A new 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. This is a joint project with Fred Nijhout of the Duke Department of Biology and Cornelia Ulrich of the Fred Hutchinson Cancer Research Center. See: M.C. Reed, H. F. Nijhout, R. Sparks, C. M. Ulrich, A Mathematical Model of the Methionine Cycle, Journal of Theoretical Biology , 226 (2004), pp. 33-43, and Nijhout, F., Reed, M., Budu, P. and N. Ulrich, A Mathematical Model of the Folate Cycle - New Insights into Folate Homeostasis, J. Biological Chemistry , 279 , 55008-55016.
A continuing 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. This is joint work with Colleen Mitchell. See, for example: M.Reed, J. Blum, and C. Mitchell, Precision of Neural Timing: Effects of Convergence and Time-windowing, J.Computational Neuroscience , 13 (2002), 35-47.
A recent research project studies the biochemical cascade by which pituitary cells produce luteinizing hormone in response to pulses of GnRH released by the hypothalmus (with J. Blum, Talitha Washington, and Michael Conn of the Oregon Health Sciences Center). See, for example: M. Reed, J. Blum, Jo Ann Janovick and M. Conn, A Mathematical Model Quantifying GnRH--induced LH Secretion from Gonadotropes, Amer. J. Physiol. Endocrinol. Metab. 278 (2000), 263-272, and T. Washington, J. Blum, M. Reed, and M. Conn, A Mathematical Model for LH Release in Response to Continuous and Pulsatile Exposure of Gonadotrophs to GnRH, Theoretical Biology and Medical Modelling , 1 (2004), 1-17.
A current 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? This is joint work with David Anderson (Ph.D., 2005) and Jonathon Mattingly.
A current research project involves the study of time-delayed partial hyperbolic differential equations. The goals are to prove global theorems about existence, propagation of singularities, and asymptotic behavior in time. See, for example, T. Laurent, B. Rider, and M. Reed, Parabolic Behavior of a Hyberbolic Delay Equation, SIAM J. Analysis , 38, 1-15, 2006.