Stability and Sensitive Analysis of a Model with Delay Quorum Sensing

This paper formulates a delay model characterizing the competition between bacteria and immune system. The center manifold reduction method and the normal form theory due to Faria and Magalhaes are used to compute the normal form of the model, and the stability of two nonhyperbolic equilibria is discussed. Sensitivity analysis suggests that the growth rate of bacteria is the most sensitive parameter of the threshold parameter R0 and should be targeted in the controlling strategies

The Role of Quorum Sensing in Bacterial Colony Dynamics

The quorum sensing (QS) signalling system allows colonies of bacteria to coordinate gene expression to optimise behaviour at low and high cell densities, giving rise to individual and group responses, respectively. The main aim of this thesis is to understand better the important roles of QS in bacterial colony dynamics. Thus a mathematical description was developed to thoroughly explore key mechanisms and parameter sensitivity. The nature of the QS system depends very much on the species. Pseudomonas aeruginosa was chosen as a model species for this study. P. aeruginosa is a Gram-negative bacterium that is responsible for a wide range of chronic infections in humans. Its QS signalling system is known to involve the las, rhl and pqs systems; this thesis focuses on the first two. The las system includes the LasR regulator and LasI synthase, which direct the synthesis of autoinducer 3O-C12-HSL. Similarly, the rhl system consists of the RhlR regulator and RhlI synthase, directing the synthesis of autoinducer C4-HSL. The mathematical model of the las system displays hysteresis phenomena and excitable dynamics. In essence, the system can have two stable steady states reflecting low and high signal molecule production, separated by one unstable steady state. This feature of the las system can give rise to excitable pulse generation with important downstream impact on the rhl system. The las system is coupled to the rhl system in two ways. First, LasR and 3O-C12-HSL activate the expression of their counterpart in the rhl system. Second, 3O-C12-HSL blocks activation of RhlR by C4-HSL. Furthermore, the las-rhl interaction provides a `quorum memory' that allows cells to trigger rhamnolipid production when they are at the edge of colony. It was demonstrated how the dynamical QS system in individual cells and with coupling between cells can affect the dynamics of the bacterial colony

Dynamical Models of Biology and Medicine

Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin

Study of Virus Dynamics by Mathematical Models

This thesis studies virus dynamics within host by mathematical models, and topics discussed include viral release strategies, viral spreading mechanism, and interaction of virus with the immune system. Firstly, we propose a delay differential equation model with distributed delay to investigate the evolutionary competition between budding and lytic viral release strategies. We find that when antibody is not established, the dynamics of competition depends on the respective basic reproduction numbers of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproduction numbers of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, lytic virus can outcompete provided that its reproductive ratio is very high. An explicit threshold is derived. Secondly, we consider model containing two modes for viral infection and spread, one is the diffusion-limited free virus transmission and the other is the direct cell-to-cell transfer of viral particles. By incorporating infection age, a rigorous analysis of the model shows that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number, which is identified explicitly. The formula for the basic reproduction number of our model reveals the effects of various model parameters including the transmission rates of the two modes, and the impact of the infection age. We show that basic reproduction number is underestimated in the existing models that only consider the cell-free virus transmission, or the cell-to-cell infection, ignoring the other. Assuming logistic growth for target cells, we find that if the basic reproduction number is greater than one, the infection can persist and Hopf bifurcation can occur from the positive equilibrium within certain parameter ranges. Thirdly, the repulsion effect of superinfecting virion by infected cells is studied by a reaction diffusion equation model for virus infection dynamics. In this model, the diffusion of virus depends not only on its concentration gradient but also on the concentration of infected cells. The basic reproduction number, linear stability of steady states, spreading speed, and existence of traveling wave solutions for the model are discussed. It is shown that viruses spread more rapidly with the repulsion effect of infected cells on superinfecting virions, than with random diffusion only. For our model, the spreading speed of free virus is not consistent with the minimal traveling wave speed. With our general model, numerical computations of the spreading speed shows that the repulsion of superinfecting vision promotes the spread of virus, which confirms, not only qualitatively but also quantitatively, some recent experimental results. Finally, the effect of chemotactic movement of CD8+ cytotoxic T lymphocytes (CTLs) on HIV-1 infection dynamics is studied by a reaction diffusion model with chemotaxis. Choosing a typical chemosensitive function, we find that chemoattractive movement of CTLs due to HIV infection does not change stability of the positive steady state of the model. However, chemorepulsion movement of CTLs destabilizes the positive steady state as the strength of the chemotactic sensitivity increases. In this case, Turing instability occurs, which can be Hopf bifurcation or steady state bifurcation, and spatial heterogeneous patterns may form

