The Cell Cycle is a Limit Cycle

C. Gérard^1, 2 ; A. Goldbeter¹

¹ Unité de Chronobiologie théorique, Faculté des Sciences, Université Libre de Bruxelles (ULB) Campus Plaine, CP 231, B-1050 Brussels, Belgium
² On leave at the Department of Biochemistry, University of Oxford, Oxford, UK

Mathematical modelling of natural phenomena, Tome 7 (2012) no. 6, pp. 126-166.

Résumé

Progression along the successive phases of the mammalian cell cycle is driven by a network of cyclin-dependent kinases (Cdks). This network is regulated by a variety of negative and positive feedback loops. We previously proposed a detailed, 39-variable model for the Cdk network and showed that it is capable of temporal self-organization in the form of sustained oscillations, which correspond to the repetitive, transient, sequential activation of the cyclin- Cdk complexes that govern the successive phases of the cell cycle [Gérard and Goldbeter (2009) Proc Natl Acad Sci 106, 21643-8]. Here we compare the dynamical behavior of three models of different complexity for the Cdk network driving the mammalian cell cycle. The first is the detailed model that counts 39 variables and is based on Michaelis-Menten kinetics for the enzymatic steps. From this detailed model, we build a version based only on mass-action kinetics, which counts 80 variables. In this version we do not need to assume that enzymes are present in much smaller amounts that their substrates, which is not necessarily the case in the cell cycle. We show that these two versions of the model for the Cdk network yield similar results. In particular they predict sustained oscillations of the limit cycle type. We show that the model for the Cdk network can be reduced to a version containing only 5 variables, which is more amenable to stochastic simulations. This skeleton version retains the dynamic properties of the more complex versions of the model for the Cdk network in regard to Cdk oscillations. The regulatory wiring of the Cdk network therefore governs its dynamic behavior, regardless of the degree of molecular detail. We discuss the relative advantages of each version of the model, all of which support the view that the mammalian cell cycle behaves as a limit cycle oscillator.

Détail

DOI : 10.1051/mmnp/20127607

Bibliographie

[1] A.W. Murray, M.W. Kirschner Nature 1989 275 280

[2] A. Murray, T. Hunt. The Cell Cycle : An Introduction. W.H. Freeman and Company (1993), New York.

[3] M.A. Félix, J.C. Labbé, M. Dorée, T. Hunt, E. Karsenti Nature 1990 379 382

[4] J.J. Tyson Proc. Natl. Acad. Sci. USA 1991 7328 7332

[5] A. Goldbeter Proc. Natl. Acad. Sci. USA 1991 9107 9111

[6] B. Novak, J.J. Tyson J. Cell. Sci. 1993 1153 1168

[7] J.E. Ferrell, E.M. Machleder Science 1998 895 898

[8] J.R. Pomerening, E.D. Sontag, J.E. Ferrell Nat. Cell. Biol. 2003 346 351

[9] W. Sha, J. Moore, K. Chen, A.D. Lassaleta, C.-S. Yi, J.J. Tyson, J.C. Sible Proc. Natl. Acad. Sci. USA 2003 975 980

[10] B. Novak, J.J. Tyson Proc. Natl. Acad. Sci. USA 1997 9147 9152

[11] K.C. Chen, L. Calzone, A. Csikasz-Nagy, F.R. Cross, B. Novak, J.J. Tyson Mol. Biol. Cell. 2004 3841 3862

[12] D. Barik, W.T. Baumann, M.R. Paul, B. Novak, J.J. Tyson Mol. Syst. Biol. 2010 405

[13] D.O. Morgan Nature 1995 131 134

[14] D.O. Morgan. The Cell Cycle : Principles of Control. Oxford Univ Press, UK, (2006).

[15] Z. Qu, J.N. Weiss, W.R. Maclellan Am. J. Physiol. Cell. Physiol. 2003 349 364

[16] M. Swat, A. Kel, H. Herzel Bioinformatics 2004 1506 1511

[17] B. Pfeuty, T. David-Pfeuty, K. Kaneko Cell Cycle 2008 3246 3257

[18] B. Novak, J.J. Tyson J. Theor. Biol. 2004 563 579

[19] E. He, O. Kapuy, R.A. Oliveira, F. Uhlmann, J.J. Tyson, B. Novak Proc. Natl. Acad. Sci. USA 2011 10016 10021

