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Caulobacter crescentus goes through a classic dimorphic cell division cycle to adapt to the stringent environment and reduce intraspecific competition. Caulobacter mother cell gives rise to two progenies with distinct morphology - a motile swarmer cell equipped with a flagellum and a sessile stalked cell equipped with a stalk. Because of the nature of dimorphic lifestyle, Caulobacter becomes a model bacterium to study the cell differentiation, signalling transduction, stress response, and asymmetry development of prokaryotes. The dimorphic cell cycle of Caulobacter is driven by the elaborate spatiotemporal organization of regulatory molecules through regulations of synthesis, degradation, phosphorelay, and localization. There is a wealth of experimental observations about gene/protein interactions and localizations accumulated in recent decades, while several mathematical models have been proposed to study the cell cycle progression in Caulobacter. However, the specific control mechanisms of stress response and spatial asymmetry establishment are yet clearly elucidated, while these mechanisms are of fundamental importance to understanding the bacterial survival strategy and developing the microbial industry. Here we utilize mathematical modeling to study the regulatory network of cell cycle control in C. crescentus, focusing on the stress response and asymmetry development. First, we investigate the starvation response of Caulobacter through the connection of phosphotransferase systems (PTS) and guanine nucleotide-based second messenger system. We have developed a mathematical model to capture the temporal dynamics of vital regulatory second messengers, c-di-GMP (cdG) and guanosine pentaphosphate or tetraphosphate (pppGpp or ppGpp), under normal and stressful conditions. This research suggests that the RelA-SpoT homolog enzymes have the potential to effectively influence the cell cycle in response to nutrition changes by regulating cdG and (p)ppGpp levels. We further integrate the second messenger network into a temporal cell cycle model to investigate molecular mechanisms underlying responses of Caulobacter to nutrition starvation. Our model suggests that the cdG-relevant starvation signal is essential but not sufficient to robustly arrest the cell cycle of Caulobacter. We also demonstrate that there may be unknown pathway(s) reducing CtrA under starvation conditions, which results in delayed cytokinesis in starved stalked cells. The cell cycle development of Caulobacter is determined by the periodical activation and deactivation of the master regulator CtrA. cdG is an essential component of the ClpXP pro- tease complex, which is specifically responsible for the degradation of CtrA. We propose a mathematical model for the hierarchical assembly of ClpXP complexes, together with modeling DNA replication, transcription, and protein interactions, to characterize the Caulobacter cell cycle. Our model suggests that the ClpXP-based proteolysis system contributes to the timing and robustness of the cell cycle progression. Furthermore, we construct a spatiotemporal model with Turing-pattern mechanism to study the morphogenesis and asymmetry establishment during the cell cycle of Caulobacter. We apply reaction-diffusion equations to capture the spatial dynamics of scaffolding proteins PodJ, PopZ, and SpmX, which organize two distinct poles of Caulobacter. The spatial regulations influence the activity and distribution of key cell cycle regulators, governing the dimorphic lifestyle of Caulobacter. Our model captures major spatiotemporal experimental observations of wild-type and mutant cells. It provides predictions of novel mutant strains and explains the spatial regulatory mechanisms of bacterial cell cycle progression. Doctor of Philosophy Cell is the basic unit of life that undergoes a process called 'cell cycle' consisting of DNA replication and cell division to exhibit various functions, abilities, and behaviors. The cell cycle is well organized by complex regulations in time and space that determine when and where changes take place. The regulations behind cell cycle development play important roles for living organisms but are not fully understood. In this dissertation, we utilize mathematical models and focus on a model bacterium, Caulobacter crescentus, to capture characteristics of cell cycle and study the underlying regulations. Caulobacter is widely distributed in freshwater, including environments with poor nutrients. It divides asymmetrically, generating a pair of daughter cells with different appearances and replicative potentials. Therefore, Caulobacter population has the flexibility to save energy by halting DNA replication and to reduce the competition with siblings by settling into different places. We utilize the nature of the asymmetrical division of Caulobacter to quantitatively investigate the control mechanisms of cell cycle development, including how cells detect and respond to external cues and develop different organelles at specific times and locations.

Estrogen receptor (ER)-positive breast cancer is responsive to a number of targeted therapies used clinically. Unfortunately, the continuous application of any targeted therapy often results in resistance to the therapy. Our ultimate goal is to use mathematical modelling to optimize alternating therapies that not only decrease proliferation but also stave off resistance. Toward this end, we measured levels of key proteins and proliferation over a 7-day time course in ER-positive MCF7 breast cancer cells. Treatments included endocrine therapy, either estrogen deprivation, which mimics the effects of an aromatase inhibitor, or fulvestrant, an ER degrader. These data were used to calibrate a mathematical model based on key interactions between ER signaling and the cell cycle. We show that the calibrated model is capable of predicting the combination treatment of fulvestrant and estrogen deprivation. Further, we show that we can add a new drug, palbociclib, to the model by measuring only two key proteins, c-Myc and hyperphosphorylated RB1, and adjusting only parameters associated with the drug. The model is then able to predict the combination treatment of estrogen deprivation and palbociclib. Then we added the dynamics of estrogen concentration in the medium into the model and extended the short-term model to a long-term model. The long-term model can simulate various mono- or combination treatments at different doses over 28 days. In addition to palbociclib, we add another Cdk4/6 inhibitor to the model, abemaciclib, which can induce apoptosis at high concentrations. Then the model can match the effects of abemaciclib treatment at two different doses and also capture the apoptosis effects induced by abemaciclib. After calibrating the model to these different treatment conditions, we used the model to explore the synergism among these different treatments. The mathematical model predicts a significant synergism between palbociclib or abemaciclib in combination with fulvestrant. And the predicted synergisms are verified by experiments. This critical synergism between these Cdk4/6 inhibitors and endocrine therapy could reflect the reason that Cdk4/6 inhibitors achieve pronounced success in clinic trails. Lastly, we used protein biomarkers (cyclinD1, cyclinE1, Cdk4, Cdk6 and Cdk2) and palbociclib dose-response proliferation assays to assess the difference between mono- and alternating therapy after 10 weeks of treatments. But neither the protein levels nor palbociclib dose-response show significant differences after 10 weeks of treatment. Therefore, we cannot conclude that alternating therapy delays palbociclib resistance compared with palbociclib mono-treatment after 10 weeks. Longer term experiments or other methods will be needed to uncover any difference. However, in this research we showed that a mechanism-based mathematical model is able to simulate and predict various effects of clinically-used treatments on ER-positive breast cancer cells at different time scales. And this mathematical model has the potential to explore ideas for potential drug treatments, optimize protocols that limit proliferation, and determine the drugs, doses, and alternating schedule for long term experiments. Doctor of Philosophy Estrogen receptors are proteins found inside breast cancer cells that are activated by the hormone estrogen. Estrogen-receptor positive breast cancer is the most common type of breast cancer and accounts for about 70% of breast cancer tumors. Endocrine therapy, which inhibits estrogen receptor signaling, and Cyclin-dependent kinase 4 and 6 (Cdk4/6) inhibitors are the preferred first-line therapy for patients with estrogen receptor-positive cancers. We built a mathematical model of MCF7 cells (an estrogen receptor-positive breast cancer cell line) in response to these standard first-line therapies. This mathematical model can capture the experimentally observed protein and cell proliferation changes in response to various treatment conditions, including different drug combinations, different doses, and different treatment durations up to 28 days. The model can then be used to look for more effective treatment possibilities. In particular, our mathematical model predicted a strong synergism between Cdk4/6 inhibitors and endocrine therapy, which could allow significant reductions in drug dosage while producing the same effect. This synergism was verified by experiments. In addition to treatment methods where one drug or combination of several drugs is used continuously, we consider alternating among various therapies in a fixed cycle. The mathematical model can help us determine which drugs and which doses might be most appropriate. Since an alternating therapy doesn't inhibit one particular target non-stop, the hope is that alternating therapies can delay the onset of drug resistance, where the drug becomes less effective or stops working completely. Unfortunately, an initial 10- week experiment to test for differences in resistance to a mono-therapy versus an alternating therapy did not show a significant difference, pointing to the need for longer experiments to see if alternating therapies can actually make a difference in resistance. Mathematical models will be important for determining the drugs, doses, and time intervals to be used in these experiments, as figuring out the best options by trial and error in such long-term experiments is not practical.

Mathematical modeling of biological systems can be useful to reveal new insights into biological observations. Here we apply mathematical modeling to study the underlying molecular networks driving observed behaviors of two systems. First, we apply systems biology and dynamic systems theory techniques to reveal new insights into the process of hematopoiesis. More specifically, we search the literature to deduce the underlying molecular mechanism that drives cell fate determination in granulocyte-monocyte progenitor (GMP) cells that are exposed to various cytokines. By converting this molecular mechanism into a set of ordinary differential equations (ODEs), we acquired new insights into the behavior of differentiating GMP cells. Next, we explore the cell cycle of the model prokaryotic organism, Caulobacter crescentus. Caulobacter is a uniquely successful oligotrophic bacterium, found abundantly in freshwater systems. While it is not a pathogenic species, Caulobacter is extremely well studied due to its distinguishable asymmetrical morphology and the ability to synchronize populations by cell cycle stage. We built a detailed mathematical model of the molecular mechanism driving the cell cycle. This research suggests a previously unknown role for the unknown form of the master regulator, CtrA, in regulating the G1-S transition. Furthermore, we incorporate a nutrient signaling model into the cell cycle model to investigate how Caulobacter responds to nutrient deprivation. We find that regulation of DivK phosphorylation is an essential component of the nutrient signaling pathway and demonstrate how starvation signals work together in synergy to manifest in observed cell cycle response. Doctor of Philosophy Every cell in the human body has the same DNA, yet there are cells of all kinds with different jobs, appearances and behaviors. This simple concept is a consequence of complex regulatory systems within cells that dictate what genes are expressed and when. This dissertation breaks down the molecular mechanisms that regulate gene expression in cells and how these mechanisms result in the interesting behaviors and morphologies that have been observed experimentally. By deriving mathematical equations to describe the molecular mechanisms, we simulate how cell behavior might change under different conditions to make novel discoveries. More specifically, we utilize these techniques to study the freshwater bacterium, Caulobacter crescentus, and human cells of the white blood cell lineage. We utilize our models to identify previously unknown aspects of the molecular mechanisms, develop explanations for mysterious cell behaviors and provide interesting predictions that have not been explored experimentally.

