Recent research in experimental and computational biology has revealed the necessity of using stochastic modeling and simulation to investigate the functionality and dynamics of gene networks. However, there is no sophisticated stochastic modeling techniques and efficient stochastic simulation algorithms (SSA) for analyzing and simulating gene networks. Therefore, the objective of this research is to design highly efficient and accurate SSAs, to develop stochastic models for certain real gene networks and to apply stochastic simulation to investigate such gene networks. To achieve this objective, we developed several novel efficient and accurate SSAs. We also proposed two stochastic models for the circadian system of Drosophila and simulated the dynamics of the system. The K-leap method constrains the total number of reactions in one leap to a properly chosen number thereby improving simulation accuracy. Since the exact SSA is a special case of the K-leap method when K=1, the K-leap method can naturally change from the exact SSA to an approximate leap method during simulation if necessary. The hybrid tau/K-leap and the modified K-leap methods are particularly suitable for simulating gene networks where certain reactant molecular species have a small number of molecules. Although the existing tau-leap methods can significantly speed up stochastic simulation of certain gene networks, the mean of the number of firings of each reaction channel is not equal to the true mean. Therefore, all existing tau-leap methods produce biased results, which limit simulation accuracy and speed. Our unbiased tau-leap methods remove the bias in simulation results that exist in all current leap SSAs and therefore significantly improve simulation accuracy without sacrificing speed. In order to efficiently estimate the probability of rare events in gene networks, we applied the importance sampling technique to the next reaction method (NRM) of the SSA and developed a weighted NRM (wNRM). We further developed a systematic method for selecting the values of importance sampling parameters. Applying our parameter selection method to the wSSA and the wNRM, we get an improved wSSA (iwSSA) and an improved wNRM (iwNRM), which can provide substantial improvement over the wSSA in terms of simulation efficiency and accuracy. We also develop a detailed and a reduced stochastic model for circadian rhythm in Drosophila and employ our SSA to simulate circadian oscillations. Our simulations showed that both models could produce sustained oscillations and that the oscillation is robust to noise in the sense that there is very little variability in oscillation period although there are significant random fluctuations in oscillation peeks. Moreover, although average time delays are essential to simulation of oscillation, random changes in time delays within certain range around fixed average time delay cause little variability in the oscillation period. Our simulation results also showed that both models are robust to parameter variations and that oscillation can be entrained by light/dark circles.

The overall goal of this research is to understand roles of gut microbiota metabolites and adipocyte transcription factor (TF) network in health and disease by developing systematic analysis methods. As microbiota can perform diverse biotransformation reactions, the spectrum of metabolites present in the gastrointestinal (GI) tract is extremely complex but only a handful of bioactive microbiota metabolites have been identified. We developed a metabolomics workflow that integrates in silico discovery with targeted mass spectrometry. A computational pathway analysis where microbiota metabolisms are modeled as a single metabolic network is utilized to predict a focused set of targets for multiple reaction monitoring (MRM) analysis. We validated our methodology by predicting, quantifying in murine cecum and feces and characterizing tryptophan (TRP)-derived metabolites as ligands for the aryl hydrocarbon receptor. The adipocyte process of lipid droplet accumulation and differentiation is regulated by multiple TFs that function together in a network. Although individual TF activation is previously reported, construction of an integrated network has been limited due to different measurement conditions. We developed an integrated network model of key TFs - PPAR, C/EBP, CREB, NFAT, FoxO1, and SREBP-1c - underlying adipocyte differentiation. A hypothetic model was determined based on literature, and stochastic simulation algorithm (SSA) was applied to simulate TF dynamics. TF activation profiles at different stages of differentiation were measured using 3T3-L1 reporter cell lines where binding of a TF to its DNA binding element drives expression of the Gaussia luciferase gene. Reaction trajectories calculated by SSA showed good agreement with experimental measurement. The TF model was further validated by perturbing dynamics of CREB using forskolin, and comparing the predicted response with experimental data. We studied the molecular recognition mechanism underlying anti-inflammatory function of a bacterial metabolite, indole in DC2.4 cells. The indole treatment attenuated the fraction of cells that were producing the pro-inflammatory cytokine, TNFα and knockdown of nuclear receptor related 1 (Nurr1; NR4A2) resulted in less indole-derived suppression of TNFα production. The first discovery of NR4A2 as a molecular mediator of the endogenous metabolite, indole is expected to provide a new strategy for treatment of inflammatory disorders.

