YY. Wang, Z. Huang, F. Antoneli and M. Golubitsky

The structure of infinitesimal homeostasis in input-output networks

J. Math. Biol. 82 (2021) https://doi.org/10.1007/s00285-021-01614-1

Homeostasis occurs when an observable of a system (such as inner body temperature remains approximately constant over a range of an external parameter (such as ambient temperature). More precisely, homeostasis refers to a phenomenon whereby the output x_o of a system is approximately constant on variation of an input I. Homeostatic phenomena are ubiquitous in biochemical networks of differential equations and these networks can be abstracted as digraphs G with a fixed input node \iota and a different fixed output node o. We assume that only the input node depends explicitly on \ and that the output is the output node value x_o(I). We then study {\em infinitesimal homeostasis}: points I_0 where dx_o/dI(I_0) = 0 by showing that there is a square homeostasis matrix H associated to G and that infinitesimal homeostasis points occur where det(H) = 0. Applying combinatorial matrix theory and graph theory to H allows us to classify types of homeostasis. We prove that the homeostasis types correspond to a set of irreducible blocks in H each associated with a subnetwork and these subnetworks divide into two classes: structural and appendage. For example, a feedforward loop motif is a structural type whereas a negative feedback loop motif is an appendage type. We give two algorithms for determining a menu of homeostasis types that are possible in G: one algorithm enumerates the structural types and one enumerates the appendage types. These subnetworks can be read directly from G without performing calculations on model equations.