I present novel solutions to two different classes of computational problems that arise in the study of complex cellular processes. The first problem arises in the context of planning large-scale genetic cross experiments that can be used to validate predictions of multigenic perturbations made by mathematical models. (i) I present CrossPlan, a novel methodology for systematically planning genetic crosses to make a set of target mutants from a set of source mutants. CrossPlan is based on a generic experimental workflow used in performing genetic crosses in budding yeast. CrossPlan uses an integer-linear-program (ILP) to maximize the number of target mutants that we can make under certain experimental constraints. I apply it to a comprehensive mathematical model of the protein regulatory network controlling cell division in budding yeast. (ii) I formulate several natural problems related to efficient synthesis of a target mutant from source mutants. These formulations capture experimentally-useful notions of verifiability (e.g., the need to confirm that a mutant contains mutations in the desired genes) and permissibility (e.g., the requirement that no intermediate mutants in the synthesis be inviable). I present several polynomial time or fixed-parameter tractable algorithms for optimal synthesis of a target mutant for special cases of the problem that arise in practice. The second problem I address is inferring gene regulatory networks (GRNs) from single cell transcriptomic (scRNA-seq) data. These GRNs can serve as starting points to build mathematical models. (iii) I present BEELINE, a comprehensive evaluation of state-of-the-art algorithms for inferring gene regulatory networks (GRNs) from single-cell gene expression data. The evaluations from BEELINE suggest that the area under the precision-recall curve and early precision of these algorithms are moderate. Techniques that do not require pseudotime-ordered cells are generally more accurate. Based on these results, I present recommendations to end users of GRN inference methods. BEELINE will aid the development of gene regulatory network inference algorithms. (iv) Based on the insights gained from BEELINE, I propose a novel graph convolutional neural network (GCN) based supervised algorithm for GRN inference form single-cell gene expression data. This GCN-based model has a considerably better accuracy than existing supervised learning algorithms for GRN inference from scRNA-seq data and can infer cell-type specific regulatory networks. Doctor of Philosophy A small number of key molecules can completely change the cell's state, for example, a stem cell differentiating into distinct types of blood cells or a healthy cell turning cancerous. How can we uncover the important cellular events that govern complex biological behavior? One approach to answering the question has been to elucidate the mechanisms by which genes and proteins control each other in a cell. These mechanisms are typically represented in the form of a gene or protein regulatory network. The resulting networks can be modeled as a system of mathematical equations, also known as a mathematical model. The advantage of such a model is that we can computationally simulate the time courses of various molecules. Moreover, we can use the model simulations to predict the effect of perturbations such as deleting one or more genes. A biologist can perform experiments to test these predictions. Subsequently, the model can be iteratively refined by reconciling any differences between the prediction and the experiment. In this thesis I present two novel solutions aimed at dramatically reducing the time and effort required for this build-simulate-test cycle. The first solution I propose is in prioritizing and planning large-scale gene perturbation experiments that can be used for validating existing models. I then focus on taking advantage of the recent advances in experimental techniques that enable us to measure gene activity at a single-cell resolution, known as scRNA-seq. This scRNA-seq data can be used to infer the interactions in gene regulatory networks. I perform a systematic evaluation of existing computational methods for building gene regulatory networks from scRNA-seq data. Based on the insights gained from this comprehensive evaluation, I propose novel algorithms that can take advantage of prior knowledge in building these regulatory networks. The results underscore the promise of my approach in identifying cell-type specific interactions. These context-specific interactions play a key role in building mathematical models to study complex cellular processes such as a developmental process that drives transitions from one cell type to another

In eukaryotes, distinct nutrient signals are integrated in order to produce robust cellular responses to fluctuations in the environment. This process of signal integration is attributed to the crosstalk between nutrient specific signaling pathways, as well as the large degree of overlap between their regulatory targets. In the budding yeast Saccharomyces cerevisiae, these distinct pathways have been well characterized. However, the significant overlap between these pathways confounds the interpretation of the overall regulatory logic in terms of nutrient-dependent cell state determination. Here, we propose a literature-curated molecular mechanism of the integrated nutrient signaling pathway in budding yeast, focussing on carbon and nitrogen signaling. We build a computational model of this pathway to reconcile the available experimental data with our proposed molecular mechanism. We evaluate the robustness of the model fit to data with respect to the variations in the values of kinetic parameters used to calibrate the model. Finally, we use the model to make novel, experimentally testable predictions of transcription factor activities in mutant strains undergoing complex nutrient shifts. We also propose a novel framework, called BoolODE for utilizing published Boolean models to generate synthetic datasets used to benchmark the performance of algorithms performing gene regulatory network inference from single cell RNA sequencing data. Doctor of Philosophy An important problem in biology is how organisms sense and adapt to ever changing environments. A good example of an environmental cue that affects animal behavior is the availability of food; scarcity of food forces animals to search for food-rich habitats, or go into hibernation. At the level of single cells, a range of behaviors are observed depending on the amount of food, or nutrients present in the environment. Moreover, different types of nutrients are important for different biological functions in single cells, and each different nutrient type will have to be available in the right quantities to support cellular growth. At the subcellular level, intricate molecular machineries exist which sense the amounts of each nutrient type, and interpret this information in order to make a decision on how best to respond. This interpretation and integration of nutrient information is a complex, poorly understood process even in a simple unicellular organism like the budding yeast. In order to understand this process, termed nutrient signaling, we propose a mathematical model of how yeasts respond to nutrient availability in the environment. Our model advances the state of knowledge by presenting the first comprehensive mathematical model of the nutrient signaling machinery, accounting for a variety of experimental observations from the last three decades of yeast nutrient signaling. We use our model to make predictions on how yeasts might behave when supplied with different combinations of nutrients, which can be verified by experiments. Finally, the cellular machinery that helps yeasts respond to nutrient availability in the environment is very similar to the machinery in cancer cells that causes them to grow rapidly. Our proposed model can serve as a stepping stone towards the construction of a model of cancer's responses to its nutritional environment.

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.

Circadian rhythm is a ~24-h mechanism that keeps our physiology and behavior in synchrony with environmental changes. PERIOD2 (PER2) is a core component of the circadian clock and a candidate tumor suppressor as its knockout expression results in a cancer-prone animal. p53 is an effector in the DNA damage response and regulates downstream effectors by trans-activation. Recent studies in our lab show that PER2 can bind to p53, and regulates the trans-activation function. This project studied the subcellular distribution of PER2 in response to DNA damage, and explored the role of p53 in the regulation of PER2 subcellular distribution. We found that PER2 accumulates in the nucleus in response to DNA damage, and such accumulation is independent of p53. In addition, we analyzed Single Nucleotide Polymorphisms (SNP) of PER2 in the 1000 Genome project to gain insight onto how missense mutations in PER2 lay at the interface of p53:PER2 binding. In a separate project, we also performed bioinformatics analysis on the iron related genes to discuss the circadian regulation of iron genes in the liver. These findings shed light on the regulation of PER2 under genotoxic stress, genetic variations of Per2 in normal human population, and expression of circadian genes under iron controlled diets. Master of Science Circadian rhythm is a ~24-h mechanism that keeps the body in synchrony with the environment. Period2 (Per2) is a gene at the core of circadian rhythm in mammals. In this work, we found that PER2 accumulates in the nucleus of cells in response to DNA damage. In addition, we analyzed the genetic variation of PER2 in general human population to gain insight onto how mutations in PER2 affect the risk of cancer that’s associated to circadian disruption. In a separate project, we performed bioinformatics analysis on the genes related to iron metabolism, and showed pattern of circadian regulation of iron genes in the liver. These findings shed light on how circadian rhythm responds to genotoxic stress, and summarized genetic variations of Per2 in normal human population, and the expression of circadian genes under iron-controlled diets.

