Browsing by Author "Tyson, John J."

Now showing 1 - 20 of 124
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    Algorithms for regulatory network inference and experiment planning in systems biology
    Pratapa, Aditya (Virginia Tech, 2020-07-17)
    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.
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    Analysis of a generic model of eukaryotic cell-cycle regulation
    Csikasz-Nagy, A.; Battogtokh, D.; Chen, Katherine C.; Novak, Bela; Tyson, John J. (CELL PRESS, 2006-06)
    We propose a protein interaction network for the regulation of DNA synthesis and mitosis that emphasizes the universality of the regulatory system among eukaryotic cells. The idiosyncrasies of cell cycle regulation in particular organisms can be attributed, we claim, to specific settings of rate constants in the dynamic network of chemical reactions. The values of these rate constants are determined ultimately by the genetic makeup of an organism. To support these claims, we convert the reaction mechanism into a set of governing kinetic equations and provide parameter values (specific to budding yeast, fission yeast, frog eggs, and mammalian cells) that account for many curious features of cell cycle regulation in these organisms. Using one-parameter bifurcation diagrams, we show how overall cell growth drives progression through the cell cycle, how cell-size homeostasis can be achieved by two different strategies, and how mutations remodel bifurcation diagrams and create unusual cell-division phenotypes. The relation between gene dosage and phenotype can be summarized compactly in two-parameter bifurcation diagrams. Our approach provides a theoretical framework in which to understand both the universality and particularity of cell cycle regulation, and to construct, in modular fashion, increasingly complex models of the networks controlling cell growth and division.
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    Another turn for p53
    Tyson, John J. (Nature Publishing Group, 2006-01-01)
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    Bifurcation Analysis and Qualitative Optimization of Models in Molecular Cell Biology with Applications to the Circadian Clock
    Conrad, Emery David (Virginia Tech, 2006-04-14)
    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.
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    Bifurcation analysis of a model of the budding yeast cell cycle
    Battogtokh, D.; Tyson, John J. (American Institute of Physics, 2004-09-01)
    We study the bifurcations of a set of nine nonlinear ordinary differential equations that describe regulation of the cyclin-dependent kinase that triggers DNA synthesis and mitosis in the budding yeast, Saccharomyces cerevisiae. We show that Clb2-dependent kinase exhibits bistability (stable steady states of high or low kinase activity). The transition from low to high Clb2-dependent kinase activity is driven by transient activation of Cln2-dependent kinase, and the reverse transition is driven by transient activation of the Clb2 degradation machinery. We show that a four-variable model retains the main features of the nine-variable model. In a three-variable model exhibiting birhythmicity (two stable oscillatory states), we explore possible effects of extrinsic fluctuations on cell cycle progression. (C) 2004 American Institute of Physics.
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    Bifurcation Analysis of a Model of the Frog Egg Cell Cycle
    Borisuk, Mark T. (Virginia Tech, 1997-04-21)
    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.
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    A Bistable Switch Mechanism for Stem Cell Domain Nucleation in the Shoot Apical Meristem
    Battogtokh, D.; Tyson, John J. (Frontiers, 2016-05-23)
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    Bridging Methodological Gaps in Network-Based Systems Biology
    Poirel, Christopher L. (Virginia Tech, 2013-10-16)
    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.
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    BubR1 recruitment to the kinetochore via Bub1 enhances spindle assembly checkpoint signaling
    Banerjee, Anand; Chen, Chu; Humphrey, Lauren; Tyson, John J.; Joglekar, Ajit P. (American Society for Cell Biology, 2022-06-29)
    During mitosis, unattached kinetochores in a dividing cell activate the spindle assembly checkpoint (SAC) and delay anaphase onset by generating the anaphase-inhibitory mitotic checkpoint complex (MCC). These kinetochores generate the MCC by recruiting its constituent proteins, including BubR1. In principle, BubR1 recruitment to signaling kinetochores should increase its local concentration and promote MCC formation. However, in human cells BubR1 is mainly thought to sensitize the SAC to silencing. Whether BubR1 localization to signaling kinetochores by itself enhances SAC signaling remains unknown. Therefore, we used ectopic SAC activation (eSAC) systems to isolate two molecules that recruit BubR1 to the kinetochore, the checkpoint protein Bub1 and the KI and MELT motifs in the kinetochore protein KNL1, and observed their contribution to eSAC signaling. Our quantitative analyses and mathematical modeling show that Bub1-mediated BubR1 recruitment to the human kinetochore promotes SAC signaling and highlight BubR1’s dual role of strengthening the SAC directly and silencing it indirectly.
