Singularly Perturbed Monotone Systems and an Application to Double Phosphorylation Cycles

Abstract

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.

No oscillations in the Michaelis-Menten approximation of the dual futile cycle under a sequential and distributive mechanism

“…Thus it is possible to discard the equations (8) to (10), and reduce the number of equations in the system from nine to six, and the resulting system is equivalent to the original nine-equation one. This six-equation system matches the ones found in [3,14,15].…”

“…Here, we do not use this simplification so that our results apply to the most general M-M reduction. A similar M-M reduction can be found in [15].…”

“…As mentioned earlier, this result was previously proven by Wang and Sontag using monotone systems theory [15].…”

“…The structure of the dual futile cycle can be seen in Figure 1. For this system, Wang and Sontag showed that within restricted parameter ranges, the system exhibits generic convergence to steady states Figure 1: The dual futile cycle but no more complicated behavior [15]. Angeli et al showed that no species tend to be eliminated for any parameter values [1].…”

“…This reduction was analyzed by Hell and Rendall to prove the bistability of the dual futile cycle [3]. More specifically, for this reduction, Wang and Sontag proved that it is non-oscillatory using monotone systems theory [15]. Here, we present a simpler proof of the non-oscillatory behavior of this system by using Bendixson's criterion and thus, we recover the M-M version of the result of Wang and Sontag mentioned earlier that states that every solution converges to some steady state.…”