# Mathematical Sciences Research Institute

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# MSRI-UP 2022: Algebraic Methods in Mathematical Biology

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The theme of the 2022 MSRI-UP was “Algebraic Methods in Mathematical Biology" and the research leader was Dr. Anne Shiu, Associate Professor of Mathematics at Texas A&M University.

MSRI-UP 2022 focused on mathematical models inspired by biology. The emphasis was on models that can be analyzed by algebraic and combinatorial methods. No background in biology was required, and all projects were accessible to undergraduate participants who have taken a course in linear algebra and a course involving proofs -- and are willing to learn, work in a team, and have fun!

The research projects were chosen from three areas of mathematical biology, with applications to biochemistry, neuroscience, and pharmacology.

Biochemical reaction networks
A reaction network can be represented by a directed graph in which each edge represents a chemical reaction such as, for instance, A+B ---> C, in which one unit of A and one of B react to form one unit of C. The focus of this REU component is on dynamical systems that arise from reaction networks taken with mass-action kinetics. The ordinary differential equations (ODEs) that govern these systems are polynomials in many variables, and therefore are amenable to algebraic techniques. Example project questions are as follows: How can a network's capacity for bistability or oscillations be predicted from its reaction diagram, and how is this capacity affected by operations on the diagram?

Convex neural codes
A starting principle in neuroscience is that "neurons that fire together, wire together''. The aim of this REU component is to clarify which sets of neurons can fire together under the assumption that each neuron of interest codes for some convex region of some Euclidean space. This assumption is valid in certain biological settings, for instance, when each neuron has a corresponding place field, a convex region in space (this space might be a tabletop on which a laboratory rat is walking) and that neuron fires precisely when the subject (rat) is in that region. This research is motivated by place cells in neuroscience, which won its discoverers the 2014 Nobel Prize in Medicine.

Linear compartmental models
Mathematical models arising in biology and other applications often involve many unknown parameters. An important question, therefore, is whether and when these parameters can be recovered from data. If recovering the parameters is possible, the model is said to be identifiable. The question of which models are identifiable, is not fully answered, even for models that involve only linear ODEs and can therefore be summarized by a directed graph. These models are called linear compartmental models, and they are often used to describe how pharmacological drugs move within and affect the body. The proposed projects will build on related work using differential algebra, which transform the identifiability problem into questions involving linear algebra and combinatorics.

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