Design, Mathematical Modelling, Construction and Testing of Synthetic Gene Network Oscillators to Establish Roseobacter Clade Bacteria and the Protozoan Trypanosoma brucei as Synthetic Biology Chassis.

The aim of this project is to establish Roseobacter marine bacteria and Trypanosoma brucei (T. brucei) protozoa as synthetic biology chassis. This work addresses the gap within synthetic biology resulting from the limited choice of host cells available for use in practice. This was done by developing synthetic bacterial and trypanosomal genetic regulatory networks (GRNs) which function as an oscillator as well as by developing the necessary protocols and set-ups to allow for the analysis of GRN dynamics within the host. Roseobacter clade bacteria are naturally found in diverse oceanic habitats and have an important ecological role in balancing global carbon levels. This makes Roseobacter an ideal chassis for future efforts to apply synthetic biology to bioremediation and geo-engineering challenges. The aim of this investigation was to establish straight-forward molecular biology procedures in Roseobacter bacteria followed by characterisation and modelling of an E. coli oscillator in Roseobacter. Results showed that Roseobacter synthetic biology is non-trivial. Protozoa are exploited as host cells for industrial production of biotherapeutics due to fast doubling times and host proteins’ mammalian-like post-translational glycosylation. As an established model organism for studying protozoa, T. brucei provided a test case for establishing synthetic biology in this phylum for the first time. T. brucei is highly divergent from eukaryotes commonly used in synthetic biology and possesses a sophisticated genomic machinery to evade host immune systems. The establishment of standard synthetic biology approaches in mathematical modelling and gene network design in T. brucei will underpin application of synthetic biology to enhance the industrial capability of the protozoa as a chassis and to probe its pathobiology. This investigation involved design and assembly of a Goodwin oscillator, followed by characterisation and modelling of the network and the development of a novel experimental set-up for live-cell imaging of single motile trypanosomes. Results showed that T. brucei is a promising novel synthetic biology chassi

Dynamics of Cellular Communities: Insights from Antibiotic-Induced Biofilms, Self-Replicating Oscillators, and Spatially-Extended Communities

Collective behavior is a fascinating phenomenon occurring at many scales in biology. From flocking of birds to synchronization in neural populations, examples abound where local interactions give rise to “macroscopic”, often counterintuitive behavior, at the level of the community. In this thesis, I investigate community behavior in three distinct systems using a combination of theoretical and experimental approaches. The work spans a broad range of topics inspired by dynamics in microbial communities. In Chapter II, we provide a comprehensive theoretical study of synchronization in coupled oscillators, a topic that is among the most widely studied in dynamical systems. However, while past work has focused almost exclusively on populations of a fixed size, I introduce a new model of self-dividing oscillator populations that exhibits a remarkable range of synchronization phenomena as growth rate is varied. Chapter III describes a largely experiment-driven effort to understand a specific and counterintuitive phenomenon: the promotion of microbial community (biofilm) growth by low doses of antibiotic drugs in a medically relevant bacterial species, E. faecalis. We show that for cell wall synthesis inhibitors–which have for decades been among the most widely prescribed classes of antibiotics–low doses stimulate cell lysis and are associated with an increase in extracellular DNA, long believed to be an important structural component of biofilms. We also develop a simple mathematical model that highlights the interplay between the toxicity of the drug and the “beneficial” effects of cell lysis and can be used to predict the impact of various chemical perturbations that impact optimal biofilm growth. Finally, Chapter IV is devoted to ongoing work on spatial pattern formation in two bacterial species, E. coli and E. faecalis, exhibiting cooperative antibiotic resistance via the production of a community good–an enzyme that targets and degrades antibiotics. The work draws on previous theoretical models to predict pattern formation in simple (non-cooperative) populations, which we quantify using customized experimental tools for quantitatively characterizing colony growth over time and space. In addition, we observe a range of new pattern-formation phenomena driven, in part, by the interplay between cell motility, cooperation, and density-dependent cell growth.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138719/1/ywenv_1.pd

