Numerous challenges arise in modeling and simulation as biochemical networks are discovered with increasing complexities and unknown mechanisms. With the improvement in experimental techniques, biologists are able to quantify genes and proteins and their dynamics in a single cell, which calls for quantitative stochastic models for gene and protein networks at cellular levels that match well with the data and account for cellular noise. This dissertation studies a stochastic spatiotemporal model of the Caulobacter crescentus cell cycle. A two-dimensional model based on a Turing mechanism is investigated to illustrate the bipolar localization of the protein PopZ. However, stochastic simulations are often impeded by expensive computational cost for large and complex biochemical networks. The hybrid stochastic simulation algorithm is a combination of differential equations for traditional deterministic models and Gillespie's algorithm (SSA) for stochastic models. The hybrid method can significantly improve the efficiency of stochastic simulations for biochemical networks with multiscale features, which contain both species populations and reaction rates with widely varying magnitude. The populations of some reactant species might be driven negative if they are involved in both deterministic and stochastic systems. This dissertation investigates the negativity problem of the hybrid method, proposes several remedies, and tests them with several models including a realistic biological system. As a key factor that affects the quality of biological models, parameter estimation in stochastic models is challenging because the amount of empirical data must be large enough to obtain statistically valid parameter estimates. To optimize system parameters, a quasi-Newton algorithm for stochastic optimization (QNSTOP) was studied and applied to a stochastic budding yeast cell cycle model by matching multivariate probability distributions between simulated results and empirical data. Furthermore, to reduce model complexity, this dissertation simplifies the fundamental cooperative binding mechanism by a stochastic Hill equation model with optimized system parameters. Considering that many parameter vectors generate similar system dynamics and results, this dissertation proposes a general α-β-γ rule to return an acceptable parameter region of the stochastic Hill equation based on QNSTOP. Different objective functions are explored targeting different features of the empirical data. / Doctor of Philosophy / Modeling and simulation of biochemical networks faces numerous challenges as biochemical networks are discovered with increased complexity and unknown mechanisms. With improvement in experimental techniques, biologists are able to quantify genes and proteins and their dynamics in a single cell, which calls for quantitative stochastic models, or numerical models based on probability distributions, for gene and protein networks at cellular levels that match well with the data and account for randomness. This dissertation studies a stochastic model in space and time of a bacterium’s life cycle— Caulobacter. A two-dimensional model based on a natural pattern mechanism is investigated to illustrate the changes in space and time of a key protein population. However, stochastic simulations are often complicated by the expensive computational cost for large and sophisticated biochemical networks. The hybrid stochastic simulation algorithm is a combination of traditional deterministic models, or analytical models with a single output for a given input, and stochastic models. The hybrid method can significantly improve the efficiency of stochastic simulations for biochemical networks that contain both species populations and reaction rates with widely varying magnitude. The populations of some species may become negative in the simulation under some circumstances. This dissertation investigates negative population estimates from the hybrid method, proposes several remedies, and tests them with several cases including a realistic biological system. As a key factor that affects the quality of biological models, parameter estimation in stochastic models is challenging because the amount of observed data must be large enough to obtain valid results. To optimize system parameters, the quasi-Newton algorithm for stochastic optimization (QNSTOP) was studied and applied to a stochastic (budding) yeast life cycle model by matching different distributions between simulated results and observed data. Furthermore, to reduce model complexity, this dissertation simplifies the fundamental molecular binding mechanism by the stochastic Hill equation model with optimized system parameters. Considering that many parameter vectors generate similar system dynamics and results, this dissertation proposes a general α-β-γ rule to return an acceptable parameter region of the stochastic Hill equation based on QNSTOP. Different optimization strategies are explored targeting different features of the observed data.

The life cycle of Caulobacter crescentus is of interest because of the asymmetric nature of cell division that gives rise to progeny that have distinct morphology and function. One daughter called the stalked cell is sessile and capable of DNA replication, while the second daughter called the swarmer cell is motile but quiescent. Advances in microscopy combined with molecular biology techniques have revealed that macromolecules are localized in a non-homogeneous fashion in the cell cytoplasm, and that dynamic localization of proteins is critical for cell cycle progression and asymmetry. However, the molecular-level mechanisms that govern protein localization, and enable the cell to exploit subcellular localization towards orchestrating an asymmetric life cycle remain obscure. There are also instances of researchers using intuitive reasoning to develop very different verbal explanations of the same biological process. To provide a complementary view of the molecular mechanism controlling the asymmetric division cycle of Caulobacter, we have developed a mathematical model of the cell cycle regulatory network. Our reaction-diffusion models provide additional insight into specific mechanism regulating different aspects of the cell cycle. We describe a molecular mechanism by which the bifunctional histidine kinase PleC exhibits bistable transitions between phosphatase and kinase forms. We demonstrate that the kinase form of PleC is crucial for both swarmer-to-stalked cell morphogenesis, and for replicative asymmetry in the predivisional cell. We propose that localization of the scaffolding protein PopZ can be explained by a Turing-type mechanism. Finally, we discuss a preliminary model of ParA- dependent chromosome segregation. Our model simulations are in agreement with experimentally observed protein distributions in wild-type and mutant cells. In addition to predicting novel mutants that can be tested in the laboratory, we use our models to reconcile competing hypotheses and provide a unified view of the regulatory mechanisms that direct the Caulobacter cell cycle. / Ph. D.