The ever-growing size and complexity of molecular network models makes them difficult to construct and understand. Modifying a model that consists of tens of reactions is no easy task. Attempting the same on a model containing hundreds of reactions can seem nearly impossible. We present the JigCell Model Connector, a software tool that supports large-scale molecular network modeling. Our approach to developing large models is to combine together smaller models, making the result easier to comprehend. At the base, the smaller models (called modules) are defined by small collections of reactions. Modules connect together to form larger modules through clearly defined interfaces, called ports. In this work, we enhance the port concept by defining different types of ports. Not all modules connect together the same way, therefore multiple connection options need to exist. Master of Science

Circadian rhythms are self-sustained, 24h, biological oscillatory processes that are present in organisms ranging from bacteria to human. Circadian rhythms, which can be synchronized by external cues, are important for organisms to adjust their behavior, physiological activity, and metabolic reactions to changes in environmental conditions. Another well-established oscillatory mechanism that shares common organizational and regulatory features with the circadian system, is the cell division cycle. Recent findings reveal that some essential regulators are common to both the cell cycle and the circadian clock. The first half of my thesis (Chapter 2-3) focuses on the function of Period 2 (Per2), a key regulatory component of the negative feedback arm of the clock and tumor suppressor protein, as a modulator of cell cycle response. We found that Per2 binds the C-terminus end of the tumor suppressor p53 thus forming a trimeric complex with p53's negative regulator Mdm2 and preventing Mdm2-mediated p53's ubiquitination and degradation. Thus, Per2 stabilizes p53 under unstressed conditions allowing for basal levels of the protein to exist and be available for a rapid response to take place in case of any stressed signals. Our experiments prove that Per2 plays an indispensible role in p53 signaling pathway. The second half of my thesis (Chapter 4-5) focuses on how Mdm2 and Per2 interplay regulate Per2 availability and its impact on circadian clock function. My research found that Mdm2 targets Per2 for ubiquitination as Mdm2 depletion stabilizes Per2 and, conversely, Mdm2 ectopic expression shorten Per2's half-life. Accordingly, association of Per2 to Mdm2 maps C-terminus of the p53 binding region in Mdm2 and thus, the RING domain remains accessible. Next, we tested the hypothesis that Mdm2-dependent ubiquitination of Per2 directly impacts circadian clock period length. Accordingly, addition of sempervirine nitrate (SN), a specific molecular inhibitor of Mdm2, to MEF cells abrogated Per2 ubiquitination leading to the accumulation of a stable pool of Per2. By recording the oscillatory behavior of the Per2:Luc reporter system in MEF cells treated with SN at different circadian times, we found that inhibition of Mdm2 E3 ligase activity promoted phase advance only when treatment took place during the degradation period. This is in agreement with our findings that radiation, but not light pulses, causes the same phase behavior. Considering the established role of both Mdm2 and p53 in the response of cells to genotoxic stress and Per2 in modulating the clock, the existence of the Mdm2-Per2-p53 complex opens the possibility of various stimuli triggering regulatory mechanisms converging in a critical node. Overall, our work provides a holistic view of how signals are integrated at multiple levels to ensure that environmental signals are sense and responses triggered timely. Ph. D.

Cell phenotypic transitions, or cell fate decision making processes, are regulated by complex regulatory networks composed of genes, RNAs, proteins and metabolites. The regulation can take place at the epigenetic, transcriptional, translational, and post-translational levels to name a few. Epigenetic histone modification plays an important role in cell phenotype maintenance and transitions. However, the underlying mechanism relating dynamical histone modifications to stable epigenetic cell memory remains elusive. Incorporating key pieces of molecular level experimental information, we built a statistical mechanics model for the inheritance of epigenetic histone modifications. The model reveals that enzyme selectivity of different histone substrates and cooperativity between neighboring nucleosomes are essential to generate bistability of the epigenetic memory. We then applied the epigenetic modeling framework to the differentiation process of olfactory sensory neurons (OSNs), where the observed 'one-neuron-one-allele' phenomenon has remained as a long-standing puzzle. Our model successfully explains this singular behavior in terms of epigenetic competition and enhancer cooperativity during the differentiation process. Epigenetic level events and transcriptional level events cooperate synergistically in the OSN differentiation process. The model also makes a list of testable experimental predictions. In general, the epigenetic modeling framework can be used to study phenotypic transitions when histone modification is a major regulatory element in the system. Post-transcriptional level regulation plays important roles in cell phenotype maintenance. Our integrated experimental and computational studies revealed such a motif regulating the differentiation of definitive endoderm. We identified two RNA binding proteins, hnRNPA1 and KSRP, which repress each other through microRNAs miR-375 and miR-135a. The motif can generate switch behavior and serve as a noise filter in the stem cell differentiation process. Manipulating the motif could enhance the differentiation efficiency toward a specific lineage one desires. Last we performed mathematical modeling on an epithelial-to-mesenchymal transition (EMT) process, which could be used by tumor cells for their migration. Our model predicts that the IL-6 induced EMT is a stepwise process with multiple intermediate states. In summary, our theoretical and computational analyses about cell phenotypic transitions provide novel insights on the underlying mechanism of cell fate decision. The modeling studies revealed general physical principles underlying complex regulatory networks. Ph. D.

Reaction Diffusion Master Equation (RDME) framework, characterized by the discretization of the spatial domain, is one of the most widely used methods in the stochastic simulation of reaction-diffusion systems. Discretization sizes for RDME have to be appropriately chosen such that each discrete compartment is "well-stirred" and the computational cost is not too expensive. An efficient discretization size based on the reaction-diffusion dynamics of each species is derived in this dissertation. Usually, the species with larger diffusion rate yields a larger discretization size. Partitioning with an efficient discretization size for each species, a multiple grid discretization (MGD) method is proposed. MGD avoids unnecessary molecular jumping and achieves great simulation efficiency improvement. Moreover, reaction-diffusion systems with reaction dynamics modeled by highly nonlinear functions, show large simulation error when discretization sizes are too small in RDME systems. The switch-like Hill function reduces into a simple bimolecular mass reaction when the discretization size is smaller than a critical value in RDME framework. Convergent Hill function dynamics in RDME framework that maintains the switch behavior of Hill functions with fine discretization is proposed. Furthermore, the application of stochastic modeling and simulation techniques to the spatiotemporal regulatory network in Caulobacter crescentus is included. A stochastic model based on Turing pattern is exploited to demonstrate the bipolarization of a scaffold protein, PopZ, during Caulobacter cell cycle. In addition, the stochastic simulation of the spatiotemporal histidine kinase switch model captures the increased variability of cycle time in cells depleted of the divJ genes. Ph. D.

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.

The innate immune system depends for its effectiveness on the function of specialized pattern recognition receptors which enable it to target pathogens for destruction on the basis of conserved molecular patterns such as flagellin or lipopolysaccharide (LPS). Specifically, LPS is recognized by the Toll-like receptor 4 (TLR4), activating a signaling pathway which triggers the production of both pro- and anti-inflammatory mediators. Very low doses of LPS, however, preferentially induce pro-inflammatory cytokines, which can lead to persistent low-grade inflammation, a contributing factor in a host of chronic diseases. The mild pro-inflammatory skewing induced by super-low-dose LPS also potentiates the inflammatory response to later challenge with a higher dose of LPS in a phenomenon known as the "Shwartzman reaction" or "endotoxin priming". We investigated the mechanisms involved in pro-inflammatory skewing by super-low-dose LPS in THP-1 cells and found it to be governed by a regulatory circuit of competitive inhibition between glycogen synthase kinase 3 (GSK3) and Akt, which promote the activity of the transcription factors FoxO1 and CREB, respectively. Super-low-dose LPS mildly activated FoxO1 and pro-inflammatory gene transcription without inducing anti-inflammatory genes or activating CREB, and this pro-inflammatory skewing could be abolished by inhibition of GSK3 or direct activation of CREB. We then examined the dynamics of the LPS response at various different dosages in murine bone-marrow-derived macrophages (BMDM). The pro-inflammatory cytokine IL-12 was most strongly induced by intermediate LPS dosages, with very low or high doses inducing less robust IL-12 production. Knockout of the inhibitory TLR4 pathway molecules Lyn or IRAK-M resulted in sustained induction of IL-12 by high doses of LPS. By activating CREB, we were able to reduce inflammation in WT BMDM, and saw that this corresponded with increased phosphorylation of CREB. Overall, we are confident that this subnetwork is an important switch regulating the resolution of inflammation in response to TLR4 stimulation. Furthermore, we propose that endotoxin priming is an example of the generalized capacity of all signaling networks to recall prior states, and that an appreciation for the history and context of exposure to stimuli is critical for the understanding of signaling behavior. Ph. D.

As a robust system, cells can wisely choose and switch between different signaling programs according to their differentiation stages and external environments. Cancer cells can hijack this plasticity to develop drug resistance. For example, breast cancers that are initially responsive to endocrine therapy often develop resistance robustly. This process is dynamically controlled by interactions of genes, proteins, RNAs and environmental factors at multiple scales. The complexity of this network cannot be understood by studying individual components in the cell. Systems biology focuses on the interactions of basic components, so as to uncover the molecular mechanism of cell physiology with a systemic and dynamical view. Mathematical modeling as a tool in systems biology provides a unique opportunity to understand the underlying mechanisms of endocrine response and resistance in breast cancer. In Chapter 2, I focused on the experimental observations that breast cancer cells can switch between estrogen receptor α (ERα) regulated and growth factor receptor (GFR) regulated signaling pathways for survival and proliferation. A mathematical model based on the signaling crosstalk between ERα and GFR was constructed. The model successfully explains several intriguing experimental findings related to bimodal distributions of GFR proteins in breast cancer cells, which had been lacking reasonable justifications for almost two decades. The model also explains how transient overexpression of ERα promotes resistance of breast cancer cells to estrogen withdrawal. Understanding the non-genetic heterogeneity associated with this survival-signaling switch can shed light on the design of more efficient breast cancer therapies. In Chapter 3, I utilized a novel strategy to model the transitions between the endocrine response and resistance states in breast cancer cells. Using the experimentally observed estrogen sensitivity phenotypes in breast cancer (sensitive, hypersensitive, and supersensitive) as example, I proposed a useful framework of modeling cell state transitions on the energy landscape of breast cancer as a dynamical system. Grounded on the most possible routes of transitions on the breast cancer landscape, a state transition model was developed. By analyzing this model, I investigated the optimum settings of two intuitive strategies, sequential and intermittent treatments, to overcome endocrine resistance in breast cancer. The method used in this study can be generalized to study treatment strategies and improve treatment efficiencies in breast cancer as well as other types of cancer. Ph. D.