Mathematical Models in Oncolytic Virotherapy and Immunology

In this thesis we build mathematical models to address fundamental questions in immunology and virotherapy. We begin with a study of T cell response to pathogens. Although the clonal expansion of T cells is well defined as a tightly regulated process, the mechanisms responsible for this control are not well understood. Guided by experimental data, we design a delay differential model to see if the CD4+ T cell response to infection is directly linked to antigen concentration. Our model successfully captures a series of experimental results, linking T cell expansion to antigen availability. Next, we turn our attention to virotherapy, a relatively novel form of cancer treatment. Introducing a spatial model, we investigate how enhancements in virus design could alter treatment outcome. Using bifurcation theory, we find that certain enhancements may cause undesirable effects in tumour dynamics, such as large oscillations. We then extend our virotherapy model to study a major barrier in the treatment of solid tumours: excess collagen, which is responsible for the lack of diffusion of oncolytic therapies. This investigation leads to a novel virus diffusion term that captures experimental observations. Importantly, we show that the classic diffusion equation, used in many virotherapy models, does not accurately capture the dispersion of virus in collagen-dense tumours, and this may ultimately result in inaccurate predictions of treatment outcome. Finally, we use our new virotherapy model to understand how different collagen-tumour configurations affect treatment outcome. We show that cell-collagen ratio, and gaps in the collagen surface need to be considered to better understand tumour response to treatment. The models developed in this thesis provide sound explanations to fundamental questions in immunology and virotherapy, highlighting key interactions that could significantly advance current therapies

