Applications of Analysis to Mathematical Biology
Poster session will be held on Tuesday, May 22, 11:35am - 1:45pm. Box lunch will be provided. John Cain, Virginia Commonwealth Analysis of Transient Alternans in a Paced Cardiac Fiber Consider a typical experimental protocol in which one end of an idealized, one-dimensional fiber of cardiac cells is periodically stimulated, or paced, resulting in a train of propagating action potentials. If the pacing period is suddenly changed, cells may exhibit alternans, an abnormal beat-to-beat alternation of action potential duration (APD). As a step towards understanding how a fiber responds to such a change, we analyze transient APD alternans in a regime without arrhythmias. We track spatial variation in APD using a sequence of integral equations that we derive from a standard, kinematic model of wave propagation in an excitable medium. The integral equations can be solved in terms of generalized Laguerre polynomials. Invoking asymptotic estimates of the behavior of Laguerre polynomials, we gain valuable physiological information concerning the the fiber's approach to steady-state following the change in the pacing period. Stanca Ciupe, Duke University T Cell Development Following Thymus Transplant in DiGeorge Patients The immune response to infectious agents involves the presence and maintenance of a large number of T cells with highly variable antigen receptors (TCR). In this study, we build a mathematical model that describes the development of T cells following thymus transplant. Our goal is to determine the mechanisms responsible for the establishment of both T cell number and TCR diversity. In particular, we study competition for limiting resources, including self peptide MHC and cytokines like IL-7 and Il-15. We fit the model to patient data measures of T cell concentration and receptor diversity and estimate the model parameters. The results help us determine the balance needed between the local and global resources for the establishment of a mature T cell repertoire. Laura Ellwein, North Carolina State University Blood Pressure and Blood Flow Variation During Hyperventilation Short-term cardio-respiratory responses to ventilation changes require complex interactions between autonomic reflexes and cerebral autoregulation. We present a model which attempts to predict dynamic changes in beat-to-beat arterial blood pressure, middle cerebral artery blood flow, and expiratory partial pressure of carbon dioxide during voluntary hyperventilation. An inverse least-sqaures problem is formulated to identify model parameters specific to an individual patient's time series in order to minimize the error between our model and the data for blood pressure, blood flow, and CO2. Zhaosheng Feng, University of Texas-Pan American Approximate Solutions to the Fisher Equation and Applications in Population Dynamics There is the widespread existence of wave phenomena in physics, chemistry and biology. This clearly necessitates a study of traveling waves in depth and of the modeling and analysis involved. In the present paper, we study a nonlinear reaction-diffusion equation, which can be regarded as a generalization of the Fisher equation. This equation is used as a density-dependent diffusion model, in the one-dimensional situation, for studying insect and animal dispersal with growth dynamics, and as a genetic model arising from the classical theory of population genetics and combustion. We focus on the analysis of traveling waves and present an approximate wave solution by means of the Adomian decomposition method and the Lie symmetry method. The results will help us to understand and anticipate how the population disperses to regions of lower density more rapidly as the population gets more crowded or environmental factors and geographical resources get changed. Paula Grajdeanu, Mathematical Biosciences Institute Effect of Tubular Inhomogeneities on the Tubuloglomerular Feedback System We use a generalized mathematical model of rat thick ascending limb (TAL) of the loop of Henle to investigate the impact of variable TAL inner radius and variable NaCl transport rate on the TGF-mediated oscillations. An analytic bifurcation analysis of the TGF minimal model with TAL backleak provides fundamental insight into how oscillatory states depend on the physiological parameters of the model. Several attempts has been made to formulate mathematical models of the TGF system that is able to reproduce both the regular oscillations (in normotensive rats), and the irregular fluctuations (in spontaneously hypertensive rats). However, in most cases the models have been successful in describing the regular oscillations, but have failed to reproduce the irregular fluctuations (but by coupling of two nephorns). We hypothesize that irregular oscillations in hypertensive rats are attributable, at least in part, to the tubule's spatial inhomogeneities. Nassar Haidar, Hariri Canadian University Embedditive Solution to the Spiral Wave Cardiac BVP We report on an extension of the embedditive solution to certain nonlinear Cauchy problems [1] to handle two-point nonlinear nonhomogeneous boundary value problems (BVP's) of one-dimensional heart reaction equations. A constructive traveling wave-type solution is shown to be derivable iteratively via solving a pertaining system of nonlinear Volterra integral equations. Ohnishi Isamu, Hiroshima University