Functioning of the living cell is controlled by a complex network of interactions among genes, proteins, and other molecules. A major goal of systems biology is to understand and explain the mechanisms by which these interactions govern the cell's response to various conditions. Molecular interaction networks have proven to be a powerful representation for studying cellular behavior. Numerous algorithms have been developed to unravel the complexity of these networks. Our work addresses the drawbacks of existing techniques. This thesis includes three related research efforts that introduce network-based approaches to bridge current methodological gaps in systems biology. i. Functional enrichment methods provide a summary of biological functions that are overrepresented in an interesting collection of genes (e.g., highly differentially expressed genes between a diseased cell and a healthy cell). Standard functional enrichment algorithms ignore the known interactions among proteins. We propose a novel network-based approach to functional enrichment that explicitly accounts for these underlying molecular interactions. Through this work, we close the gap between set-based functional enrichment and topological analysis of molecular interaction networks. ii. Many techniques have been developed to compute the response network of a cell. A recent trend in this area is to compute response networks of small size, with the rationale that only part of a pathway is often changed by disease and that interpreting small subnetworks is easier than interpreting larger ones. However, these methods may not uncover the spectrum of pathways perturbed in a particular experiment or disease. To avoid these difficulties, we propose to use algorithms that reconcile case-control DNA microarray data with a molecular interaction network by modifying per-gene differential expression p-values such that two genes connected by an interaction show similar changes in their gene expression values. iii. Top-down analyses in systems biology can automatically find correlations among genes and proteins in large-scale datasets. However, it is often difficult to design experiments from these results. In contrast, bottom-up approaches painstakingly craft detailed models of cellular processes. However, developing the models is a manual process that can take many years. These approaches have largely been developed independently. We present Linker, an efficient and automated data-driven method that analyzes molecular interactomes. Linker combines teleporting random walks and k-shortest path computations to discover connections from a set of source proteins to a set of target proteins. We demonstrate the efficacy of Linker through two applications: proposing extensions to an existing model of cell cycle regulation in budding yeast and automated reconstruction of human signaling pathways. Linker achieves superior precision and recall compared to state-of-the-art algorithms from the literature. Ph. D.

Autophagy is a conserved biological stress response in mammalian cells that is responsible for clearing damaged proteins and organelles from the cytoplasm and recycling their contents via the lysosomal pathway. In cases where the stress is not too severe, autophagy acts as a survival mechanism. In cases of severe stress, it may lead to programmed cell death. Autophagy is abnormally regulated in a wide-range of diseases, including cancer. To integrate the existing knowledge about this decision process into a rigorous, analytical framework, we built a mathematical model of cell fate decision mediated by autophagy. The model treats autophagy as a gradual response to stress that delays the initiation of apoptosis to give the cell an opportunity to survive. We show that our dynamical model is consistent with existing quantitative measurements of time courses of autophagic responses to cisplatin treatment. To understand the function of this response in cancer cells we have provided a systems biology experimental framework to study dynamical aspects of autophagy in single cancer cells using live-cell imaging and quantitative uorescence microscopy. This framework can provide new insights on function of autophagic response in cancer cells. Ph. D.

CD4+ T cells are a group of lymphocytes that play critical roles in the immune system. By releasing cytokines, CD4+ T cells regulate other immune cells for maximizing the efficiency of the system. Naive CD4+ T cells are activated and become mature upon engagement with antigens, and the mature CD4+ T cells have several subsets, which play diverse regulatory functions. For the past two decades, our understanding of CD4+ T cells has been advanced through the studies on the differentiation process and the lineage specification of various subsets of these cells. Although in most experimental studies of CD4+ T cells, researchers focused on how transcription factors and signaling molecules influence the differentiation of a particular subset of these cells, many evidence have shown that the differentiation of CD4+ T cells can be heterogeneous in terms of the phenotypes of the cells involved. This dissertation describes a framework that uses mathematical models of the dynamics of the signaling pathways to explain heterogeneous differentiation. We show that the mutual inhibitions among the master regulators govern the formation of multi-stability behavior, which in turn gives rise to heterogeneous differentiation. The framework can be applied to systems with two or more master regulators, and models based on the framework can make specific predictions about heterogeneous differentiations. In addition, this dissertation describes an experimental study on CD4+ T cell differentiation. Being part of the adaptive immune system, the differentiation of CD4+ T cells was previously known to be induced by the signals from the innate immune cells. However, the expression of Toll-like receptor in CD4+ T cells suggests that microbial products can also influence the differentiation directly. Using an in vitro cell differentiation approach, we show that the differentiation and proliferation of CD4+ T cells can be influenced by lipopolysaccharide under the condition that would favor the differentiation of induced regulatory T cells. These theoretical and experimental studies give novel insights on how CD4+ T cells differentiate in response to pathogenic challenges, and help to gain deeper understanding of regulatory mechanisms of the complex immune system. Ph. D.

Synthetic biology endeavors to forward engineer genetic circuits with novel function. A major inspiration for the field has been the enormous success in the engineering of digital electronic circuits over the past half century. This dissertation approaches synthetic biology from the perspective of the engineering design cycle, a concept ubiquitous across many engineering disciplines. First, an analysis of the state of the engineering design cycle in synthetic biology is presented, pointing out the most limiting challenges currently facing the field. Second, a principle commonly used in electronics to weigh the tradeoffs between hardware and software implementations of a function, called co-design, is applied to synthetic biology. Designs to implement a specific logical function in three distinct domains are proposed and their pros and cons weighed. Third, automatic transitioning between an abstract design, its physical implementation, and accurate models of the corresponding system are critical for success in synthetic biology. We present a framework for accomplishing this task and demonstrate how it can be used to explore a design space. A major limitation of the aforementioned approach is that adequate parameter values for the performance of genetic components do not yet exist. Thus far, it has not been possible to uniquely attribute the function of a device to the function of the individual components in a way that enables accurate prediction of the function of new devices assembled from the same components. This lack presents a major challenge to rapid progression through the design cycle. We address this challenge by first collecting high time-resolution fluorescence trajectories of individual cells expressing a fluorescent protein, as well as snapshots of the number of corresponding mRNA molecules per cell. We then leverage the information embedded in the cell-cell variability of the population to extract parameter values for a stochastic model of gene expression more complex than typically used. Such analysis opens the door for models of genetic components that can more reliably predict the function of new combinations of these basic components. Ph. D.

In the past two decades, molecular cell biology has transitioned from a traditional descriptive science into a quantitative science that systematically measures cellular dynamics on different levels of genome, transcriptome and proteome. Along with this transition emerges the interdisciplinary field of systems biology, which aims to unravel complex interactions in biological systems through integrating experimental data into qualitative or quantitative models and computer simulations. In this dissertation, we applied various systems biology tools to investigate two important problems with respect to cellular activation dynamics and reprograming. Specifically, in the first section of the dissertation, we focused on lipopolysaccharide (LPS)-mediated priming and tolerance: a reprogramming in cytokine production in macrophages pretreated with specific doses of LPS. Though both priming and tolerance are important in the immune system's response to pathogens, the molecular mechanisms still remain unclear. We computationally investigated all network topologies and dynamics that are able to generate priming or tolerance in a generic three-node model. Accordingly, we found three basic priming mechanisms and one tolerance mechanism. Existing experimental evidence support these in silico found mechanisms. In the second part of the dissertation, we applied stochastic modeling and simulations to investigate the phenotypic transition of bacteria E.coli between normally-growing cells and persister cells (growth-arrested phenotype), and how this process can contribute to drug resistance. We built up a complex computational model capturing the molecular mechanism on both single cell level and population level. The paper also proposed a novel way to accelerate the phenotypic transition from persister cells to normally growing cell under resonance activation. The general picture of phenotypic transitions should be applicable to a broader context of biological systems, such as T cell differentiation and stem cell reprogramming. Ph. D.

Cellular processes critical to sustaining physiology, including growth, division and differentiation, are carefully governed by intricate control systems. Deregulations in these systems often result in complex diseases such as cancer. Hence, it is crucial to understand the interactions between molecular players of these control systems, their emergent network dynamics, and, ultimately, the overall contribution to cellular physiology. In this dissertation, we have developed a mathematical framework to understand two such cellular systems: an early checkpoint (START) in the budding yeast cell cycle (Chapter 1), and the canonical Wnt signaling pathway involved in cell proliferation and differentiation (Chapter 2). START transition is an important decision point where the cell commits to one round DNA replication followed by cell division. Several years of experimental research have gone into uncovering molecular details of this process, but a unified understanding is yet to emerge. In chapter one, we have developed a comprehensive mathematical model of START transition that incorporates several findings including information about the phosphorylation state of key START proteins and their subcellular localization. In the second chapter, we focus on modeling the canonical Wnt signaling pathway, a cellular circuit that plays a key role in cell proliferation and differentiation. The Wnt pathway is often deregulated in colon cancers. Based on some evidence of bistability in the Wnt signaling pathway, we proposed the existence of a positive feedback loop underlying the activation and inactivation of the core protein complex of the pathway. Bistability is a common feature of biological systems that toggle between ON and OFF states because it ensures robust switching back and forth between the two states. To study and explain the behavior of this dynamical system, we developed a mathematical model. Based on experimentally determined interactions, our simple model recapitulates the observed phenomena of bimodality (bistability) and hysteresis under the effects of the physiological signal (Wnt), a Wnt-mimic (LiCl), and a stabilizer of one of the key members of core complex (IWR-1). Overall, we believe that cell biologists and molecular geneticists can benefit from our work by using our model to make novel quantitative predictions for experimental verification. Ph. D.