Dynamical Models of biological networks

In der Molekularbiologie sind mathematische Modelle von regulatorischen und metabolischen Netzwerken essentiell, um von einer Betrachtung isolierter Komponenten und Interaktionen zu einer systemischen Betrachtungsweise zu kommen. Genregulatorische Systeme eignen sich besonders gut zur Modellierung, da sie experimentell leicht zugänglich und manipulierbar sind. In dieser Arbeit werden verschiedene genregulatorische Netzwerke unter Zuhilfenahme von mathematischen Modellen analysiert. Weiteres wird ein Modell einer in silico Zelle vorgestellt und diskutiert. Zunächst werden zwei zyklische genregulatorische Netzwerke - der klassische Repressilator und ein Repressilator mit zusätzlicher Autoaktivierung – im Detail mit analytischen Methoden untersucht. Um den Einfluß zufällig schwankender Molekülzahlen auf die Dynamik der beiden Systeme zu untersuchen, werden stochastische Modelle erstellt und die beiden oszillierenden Systeme verglichen. Weiteres werden mögliche Auswirkungen von Genduplikationen auf ein einfaches genregulatorisches Netzwerk untersucht. Dazu wird zunächst ein kleines Netzwerk von GATA Transkriptionsfaktoren, das eine zentrale Rolle in der Regulation des Stickstoffmetabolismus in Hefe spielt, modelliert und das Modell mit experimentellen Daten verglichen, um Parameterregionen einschränken zu können. Außerdem werden potentielle Topologien genregulatorischer Netzwerke von GATA Transkriptionsfaktoren in verwandten Fungi mittels sequenzbasierender Methoden gesucht und verglichen. Im letzten Teil der Arbeit wird MiniCellSim vorgestellt, ein Modell einer selbständigen in silico Zelle. Es erlaubt ein dynamisches System, das eine Protozelle mit einem genregulatorischen Netzwerk, einem einfachen Metabolismus und einer Zellmembran beschreibt, aus einer Sequenz abzuleiten. Nachdem alle Parameter, die zur Berechnung des dynamischen Systems benötigt werden, ohne zusätzliche Eingabe nur aus der Sequenzinformation abgeleitet werden, kann das Modell für Studien zur Evolution von genregulatorischen Netzwerken verwendet werden.In this thesis different types of gene regulatory networks are analysed using mathematical models. Further a computational framework of a novel, self-contained in silico cell model is described and discussed. At first the behaviour of two cyclic gene regulatory systems - the classical repressilator and a repressilator with additional auto-activation - are inspected in detail using analytical bifurcation analysis. To examine the behaviour under random fluctuations, stochastic versions of the systems are created. Using the analytical results sustained oscillations in the stochastic versions are obtained, and the two oscillating systems compared. In the second part of the thesis possible implications of gene duplication on a simple gene regulatory system are inspected. A model of a small network formed by GATA-type transcription factors, central in nitrogen catabolite repression in yeast, is created and validated against experimental data to obtain approximate parameter values. Further, topologies of potential gene regulatory networks and modules consisting of GATA-type transcription factors in other fungi are derived using sequence-based approaches and compared. The last part describes MiniCellSim, a model of a self-contained in silico cell. In this framework a dynamical system describing a protocell with a gene regulatory network, a simple metabolism, and a cell membrane is derived from a string representing a genome. All the relevant parameters required to compute the time evolution of the dynamical system are calculated from within the model, allowing the system to be used in studies of evolution of gene regulatory and metabolic networks

SCALABLE MODELING APPROACHES IN SYSTEMS IMMUNOLOGY

Systems biology seeks to build quantitative predictive models of biological system behavior. Biological systems, such as the mammalian immune system, operate across multiple spatiotemporal scales with a myriad of molecular and cellular players. Thus, mechanistic, predictive models describing such systems need to address this multiscale nature. A general outstanding problem is to cope with the high-dimensional parameter space arising when building reasonably detailed models. Another challenge is to devise integrated frameworks incorporating behavioral characteristics manifested at various organizational levels seamlessly. In this dissertation, I present two research projects addressing problems in immunological, or biological systems in general, using quantitative mechanistic models and machine learning, touching on the aforementioned challenges in scalable modeling. First, I aimed to understand how cell-to-cell heterogeneities are regulated through gene expression variations and their propagation at the single-cell level. To better understand detailed gene regulatory circuit models with many parameters without analytical solutions, I developed a framework called MAchine learning of Parameter-Phenotype Analysis (MAPPA). MAPPA combines machine learning approaches and stochastic simulation methods to dissect the mapping between high- dimensional parameters and phenotypes. MAPPA elucidated regulatory features of stochastic gene-gene correlation phenotypes. Next, I sought to quantitatively dissect immune homeostasis conferring tolerance to self-antigens and responsiveness to foreign antigens. Towards this goal, I built a series of models spanning from intracellular to organismal levels to describe the recurrent reciprocal relationships between self-reactive T cells and regulatory T cells in collaboration with an experimentalist. This effort elucidated critical immune parameters regulating the circuitry enabling the robust suppression of self-reactive T cells, followed by experimental validation. Moreover, by bridging these models across organizational scales, I derived a framework describing immune homeostasis as a dynamical equilibrium between self-activated T cells and regulatory T cells, typically operating well below thresholds that could result in clonal expansion and subsequent autoimmune diseases. I start with an introduction with a perspective linking seemingly contradictory behaviors of the immune system at different scales: microscopic “noise” and macroscopic deterministic outcomes. By connecting these aspects in the adaptive immune system analogously with an ansatz from statistical physics, I introduced a view on how robust immune homeostasis ensues