×

Oscillations in multi-stable monotone systems with slowly varying feedback. (English) Zbl 1331.37142

Summary: The study of dynamics of gene regulatory networks is of increasing interest in systems biology. A useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems. Key features of the dynamics of the original system are then deduced from the input-output characteristics of the open loop system and the sign of the feedback. This paper extends these results, showing how to use the same framework of input-output systems in order to prove existence of oscillations, if the slowly varying strength of the feedback depends on the state of the system.

MSC:

37N35 Dynamical systems in control
34C26 Relaxation oscillations for ordinary differential equations
37N25 Dynamical systems in biology
93B52 Feedback control
93C70 Time-scale analysis and singular perturbations in control/observation systems
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Angeli, D.; Sontag, E. D., Monotone control systems, IEEE Trans. Automat. Control, 48, 10, 1684-1698 (2003) · Zbl 1364.93326
[2] Angeli, D.; Sontag, E. D., Multi-stability in monotone input/output systems, Systems Control Lett., 51, 185-202 (2004) · Zbl 1157.93509
[3] Angeli, D.; Sontag, E. D., An analysis of a circadian model using the small-gain approach to monotone systems, (Proc. IEEE Conf. Decision and Control. Proc. IEEE Conf. Decision and Control, Paradise Island, Bahamas, December 2004 (2004), IEEE Publications), 575-578
[4] Angeli, D.; De Leenheer, P.; Sontag, E. D., A small-gain theorem for almost global convergence of monotone systems, Systems Control Lett., 52, 407-414 (2004) · Zbl 1157.93499
[5] Angeli, D.; Ferrell, J. E.; Sontag, E., Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Natl. Acad. Sci., 101, 7, 1822-1827 (2004)
[6] Bagowski, C. P.; Ferrell, J. E., Bistability in the JNK cascade, Curr. Biol., 11, 1176-1182 (2001)
[7] Bhalla, U. S.; Ram, P. T.; Iyengar, R., MAP kinase phosphatase as a locus of flexibility in a mitogen-activated protein kinase signaling network, Science, 297, 1018-1023 (2002)
[8] Boczko, E.; Cooper, T. G.; Gedeon, T.; Mischaikow, K.; Murdock, D.; Pratap, S.; Wells, S., Structure theorems and the dynamics of nitrogen catabolite repression in yeast, Proc. Natl. Acad. Sci., 102, 5647-5652 (2005)
[9] P. Brunovsky, Controlling nonuniqueness of local invariant manifolds, Comenius University Preprint M6-91, 1991; P. Brunovsky, Controlling nonuniqueness of local invariant manifolds, Comenius University Preprint M6-91, 1991 · Zbl 0783.58061
[10] Conley, C., Isolated Invariant Sets in Compact Metric Spaces, CBMS Reg. Conf. Ser. Math., vol. 38 (1978), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[11] De Leenheer, P.; Angeli, D.; Sontag, E. D., On predator-prey systems and small-gain theorems, J. Math. Biosci. Eng., 2, 25-42 (2005) · Zbl 1079.34040
[12] De Leenheer, P.; Malisoff, M., A small-gain theorem for monotone systems with multi-valued input-state characteristics, IEEE Trans. Automat. Control, 41, 2, 287-292 (2006) · Zbl 1366.93258
[13] Elowitz, M. B.; Leibler, S., A synthetic oscillatory network of transcriptional regulators, Nature, 403, 335-338 (2000)
[14] Enciso, G.; Sontag, E. D., Monotone systems under positive feedback: Multi-stability and a reduction theorem, Systems Control Lett., 54, 159-168 (2005) · Zbl 1129.93398
[15] Enciso, G.; Sontag, E. D., Global attractivity, I/O monotone small-gain theorems, and biological delay systems, Discrete Contin. Dyn. Syst., 14, 3, 549-578 (2006) · Zbl 1111.93071
[16] Enciso, G.; Smith, H. L.; Sontag, E. D., Non-monotone systems decomposable into monotone systems with negative feedback, J. Differential Equations, 224, 1, 205-227 (2006) · Zbl 1103.34021
[17] Ferrell, J. E.; Machleder, E. M., The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes, Science, 280, 895-898 (1998)
[18] Golubitskii, M.; Schaeffer, D., Singularities and Groups in Bifurcation Theory (1984), Springer-Verlag
[19] Hale, J., Ordinary Differential Equations (1980), John Wiley & Sons Inc.
[20] C.K.R.T. Jones, A geometric approach to applied dynamics and differential equations, Lecture Notes, 1996; C.K.R.T. Jones, A geometric approach to applied dynamics and differential equations, Lecture Notes, 1996
[21] Lefschetz, S., Differential Equations: Geometric Theory (1977), Dover · Zbl 0107.07101
[22] Katok, A.; Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems (1995), Cambridge University Press · Zbl 0878.58020
[23] Kholodenko, B. N., Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades, Eur. J. Biochem., 267, 1583-1588 (2000)
[24] McCord, Ch.; Mischaikow, K.; Mrozek, M., Zeta functions, periodic trajectories, and the Conley index, J. Differential Equations, 121, 258-292 (1995) · Zbl 0833.34045
[25] Novick, A.; Wiener, M., Enzyme induction as an all-or-none phenomena, Proc. Natl. Acad. Sci., 43, 553-566 (1957)
[26] Pomerening, J. R.; Sontag, E. D.; Ferrell, J. R., Building a cell cycle oscillator: Hysteresis and bistability in the activation of Cdc2, Nat. Cell Biol., 5, 346-351 (2003)
[27] Ptashne, M., A Genetic Switch: Phage and Higher Organisms (1992), Blackwell: Blackwell Oxford
[28] Sha, W.; Moore, J.; Chen, K.; Lassaletta, Y. D.; Yi, C. S.; Tyson, J. J.; Sible, I. C., Hysteresis drives cell-cycle transitions in Xenopus Laevis egg extracts, Proc. Natl. Acad. Sci., 100, 975-980 (2002)
[29] Smith, H., Monotone Dynamical Systems, Math. Surveys Monogr., vol. 41 (1995), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[30] Sohaskey, M. L.; Ferrell, J. E., Distinct, constitutively active MAPK phosphatases function in Xenopus oocytes: Implications for p42 MAPK regulation in vivo, Mol. Biol. Cell, 10, 3729-3743 (1999)
[31] Sontag, E. D., Some new directions in control theory inspired by systems biology, IEEE Systems Biology, 1, 9-18 (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.