Symmetry invariance for adapting biological systems

TL;DR

This paper provides a theoretical characterization of symmetry invariance in biological dynamical systems, especially focusing on fold-change detection, using control theory and group symmetry concepts.

Contribution

It introduces a necessary and sufficient condition for adapting systems to exhibit invariance under symmetry groups, extending equivariance concepts with control theoretic methods.

Findings

Characterization of invariant transient behaviors in biological systems

Extension of equivariance to non-compact symmetry groups

Application of control theory to biological symmetry analysis

Abstract

We study in this paper certain properties of the responses of dynamical systems to external inputs. The motivation arises from molecular systems biology. and, in particular, the recent discovery of an important transient property, related to Weber's law in psychophysics: "fold-change detection" in adapting systems, the property that scale uncertainty does not affect responses. FCD appears to play an important role in key signaling transduction mechanisms in eukaryotes, including the ERK and Wnt pathways, as well as in E.coli and possibly other prokaryotic chemotaxis pathways. In this paper, we provide further theoretical results regarding this property. Far more generally, we develop a necessary and sufficient characterization of adapting systems whose transient behaviors are invariant under the action of a set (often, a group) of symmetries in their sensory field. A particular instance is FCD, which amounts to invariance under the action of the multiplicative group of positive real numbers. Our main result is framed in terms of a notion which extends equivariant actions of compact Lie groups. Its proof relies upon control theoretic tools, and in particular the uniqueness theorem for minimal realizations.