling framework, this aspect of the methods offers the capacity to address sophisticated intracellular spatial effects, such as macromolecular crowding, active transport along cytoskeleton structures, and reactions on heterogeneous surfaces, as well as in porous media. Taken together, the work in this dissertation successfully developed theories and simulation methods which extend the deterministic, continuous framework of Biochemical Systems Theory to allow the account of delays, stochasticity, discrete features or events, and spatial effects for the modeling of biological systems, which are hybrid and multiscale by nature.

As an important endogenous source of reactive oxygen species (ROS), NADPH oxidase 1 (Nox1) has received tremendous attention in the past few decades. It has been identified to play a key role as the initial "kindle," whose activation is crucial for amplifying ROS production through several propagation mechanisms in the vascular system. As a consequence, Nox1 has been implicated in the initiation and genesis of many cardiovascular diseases and has therefore been the subject of detailed investigations. The literature on experimental studies of the Nox1 system is extensive. Numerous investigations have identified essential features of the Nox1 system in vasculature and characterized key components, possible regulatory signals and/or signaling pathways, potential activation mechanisms, a variety of Nox1 stimuli, and its potential physiological and pathophysiological functions. While these experimental studies have greatly enhanced our understanding of the Nox1 system, many open questions remain regarding the overall functionality and dynamic behavior of Nox1 in response to specific stimuli. Such questions include the following. What are the main regulatory and/or activation mechanisms of Nox1 systems in different types of vascular cells? Once Nox1 is activated, how does the system return to its original, unstimulated state, and how will its subunits be recycled? What are the potential disassembly pathways of Nox1? Are these pathways equally important for effectively reutilizing Nox1 subunits? How does Nox1 activity change in response to dynamic signals? Are there generic features or principles within the Nox1 system that permit optimal performance? These types of questions have not been answered by experiments, and they are indeed quite difficult to address with experiments. I demonstrate in this dissertation that one can pose such questions and at least partially answer them with mathematical and computational methods. Two specific cell types, namely endothelial cells (ECs) and vascular smooth muscle cells (VSMCs), are used as "templates" to investigate distinct modes of regulation of Nox1 in different vascular cells. By using a diverse array of modeling methods and computer simulations, this research identifies different types of regulation and their distinct roles in the activation process of Nox1. In the first study, I analyze ECs stimulated by mechanical stimuli, namely shear stresses of different types. The second study uses different analytical and simulation methods to reveal generic features of alternative disassembly mechanisms of Nox1 in VSMCs. This study leads to predictions of the overall dynamic behavior of the Nox1 system in VSMCs as it responds to extracellular stimuli, such as the hormone angiotensin II. The studies and investigations presented here improve our current understanding of the Nox1 system in the vascular system and might help us to develop potential strategies for manipulation and controlling Nox1 activity, which in turn will benefit future experimental and clinical studies.