Abstract
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 matrixH({{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). 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).
Highlights
1.1 Overview and perspectiveThis paper divides into three parts
The degree of the homeostasis type is defined as the size k of the square block Bη and we prove that each block Bη has either k or k −1 self-couplings
The input–output function associated with Gc has a point of infinitesimal homeostasis at I0 if and only if the input–output function associated with G has a point of infinitesimal homeostasis at I0
Summary
Golubitsky and Wang (2020) classified the ‘homeostasis types’ that can occur in three-node input–output networks based on the notion of infinitesimal homeostasis (Golubitsky and Stewart 2017) (see Definition 1.2). Using this approach, they were able to reproduce the classification results in Ma et al (2009) and Reed et al. (2017), within a broader class of systems including, but not limited to, specific model systems based on mass action or Michaelis-Menten kinetics They showed that threenode networks that can exhibit infinitesimal homeostasis are, up to core equivalence (see Definition 1.9), the three network topologies mentioned above. As an aside: In our math networks arrows are identical and represent couplings and nodes are identical and represent
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