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    Cell cycle control and environmental response by second messengers in Caulobacter crescentus
    Xu, Chunrui; Weston, Bronson R.; Tyson, John J.; Cao, Yang (2020-09-30)
    Background Second messengers, c-di-GMP and (p)ppGpp, are vital regulatory molecules in bacteria, influencing cellular processes such as biofilm formation, transcription, virulence, quorum sensing, and proliferation. While c-di-GMP and (p)ppGpp are both synthesized from GTP molecules, they play antagonistic roles in regulating the cell cycle. In C. crescentus, c-di-GMP works as a major regulator of pole morphogenesis and cell development. It inhibits cell motility and promotes S-phase entry by inhibiting the activity of the master regulator, CtrA. Intracellular (p)ppGpp accumulates under starvation, which helps bacteria to survive under stressful conditions through regulating nucleotide levels and halting proliferation. (p)ppGpp responds to nitrogen levels through RelA-SpoT homolog enzymes, detecting glutamine concentration using a nitrogen phosphotransferase system (PTS Ntr). This work relates the guanine nucleotide-based second messenger regulatory network with the bacterial PTS Ntr system and investigates how bacteria respond to nutrient availability. Results We propose a mathematical model for the dynamics of c-di-GMP and (p)ppGpp in C. crescentus and analyze how the guanine nucleotide-based second messenger system responds to certain environmental changes communicated through the PTS Ntr system. Our mathematical model consists of seven ODEs describing the dynamics of nucleotides and PTS Ntr enzymes. Our simulations are consistent with experimental observations and suggest, among other predictions, that SpoT can effectively decrease c-di-GMP levels in response to nitrogen starvation just as well as it increases (p)ppGpp levels. Thus, the activity of SpoT (or its homologues in other bacterial species) can likely influence the cell cycle by influencing both c-di-GMP and (p)ppGpp. Conclusions In this work, we integrate current knowledge and experimental observations from the literature to formulate a novel mathematical model. We analyze the model and demonstrate how the PTS Ntr system influences (p)ppGpp, c-di-GMP, GMP and GTP concentrations. While this model does not consider all aspects of PTS Ntr signaling, such as cross-talk with the carbon PTS system, here we present our first effort to develop a model of nutrient signaling in C. crescentus.
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    Cell Cycle Control by a Minimal Cdk Network
    Gerard, Claude; Tyson, John J.; Coudreuse, Damien; Novak, Bela (PLOS, 2015-02-01)
    In present-day eukaryotes, the cell division cycle is controlled by a complex network of interacting proteins, including members of the cyclin and cyclin-dependent protein kinase (Cdk) families, and the Anaphase Promoting Complex (APC). Successful progression through the cell cycle depends on precise, temporally ordered regulation of the functions of these proteins. In light of this complexity, it is surprising that in fission yeast, a minimal Cdk network consisting of a single cyclin-Cdk fusion protein can control DNA synthesis and mitosis in a manner that is indistinguishable from wild type. To improve our understanding of the cell cycle regulatory network, we built and analysed a mathematical model of the molecular interactions controlling the G1/S and G2/M transitions in these minimal cells. The model accounts for all observed properties of yeast strains operating with the fusion protein. Importantly, coupling the model’s predictions with experimental analysis of alternative minimal cells, we uncover an explanation for the unexpected fact that elimination of inhibitory phosphorylation of Cdk is benign in these strains while it strongly affects normal cells. Furthermore, in the strain without inhibitory phosphorylation of the fusion protein, the distribution of cell size at division is unusually broad, an observation that is accounted for by stochastic simulations of the model. Our approach provides novel insights into the organization and quantitative regulation of wild type cell cycle progression. In particular, it leads us to propose a new mechanistic model for the phenomenon of mitotic catastrophe, relying on a combination of unregulated, multi-cyclin-dependent Cdk activities.
