A classification of homeostasis types in four-node input-output networks

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The Ohio State University

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Homeostasis refers to a living system's ability to maintain the steady-state of its internal, physical, or chemical conditions. A system is said to be homeostatic if its output is insensitive to the variations of its input parameter $I$ over some interval. Homeostasis in a sensory system is often called adaptation. One example of a homeostatic system is the thermoregulation of mammals, in which the body temperatures of mammals are approximately constant over a range of environmental temperatures. [Golubitsky and Stewart (2017), Antoneli et al. (2018), Golubitsky and Wang (2020),Wang et al. (2021)] studied infinitesimal homeostasis, where the input-output function $x_o(I)$ has derivative zero at an isolated point $I_0$. [Ma et al. (2009)] pointed out that if the number of nodes is fixed, there could be a small number of network topologies leading to infinitesimal homeostasis. For example, [Reed et al. (2017)] identified two three-node mechanisms which exhibit infinitesimal homeostasis: feed-forward excitation and kinetic homeostasis. Using graph theoretical approach, [Golubitsky and Wang (2020)] assumed the network $G$ has a designated input-node $\iota$, a designated output-node $o$, and a third node $\rho$; they showed the network must have one path (or simple path) from the input node $\iota$ to the output node $o$ for infinitesimal homeostasis to exist. Such networks are called input-output networks. Up to core equivalence, they classified 3 three-node input-output networks with distinct types of infinitesimal homeostasis: feed-forward loop, Haldane homeostasis, and null-degradation homeostasis. In this thesis, we extend the results of [Golubitsky and Wang (2020)] to four-node input-output networks. We introduce the concepts of core network and core equivalence, which give a class of minimal networks that can achieve infinitesimal homeostasis. Despite there exists 199 four-node input-output networks, we show up to core equivalence, there are 20 four-node input-output networks. Furthermore, we partitioned the 20 networks (as shown in networks 1-20) into three categories: irreducible networks (containing one type of infinitesimal homeostasis), networks with three degree 1 factor (has three different types of infinitesimal homeostasis), and networks with one degree 1 factor and one degree 2 factor (has two different types of infinitesimal homeostasis). We provide a description of how infinitesimal homeostasis can arise in each of the 20 networks and the stability conditions for the steady-state points (a necessary assumption of infinitesimal homeostasis). For instance, network 1 of 3 has three different Haldane factors (neutral couplings) and the stability condition requires all the internal dynamic terms of the form $f_{\ell,x_\ell}$ to be zero. Lastly, we demonstrate our classification theorem with three biochemical networks: intracellular copper regulation, E. coli chemotaxis and allosteric regulation of PFKL/M.