# Propagation of fluctuations in biochemical systems, I: Linear SSC networks

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2007-08-01

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We investigate the propagation of random fluctuations through biochemical networks in which the number of molecules of each species is large enough so that the concentrations are well modeled by differential equations. We study the effect of network topology on the emergent properties of the reaction system by characterizing the behavior of variance as fluctuations propagate down chains and studying the effect of side chains and feedback loops. We also investigate the asymptotic behavior of the system as one reaction becomes fast relative to the others. © 2007 Springer Science+Business Media, Inc.

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Anderson, David F, Jonathan C Mattingly, H Frederik Nijhout and Michael C Reed (2007). Propagation of fluctuations in biochemical systems, I: Linear SSC networks. *Bulletin of Mathematical Biology*, 69(6). pp. 1791–1813. 10.1007/s11538-007-9192-2 Retrieved from https://hdl.handle.net/10161/9281.

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### Scholars@Duke

#### Jonathan Christopher Mattingly

Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave elementary and Charlotte Country Day. He graduated from the NC School of Science and Mathematics and received a BS is Applied Mathematics with a concentration in physics from Yale University. After two years abroad with a year spent at ENS Lyon studying nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to attend Princeton University where he obtained a PhD in Applied and Computational Mathematics in 1998. After 4 years as a Szego assistant professor at Stanford University and a year as a member of the IAS in Princeton, he moved to Duke in 2003. He is currently a Professor of Mathematics and of Statistical Science.

His expertise is in the longtime behavior of stochastic system including randomly forced fluid dynamics, turbulence, stochastic algorithms used in molecular dynamics and Bayesian sampling, and stochasticity in biochemical networks.

Since 2013 he has also been working to understand and quantify gerrymandering and its interaction of a region's geopolitical landscape. This has lead him to testify in a number of court cases including in North Carolina, which led to the NC congressional and both NC legislative maps being deemed unconstitutional and replaced for the 2020 elections.

He is the recipient of a Sloan Fellowship and a PECASE CAREER award. He is also a fellow of the IMS and the AMS. He was awarded the Defender of Freedom award by Common Cause for his work on Quantifying Gerrymandering.

#### H. Frederik Nijhout

Fred Nijhout is broadly interested in developmental physiology and in the interactions between development and evolution. He has several lines of research ongoing in his laboratory that on the surface may look independent from one another, but all share a conceptual interest in understanding how complex traits arise through, and are affected by, the interaction of genetic and environmental factors. 1) The control of polyphenic development in insects. This work attempts to understand how the insect developmental hormones, ecdysone and juvenile hormone, act to control alternative developmental pathways within a single individual. His studies and those of his students have dealt with the control of sequential polyphenism in metamorphosis, of alternate polyphenisms in caste determination of social insects and the many seasonal forms of insects. 2) The regulation of organ and body size in insects. Ongoing research deals with the mechanism by which insects asses their body size and stop growing when they have achieved a characteristic size. Other studies deal with the control of growth and size of imaginal disks. This work is revealing that the control of body and organ size does not reside in any specific cellular or molecular mechanism but that it is a systems property in which cellular, physiological and environmental signals all contribute in inextricable ways to produce the final phenotype. 3) The development and evolution of color patterns in Lepidoptera. Ongoing research attempts to elucidate the evolution of mimicry using genetic and genomic approaches. 4) The development, genetics and evolution of complex traits. Complex traits are those whose variation is affected by many genes and environmental factors and whose inheritance does not follow Mendel’s laws. In practice this involves understanding how genetic and developmental networks operate when there is allelic variation in their genes. This work attempts to reconstruct complex traits through mathematical models of the genetic and developmental processes by which they originate, and uses these models to study the effects of mutation and selection. Currently metabolic networks are being used to develop a deeper understanding of the functional relationships between genetic variation and trait variation, and of the mechanisms by which genetic and environmental variables interact to produce phenotypes. More on web page: http://www.biology.duke.edu/nijhout/

#### Michael C. Reed

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.

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