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    Cell cycle regulation by feed-forward loops coupling transcription and phosphorylation
    Csikasz-Nagy, Attila; Kapuy, Orsolya; Toth, Attila; Pal, Csaba; Jensen, Lars Juhl; Uhlmann, Frank; Tyson, John J.; Novak, Bela (Nature Publishing Group, 2009-01-01)
    The eukaryotic cell cycle requires precise temporal coordination of the activities of hundreds of ‘executor’ proteins (EPs) involved in cell growth and division. Cyclin-dependent protein kinases (Cdks) play central roles in regulating the production, activation, inactivation and destruction of these EPs. From genome-scale data sets of budding yeast, we identify 126 EPs that are regulated by Cdk1 both through direct phosphorylation of the EP and through phosphorylation of the transcription factors that control expression of the EP, so that each of these EPs is regulated by a feed-forward loop (FFL) from Cdk1. By mathematical modelling, we show that such FFLs can activate EPs at different phases of the cell cycle depending of the effective signs (+ or -) of the regulatory steps of the FFL.We provide several case studies of EPs that are controlled by FFLs exactly as our models predict. The signal-transduction properties of FFLs allow one (or a few) Cdk signal(s) to drive a host of cell cycle responses in correct temporal sequence.
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    Cell-cycle transitions: a common role for stoichiometric inhibitors
    Hopkins, Michael; Tyson, John J.; Novak, Bela (2017-11-07)
    The cell division cycle is the process by which eukaryotic cells replicate their chromosomes and partition them to two daughter cells. To maintain the integrity of the genome, proliferating cells must be able to block progression through the division cycle at key transition points (called "checkpoints") if there have been problems in the replication of the chromosomes or their biorientation on the mitotic spindle. These checkpoints are governed by protein-interaction networks, composed of phase-specific cell-cycle activators and inhibitors. Examples include Cdk1: Clb5 and its inhibitor Sic1 at the G1/S checkpoint in budding yeast, APC: Cdc20 and its inhibitor MCC at the mitotic checkpoint, and PP2A: B55 and its inhibitor, alpha-endosulfine, at the mitotic-exit checkpoint. Each of these inhibitors is a substrate as well as a stoichiometric inhibitor of the cell-cycle activator. Because the production of each inhibitor is promoted by a regulatory protein that is itself inhibited by the cell-cycle activator, their interaction network presents a regulatory motif characteristic of a " feedback-amplified domineering substrate" (FADS). We describe how the FADS motif responds to signals in the manner of a bistable toggle switch, and then we discuss how this toggle switch accounts for the abrupt and irreversible nature of three specific cell-cycle checkpoints.
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    Cellular automata models for excitable media
    Weimar, Jörg Richard (Virginia Tech, 1991-05-15)
    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.
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    Comparison of Domain Nucleation Mechanisms in a Minimal Model of Shoot Apical Meristem
    Battogtokh, D.; Tyson, John J. (2016-04-20)
    Existing mathematical models of the shoot apical meristem (SAM) explain nucleation and confinement of a stem cell domain by Turing's mechanism, assuming that the diffusion coefficients of the activator (WUSCHEL) and inhibitor (CLAVATA) are significantly different. As there is no evidence for this assumption of differential diffusivity, we recently proposed a new mechanism based on a bistable switch model of the SAM. Here we study the bistable-switch mechanism in detail, demonstrating that it can be understood as localized switches of WUSHEL activity in individual cells driven by a non-uniform field of a peptide hormone. By comparing domain formation by Turing and bistable-switch mechanisms on a cell network, we show that the latter does not require the assumptions needed by the former, which are not supported by biological evidences.
