Singularly Perturbed Monotone Systems and an Application to Double Phosphorylation Cycles

The theory of monotone dynamical systems has been found very useful in the modeling of some gene, protein, and signaling networks. In monotone systems, every net feedback loop is positive. On the other hand, negative feedback loops are important features of many systems, since they are required for adaptation and precision. This paper shows that, provided that these negative loops act at a comparatively fast time scale, the main dynamical property of (strongly) monotone systems, convergence to steady states, is still valid. An application is worked out to a double-phosphorylation ``futile cycle'' motif which plays a central role in eukaryotic cell signaling.Comment: 21 pages, 3 figures, corrected typos, references remove

A petri net approach to persistence analysis in chemical reaction networks

Summary. A positive dynamical system is said to be persistent if every solution that starts in the interior of the positive orthant does not approach the boundary of this orthant. For chemical reaction networks and other models in biology, persistence represents a non-extinction property: if every species is present at the start of the reaction, then no species will tend to be eliminated in the course of the reaction. This paper provides checkable necessary as well as sufficient conditions for persistence of chemical species in reaction networks, and the applicability of these conditions is illustrated on some examples of relatively high dimension which arise in molecular biology. More specific results are also provided for reactions endowed with mass-action kinetics. Overall, the results exploit concepts and tools from Petri net theory as well as ergodic and recurrence theory.