By increasing the level of abstraction in the representation of regulatory network models, we can hope to allow modelers to create models that are beyond the threshold of what can currently be expressed reliably. As hundreds of reactions are difficult to understand, maintain, and extend, thousands of reactions become next to impossible without any automation or aid. Using the multistate-species concept we can reduce the number of reactions needed to represent certain systems and thus, lessen the cognitive load on modelers. A multistate species is an entity with a defined range for state variables, which refers to a group of different forms for a specific species. A multistate reaction involves one or more multistate species and compactly represents a group of similar single reactions. In this work, we have extended JCMB (the JigCell Model Builder) to comply with multistate species and reactions modeling and presented a proposal for enhancing SBML (the Systems Biology Markup Language) standards to support multistate models. Master of Science

Mathematical modeling has become increasingly popular as a tool to study regulatory interactions within gene-protein networks. From the modeler's perspective, two challenges arise in the process of building a mathematical model. First, the same regulatory network can be translated into different types of models at different levels of detail, and the modeler must choose an appropriate level to describe the network. Second, realistic regulatory networks are complicated due to the large number of biochemical species and interactions that govern any physiological process. Constructing and validating a realistic mathematical model of such a network can be a difficult and lengthy task. To confront the first challenge, we develop a new modeling approach that classifies components in the networks into three classes of variables, which are described by different rate laws. These three classes serve as "building blocks" that can be connected to build a complex regulatory network. We show that our approach combines the best features of different types of models, and we demonstrate its utility by applying it to the budding yeast cell cycle. To confront the second challenge, modelers have developed rule-based modeling as a framework to build complex mathematical models. In this approach, the modeler describes a set of rules that instructs the computer to automatically generate all possible chemical reactions in the network. Building a mathematical model using rule-based modeling is not only less time-consuming and error-prone, but also allows modelers to account comprehensively for many different mechanistic details of a molecular regulatory system. We demonstrate the potential of rule-based modeling by applying it to the generation of circadian rhythms in cyanobacteria. Ph. D.

Low dose endotoxemia is caused by several health conditions including smoking, alcohol abuse, high fat diets, and aging. Several studies have correlated low dose endotoxemia with increased risks of atherosclerosis, diabetes, and Parkinson's disease. Unlike high doses of endotoxin which induce a strong but transient induction of pro-inflammatory mediators, low doses of endotoxin result in a mild but chronic induction of pro-inflammatory genes. The central hypothesis of our study was that if low doses of endotoxin are capable of inducing mild prolonged inflammation, then a unique signaling circuit must be utilized. In the first study, the molecular mechanisms for the persistent induction of lipocalin 2 (LCN2) in response to 100 ng/mL of lipopolysaccharide (LPS) in kidney fibroblasts was examined. It appears that the intracellular signaling network responsible for the persistent induction of LCN2 requires both activator protein-1 (AP-1) and CCAAT/enhancer binding protein delta (C/ebpδ). Interleukin-1 receptor-associated kinase 1 (IRAK-1) is critical for LCN2 expression. In the second study, the molecular mechanisms governing the persistent induction of interleukin 6 (IL-6) upon a 50 pg/mL challenge of LPS in macrophages was examined. At this dose, only the persistent activation of cJun N-terminal kinase (JNK) and C/ebpδ was observed. IL-6 transcription requires the transient recruitment of activating transcription factor 2 (ATF2) and the persistent recruitment of C/ebpδ to the IL-6 promoter. In the third study, the molecular mechanisms that mediate LPS-induced priming was examined. The results demonstrate that macrophages are able to sense their prior history of exposure to LPS that result in either a priming or tolerance phenotype upon a secondary challenge of LPS. Results suggest that this sensing mechanism involves cross-talk between IRAK-1 and phosphoinositide-3-kinase (PI3K). Collectively, these studies indicate that JNK and C/ebpδ are the primary players responsible for the persistent expression of pro-inflammatory genes during low dose endotoxemia. IRAK-1 is a key intracellular signaling kinase that mediates signaling at low doses of LPS. IRAK-1 is not only critical for low dose induced expression, but also for LPS-induced priming. This research has revealed a novel signaling pathway that could provide new molecular targets for drug development against chronic inflammatory diseases. Ph. D.

Protein regulatory networks are the hallmark of many important biological functionalities. Two of these functionalities are mammalian cell cycle progression and near-perfect adaptive responses. Modeling and simulating these functionalities are crucial stages to understanding and predicting them as systems-level properties of cells. In the context of the mammalian cell cycle, the timing of DNA synthesis, mitosis and cell division is regulated by a complex network of biochemical reactions that control the activities of a family of cyclin-dependent kinases. The temporal dynamics of this reaction network is typically modeled by nonlinear differential equations describing the rates of the component reactions. This approach provides exquisite details about molecular regulatory processes but is hampered by the need to estimate realistic values for the many kinetic constants that determine the reaction rates. To avoid this problem, modelers often resort to "qualitative" modeling strategies, such as Boolean switching networks, but these models describe only the coarsest features of cell cycle regulation. In this work, we describe a hybrid approach that combines features of continuous and discrete networks. The model is evaluated in terms of flow cytometry measurements of cyclin proteins in asynchronous populations of human cell lines. Using our hybrid approach, modelers can quickly create quantitatively accurate, computational models of protein regulatory networks found in various contexts within cells. Large-scale protein regulatory networks, such as the one that controls the progression of the mammalian cell cycle, also contain small-scale motifs or modules that carry out specific dynamical functions. Systematic characterization of smaller, interacting, network motifs whose individual behavior is well known under certain conditions is therefore of great interest to systems biologists. We model and simulate various 3-node network motifs to find near-perfect adaptation behavior. This behavior entails that a system responds to a change in its environmental cues, or signals, by coming back nearly to its pre-signal state even in the continued presence of the signal. We let various topologies evolve in their parameter space such that they eventually stumble upon a region where they score well under a pre-defined scoring metric. We find many such parameter sample sets across various classes of topologies. Ph. D.

Caulobacter crescentus is an important model organism for studying regulation of cell growth, division and differentiation in prokaryotes. C. crescentus undergoes asymmetric division producing two progeny cells with identical genome but different developmental programs: the "swarmer" cell is flagellated and motile, and the "stalked" cell is sessile (attached to a surface by its stalk and holdfast). Only stalked cells undergo chromosome replication and cell division. A swarmer cell must shed its flagellum and grow a stalk before it can enter the replication-division cycle. Based on published experimental evidence, we propose a molecular mechanism controlling the cell division cycle in this bacterium. Our quantitative model of the mechanism illustrates detailed temporal dynamics of regulatory proteins and corresponding physiological changes during the process of cell cycle progression and differentiation of wild-type cells (both stalked cells and swarmer cells) and of a number of known and novel mutant strains. Our model presents a unified view of temporal regulation of protein activities during the asymmetric cell division cycle of C. crescentus and provides an opportunity to study and analyze the system dynamics of the Caulobacter cell cycle (as opposed to the dynamics of individual steps). The model can serve as a starting point for investigating molecular regulations of cell division and differentiation in other genera of alpha-proteobacteria, such as Brucella and Rhizobium, because recent experimental data suggest that these alpha-proteobacteria share similar genetic mechanisms for cell cycle control. Ph. D.

Mathematical models of regulatory networks become more difficult to construct and understand as they grow in size and complexity. Large regulatory network models can be built up from smaller models, representing subsets of reactions within the larger network. This dissertation focuses on novel model construction techniques that extend the ability of biological modelers to construct larger models by supplying them with tools for decomposing models and using the resulting components to construct larger models. Over the last 20 years, molecular biologists have amassed a great deal of information about the genes and proteins that carry out fundamental biological processes within living cells --- processes such as growth and reproduction, movement, signal reception and response, and programmed cell death. The full complexity of these macromolecular regulatory networks is too great to tackle mathematically at the present time. Nonetheless, modelers have had success building dynamical models of restricted parts of the network. Systems biologists need tools now to support composing "submodels" into more comprehensive models of integrated regulatory networks. We have identified and developed four novel processes (fusion, composition, flattening, and aggregation) whose purpose is to support the construction of larger models. Model Fusion combines two or more models in an irreversible manner. In fusion, the identities of the original (sub)models are lost. Beyond some size, fused models will become too complex to grasp and manage as single entities. In this case, it may be more useful to represent large models as compositions of distinct components. In Model Composition one thinks of models not as monolithic entities but rather as collections of smaller components (submodels) joined together. A composed model is built from two or more submodels by describing their redundancies and interactions. While it is appealing in the short term to build larger models from pre-existing models, each developed independently for their own purposes, we believe that ultimately it will become necessary to build large models from components that have been designed for the purpose of combining them. We define Model Aggregation as a restricted form of composition that represents a collection of model elements as a single entity (a "module"). A module contains a definition of pre-determined input and output ports. The process of aggregation (connecting modules via their interface ports) allows modelers to create larger models in a controlled manner. Model Flattening converts a composed or aggregated model with some hierarchy or connections to one without such connections. The relationships used to describe the interactions among the submodels are lost, as the composed or aggregated model is converted into a single large (flat) model. Flattening allows us to use existing simulation tools, which have no support for composition or aggregation. Ph. D.