Mathematical Analysis to adaptive network of Plasmodium system In this talk, we consider about a mathematical model for adaptive network made by the plasmodium. The organism contains a tube network by means of which nutrients and signals circulate through the body. The tube network changes its shape to connect two exits through the shortest path when the organism is put in a maze and food is placed at two exits. Recently, a mathematical model for this adaptation process of the plasmodium has been proposed. Here we analyze it mathematically rigorously. In ring-shaped network and Wheatstone bridge-shaped network, we mainly show that the globally asymptotically stable equilibrium point of the model corresponds to the shortest path connecting two special points on the network in the case where the shortest path is determined uniquely. From the viewpoint of mathematical technique, especially in the case of Wheatstone bridge-shaped network, we show that there is a simple but novel device used here by which we prove the global asymptotic stability, even when Lyapunov function cannot be constructed. This is a joint work with Tomoyuki Miyaji (my D1 student). Yun Kang, Arizona State University A Model on Interaction Between Plant and Herbivore: Modeling on Gypsy Moth The interaction between a forest pest(gypsy moth) and the tree cover is moderated by availability and recycling nutrients in the soil. A non-overlapping model is developed that determines the dynamics of this interaction as a function of the total available nutrients and the nutrient uptake rate. The most interesting phenomena observed involve bistability between chaotic and stationary dynamics. We also found the existence of 8-periodical cycle is matching the fact that gypsy moth has outbreak every 8-10 years. Tim Lucas, Duke University Numerical Solutions of an Immunology Model Using Reaction-Diffusion Equations with Stochastic Source Terms When immune cells detect foreign molecules, they secrete soluble factors that attract other immune cells to the site of the infection. In this thesis, I study numerical solutions to a model this behavior proposed by Thomas Kepler. The soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the cells is stochastic, but biased toward the gradient of the soluble factors. The solution to this system exists for all time and remains positive, the supremum is a priori bounded and the derivatives are bounded for finite time. I have developed a numerical method for solving the reaction-diffusion stochastic system based on a first order splitting scheme. This method makes use of known first order schemes for solving the diffusion, the reaction and the stochastic differential equations separately. The domain is discretized using finite elements and the diffusion is solved using a backward Euler scheme combined with multigrid. The reaction and stochastic differential equations are solved using standard first order schemes. Garrett Mitchener, College of Charleston A Model of Language Variation and Change: Change in an age-structured population in which the change is caused by a prediction-driven instability. To model language change in a population, one might construct a simple one-dimensional differential equation for the fraction of the population using the new form of the language. However, this is unsatisfactory for modeling a population whose language is perpetually in flux: Generically, solutions converge to a stable fixed point representing a population dominated by one language. Furthermore, converting a one-dimensional model to a stochastic differential equation is not much of an improvement. Solutions tend to hover around one fixed point and only rarely change to another, and a change in progress is likely to reverse itself. I will present an improved model, in which an age-structured population learns language using a predictive algorithm: Children are sensitive to trends in the population, and will adjust their speech to avoid a language they predict will soon be outdated. In its deterministic form, the model is two-dimensional with two stable fixed points representing two languages, both of which are very close to the separatrix between their basins of attraction. In the stochastic form, random fluctuations cause the population to alternate between the two stable fixed points, and once a new language begins to take over, the change runs to completion in harmony with data from historical linguistics. Intuitively, the model works by supposing that children will occasionally detect an accidental correlation between age and speech patterns, then predict which speech patterns are becoming outdated. Their speech then accelerates the change, driving the population over the nearby separatrix and over to the other stable fixed point, where the process can repeat itself. Sarah D. Olson, North Carolina State University Modeling Articular Cartilage Regeneration Using Hydrogel Scaffolds Osteochondral defects are ``holes'' in articular cartilage that result from degradation of the tissue extracellular matrix due to osteoarthritis. Biocompatible hydrogel scaffolds have potential utility for filling such defects to restore mechanical integrity and facilitate cell proliferation and tissue regeneration in the defect region. We model the process of cartilage regeneration in a hydrogel scaffold seeded with cartilage cells (chondrocytes). We formulate and analyze reaction