The Structure of Infinitesimal Homeostasis in Input-Output Networks

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 $\mathcal{I}$. Homeostatic phenomena are ubiquitous in biochemical networks of differential equations and these networks can be abstracted as digraphs $\mathcal{G}$ with a fixed input node $\iota$ and a different fixed output node $o$. We assume that only the input node depends explicitly on $\mathcal{I}$ and that the output is the output node value $x_o(\mathcal{I})$. We then study infinitesimal homeostasis: points $\mathcal{I}_0$ where $\frac{dx_o}{d\mathcal{I}}(\mathcal{I}_0)=0$ by showing that there is a square homeostasis matrix $H$ associated to $\mathcal{G}$ 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 $\mathcal{G}$: one algorithm enumerates the structural types and one enumerates the appendage types. These subnetworks can be read directly from $\mathcal{G}$ without performing calculations on model equations.