Continuous time Markov chains are often used in the literature to model the dynamics of a system with low species count and uncertainty in transitions. In this paper, we investigate three particular algorithms that can be used to numerically simulate continuous time Markov chain models (a stochastic simulation algorithm, explicit and implicit tau-leaping algorithms). To compare these methods, we used them to analyze two stochastic infection models with different level of complexity. One of these models describes the dynamics of Vancomycin-Resistant Enterococcus (VRE) infection in a hospital, and the other is for the early infection of Human Immunodeficiency Virus (HIV) within a host. The relative efficiency of each algorithm is determined based on computational time and degree of precision required. The numerical results suggest that all three algorithms have similar computational efficiency for the VRE model due to the low number of species and small number of transitions. However, we found that with the larger and more complex HIV model, implementation and modification of tau-Leaping methods are preferred.

Stochastic effects in cellular systems are usually modeled and simulated with Gillespie's stochastic simulation algorithm (SSA), which follows the same theoretical derivation as the chemical master equation (CME), but the low efficiency of SSA limits its application to large chemical networks. To improve efficiency of stochastic simulations, Haseltine and Rawlings proposed a hybrid of ODE and SSA algorithm, which combines ordinary differential equations (ODEs) for traditional deterministic models and SSA for stochastic models. In this dissertation, accuracy analysis, efficient implementation strategies, and application of of Haseltine and Rawlings's hybrid method (HR) to a budding yeast cell cycle model are discussed. Accuracy of the hybrid method HR is studied based on a linear chain reaction system, motivated from the modeling practice used for the budding yeast cell cycle control mechanism. Mathematical analysis and numerical results both show that the hybrid method HR is accurate if either numbers of molecules of reactants in fast reactions are above certain thresholds, or rate constants of fast reactions are much larger than rate constants of slow reactions. Our analysis also shows that the hybrid method HR allows for a much greater region in system parameter space than those for the slow scale SSA (ssSSA) and the stochastic quasi steady state assumption (SQSSA) method. Implementation of the hybrid method HR requires a stiff ODE solver for numerical integration and an efficient event-handling strategy for slow reaction firings. In this dissertation, an event-handling strategy is developed based on inverse interpolation. Performances of five wildly used stiff ODE solvers are measured in three numerical experiments. Furthermore, inspired by the strategy of the hybrid method HR, a hybrid of ODE and SSA stochastic models for the budding yeast cell cycle is developed, based on a deterministic model in the literature. Simulation results of this hybrid model match very well with biological experimental data, and this model is the first to do so with these recently available experimental data. This study demonstrates that the hybrid method HR has great potential for stochastic modeling and simulation of large biochemical networks. / Ph. D.

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.

An accelerated pace of discovery in biological sciences is made possible by a new generation of computational biology and bioinformatics tools. In this dissertation we develop novel computational, analytical, and high performance simulation techniques for biological problems, with applications to the yeast cell division cycle, and to the RNA-Sequencing of the yellow fever mosquito. Cell cycle system evolves stochastic effects when there are a small number of molecules react each other. Consequently, the stochastic effects of the cell cycle are important, and the evolution of cells is best described statistically. Stochastic simulation algorithm (SSA), the standard stochastic method for chemical kinetics, is often slow because it accounts for every individual reaction event. This work develops a stochastic version of a deterministic cell cycle model, in order to capture the stochastic aspects of the evolution of the budding yeast wild-type and mutant strain cells. In order to efficiently run large ensembles to compute statistics of cell evolution, the dissertation investigates parallel simulation strategies, and presents a new probabilistic framework to analyze the performance of dynamic load balancing algorithms. This work also proposes new accelerated stochastic simulation algorithms based on a fully implicit approach and on stochastic Taylor expansions. Next Generation RNA-Sequencing, a high-throughput technology to sequence cDNA in order to get information about a sample's RNA content, is becoming an efficient genomic approach to uncover new genes and to study gene expression and alternative splicing. This dissertation develops efficient algorithms and strategies to find new genes in Aedes aegypti, which is the most important vector of dengue fever and yellow fever. We report the discovery of a large number of new gene transcripts, and the identification and characterization of genes that showed male-biased expression profiles. This basic information may open important avenues to control mosquito borne infectious diseases. / Ph. D.

The chemical master equation, formulated on the Markov assumption of underlying chemical kinetics, offers an accurate stochastic description of general chemical reaction systems on the mesoscopic scale. The chemical master equation is especially useful when formulating mathematical models of gene regulatory networks and protein-protein interaction networks, where the numbers of molecules of most species are around tens or hundreds. However, solving the master equation directly suffers from the so called "curse of dimensionality" issue. This thesis first tries to study the numerical properties of the master equation using existing numerical methods and parallel machines. Next, approximation algorithms, namely the adaptive aggregation method and the radial basis function collocation method, are proposed as new paths to resolve the "curse of dimensionality". Several numerical results are presented to illustrate the promises and potential problems of these new algorithms. Comparisons with other numerical methods like Monte Carlo methods are also included. Development and analysis of the linear Shepard algorithm and its variants, all of which could be used for high dimensional scattered data interpolation problems, are also included here, as a candidate to help solve the master equation by building surrogate models in high dimensions. / Ph. D.

