1. D. Angeli and E.D. Sontag. Oscillations in I/O monotone systems. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:166-176, 2008. Note: Preprint version in arXiv q-bio.QM/0701018, 14 Jan 2007. [PDF] Keyword(s): monotone systems, hopf bifurcations, circadian rhythms, tridiagonal systems, nonlinear dynamics, systems biology, biochemical 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.




2. D. Angeli, J. E. Ferrell, and E.D. Sontag. Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems.. Proc Natl Acad Sci USA, 101(7):1822-1827, 2004. Note: A revision of Suppl. Fig. 7(b) is here: http://sontaglab.org/FTPDIR/nullclines-f-g-REV.jpg; and typos can be found here: http://sontaglab.org/FTPDIR/angeli-ferrell-sontag-pnas04-errata.txt. [WWW] [PDF] [doi:10.1073/pnas.0308265100] Keyword(s): MAPK cascades, multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems. Abstract:

   Multistability is an important recurring theme in cell signaling, of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex than this. Here we show that for a class of feedback systems of arbitrary order, the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this "open loop," feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems, and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable MAPK cascade.




3. L. Moreau and E.D. Sontag. Balancing at the border of instability. Phys. Rev. E (3), 68(2):020901, 4, 2003. [PDF] Keyword(s): bifurcations, adaptive control. Abstract:

   Some biological systems operate at the critical point between stability and instability and this requires a fine-tuning of parameters. We bring together two examples from the literature that illustrate this: neural integration in the nervous system and hair cell oscillations in the auditory system. In both examples the question arises as to how the required fine-tuning may be achieved and maintained in a robust and reliable way. We study this question using tools from nonlinear and adaptive control theory. We illustrate our approach on a simple model which captures some of the essential features of neural integration. As a result, we propose a large class of feedback adaptation rules that may be responsible for the experimentally observed robustness of neural integration. We mention extensions of our approach to the case of hair cell oscillations in the ear.




4. L. Moreau, E.D. Sontag, and M. Arcak. Feedback tuning of bifurcations. Systems Control Lett., 50(3):229-239, 2003. [PDF] Keyword(s): bifurcations, adaptive control. Abstract:

   This paper studies a feedback regulation problem that arises in at least two different biological applications. The feedback regulation problem under consideration may be interpreted as an adaptive control problem for tuning bifurcation parameters, and it has not been studied in the control literature. The goal of the paper is to formulate this problem and to present some preliminary results.

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