The statistical mechanics of complex signaling networks : nerve growth factor signaling

It is becoming increasingly appreciated that the signal transduction systems used by eukaryotic cells to achieve a variety of essential responses represent highly complex networks rather than simple linear pathways. While significant effort is being made to experimentally measure the rate constants for individual steps in these signaling networks, many of the parameters required to describe the behavior of these systems remain unknown, or at best, estimates. With these goals and caveats in mind, we use methods of statistical mechanics to extract useful predictions for complex cellular signaling networks. To establish the usefulness of our approach, we have applied our methods towards modeling the nerve growth factor (NGF)-induced differentiation of neuronal cells. Using our approach, we are able to extract predictions that are highly specific and accurate, thereby enabling us to predict the influence of specific signaling modules in determining the integrated cellular response to the two growth factors. We show that extracting biologically relevant predictions from complex signaling models appears to be possible even in the absence of measurements of all the individual rate constants. Our methods also raise some interesting insights into the design and possible evolution of cellular systems, highlighting an inherent property of these systems wherein particular ''soft'' combinations of parameters can be varied over wide ranges without impacting the final output and demonstrating that a few ''stiff'' parameter combinations center around the paramount regulatory steps of the network. We refer to this property -- which is distinct from robustness -- as ''sloppiness.''Comment: 24 pages, 10 EPS figures, 1 GIF (makes 5 multi-panel figs + caption for GIF), IOP style; supp. info/figs. included as brown_supp.pd

Optimal and Model Free Control of Tumor Immune Interaction Dynamic to Schedule Cancer Treatments

Cancer is an intricate disease that can attack different parts of the human body. In the most common types of cancer, abnormal cells divide uncontrollably and impair body tissue. Cross disciplinary research has long aided expansion of our knowledge and ability to approach problems with a different perspective. Engineers and clinicians can collaborate to solve mysteries surrounding cancer cells function and responses. Engineers have contributed to cancer treatment, by studying new ways to diagnose and treat cancer. According to a study by John Hopkins university engineered Nano-particles can induce immune reaction and kill cancer cells. In addition, new ways of delivering cancer therapy to actuate the immune system to kill cancerous cells were found through engineering research. The goal behind modelling biological systems is to drive the states to a desirable outcome using control elements in the dynamic system. In this thesis, we explore the effects of an Intelligent Proportional Integral Derivative (iPID) controllers; using optimal and model free control to improve the state of a cancer patient using recommended safe dosages. A non-linear mathematical model of Ordinary Differential Equations (ODE) is used to simulate a virtual cancer patient using realistic valued parameters. It is important to bear in mind that the human body is complex and variable. In particular, the immune response can vary from one patient to another. The parameters used to model the cancer have been deduced by clinicians and engineers to represent the tumor immune interaction using mathematical equations. The dissertation will begin by exploring the cancer therapy\u27s mathematical model, the controllability and observability to assure that the model is controllable, and explore different control methods and compare the results

Travelling waves in hyperbolic chemotaxis equations

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s [Keller and Segel, J. Theor. Biol., 1971]. The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically

Impact of Interdisciplinary Undergraduate Research in Mathematics and Biology on the Development of a New Course Integrating Five STEM Disciplines

Funded by innovative programs at the National Science Foundation and the Howard Hughes Medical Institute, University of Richmond faculty in biology, chemistry, mathematics, physics, and computer science teamed up to offer first- and second-year students the opportunity to contribute to vibrant, interdisciplinary research projects. The result was not only good science but also good science that motivated and informed course development. Here, we describe four recent undergraduate research projects involving students and faculty in biology, physics, mathematics, and computer science and how each contributed in significant ways to the conception and implementation of our new Integrated Quantitative Science course, a course for first-year students that integrates the material in the first course of the major in each of biology, chemistry, mathematics, computer science, and physics

Nonlinear Models of Neural and Genetic Network Dynamics:\ud \ud Natural Transformations of Łukasiewicz Logic LM-Algebras in a Łukasiewicz-Topos as Representations of Neural Network Development and Neoplastic Transformations \ud

A categorical and Łukasiewicz-Topos framework for Algebraic Logic models of nonlinear dynamics in complex functional systems such as Neural Networks, Cell Genome and Interactome Networks is introduced. Łukasiewicz Algebraic Logic models of both neural and genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable next-state/transfer functions is extended to a Łukasiewicz Topos with an N-valued Łukasiewicz Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis.\u

Biomathematics of Chlamydia

Chlamydia trachomatis (C. trachomatis) related sexually transmitted infections are a major global public health concern. C. trachomatis afflict millions of men, women, and children worldwide and frequently result in serious medical diseases. In this thesis, mathematical modeling is applied in order to comprehend the dynamics of Chlamydia pathogens within host, their interactions with the immune systems, behavior in the presence of other pathogens, transmission dynamics in a human population, and the efficacy of control measures. The thesis begins with a brief introduction of the bacteria Chlamydia in Chapter 1. In Chapter 2, we give a brief detail of the mathematical modeling of infectious diseases, and its specific application to study the pathogen. In Chapter 3, a linear delay differential compartmental model is developed, and its special application is shown for a laboratory experiment conducted to study the intracellular development cycle of Chlamydia. The delay accounts for the time spent by bacteria in their various forms and for the time taken to go through the replication cycle. The mathematical model tracks the number of Chlamydia infected cells at each stage of the cell division cycle. Moreover, the formula for the final size of each compartment is derived. With initial conditions taken from the experiment, the model is fitted to results from the laboratory data. This simple linear model is capable of reflecting the outcomes of the laboratory experiment. In Chapter 4, at a population level, a novel mathematical model is introduced to study the dynamics of the co-infection between C. trachomatis, and herpes simplex virus (HSV). The concept of the model is based on the observation that in an individual simultaneously infected with both pathogens, the presence of HSV will make the Chlamydia persistent. In its persistent phase, Chlamydia is not replicating and is non-infectious. Important threshold parameters are obtained for the persistence of both infections. We prove global stability results for the disease-free and the boundary equilibria by applying the theory of asymptotically autonomous systems. Further, the model is calibrated to disease parameters to determine the population prevalence of both diseases and compare it with epidemiological findings. In Chapter 5, a compartmental maturity structured model is developed to investigate an optimal control problem for the treatment of chronic Chlamydia infection. The model takes into account the interaction of the pathogens with the immune system and its effects on the formation of persistent Chlamydia particles. As the system takes the form of a mixed ODE-PDE system, the results of the conventional form of Pontryagin’s maximal principle for ordinary differential equations are not suitable. For our purpose, we construct an optimal control problem for a general maturity compartmental model, and hence it consists of ordinary and partial differential equations, moreover, the boundary conditions are also nonlinear. For a fixed control, we verify the existence, uniqueness, and boundedness of the solutions. The system is numerically simulated for a variety of cost functions in order to calculate the optimal treatment for curing Chlamyida infection. We believe that since our findings were validated for a general model with maturity structure, they may be applied to any specific compartmental model that is compatible with the established system