MtDNA mutations in mammalian cells are implicated in cellular ageing and encephalomyopathies, although mechanisms involved are not completely understood. The mitochondrial genetic bottleneck has puzzled biologists for a long time. Approximate models of genetic bottleneck proposed in the literature do not accurately model underlying biology. Recent studies indicate mitochondrial morphology changes during cellular aging in culture. In particular, the rates of mitochondrial fission and fusion are shown to be in tight balance, though this rate decreases with age. Some proteins involved in mitochondrial morphology maintenance are implicated in apoptosis. Hence, mitochondrial genetic and morphologic dynamics are critical to the life and death of cells. By working closely with experimental collaborators and by utilizing data derived from literature, we have developed stochastic simulation models of mitochondrial genetic and morphologic dynamics. Hypotheses from the mitochondrial genetic dynamics model include: (1) the decay of mtDNA heteroplasmy in blood is exponential and not linear as reported in literature. (2) Blood heteroplasmy measurements are a good proxy for the blood stem cell heteroplasmy. (3) By analyzing our simulation results in tandem with published longitudinal clinical data, we propose for the first time, a way to correct for the patient's age in the analysis of heteroplasmy data. (4) We develop a direct model of the genetic bottleneck process during mouse embryogenesis. (5) Partitioning of mtDNA into daughter cells during blastocyst formation and relaxed replication of mtDNA during the exponential growth phase of primordial germ cells leads to the variation in heteroplasmy inherited by offspring from the same mother. (6) We develop a “simulation control” for experimental studies on mtDNA heteroplasmy variation in cell cultures. Hypothesis from the mitochondrial morphologic dynamics model: (7) A cell adjusts the mitochondrial fusion rate to compensate for the fluctuations in the fission rate, but not vice versa. A deterministic model for this control is proposed. Contributions: extensible simulation models of mitochondrial genetic and morphologic dynamics to aide in the powerful analysis of published and new experimental data. Our results have direct relevance to cell biology and clinical diagnosis. The work also illustrates scientific success by tight integration of theory with practice. Ph. D.

Two parallel deterministic direct search algorithms are combined to find improved parameters for a system of differential equations designed to simulate the cell cycle of budding yeast. Comparing the model simulation results to experimental data is difficult because most of the experimental data is qualitative rather than quantitative. An algorithm to convert simulation results to mutant phenotypes is presented. Vectors of the 143 parameters defining the differential equation model are rated by a discontinuous objective function. Parallel results on a 2200 processor supercomputer are presented for a global optimization algorithm, DIRECT, a local optimization algorithm, MADS, and a hybrid of the two. A second formulation is presented that uses a system of smooth inequalities to evaluate the phenotype of a mutant. Preliminary results of this formulation are given. Master of Science

Oxidative stress is a harmful condition in a cell, tissue, or organ, caused by an imbalance between reactive oxygen species or other oxidants and the capacity of antioxidant defense systems to remove them. These oxidants cause wide-ranging damage to macromolecules, including proteins, lipids, DNA and carbohydrates. Oxidative stress is an important pathophysiologic component of a number of diseases, such as Alzheimer's disease, diabetes and certain cancers. Cells contain effective defense mechanisms to respond to oxidative stress. Despite much accumulated knowledge about these responses, their kinetics, especially the kinetics of early responses is still not clearly understood. The Yap1 transcription factor is crucial for the normal response to a variety of stress conditions including oxidative stress. Previous studies on Yap1 regulation started to measure gene expression profile at least 20 minutes after the induction of oxidative stress. Genes and pathways regulated by Yap1 in early oxidative stress response (within 20 minutes) were not identified in these studies. Here we study the kinetics of early oxidative stress response induced by the cumene hydroperoxide (CHP) in Saccharomyces cerevisiae wild type and yap1 mutant. Gene expression profiles after exposure to CHP were obtained in controlled conditions using Affymetrix Yeast Genome S98 arrays. The oxidative stress response was measured at 8 time points along 120 minutes after the addition of CHP, with the earliest time point at 3 minute after the exposure. Statistical analysis methods, including ANOVA, k-means clustering analysis, and pathway analysis were used to analyze the data. The results from this study provide a dynamic resolution of the oxidative stress responses in S. cerevisiae, and contribute to a richer understanding of the antioxidant defense systems. It also provides a global view of the roles that Yap1 plays under normal and oxidative stress conditions. Ph. D.

Circadian rhythms are the endogenous, roughly 24-hour rhythms that coordinate an organism's interaction with its cycling environment. The molecular mechanism underlying this physiological process is a cell-autonomous oscillator comprised of a complex regulatory network of interacting DNA, RNA and proteins that is surprisingly conserved across many different species. It is not a trivial task to understand how the positive and negative feedback loops interact to generate an oscillator capable of a) maintaining a 24-hour rhythm in constant conditions; b) entraining to external light and temperature signals; c) responding to pulses of light in a rather particular, predictable manner; and d) compensating itself so that the period is relatively constant over a large range of temperatures, even for mutations that affect the basal period of oscillation. Mathematical modeling is a useful tool for dealing with such complexity, because it gives us an object that can be quickly probed and tested in lieu of the experiment or actual biological system. If we do a good job designing the model, it will help us to understand the biology better by predicting the outcome of future experiments. The difficulty lies in properly designing a model, a task that is made even more difficult by an acute lack of quantitative data. Thankfully, our qualitative understanding of a particular phenomenon, i.e. the observed physiology of the cell, can often be directly related to certain mathematical structures. Bifurcation analysis gives us a glimpse of these structures, and we can use these glimpses to build our models with greater confidence. In this dissertation, I will discuss the particular problem of the circadian clock and describe a number of new methods and tools related to bifurcation analysis. These tools can effectively be applied during the modeling process to build detailed models of biological regulatory with greater ease. Ph. D.

The cell is a highly ordered and intricate machine within which a wide variety of chemical processes take place. The full scientific understanding of cellular physiology requires accurate mathematical models that depict the temporal dynamics of these chemical processes. Modelers build mathematical models of chemical processes primarily from systems of differential equations. Although developing new biological ideas is more of an art than a science, constructing a mathematical model from a biological idea is largely mechanical and automatable. This dissertation describes the practices and processes that biological modelers use for modeling and simulation. Computational biologists struggle with existing tools for creating models of complex eukaryotic cells. This dissertation develops new processes for biological modeling that make model creation, verification, validation, and testing less of a struggle. This dissertation introduces computational software that automates parts of the biological modeling process, including model building, transformation, execution, analysis, and evaluation. User and methodological requirements heavily affect the suitability of software for biological modeling. This dissertation examines the modeling software in terms of these requirements. Intelligent, automated model evaluation shows a tremendous potential to enable the rapid, repeatable, and cost-effective development of accurate models. This dissertation presents a case study that indicates that automated model evaluation can reduce the evaluation time for a budding yeast model from several hours to a few seconds, representing a more than 1000-fold improvement. Although constructing an automated model evaluation procedure requires considerable domain expertise and skill in modeling and simulation, applying an existing automated model evaluation procedure does not. With this automated model evaluation procedure, the computer can then search for and potentially discover models superior to those that the biological modelers developed previously. Ph. D.

In somatic cells, checkpoint pathways trigger cell cycle arrest in response to unreplicated or damaged DNA by inhibiting the activity of cyclin-dependent kinases (Cdks). In the Xenopus laevis embryo, checkpoints are not operational until the midblastula transition (MBT). Studies in cell-free egg extracts indicate that a threshold concentration of nuclei, which approximates the MBT concentration, is required to elicit a checkpoint. The checkpoint response to unreplicated DNA in the extract prevents transition into mitosis by inhibiting Cdk1/cyclin B, causing an increase in the minimum amount of cyclin B necessary to enter mitosis, termed the cyclin threshold. Once the threshold of cyclin is maintained or exceeded, the system will proceed into mitosis after a lag time. We have investigated the relationship between nuclear concentration and cell cycle regulation in the extract. By precisely regulating the concentration of cyclin B and nuclear content in extract samples, we have found 1) the concentration of nuclei affects cyclin B thresholds and lag time of entry into mitosis, 2) elevated cyclin thresholds caused by DNA replication blocks are further increased by increasing the concentration of nuclei, and 3) double-stranded DNA breaks in the extract system do not affect cyclin thresholds or lag time of entry into mitosis within the range of nuclear concentrations that can be efficiently replicated. This data provides evidence of the importance of the nucleocytoplasmic ratio in normal cell cycle progression and its importance for checkpoint acquisition during early Xenopus laevis development. Master of Science

The cell cycle of eukaryotes consists of alternation between growth and DNA replication (interphase), and DNA distribution and cell-division (mitosis or M-phase). This process is regulated by a complex network of biochemical reactions. A core part of this network, called the "Cell Cycle engine" is evolutionarily conserved. The dimer of CDK1 (a protein kinase) and Cyclin proteins (the regulatory components), called M-phase Promoting Factor (MPF), and its key regulatory proteins Cdc25 and Wee1, are central parts of this cell cycle engine. Maintaining the fidelity of the DNA during the cell cycle is critical for successful propagation of the cell lineage. In the presence of unreplicated DNA, the cell cycle engine''s progress into mitosis is slowed down (or halted) by regulation of MPF activity through Cdc25 and Wee1. This regulatory event, called the unreplicated DNA checkpoint, was modeled in a rudimentary fashion in the Novak and Tyson (1993) model of frog eggs. Since then, many new experiments have uncovered relevant parts of this network. Here, we include these parts into a detailed model of the unreplicated DNA checkpoint in the cell cycle of frog-egg extracts. This work and future studies of the unreplicated DNA checkpoint will lead to its better understanding and hopefully to some strategies for tackling cancer. Master of Science

