The Structure of Infinitesimal Homeostasis in Input-Output Networks

Homeostasis refers to a phenomenon whereby the output $x_o$ of a system is approximately constant on variation of an input $\mathcal{I}$. 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 $\mathcal{G}$ with a distinguished input node $\iota$, a different distinguished output node $o$, and a number of regulatory nodes $\rho_1,\ldots,\rho_n$. In these models the input-output map $x_o(\mathcal{I})$ is defined by a stable equilibrium $X_0$ at $\mathcal{I}_0$. Stability implies that there is a stable equilibrium $X(\mathcal{I})$ for each $\mathcal{I}$ near $\mathcal{I}_0$ and infinitesimal homeostasis occurs at $\mathcal{I}_0$ when $(dx_o/d\mathcal{I})(\mathcal{I}_0) = 0$. We show that there is an $(n+1)\times(n+1)$ homeostasis matrix $H(\mathcal{I})$ for which $dx_o/d\mathcal{I} = 0$ if and only if $\det(H) = 0$. We note that the entries in $H$ are linearized couplings and $\det(H)$ is a homogeneous polynomial of degree $n+1$ in these entries. We use combinatorial matrix theory to factor the polynomial $\det(H)$ and thereby determine a menu of different types of possible homeostasis associated with each digraph $\mathcal{G}$. Specifically, we prove that each factor corresponds to a subnetwork of $\mathcal{G}$. 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 $\det(H)$ without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of $\det(H)$.