Fluctuations, or "noise", can play a key role in determining the behaviour of living systems. The molecular-level fluctuations that occur in genetic networks are of particular importance. Here, noisy gene expression can result in genetically identical cells displaying significant variation in phenotype, even in identical environments. This variation can act as a basis for natural selection and provide a fitness benefit to cell populations under stress. This thesis focuses on the development of new conceptual knowledge about how gene expression noise and gene network topology influence drug resistance, as well as new simulation techniques to better understand cell population dynamics. Network topology may at first seem disconnected from expression noise, but genes in a network regulate each other through their expression products. The topology of a genetic network can thus amplify or attenuate noisy inputs from the environment and influence the expression characteristics of genes serving as outputs to the network. The main body of the thesis consists of five chapters: 1. A published review article on the physical basis of cellular individuality. 2. A published article presenting a novel method for simulating the dynamics of cell populations. 3. A chapter on modeling and simulating replicative aging and competition using an object-oriented framework. 4. A published research article establishing that noise in gene expression can facilitate adaptation and drug resistance independent of mutation. 5. An article submitted for publication demonstrating that gene network topology can affect the development of drug resistance. These chapters are preceded by a comprehensive introduction that covers essential concepts and theories relevant to the work presented.

Fluctuations, or "noise", can play a key role in determining the behaviour of living systems. The molecular-level fluctuations that occur in genetic networks are of particular importance. Here, noisy gene expression can result in genetically identical cells displaying significant variation in phenotype, even in identical environments. This variation can act as a basis for natural selection and provide a fitness benefit to cell populations under stress. This thesis focuses on the development of new conceptual knowledge about how gene expression noise and gene network topology influence drug resistance, as well as new simulation techniques to better understand cell population dynamics. Network topology may at first seem disconnected from expression noise, but genes in a network regulate each other through their expression products. The topology of a genetic network can thus amplify or attenuate noisy inputs from the environment and influence the expression characteristics of genes serving as outputs to the network. The main body of the thesis consists of five chapters: 1. A published review article on the physical basis of cellular individuality. 2. A published article presenting a novel method for simulating the dynamics of cell populations. 3. A chapter on modeling and simulating replicative aging and competition using an object-oriented framework. 4. A published research article establishing that noise in gene expression can facilitate adaptation and drug resistance independent of mutation. 5. An article submitted for publication demonstrating that gene network topology can affect the development of drug resistance. These chapters are preceded by a comprehensive introduction that covers essential concepts and theories relevant to the work presented.

This thesis is concerned with mathematical modelling of oncolytic virotherapy: the use of genetically modified viruses to selectively spread, replicate and destroy cancerous cells in solid tumours. Traditional spatially-dependent modelling approaches have previously assumed that virus spread is due to viral diffusion in solid tumours, and also neglect the time delay introduced by the lytic cycle for viral replication within host cells. A deterministic, age-structured reaction-diffusion model is developed for the spatially-dependent interactions of uninfected cells, infected cells and virus particles, with the spread of virus particles facilitated by infected cell motility and delay. Evidence of travelling wave behaviour is shown, and an asymptotic approximation for the wave speed is derived as a function of key parameters. Next, the same physical assumptions as in the continuum model are used to develop an equivalent discrete, probabilistic model for that is valid in the limit of low particle concentrations. This mesoscopic, compartment-based model is then validated against known test cases, and it is shown that the localised nature of infected cell bursts leads to inconsistencies between the discrete and continuum models. The qualitative behaviour of this stochastic model is then analysed for a range of key experimentally-controllable parameters. Two-dimensional simulations of in vivo and in vitro therapies are then analysed to determine the effects of virus burst size, length of lytic cycle, infected cell motility, and initial viral distribution on the wave speed, consistency of results and overall success of therapy. Finally, the experimental difficulty of measuring the effective motility of cells is addressed by considering effective medium approximations of diffusion through heterogeneous tumours. Considering an idealised tumour consisting of periodic obstacles in free space, a two-scale homogenisation technique is used to show the effects of obstacle shape on the effective diffusivity. A novel method for calculating the effective continuum behaviour of random walks on lattices is then developed for the limiting case where microscopic interactions are discrete.