Joe Huang – Applied Math

The structure of Infinitesimal Homeostasis in Input-Output Networks

Homeostasis refers to a phenomenon whereby the output of a system is approximately constant on a variation of an input. Our project follows [Golubtisky and Steward (2017)], and considers infinitesimal homeostasis, a mathematical concept. Specifically, we say that an input-output map xo(I) has infinitesimal homeostasis at I0 if x'(I0) = 0. A consequence of infinitesimal homeostasis is that xo(I) is approximately constant in a neighborhood of I0. An input-output network is a network that has a designated input node i, a designated output node o, and a set of regulatory nodes. We assume that the system of network differential equations X’ = F(X, I) has a stable equilibrium at X0. The implicit function theorem implies that there exists a family of equilibria, where xo(I) is the network input-output map. We use the network architecture of input-output networks to classify infinitesimal homeostasis into six types: Haldane, null-degradation, structural of degree 2, appendage, structural of degree >2, and mixed. We also show that the first four types are related to specific network topology properties. This research is a joint project with Prof. Martin Golubitsky, Yangyang Wang, Ian Stewart, and Fernando Antoneli.

References

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[2] [Golubitsky and Stewart (2017)] M. Golubitsky and I. Stewart. Homeostasis, singularities and networks. J. Math. Biol. 74 (2017) 387–407.

[3] [Golubitsky and Wang (2020)] Homeostasis in three-node networks. Submitted.

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[5] [Reed et al. (2017)] M. Reed, J. Best, M. Golubitsky, I. Stewart and H.F. Nijhout. Analysis of homeostatic mechanisms in biochemical networks. Bull. Math. Biol. 79 (9) (2017) 1–24.

[6] [Reed et al. (2010)] M.C. Reed, A. Lieb and H.F. Nijhout. The biological significance of substrate inhibition: a mechanism with diverse functions. Bioessays 32 (5) (2010) 422–429.