Logo image
Back
The structure of infinitesimal homeostasis in input–output networks
Journal article   Open access   Peer reviewed

The structure of infinitesimal homeostasis in input–output networks

Yangyang Wang, Zhengyuan Huang, Fernando Antoneli and Martin Golubitsky
Journal of mathematical biology, Vol.82(7), 62
2021
DOI: 10.1007/s00285-021-01614-1
PMCID: PMC8139887
PMID: 34021398
url
https://doi.org/10.1007/s00285-021-01614-1View
Published (Version of record) Open Access

Abstract

Homeostasis refers to a phenomenon whereby the output xo of a system is approximately constant on variation of an input 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 G with a distinguished input node ι, a different distinguished output node o, and a number of regulatory nodes ρ1,…,ρn. In these models the input–output map xo(I) is defined by a stable equilibrium X0 at I0. Stability implies that there is a stable equilibrium X(I) for each I near I0 and infinitesimal homeostasis occurs at I0 when (dxo/dI)(I0)=0. We show that there is an (n+1)×(n+1) homeostasis matrix H(I) for which dxo/dI=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 G. Specifically, we prove that each factor corresponds to a subnetwork of 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).
 Biochemical networks Coupled systems 92C42 34C99 94C15 Homeostasis Input–output networks Combinatorial matrix theory 92C40 Perfect adaptation

Details

Metrics

12 Record Views
21 Times Cited - Web of Science
Logo image