On structural contraction of biological interaction networks

TL;DR

This paper introduces the concept of structural contractivity for biological interaction networks, demonstrating that such networks exhibit nonexpansiveness and strict contractivity, which explains their robustness and ability to entrain to periodic inputs.

Contribution

The paper develops a new notion of structural contractivity using non-standard norms, extending previous work and providing verifiable conditions for network stability and entrainment.

Findings

Biological networks are nonexpansive under certain conditions.

Networks can be strictly contractive over positive compact sets.

Networks entrain to periodic inputs, explaining robustness.

Abstract

Biological networks are customarily described as structurally robust. This means that they often function extremely well under large forms of perturbations affecting both the concentrations and the kinetic parameters. In order to explain this property, various mathematical notions have been proposed in the literature. In this paper, we propose the notion of structural contractivity, building on the previous work of the authors. That previous work characterized the long-term dynamics of classes of Biological Interaction Networks (BINs), based on "rate-dependent Lyapunov functions". Here, we show that stronger notions of convergence can be established by proving structural contractivity with respect to non-standard polyhedral $\ell_\infty$-norms. In particular, we show that such networks are nonexpansive. With additional verifiable conditions, we show that they are strictly contractive over arbitrary positive compact sets. In addition, we show that such networks entrain to periodic inputs. We illustrate our theory with examples drawn from the modeling of intracellular signaling pathways.