The quorum sensing signaling system based on intercellular exchange of N-acyl-homoserine lactones is used by many proteobacteria to regulate the transcription of essential genes in a signal density-dependent manner. It is involved in a number of processes including the development of highly organized bacterial communities, e.g., biofilms, the regulation of expression of virulence factors, production of antibiotics, and bioluminescence. The extensive genetic and biochemical data available on the quorum sensing system in Vibrio fischeri allows the development of a systems biology approach to undertake a spatial and dynamical analysis of the regulation throughout the population. The quorum sensing regulated lux genes are organized in two divergent transcriptional units: luxR and luxICDABEG. The latter contains the genes required for luminescence and the luxI gene necessary for synthesis of an N-acyl-homoserine lactone commonly called autoinducer (AI). The luxR gene codes for a transcriptional regulatory protein that activates the transcription of both operons at a threshold concentration of AI. The positive feedback loop induces a rapid increase of transcription level of the lux genes when a critical population density is reached (reflected by the concentration of AI in the environment). With a combination of molecular biology tools, physiological analysis, and mathematical modeling we identified critical characteristics of the system and expect to assign parameter values in order to achieve a comprehensive understanding of the dynamics. An ordinary differential equation mathematical model is used to investigate the dynamics of the system and derive parameter values. In parallel a novel microfluidic cell culture experimental set-up is used to carefully control environmental parameters as well as to achieve chemostatic conditions for high-density cell populations. An unstable variant of the green fluorescent protein was used as a reporter to follow the time response at a single cell level. Thus spatial organization and noise across the population can be analyzed. Plasmids carrying different genetic constructs were transformed in a recombinant Escherichia coli strain to specifically identify genetic and biochemical elements involved in the regulation of the lux genes under diverse conditions. Then the quantitative data extracted from batch culture and microfluidic assays were used to assign parameter values in the models. The particular question being investigated first is the nature of the regulation to increasing concentration of the signal. The hypothesis tested is that the regulation of the production of the signal by individual cells is biphasic and, therefore, quorum sensing should be robust to global and local variations in cell density. Master of Science

Circadian rhythms are periodic physiological events that recur about every 24 hours. The word circadian derives from the Latin words circa "about" and dies "day". The importance of circadian rhythms is well recognized in many different organisms' survival as well as in human physiology. It was in the 1950's that scientists demonstrated the existence of an endogenous biological clock, and that the clock is temperature compensated. However, the molecular mechanism of circadian rhythms began to come clear only after the discovery of the period (per) gene in Drosophila melanogaster in 1971, and the frequency (frq) gene in Neurospora crassa in 1973. Since the breakthrough discoveries of the per and frq genes and their mutants (short period mutants, perS or frq1, frq2; and long period mutants perL or frq3, frq7), molecular biologists have discovered other crucial components of the mechanism of circadian rhythms. Currently, there are about a dozen identified circadian genes in Drosophila melanogaster. The consensus idea of the mechanism is that it involves two-interlocked feedback loops largely based on transcription-translation controls. However, based on our mathematical models and analysis, we propose that there is also an autocatalytic effect based on proteolysis and stabilization of PER proteins. Based on the dynamics of multiple steady states and limit cycle oscillation, we propose an alternative mechanism for robust temperature compensation. We start with a simple model in order to understand the core dynamics of the clock mechanism, and move to a more comprehensive model. In both cases, we use bifurcation analysis as a tool to understand the dynamics of the system. With our model, we propose hypotheses to be tested in Neurospora crassa. Ph. D.

The budding yeast cell cycle has attracted attention from many experimentalists over the years for its simplicity and amenability to genetic manipulation. Moreover, the regulatory components described in budding yeast, Saccharomyces cerevisiae, are conserved in higher eukaryotes. The budding yeast cell cycle is governed by a complex network of chemical reactions controlling the activity of the cyclin-dependent kinases (CDKs), proteins that drive the major events of the cell cycle. The presence of these proteins is required for the transition from G1 to S phase (Start) whereas their absence permits the transition from S/M to G1 phase (Finish). The cell cycle of budding yeast is based on alternation between these two states. To test the accuracy of this theory against experiments, we built a hypothetical molecular mechanism of the budding yeast cell cycle and transcribed it into differential equations. With a proper choice of kinetic parameters, the differential equations reproduce the main events of the cell cycle such as: the synthesis of cyclins (Cln1,2; Cln3; Clb1,2; Clb5,6) by their transcription factors (SBF, Mcm1, MBF); their association with stoichiometric inhibitors (Sic1, Cdc6); their degradation by SCF and adaptors of the APC (Cdc20, Cdh1). The emphasis was put on mechanisms responsible for the release of Cdc14 from the RENT complex, Cdc14 being a major player in exit from mitosis. Simulations of the wild type strain and more than 100 mutants showed phenotypes in accordance with experimental observations. Some mutants defective in the Start and Finish transitions and the different ways to rescue them will be presented. Ph. D.

In 1993, Novak and Tyson published a comprehensive mathematical model of the regulation of M-phase promoting factor (MPF) activity in Xenopus laevis eggs and egg extracts. Although this model was in agreement with existing and subsequent experimental data, fundamental predictions that the cell cycle is driven by a hysteresis loop have never been validated experimentally. The model's predictions of bifurcations that create and destroy MPF activity, indicative of hysteresis, were tested in this study. Prediction 1: The threshold concentration of cyclin B required to activate MPF is measurably higher than the threshold concentration required to inactivate MPF. The difference in thresholds implies that the MPF control system is hysteretic and bistable. To measure these thresholds, extracts in interphase or M-phase were supplemented with varying concentrations of non-degradable human cyclin B1 protein. MPF activity was determined by the morphology of sperm nuclei and by assays of histone H1 kinase activity. Consistent with the model, the activation threshold was determined to be 40 nM, which is two-fold higher than the inactivation threshold, 20 nM. Prediction 2: For cyclin levels marginally above the activation threshold concentration of cyclin B, there is a dramatic "slowing-down" in the rate of MPF activation. Supra-threshold concentrations of nondegradable cyclin B1 were added to cycloheximide-treated CSF-released extracts, and samples taken at various time-points were analyzed for MPF activity. At 40 nM cyclin B1, just above the activation threshold, the lag time for MPF activation was 45 - 60 minutes; at 50 nM cyclin B1, the lag time was between 30 - 45 minutes; and at 60 nM or higher concentrations of cyclin B1, the lag time was 20 - 30 minutes, thus confirming the prediction of the Novak-Tyson model. Prediction 3: DNA replication checkpoint increases the activation threshold concentration of cyclin B by increasing the hysteresis loop. Cycloheximide-treated, CSF-released extracts containing 1200 sperm nuclei/μl were treated with aphidicolin, then supplemented with varying concentrations of nondegradable cyclin B1. The activation threshold was 100 nM, 2.5 fold higher than in extracts lacking aphidicolin. Conclusions: These studies confirm three predictions of the Novak-Tyson model and indicate that hysteresis underlies cell cycle control in Xenopus egg extracts. These experiments validate use of mathematical models to study complex biological control systems such as the eukayotic cell cycle. Master of Science

During the early development of Xenopus laevis embryos, cells divide without checkpoints. At the midblastula transition (MBT), the cell cycle is remodeled as the division time lengthens and checkpoints are acquired. Initiation of the MBT depends upon the degradation of maternally supplied cyclin E, which is the regulatory partner of the cyclin dependent kinase, Cdk2. To study the program that drives cyclin E degradation and cell cycle remodeling at the MBT, embryos were treated with two cell cycle inhibitors, GST-D34Xic1 and XChk1. Injection of embryos with GST-D34Xic1, a stoichiometric inhibitor of cyclin E/Cdk2, delays degradation of cyclin E and onset of the MBT. GST-D34Xic1 lowers Wee1 level, a kinase that maintains Cdks in an inactivate state. Eventual degradation of cyclin E is preceded by degradation of GST-D34Xic1. The mathematical modelers, Andrea Ciliberto and John Tyson, incorporated the data into a kinetic model and set of ordinary differential equations. The model accurately described the experimental data and made additional predictions, which were tested experimentally. Additionally, embryos were injected with mRNA encoding XChk1, a kinase that activates Wee1 and inhibits Cdc25, the phosphatase opposing Wee1. Like GST-D34Xic1, XChk1 inhibits cyclin E/Cdk2 and delays the degradation of cyclin E. In contrast to GST-D34Xic1, XChk1 elevates the level of Wee1 at a time when sibling controls begin the MBT, despite cell cycle arrest. Since XChk1 inhibits both Cdk1 and Cdk2, and GST-D34Xic1 inhibits only Cdk2, we propose Cdk1 destabilizes Wee1, whereas Cdk2 elevates Wee1 level. Prior to the MBT, when cyclin E/Cdk2 is active, Wee1 is maintained. After cyclin E/Cdk2 is destroyed at the MBT, Wee1 is degraded. Master of Science

Cell cycle models used in biology can be very complex. These models have parameters with initially unknown values. The values of the parameters vastly aect the accuracy of the models in representing real biological cells. Typically people search for the best parameters to these models using computers only as tools to run simulations. In this thesis methods and results are described for a computer program that searches for parameters to a series of related models using well tested algorithms. The code for this program uses ODRPACK for parameter estimation and LSODAR to solve the dierential equations that comprise the model. Master of Science

