### A qualitative/quantitative view of biomolecular networks

24 February 2006

Eduardo Sontag

Department of Mathematics and BioMAPS

Rutgers

#### Abstract

Biomolecular networks, while exhibiting a rich variety of behaviors in signaling and regulation, would appear to be fairly well behaved as dynamical systems. Their (mathematical) models have solutions that tend to settle into well-defined steady states or periodic, but not "chaotic", behavior. This presents one major challenge to theoreticians: what is special about such networks, vis a vis general dynamical systems?

A second challenge arises in the mathematical analysis itself: while on the one hand good qualitative, graph-theoretic, knowledge is frequently available, on the other hand it is often hard to experimentally validate the form of the nonlinearities used in reaction terms, and even when such forms are known, to accurately estimate coefficients (parameters). This "data-rich/data-poor" dichotomy seems to be pervasive in systems biology.

The present talk is concerned with both challenges. I will describe a class of nonlinear dynamical systems, which are decomposable into order-preserving ("monotone") input/output systems. This class arose originally from the study of possible multi-stability or oscillations in feedback loops in cell signal transduction, but turned out to be of more general applicability. An approach to their study employs a blend of qualitative and (relatively sparse) quantitative information, allowing one to draw conclusions about global dynamical behavior and the location of steady states. I will also discuss some evidence suggesting that certain signaling and transcriptional regulation networks may be profitably viewed in this manner.

#### References

E. D. Sontag, *"Some new directions in control theory inspired by systems biology"*, Systems Biology 1:9-18 2004. PDF

E. D. Sontag, *"Molecular systems biology and control"*, Euro Jour Control 11:396-435 2005. PDF

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