Laboratory for Control, Learning, and Systems Biology

circadian rhythms

2008
  1. ▪D. Angeli, E.D. Sontag, "Oscillations in I/O monotone systems", IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, vol. 55, pp. 166-176, 2008. pdf
    Preprint version in arXiv q-bio.QM/0701018, 14 Jan 2007
    monotone systems · hopf bifurcations · circadian rhythms · tridiagonal systems · nonlinear dynamics · systems biology · reaction networks · oscillations · periodic behavior · delay-differential systems
    Abstract

    In this note, we show how certain properties of Goldbeter's 1995 model for circadian oscillations can be proved mathematically, using techniques from the recently developed theory of monotone systems with inputs and outputs. The theory establishes global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, based on the application of a tight small gain condition. This stability persists even under arbitrary delays in the feedback loop. On the other hand, when the condition is violated a Poincare'-Bendixson result allows to conclude existence of oscillations, for sufficiently high delays.

2004
  1. ▪D. Angeli, E.D. Sontag, "An analysis of a circadian model using the small-gain approach to monotone systems", In Proc.\ IEEE Conf.\ Decision and Control, Paradise Island, Bahamas, Dec.\ 2004, IEEE Publications, pp. 575–578, 2004. pdf
    circadian rhythms · tridiagonal systems · nonlinear dynamics · systems biology · reaction networks · oscillations · periodic behavior · monotone systems · delay-differential systems
    Abstract

    We show how certain properties of Goldbeter's original 1995 model for circadian oscillations can be proved mathematically. We establish global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, but, on the other hand, this stability persists even under arbitrary delays in the feedback loop. We are mainly interested in illustrating certain mathematical techniques, including the use of theorems concerning tridiagonal cooperative systems and the recently developed theory of monotone systems with inputs and outputs.