Symmetry invariance for adapting biological systems

For a general class of dynamical systems, this paper presents a necessary and sufficient characterization of invariance of transient responses to symmetries in inputs. A particular example of this property, scale-invariance or "fold-change detection" (FCD), has been shown to be exhibited in biological sensory systems ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes. The characterization is in terms of a notion of equivariance and amounts to the solvability of an associated partial differential equation. For several simple system motifs that are recurrent in biology, the solvability criterion may be checked explicitly.