# Michael C. Reed

- Professor of Mathematics
- Bass Fellow

**External address:**237 Physics Bldg., Duke University, Box 90320, Durham, NC 27708

**Internal office address:**Box 90320, Duke University 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2808

Professor Reed is engaged in a large number of research projects that involve the application of mathematics to questions in physiology and medicine. He also works on questions in analysis that are stimulated by biological questions. For recent work on cell metabolism and public health, go to sites@duke.edu/metabolism.

Since 2003, Professor Reed has worked with Professor Fred Nijhout (Duke Biology) to use mathematical methods to understand regulatory mechanisms in cell metabolism. Most of the questions studied are directly related to public health questions. A primary topic of interest has been liver cell metabolism where Reed and Nijhout have created mathematical models for the methionine cycle, the folate cycle, and glutathione metabolism. The goal is to understand the system behavior of these parts of cell metabolism. The models have enabled them to answer biological questions in the literature and to give insight into a variety of disease processes and syndromes including: neural tube defects, Down’s syndrome, autism, vitamin B6 deficiency, acetaminophen toxicity, and arsenic poisoning.

A second major topic has been the investigation of dopamine and serotonin metabolism in the brain; this is collaborative work with Professor Nijhiout and with Janet Best, a mathematician at The Ohio State University. The biochemistry of these neurotransmitters affects the electrophysiology of the brain and the electrophysiology affects the biochemistry. Both affect gene expression, the endocrine system, and behavior. In this complicated situation, especially because of the difficulty of experimentation, mathematical models are an essential investigative tool that can shed like on questions that are difficult to get at experimentally or clinically. The models have shed new light on the mode of action of selective serotonin reuptake inhibitors (used for depression), the interactions between the serotonin and dopamine systems in Parkinson’s disease and levodopa therapy, and the interactions between histamine and serotonin.

Recent work on homeostatic mechanisms in cell biochemistry in health and disease have shown how difficult the task of precision medicine is. A gene polymorphism may make a protein such as an enzyme less effective but often the system compensates through a variety of homeostatic mechanisms. So even though an individual's genotype is different, his or her phenotype may not be different. The individuals with common polymorphisms tend tend to live on homeostatic plateaus and only those individuals near the edges of the plateau are at risk for disease processes. Interventions should try to enlarge the homeostatic plateau around the individual's genotype.

Other areas in which Reed uses mathematical models to understand physiological questions include: axonal transport, the logical structure of the auditory brainstem, hyperacuity in the auditory system, models of pituitary cells that make luteinizing hormone and follicle stimulating hormone, models of maternal-fetal competition, models of the owl’s optic tectum, and models of insect metabolism.

For general discussions of the connections between mathematics and biology, see his articles: ``Why is Mathematical Biology so Hard?,'' 2004, Notices of the AMS, 51, pp. 338-342, and ``Mathematical Biology is Good for Mathematics,'' 2015, Notices of the AMS, 62, pp., 1172-1176.

Often, problems in biology give rise to new questions in pure mathematics. Examples of work with collaborators on such questions follow:

Laurent, T, Rider, B., and M. Reed (2006) Parabolic Behavior of a Hyberbolic Delay Equation, SIAM J. Analysis, 38, 1-15.

Mitchell, C., and M. Reed (2007) Neural Timing in Highly Convergent Systems, SIAM J. Appl. Math. 68, 720-737.

Anderson,D., Mattingly, J., Nijhout, F., and M. Reed (2007) Propagation of Fluctuations in Biochemical Systems, I: Linear SSC Networks, Bull. Math. Biol. 69, 1791-1813.

McKinley S, Popovic L, and M. Reed M. (2011) A Stochastic compartmental model for fast axonal transport, SIAM J. Appl. Math. 71, 1531-1556.

Lawley, S. Reed, M., Mattingly, S. (2014), Sensitivity to switching rates in stochastically switched ODEs,'' Comm. Math. Sci. 12, 1343-1352.

Lawley, S., Mattingly, J, Reed, M. (2015), Stochastic switching in infinite dimensions with applications to parabolic PDE, SIAM J. Math. Anal. 47, 3035-3063.

### Selected Grants

Mathematical Sciences/GIG: Applications of Mathematics to Physiology awarded by National Science Foundation (Principal Investigator). 1997 to 2001

(97-0521) Applications of Mathematics to Physiology awarded by National Science Foundation (Principal Investigator). 1997 to 1999

(95-0205) Application of Mathematics in Neurobiology awarded by National Science Foundation (Principal Investigator). 1995 to 1997

(96-0881) Applications of Mathematics in Neurobiology awarded by National Science Foundation (Principal Investigator). 1995 to 1997

(94-0062) Mathematical Models of Neural Processing in the Auditory Brainstem awarded by National Science Foundation (Principal Investigator). 1992 to 1995

(95-0033) Mathematical Models of Neural Processing in the Auditory Brainstem awarded by National Science Foundation (Principal Investigator). 1992 to 1995

(92-0227) Mathematical Models of Neural Processing in the Auditory Brainstem awarded by National Science Foundation (Principal Investigator). 1992 to 1994

