Abstract
Using computer algebra methods to prove that a gene regulatory network cannot oscillate appears to be easier than expected. We illustrate this claim with a family of models related to historical examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Morant, P.E., Vandermoere, C., Parent, B., Lemaire, F., Corellou, F., Schwartz, C., Bouget, F.Y., Lefranc, M.: Oscillateurs génétiques simples. Applications à l’horloge circadienne d’une algue unicellulaire. In: Proceedings of the Rencontre du non linéaire, Paris (2007), http://nonlineaire.univ-lille1.fr
McClung, C.R.: Plant Circadian Rhythms. The Plant Cell 18, 792–803 (2006)
Fall, C.P., Marland, E.S., Wagner, J.M., Tyson, J.J. (eds.): Computational Cell Biology. Interdisciplinary Applied Mathematics, vol. 20. Springer, Heidelberg (2002)
Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge University Press, Cambridge (2004)
Françoise, J.P.: Oscillations en biologie. Mathématiques et Applications, vol. 46. Springer, Heidelberg (2005)
Hale, J.K., Koçak, H.: Dynamics and Bifurcations. Texts in Applied Mathematics, vol. 3. Springer, New York (1991)
Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations I. Nonstiff problems, 2nd edn. Springer Series in Computational Mathematics, vol. 8. Springer, New York (1993)
Doedel, E.: AUTO software for continuation and bifurcation problems in ODEs (1996), http://indy.cs.concordia.ca/auto
Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. Software, Environments, and Tools, vol. 14, SIAM (2002)
El Kahoui, M., Weber, A.: Deciding Hopf bifurcations by quantifier elimination in a software–component architecture. Journal of Symbolic Computation 30(2), 161–179 (2000)
Wang, D., Xia, B.: Stability Analysis of Biological Systems with Real Solution Classification. In: proceedings of ISSAC 2005, Beijing, China 354–361 (2005)
Gatermann, K., Hosten, S.: Computational algebra for bifurcation theory. Journal of Symbolic Computation 40, 1180–1207 (2005)
Gatermann, K., Eiswirth, M., Sensse, A.: Toric Ideals and graph theory to analyze Hopf bifurcations in mass action systems. Journal of Symbolic Computation 40, 1361–1382 (2005)
Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. SIGSAM Bulletin 37(4), 97–108 (2003)
Dolzmann, A., Sturm, T.: Redlog: computer algebra meets computer logic. SIGSAM Bulletin 31(2), 2–9 (1997)
El Din, M.S.: RAGLib (Real Algebraic Library Maple package) (2003), http://www-calfor.lip6.fr/~safey/RAGLib
Goodwin, B.C.: Temporal Organization in Cells. Academic Press, London (1963)
Goodwin, B.C.: In: Advances in Enzyme Regulation, vol. 3, p. 425. Pergamon Press, Oxford (1965)
Griffith, J.S.: Mathematics of Cellular Control Processes. I. Negative Feedback to One Gene. Journal of Theoretical Biology 20, 202–208 (1968)
Selleck, W., Howley, R., Fang, Q., Podolny, V., Fried, M.G., Buratowski, S., Tan, S.: A histone fold TAF octamer within the yeast TFIID transcriptional coactivator. Nature Structural Biology 8(8), 695–700 (2001)
Ruoff, P., Rensing, L.: The Temperature–Compensated Goodwin Model Simulates Many Circadian Clock Properties. Journal of Theoretical Biology 179, 275–285 (1996)
Ruoff, P., Vinsjevik, M., Mohsenzadeh, S., Rensing, L.: The Goodwin Model: Simulating the Effet of Cycloheximide and Heat Shock on the Sporulation Rhythm of Neurospora crassa. Journal of Theoretical Biology 196, 483–494 (1999)
Kurosawa, G., Mochizuki, A., Iwasa, Y.: Comparative Study of Circadian Clock Models, in Search of Processes Promoting Oscillation. Journal of Theoretical Biology 216, 193–208 (2002)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. An introduction to computational algebraic geometry and commutative algebra. Undergraduate Texts in Mathematics. Springer, New York (1992)
Becker, T., Weispfenning, V.: Gröbner Bases: a computational approach to commutative algebra. Graduate Texts in Mathematics, vol. 141. Springer, Heidelberg (1991)
Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Heidelberg (2003)
Collins, G.E., Akritas, A.G.: Polynomial real root isolation using Descartes’rule of signs. In: proceedings of ISSAC 1976, Yorktown Heights, NY, pp. 272–275 (1976)
Collins, G.E.: Quantifier Elimination for the Elementary Theory of Real Closed Fields by Cylindrical Algebraic Decomposition. In: Brakhage, H. (ed.) Automata Theory and Formal Languages. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Boulier, F., Lefranc, M., Lemaire, F., Morant, PE., Ürgüplü, A. (2007). On Proving the Absence of Oscillations in Models of Genetic Circuits. In: Anai, H., Horimoto, K., Kutsia, T. (eds) Algebraic Biology. AB 2007. Lecture Notes in Computer Science, vol 4545. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73433-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-73433-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73432-1
Online ISBN: 978-3-540-73433-8
eBook Packages: Computer ScienceComputer Science (R0)