Insufficient supply and pathogen safety concerns regarding plasma-derived therapeutic proteins, such as fibrinogen and immunoglobulins, have been the impetus for the development of genetic engineering techniques and separations methods for the economical and safe production of these proteins. This study is concerned with the isolation of these important therapeutics from complex media. Immunoaffinity chromatography has been an important method in the isolation of these products, typically being implemented as the final cleanup step yielding an extremely pure, homogenous final product. However, the use of immunoaffinity chromatography in large-scale purification processes have been precluded due to high capital costs and the inherent lability of immunosorbents. Peptide-based affinity sorbents are being developed in order to surmount the inherent limitations posed by monoclonal antibodies that are used as ligands in immunosorbents. The objective of this research is to quantitatively assess the impact of affinity ligand orientation, local density and transport phenomena on peptide-based affnity sorbent performance. The peptides under study herein can form high-affinity complexes with their protein targets, thus these ligands are one of the newest technologies arising from combinatorial chemistry with applications to the difficult problem of purifying high-molecular weight proteins from complex mixtures. Two types of structural motifs which are common to small peptide affinity ligands derived from combinatorial chemistry are studied here: a linear peptide which is comprised of the affinity recognition sequence in its entirety and a chain structure which displays multiple branches of the recognition sequence emanating from a central lysinic core structure. Two recognition sequences are studied here which bind plasma proteins. One peptide recognition sequence forms a high affinity complex with fibrinogen. Another peptide recognition sequence binds the Fc region of immunoglobulins. Immunglobulins are plasma proteins which range in molecular weight from 155 to 900-kDa and are valuable for therapeutic uses for imparting passive immunity. This study seeks to identify factors analogous to those manifested in immunosorbent performance that may also be important in the optimal design of peptide-based affinity sorbents. In general, previous research with the design of immunosorbents have found that immunosorbent performance, i.e., target-binding efficiency or activity, is substantially dependent upon several factors which include effects associated with ligand orientation, and local density as related to steric incumbrance of target binding sites, and transport phenomena as related to under utilization of intra matrix volume. In summary, this study asks the questions: (1) What factors regarding ligand orientation, local ligand density, and intraparticle transport phenomena, are important in the optimal design of peptide affinity sorbents?; and (2) Are these effects analogous to those manifested in immunosorbent performance? This study seeks to investigate the use of techniques used to mitigate the effects associated with these negative factors upon immunosorbent performance in order to elucidate the nature of these same effects upon peptide-based affinity sorbents. For example, oriented ligand immobilization can be facilitated through selective coupling chemistries and the premasking of ligand binding domains prior to immobilization. In addition, the manipulation of local ligand density using novel spatially controlled matrix activation and ligand immobilization methods can be assessed and implemented for the optimization of the performance and design of peptide-based affinity sorbents. This study has found that enhanced transport phenomena into the matrix interior volume can be achieved by using low solids content cellulose matrices having a low extent of crosslinking. This study demonstrates the effective use of these large-particle diameter, low-solids content cellulose hydrogel matrices in immunoaffinity, peptide-based affinity and ion exchange chromatography in the separation of high-molecular weight therapeutic proteins. Ph. D. This report presents an evaluation of the design of low-solids content, large-particle diameter beaded cellulose supports for column-mode protein purification. The study presented here optimizes the molecular accessibility of the cellulose support to high molecular weight proteins relative to the mechanical stability of the support at high operating linear velocities by the manipulation of bead particle diameter and solids content. A novel epoxidegradient activation method (epoxy-GAM) is developed for creating a gradient of support activation for support crosslinking and affinity ligand installation in the preparation of DEAE hydrogel matrices. These cellulose hydrogel supports were evaluated with regards to structure, dynamic and static binding capacity, pressure-flow stability, chemical stability and intraparticle transport phenomena. The utility of the low-solids content, large-particle diameter DEAE hydrogel matrices was demonstrated in a column-mode protein purification using albumin and fibrinogen mixtures.

The cell cycle of the budding yeast, Saccharomyces cerevisiae, is regulated by a complex network of chemical reactions controlling the activity of the cyclin-dependent kinases (CDKs), a family of protein kinases that drive the major events of the cell cycle. A previous mathematical model by Chen et al. (2000) described a molecular mechanism for the Start transition (passage from G1 phase to S/M phase) in budding yeast. In this thesis, my main goal is to extend Chen's model to include new information about the mechanism controlling Finish (passage from S/M phase to G1 phase). Using laws of biochemical kinetics, I transcribed the hypothetical molecular mechanism into a set of differential equations. Simulations of the wild-type cell cycle and the phenotypes of more than 60 mutants provide a thorough understanding of how budding yeast cells exit mitosis. Master of Science

The goal of this paper is to explain the mathematics involved in modeling biochemical oscillations. We first discuss several important biochemical concepts fundamental to the construction of descriptive mathematical models. We review the basic theory of differential equations and stability analysis as it relates to two-variable models exhibiting oscillatory behavior. The importance of the Hopf Bifurcation will be discussed in detail for the central role it plays in limit cycle behavior and instability. Once we have exposed the necessary mathematical framework, we consider several specific models of biochemical oscillators in three or more variables. This will include a detailed analysis of Goodwin's equations and their modification first studied by Painter. Additionally, we consider the consequences of introducing both distributed and discrete time delay into Goodwin's model. We will show that the presence of distributed time lag modifies Goodwin's model in no significant way. The final section of the paper will discuss discrete time lag in the context of a minimal model of the circadian rhythm. In the main, this paper will address mathematical, as opposed to biochemical, issues. Nevertheless, the significance of the mathematics to the biochemistry will be considered throughout. Master of Science

Circadian rhythms are periodic physiological cycles that recur about every 24 hours, by means of which organisms integrate their physiology and behavior to the daily cycle of light and temperature imposed by the rotation of the earth. Circadian derives from the Latin word circa "about" and dies "day". Circadian rhythms have three noteworthy properties. They are endogenous, that is, they persist in the absence of external cues (in an environment of constant light intensity, temperature, etc.). Secondly, they are temperature compensated, that is, the nearly 24 hour period of the endogenous oscillator is remarkably independent of ambient temperature. Finally, they are phase shifted by light. The circadian rhythm can be either advanced or delayed by applying a pulse of light in constant darkness. Consequently, the circadian rhythm will synchronize to a periodic light-dark cycle, provided the period of the driving stimulus is not too far from the period of the endogenous rhythm. A window on the molecular mechanism of 24-hour rhythms was opened by the identification of circadian rhythm mutants and their cognate genes in Drosophila, Neurospora, and now in other organisms. Since Konopka and Benzer first discovered the period mutant in Drosophila in 1971 (Konopka and Benzer, 1971), there have been remarkable developments. Currently, the consensus opinion of molecular geneticists is that the 24-hour period arises from a negative feedback loop controlling the transcription of clock genes. However, a better understanding of this mechanism requires an approach that integrates both mathematical and molecular biology. From the recent discoveries in molecular biology and through a mathematical approach, we propose that the mechanism of circadian rhythm is based upon the combination of both negative and positive feedback. Master of Science

Fertilized frog eggs (and cell-free extracts) undergo periodic oscillations in the activity of "M-phase promoting factor" (MPF), the crucial triggering enzyme for mitosis (nuclear division) and cell division. MPF activity is regulated by a complex network of biochemical reactions. Novak and Tyson, and their collaborators, have been studying the qualitative and quantitative properties of a large system of nonlinear ordinary differential equations that describe the molecular details of this system as currently known. Important clues to the behavior of the model are provided by bifurcation theory, especially characterization of the codimension-1 and -2 bifurcation sets of the differential equations. To illustrate this method, I have been studying a system of 9 ordinary differential equations that describe the frog egg cell cycle with some fidelity. I will describe the bifurcation diagram of this system in a parameter space spanned by the rate constants for cyclin synthesis and cycling degradation. My results suggest either that the cell cycle control system should show dynamical behavior considerably more complex than the limit cycles and steady states reported so far, or that the biochemical rate constants of the system are constrained to avoid regions of parameter space where complex bifurcation points unfold. Ph. D.

In the current study we tested the hypothesis that tumorigenic transformation of T cells may result from aberrant regulation of autocrine growth factor production. We describe for the first time characterization of several normal T cell lines that underwent spontaneous transformation resulting from constitutive expression of interleukin-2 (IL-2) and interleukin 2 receptor (IL-2R) genes. Such cell lines could induce tumors in vivo and furthermore, the growth of such transformed cells both ex vivo and in vivo could be inhibited by monoclonal antibodies (mAbs) or antisense oligonucleotides specific for IL-2 or IL-2R. Interestingly, an in vivo originated T cell lymphoma, LSA, was also found to be dependent on constitutive production of IL-2 and Interleul(in-4 (IL-4). Together, these data demonstrated that dysregulation in the production or responsiveness to autocrine T cell growth factors, plays an important role in T cell transformation and tumorigenicity. Due to their ability to produce T cell growth factors, such tumor cells were found to be highly immunogenic. Thus it was surprising that some T cell lymphomas could still grow in an immunocompetent host and kill the host. To this effect, we investigated additional mechanisms used by such tumor cell lines to grow in an immunocompetent host. It was noted that the tumor cells used mechanisms such as failure to express MHC and production of immunosuppressive cytokines, such as, transforming growth factor-β (TGF-β) and interleukin-10 (IL-10), to evade the action of the immune system. In addition, the T cell lymphomas constitutively expressed F as-ligand and triggered apoptosis of host T cells that expressed Fas. Together, the current study suggests that cytokines produced by the tumor cells play an important role in tumorigenic transformation as vvell as maintenance of tumor growth in an immunocompetent host. Ph. D.

A cellular automaton is developed for simulating excitable media. First, general "masks" as discrete approximations to the diffusion equation are examined, showing how to calculate the diffusion coefficient from the elements of the mask. The mask is then combined with a thresholding operation to simulate the propagation of waves (shock fronts) in excitable media, showing that (for well-chosen masks) the waves obey a linear "speedcurvature" relation with slope given by the predicted diffusion coefficient. The utility of different masks in terms of computational efficiency and adherence to a linear speed-curvature relation is assessed. Then, a cellular automaton model for wave propagation in reaction diffusion systems is constructed based on these "masks" for the diffusion component and on singular perturbation analysis for the reaction component. The cellular automaton is used to model spiral waves in the Belousov-Zhabotinskii reaction. The behavior of the spiral waves and the movement of the spiral tip are analyzed. By comparing these results to solutions of the Oregonator PDE model, the automaton is shown to be a useful and efficient replacement for the standard numerical solution of the PDE's. Master of Science