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    Computational Analysis of Dynamical Responses to the Intrinsic Pathway of Programmed Cell Death
    Zhang, T. L.; Brazhnik, P.; Tyson, John J. (CELL PRESS, 2009-07)
    Multicellular organisms shape development and remove aberrant cells by programmed cell death ("apoptosis"). Because defective cell death (too little or too much) is implicated in various diseases (like cancer and autoimmunity), understanding how apoptosis is regulated is an important goal of molecular cell biologists. To this end, we propose a mathematical model of the intrinsic apoptotic pathway that captures three key dynamical features: a signal threshold to elicit cell death, irreversible commitment to the response, and a time delay that is inversely proportional to signal strength. Subdividing the intrinsic pathway into three modules (initiator, amplifier, executioner), we use computer simulation and bifurcation theory to attribute signal threshold and time delay to positive feedback in the initiator module and irreversible commitment to positive feedback in the executioner module. The model accounts for the behavior of mutants deficient in various genes and is used to design experiments that would test its basic assumptions. Finally, we apply the model to study p53-induced cellular responses to DNA damage. Cells first undergo cell cycle arrest and DNA repair, and then apoptosis if the damage is beyond repair. The model ascribes this cell-fate transition to a transformation of p53 from "helper" to "killer" forms.
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    Computational modeling of unphosphorylated CtrA:Cori binding in the Caulobacter cell cycle
    Weston, Bronson R.; Tyson, John J.; Cao, Yang (Cell Press, 2021-12-17)
    In the alphaproteobacterium, Caulobacter crescentus, phosphorylated CtrA (CtrA similar to P), a master regulatory protein, binds directly to the chromosome origin (Cori) to inhibit DNA replication. Using a mathematical model of CtrA binding at Cori site [d], we provide computational evidence that CtrA(U) can disc ace CtrA similar to P from Cori at the G1-S transition. Investigation of this interaction within a detailed model of the C. crescentus cell cycle suggests that CckA phosphatase may clear Cori of CtrA similar to P by altering the [CtrA(U)]/[CtrA similar to P] ratio rather than by completely depleting CtrA similar to P. Model analysis reveals that the mechanism allows for a speedier transition into S phase, stabilizes the timing of chromosome replication under fluctuating rates of CtrA proteolysis, and may contribute to the viability of numerous mutant strains. Overall, these results suggest that CtrA(U) enhances the robustness of chromosome replication. More generally, our proposed regulation of CtrA:Cori dynamics may represent a novel motif for molecular signaling in cell physiology.
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    Computational Software for Building Biochemical Reaction Network Models with Differential Equations
    Allen, Nicholas A. (Virginia Tech, 2005-11-11)
    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.
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    Computational Systems Biology Analysis of Cell Reprogramming and Activation Dynamics
    Fu, Yan (Virginia Tech, 2012-07-17)
    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.
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    Computational Tools for Molecular Networks in Biological Systems
    Zwolak, Jason W. (Virginia Tech, 2004-12-15)
    Theoretical molecular biologists try to understand the workings of cells through mathematics. Some theoreticians use systems of ordinary differential equations (ODEs) as the basis for mathematical modelling of molecular networks. This thesis develops algorithms for estimating molecular reaction rate constants within those mathematical models by fitting the models to experimental data. An additional step is taken to fit non-timecourse experimental data (e.g., transformations must be performed on the ODE solutions before the experimental and simulation data are similar, and therefore, comparable). VTDIRECT is used to perform (a deterministic direct search) global estimation and ODRPACK is used to perform (a trust region Levenberg-Marquardt based) local estimation of rate constants. One such transformation performed on the ODE solutions determines the value of the steady state of the ODE solutions. A new algorithm was developed that finds all steady state solutions of the ODE system given that the system has a special structure (e.g., the right hand sides of the ODEs are rational functions). Also, since the rate constants in the models cannot be negative and may have other restrictions on the values, ODRPACK was modified to address this problem of bound constraints. The new Fortran 95 version of ODRPACK is named ODRPACK95.