The structure of infinitesimal homeostasis in input-output networks

Homeostasis refers to a phenomenon whereby the output [Formula: see text] of a system is approximately constant on variation of an input [Formula: see text]. Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs [Formula: see text] with a distinguished input node [Formula: see text], a different distinguished output node o, and a number of regulatory nodes [Formula: see text]. In these models the input-output map [Formula: see text] is defined by a stable equilibrium [Formula: see text] at [Formula: see text]. Stability implies that there is a stable equilibrium [Formula: see text] for each [Formula: see text] near [Formula: see text] and infinitesimal homeostasis occurs at [Formula: see text] when [Formula: see text]. We show that there is an [Formula: see text] homeostasis matrix [Formula: see text] for which [Formula: see text] if and only if [Formula: see text]. We note that the entries in H are linearized couplings and [Formula: see text] is a homogeneous polynomial of degree [Formula: see text] in these entries. We use combinatorial matrix theory to factor the polynomial [Formula: see text] and thereby determine a menu of different types of possible homeostasis associated with each digraph [Formula: see text]. Specifically, we prove that each factor corresponds to a subnetwork of [Formula: see text]. The factors divide into two combinatorially defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of [Formula: see text] without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of [Formula: see text]. There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).