Duke University
Mathematical Biology Projects 

Mathematical modeling of renal physiology
Harold Layton, Anita Layton, Paula Budu

This project models renal function at the level of the nephron (the functional unit of the kidney) and at the level of nephron populations. In particular, we are studying tubuloglomerular feedback (TGF), the urine concentrating mechanism, and the hemodynamics of the afferent arteriole. Dynamic models for TGF involve small systems of semilinear hyperbolic partial differential equations (PDEs) with time-delays, which are solved numerically for cases of physiological interest, or which are linearized for qualitative analytical investigation. Dynamic models for the concentrating mechanism involve large systems of coupled hyperbolic PDEs that describe tubular convection and epithelial transport. Numerical solutions of these PDEs help to integrate and interpret quantities determined by physiologists in many separate experiments. To study the fluid dynamics in the afferent arteriole, low Reynolds number flow (Re ~ 0.1) is simulated in a two-dimensional elastic-contractile tube. The tubular walls are included by means of the immersed boundary method. Current work is directed to quantifying the regulatory role of the vasodilator nitric oxide.

Mathematical modeling of cell metabolism
Michael Reed, Fred Nijhout, Cornelia Ulrich

This research area involves the applications of mathematics to the study of various aspects of cell metabolism, in particular, folate and methionine metabolism. The folic acid cycle plays a central role in cell metabolism. Among the important functions of the folate cycle are the synthesis of pyrimidines and purines and the delivery of one carbon units to the methionine cycle for use in methylation reactions. Dietary folate deficiencies as well as mutations in enzymes of the folate cycle are associated with megaloblastic anemia, cancers of the colon, breast and cervix, affective disorders, cleft palate, neural tube defects, Alzheimers disease, Down's syndrome, preeclampsia and early pregnancy loss and several enzymes in the cycle are the targets of anti-cancer drugs. The methionine cycle is important for the regulation of homocysteine, an important risk factor for heart disease, and for the control of DNA methylation. Both hyper- and hypomethylation have been proposed as crucial steps in chains of events that turn normal cells into cancerous cells. The purpose of the project is to use mathematics to understand normal folate and methionine metabolism, DNA methylation, and purine and pyrimidine synthesis and then to understand how they are affected by alterations in diet and gene abnormalities.

Mammalian auditory brainstem
Michael Reed and Colleen Mitchell

This research area is the study of information processing in the mammalian auditory brainstem by the use of mathematical and computational models. The purpose is to understand what the nuclei in the brainstem (and midbrain) are computing and how they do it. This is done by creating mathematical and computational models, based on known (partial) information about physiology and anatomy, which incorporate hypotheses about the details of the anatomy and physiology of the nuclei and the ways in which the nuclei communicate with each other. By investigating these models and comparing the results to experimental findings one can (one hopes) confirm or reject the hypotheses and thus contribute to understanding of the brainstem. Recent work has utilized probabilistic methods and has focused on hyperacuity and the mechanism of sharpening timing as information progresses from the auditory periphery up the brainstem.

ODEs arising in chemical reactions
Michael Reed and David Anderson

This research project involves the study of large systems of ordinary differential equations that arise from chemical reactions, for example in cell metabolism and cell signalling processes. What properties of the system depend only on the geometry and topology of the reaction diagram? What classes of reaction diagrams guarantee certain kinds of system behavior? How can large systems be simplified and yet keep their essential behavior? How do stochastic variations of one component of the system affect the other components?

BioGeometry
Herbert Edelsbrunner, Pankaj Agarwal, Homme Hellinga, Johannes Ruldolph

We have assembled an interdisciplinary team to address fundamental computational problems in the representation of molecular structures and the simulation of biochemical processes important to life. Among these are ligand-to-protein docking, ab initio structure prediction, and protein folding. We also plan to consider engineering tasks, such as drug and protein design. Through a novel focus on geometric and topological representations, we have an opportunity to enable new scientific breakthroughs and more generally to transform the way we represent, analyze, communicate, and teach those fundamental structures and processes in molecular biology. In the process of doing so we will need to make advances in several areas of information technology that will be scientific accomplishments in themselves as well as being potentially relevant to other natural sciences and engineering disciplines dealing with computer modeling of the physical world.

Mathematical approaches to linguistics
Garrett Mitchener

My mathematical interests are in dynamical systems and probability. My application is to linguistics: I'm interested in language change, and understanding why it occurs, how it spreads, and what this can tell us about the human mind. Much of language change happens inside a black box, because the mental and social processes that drive it are not directly accessible. Often, the only available information is a limited corpus of the old manuscripts that happened to survive the ages. Mathematical models and simulations can help fill in the missing information and allow researchers to test and improve theories about how we speak, write, learn, and interact. Math can also help us make the most of the limited data we have and properly connect it to theory.

Characterizing and controlling cardiac dynamics
David Schaeffer, Daniel Gauthier, John Cain

One intriguing application of the chaos control methods we have developed is in the biological area. We have initiated a program to characterize in vitro the dynamics of small pieces of rapidly paced cardiac muscle and to use feedback methods to suppress or control the observed bifurcations by applying small perturbations to the tissue. We find that there are only a small number of classes of bifurcations in the tissue, but that there is significant variation in the prevalence of these behaviors from animal to animal. In addition, we are using similar methods to control in vivo a fibrillating sheep atrium. The eventual long term goal of this project is to develop an implantable defibrillator that will maintain a healthy rhythm in humans prone to the onset of atrial fibrillation using only small electrical shocks. In our current experiments with sheep, we use a high-density mapping system to record the spatial-temporal complexity occurring on the surface of the heart during atrial fibrillation. Small control shocks are applied to a single electrode attached to the surface of the heart based on real-time measurements of the cardiac dynamics at a nearby spatial location.

Electrical wave propagation in the heart
John Trangenstein, Wenjun Ying

We apply second-order operator splitting to the Luo-Rudy model for electrical wave propagation in the heart. This approach allows us to use local nonlinear iterations for the stiff nonlinear reactions, and to solve global linear systems for the implicit treatment of diffusion. For computational efficiency, we use dynamically adaptive mesh refinement (AMR), involving hierarchies of unions of grid patches on distinct levels of refinement. The linear system for discretization of the diffusion on the composite AMR grid is formulated via standard conforming finite elements on unions grid patches within a level of refinement, and aligned mortar elements along interfaces between levels of refinement. The linear system are solved iteratively by preconditioned conjugate gradient. Our preconditioner uses multiplicative domain decomposition between levels of refinement; the smoother involves algebraic additive domain decomposition between patches within a level of refinement, and Gauss-Seidel iteration within grid patches.

Continuum models of population dynamics
Tom Witelski

The dynamics of large populations (of bacteria, insects, animals, ...) can be modeled by continuum models. The behaviors of individuals are appropriately averaged to yield smooth density functions describing the mean behavior in the population. The governing equations are nonlinear partial differential equations. Numerical simulations, analysis, and perturbation theory are used to study special solutions modeling different population dynamics: traveling waves for migration and other "driven" behavior, similarity solutions for spreading and merging populations, blow-up solutions for "population explosions", and the influence of landscape boundaries via boundary layers and matched asymptotic expansions.

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