models for dynamical interactions between nutrients, hydrogel and extracellular matrix constituents in an evolving gel-tissue construct. Angela Reynolds, University of Pittsburgh A Model of the Acute Inflammatory Response Containing Separate Compartments for Blood and Tissue During an inflammatory response to an infection within the tissue, local macrophages are activated. These activated macrophages produce cytokines that diffuse into the blood and trigger a local influx of activated neutrophils. The interplay between compartments is essential in eliminating infection and restoring homeostasis. Therefore, we expanded our previous acute inflammation model (Reynolds, 2006) to incorporate a distinction between immune components that are in the blood and those that are in the tissue. We derived a model for the acute inflammatory response consisting of differential equations for immune system components in both the tissue and/or blood, depending on their role in the immune response. We also included variables that track the level of inflammation, to determine its effects on diffusion, and the level of tissue integrity. This model was developed via a modular approach in which multiple subsystems were analyzed. This approach allowed us to develop a model in which known dynamics are present, such as bifurcation structures, bistability, and excitability. Three subsystems were combined to form the full model and each was first analyzed independent of the others. The first subsystem consists of the immune components within the tissue, which are isolated from the blood. This subsystem is excitable given an immune response triggered by an insult; since, without the ability to communication to neutrophils in the blood, there is no significant tissue damage to sustain an immune response. The second subsystem explores the role of diffusion in the immune response. It considers the response of the pro-inflammatory mediators from both the blood and tissue compartments and their diffusion in the absence of anti-inflammatory mediators. In this subsystem the immune response is sustained after a significant insult, yet low levels resolve in health. Hence, this subsystem is bistable. The final subsystem isolates tissue integrity along with activated neutrophils and radicals, both of which are harmful to tissue. This subsystem, as with the previous subsystem, is also bistable: given that the tissue is damaged, there is a regime for which it will recover, however, significant damage will lead to tissue death. The combination of these subsystems forms our compartmental model. This model is the first step to linking multiple tissue compartments, including a lung compartment, to the same blood source, and exploring the conditions which lead to a secondary infection in the lung. Reynolds, A., Rubin, J., Clermont, G., Day, J., Ermentrout, B., 2006. A reduced mathematical model of the acute inflammatory response: I Derivation of the model and analysis of anti-inflammation. J Theor Biol., 242, 220-236. Xiaohui Wang, University of Texas-Pan American Statistical Methods for Biological Data A lot of biological studies involve functional data. Regression procedure is usually conducted at the first place. Further in some scenarios, classification of those functional data with respect to binary or multi-level responses is particular of interest. This problem is very challenging because of the features of different data. We propose classification models for binary and multi-level cases based on functional regression model and logistic classification model. Our methodology is Bayesian, using wavelet basis functions which have nice approximation properties over a large class of functional spaces and can accommodate a variety of functional forms observed in biological applications. Richard Yamada, Cornell University A Chemical Kinetic Model of Transcriptional Elongation Transcription is the first step in gene expression, and the step at which 99% of gene regulation occurs. Transcription consists of 3 distinct stages: initiation, elongation, and termination. Out of all these steps, transcriptional elongation is the step most amenable to a quantitative description; experimental results in the past 5 years have made it possible to test predictions from quantitative models. Thus, this research area is an exciting topic to apply the tools of probability, analysis, and computation to biology. In this talk, a chemical kinetic model of the transcriptional elongation dynamics of RNA polymerase along a DNA sequence is introduced. The proposed model governs the discrete movement of the RNA polymerase along a DNA template, with no consideration given to elastic effects. The model's novel concept is a ``look-ahead'' feature, in which nucleotides bind reversibly to the DNA prior to being incorporated covalently into the nascent RNA chain. Analytical AND computational results for the proposed model are presented for specific DNA sequences used in actual single-molecule experiments of RNA polymerase along DNA. By replicating the data analysis algorithm from the experimental procedure, the computational model produces velocity histograms, enabling direct comparison with these published experimental results. Parameter estimation techniques to find an optimal set of the model's parameters, along with their interpretation, are also discussed This research is joint work with Prof. Charles Peskin (CIMS/NYU).