[20] C. Gérard, A. Goldbeter Proc. Natl. Acad. Sci. USA 2009 21643 21648

[21] C. Gérard, A. Goldbeter Interface Focus 2011 24 35

[22] C. Gérard, D. Gonze, A. Goldbeter FEBS J. 2012 3411 3431

[23] A. Chauhan, S. Lorenzen, H. Herzel, S. Bernard J. Theor. Biol. 2011 103 112

[24] C. Gérard, A. Goldbeter PLoS Comput. Biol. 2012

[25] E. Filipski, V.M. King, X.M. Li, T.G. Granda, M.C. Mormont, X. Liu, B. Claustrat, M.H. Hastings, F. Lévi J. Natl. Cancer Inst. 2002 690 697

[26] L. Fu, C.C. Lee Nature 2003 350 361

[27] J.S. Pendergast, M. Yeom, B.A. Reyes, Y. Ohmiya, S. Yamazaki Commun. Integr. Biol. 2010 536 539

[28] L.A. Segel Bull. Math. Biol. 1988 579 593

[29] J.A. Borghans, R.J. De Boer, L.A. Segel Bull. Math. Biol. 1996 43 63

[30] A. Ciliberto, F. Capuani, J.J. Tyson PLoS Comput. Biol. 2007

[31] W. Zachariae, K. Nasmyth Genes Dev. 1999 2039 2058

[32] E.R. Kramer, N. Scheuringer, A.V. Podtelejnikov, M. Mann, J.M. Peters Mol. Biol. Cell. 2000 1555 1569

[33] I. Hoffmann, P.R. Clarke, M.J. Marcote, E. Karsenti, G. Draetta EMBO J. 1993 53 63

[34] M. Sabouri-Ghomi, A. Ciliberto, S. Kar, B. Novak, J.J. Tyson J. Theor. Biol. 2008 209 218

[35] A. Goldbeter, D.E. Koshland Proc. Natl. Acad. Sci. USA 1981 6840 6844

[36] H. Matsushime, D.E. Quelle, S.A. Shurtleff, M. Shibuya, C.J. Sherr, J.-Y. Kato Mol. Cell. Biol. 1994 2066 2076

[37] A. Goldbeter, C. Gérard, J.-C. Leloup Médecine/Sciences 2010 49 56

[38] A. Goldbeter, C. Gérard, J.-C. Leloup, D. Gonze, G. Dupont FEBS Lett. 2012 2955 2965

[39] C. Gérard, A. Goldbeter Chaos 2010 045109

[40] S. Mittnacht Curr. Opin. Genet. Dev. 1998 21 27

[41] J.W. Harbour, D.C. Dean Genes Dev. 2000 2393 2409

[42] J.-H. Dannenberg, A. Van Rossum, L. Schuijff, H. Te Riele Genes Dev. 2000 3051 3064

[43] J. Sage, G.J. Mulligan, L.D. Attardi, A. Miller, S. Chen, B. Williams, E. Theodorou, T. Jacks Genes Dev. 2000 3037 3050

[44] J.R. Pomerening, S.Y. Kim, J.E. Ferrell Cell 2005 565 578

[45] D. Gonze, M. Hafner. Positive feedbacks contribute to the robustness of the cell cycle with respect to molecular noise. Adv. in theory of control, signals. LNCIS 407, (2010) pp. 283–295 (Lévine J Müllhaupt, eds), Springer-Verlag Berlin Heidelberg, Germany.

[46] C. Gérard, A. Goldbeter Front. Physiol. 2012 413

[47] A. Altinok, D. Gonze, F. Lévi, A. Goldbeter Interface Focus 2011 36 47

[48] A. Altinok, F. Lévi, A. Goldbeter Adv. Drug Deliv. Rev. 2007 1036 1053

[49] A.T. Winfree. Discontinuities and singularities in the timing of nuclear division. In : Cell Cycle Clocks. L.N. Edmunds Jr, ed. Marcel Dekker, New York and Basel, (1984) pp. 63–80.

[50] L.N. Jr. Edmunds. Cellular and Molecular Bases of Biological Clocks. Models and Mechanisms for Circadian Time- keeping. Springer, New York (1988).

[51] A.T. Winfree. The Geometry of Biological Time. Springer, New York (Reprinted as Springer Study Edition, 1990, Springer, Berlin, 1980).

[52] J.-C. Leloup, A. Goldbeter Am. J. Physiol. Reg. Integr. Comp. Physiol. 2001 R1206 R1212

[53] D. Gonze, A. Goldbeter J Theor Biol 2001 167 186

[54] I. Conlon, M. Raff J. Biol. 2003 7