(92-0043) Mathematical Sciences: Applications of Mathematics to Physiology awarded by National Science Foundation (Principal Investigator). 1989 to 1992

(90-0657) Mathematical Sciences: Applications of Mathematics to Physiology awarded by National Science Foundation (Principal Investigator). 1989 to 1991

(89-0115) Applications of Mathematics to Physiology awarded by National Science Foundation (Principal Investigator). 1989 to 1990

## Pages

Reed, MC, and Blum, J. "A reaction-hyperbolic system in Physiology." *Non-linear Semigroups, Partial Differential Equations, and Attractors.* Ed. T Gill and W Zachary. New York: Springer, 1987. 127-133. (Chapter)

Reed, MC, and Rauch, J. "On the absorption of singularities in dissipative nonlinear equations." *Differential Equations and Mathematical Physics.* Ed. I Knowles and Y Saito. New York: Springer, 1987. 403-407. (Chapter)

Reed, MC, and Rauch, J. "Ultrasingularities in nonlinear waves." *Non-linear Semigroups, Partial Differential Equations, and Attractors.* Ed. T Gill and W Zachary. New York: Springer, 1985. 134-141. (Chapter)

Reed, MC, and Rauch, J. "Geometric structures in the singularity theory of hyperbolic equations." *Advances in Nonlinear Waves, I.* Ed. E Debnath. Cambridge UK: Cambridge University Press, 1984. 273-280. (Chapter)

Reed, MC. "Geometry and discrete velocity approximations to the Boltzmann equation." *Differential Equations.* Ed. I Knowles and R Lewis. Amsterdam, NE: North-Holland, 1984. (Chapter)

Reed, MC. "Propagation of singularities." *Many Degrees of Freedom in Particle Physics and Field Theory.* Ed. L Streit. New York: Plenum Press, 1978. (Chapter)

Reed, MC. "Propagation of singularities in wave equations of Klein-Gordon type." *Non-linear Partial Differential Equations and Applications.* Ed. J Chadam. 1978. (Chapter)

Reed, MC. "The GNS Construction--a pedagogical example." *Lectures on Elementary Particles and Quantum Field Theory.* Cambridge, MA: MIT Press, 1970. 415-438. (Chapter)

Reed, MC. "Functional analysis and probability theory." *Constructive Quantum Field Theory.* Ed. A Wightman. (Chapter)

Reed, MC. "Unsolved problems in the theory of non-linear wave equations." *Many Degrees of Freedom in Particle Physics and Field Theory.* Ed. L Streit. New York: Plenum Press. (Chapter)

## Pages

Lawley, SD, Mattingly, JC, and Reed, MC. "Stochastic Switching in Infinite Dimensions with Applications to Random Parabolic PDE." *SIAM Journal on Mathematical Analysis* 47.4 (January 2015): 3035-3063.
Full Text Open Access Copy

Nijhout, HF, Best, JA, and Reed, MC. "Using mathematical models to understand metabolism, genes, and disease." *BMC biology* 13 (January 2015): 79-.
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Nijhout, HF, Best, J, and Reed, MC. "Escape from homeostasis." *Mathematical biosciences* 257 (November 2014): 104-110.
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Wood, KM, Zeqja, A, Nijhout, HF, Reed, MC, Best, J, and Hashemi, P. "Voltammetric and mathematical evidence for dual transport mediation of serotonin clearance in vivo." *Journal of neurochemistry* 130.3 (August 2014): 351-359.
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Nijhout, HF, and Reed, MC. "Homeostasis and dynamic stability of the phenotype link robustness and plasticity." *Integrative and comparative biology* 54.2 (July 2014): 264-275.
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da Silva, VR, Ralat, MA, Quinlivan, EP, DeRatt, BN, Garrett, TJ, Chi, Y-Y, Frederik Nijhout, H, Reed, MC, and Gregory, JF. "Targeted metabolomics and mathematical modeling demonstrate that vitamin B-6 restriction alters one-carbon metabolism in cultured HepG2 cells." *American journal of physiology. Endocrinology and metabolism* 307.1 (July 2014): E93-101.
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Lawley, SD, Yun, J, Gamble, MV, Hall, MN, Reed, MC, and Nijhout, HF. "Mathematical modeling of the effects of glutathione on arsenic methylation." *Theoretical biology & medical modelling* 11 (May 16, 2014): 20-.
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Wood, KM, Zeqja, A, Nijhout, HF, Reed, MC, Best, J, and Hashemi, P. "Voltammetric and mathematical evidence for dual transport mediation of serotonin clearance in vivo." *Journal of Neurochemistry* 130.3 (January 1, 2014): 351-359.
Full Text

Reed, M, Nijhout, HF, and Best, J. "Projecting Biochemistry Over Long Distances." Ed. L Pujo-Menjouet. *Mathematical Modelling of Natural Phenomena* 9.1 (2014): 133-138.
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Lawley, SD, Mattingly, JC, and Reed, MC. "Sensitivity to switching rates in stochastically switched odes." *Communications in Mathematical Sciences* 12.7 (2014): 1343-1352.
Full Text Open Access Copy