Some transient properties of adapting biological systems

10 June 2011

Eduardo Sontag
Department of Mathematics
Rutgers University


Often, the ultimate goal of regulation is to maintain a narrow range of concentration levels of vital quantities (homeostasis, adaptation) while at the same time appropriately reacting to changes in the environment (signal detection or sensitivity). Much theoretical, modeling, and analysis effort has been devoted to the understanding of these questions, traditionally in the context of steady-state responses to constant or step-changing stimuli. In this talk, we present a new theorem that provides a necessary and sufficient characterization of invariance of transient responses to symmetries in inputs. A particular example of this property, scale-invariance (a.k.a. "fold change detection"), appears to be exhibited by biological sensory systems ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes. The new characterization amounts to the solvability of an associated partial differential equation. It is framed in terms of a notion which considerably extends equivariant actions of compact Lie groups. For several simple system motifs that are recurrent in biology, the solvability criterion may be checked explicitly.

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