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Original Research Paper

Graphic Requirements for Multistationarity

Soulé C.

Author affiliations

CNRS et Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

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ComPlexUs 2003;1:123–133

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Article / Publication Details

First-Page Preview
Abstract of Original Research Paper

Received: February 17, 2003
Accepted: June 30, 2003
Published online: March 10, 2004
Issue release date: 2003

Number of Print Pages: 11
Number of Figures: 0
Number of Tables: 0

ISSN: 1424-8492 (Print)
eISSN: 1424-8506 (Online)

For additional information: https://www.karger.com/CPU

Abstract

We discuss properties which must be satisfied by a genetic network in order for it to allow differentiation. These conditions are expressed as follows in mathematical terms. Let F be a differentiable mapping from a finite dimensional real vector space to itself. The signs of the entries of the Jacobian matrix of F at a given point a define an interaction graph, i.e. a finite oriented finite graph G(a) where each edge is equipped with a sign. René Thomas conjectured 20 years ago that if F has at least two nondegenerate zeroes, there exists a such that G(a) contains a positive circuit. Different authors proved this in special cases, and we give here a general proof of the conjecture. In particular, in this way we get a necessary condition for genetic networks to lead to multistationarity, and therefore to differentiation. We use for our proof the mathematical literature on global univalence, and we show how to derive from it several variants of Thomas rule, some of which had been anticipated by Kaufman and Thomas.

© 2004 S. Karger AG, Basel


References

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  2. Cinquin O, Demongeot J: Positive and negative feedback: Striking a balance between necessary antagonists. J Theor Biol 2002;216:229–241.
  3. Gouzé J-L: Positive and negative circuits in dynamical systems. J Biol Syst 1998;6:11–15.
    External Resources
  4. Plahte E, Mestl T, Omholt WS: Feedback circuits, stability and multistationarity in dynamical systems. J Biol Syst 1995;3:409–413.
  5. Snoussi EH: Necessary conditions for multistationarity and stable periodicity. J Biol Syst 1998;6:3–9.
    External Resources
  6. Gale D, Nikaido H: The Jacobian matrix and global univalence of mappings. Math Ann 1965;159:81–93.
  7. Parthasarathy T: On Global Univalence Theorems, Lecture Notes in Mathematics. Berlin, Springer, 1983, p 977.
  8. Garcia CB, Zangwill WI: On univalence and P-matrices. Linear Algebra Appl 1979;24:239–250.
  9. Thomas R, Kaufman M: Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behaviour. Chaos 2001;11:170–179.
  10. Eisenfeld J, DeLisi C: On conditions for qualitative instability of regulatory circuits with application to immunological control loops; in Eisenfeld J, DeLisi C (eds): Mathematics and Computers in Biomedical Applications. Amsterdam, Elsevier, 1985, pp 39–53.
  11. Gowda MS, Ravindran G: Algebraic univalence theorems for nonsmooth functions. J Math Anal Appl 2000;252:917–935.
    External Resources
  12. Alexandroff P, Hopf H: Topologie, vol 45: Berichtigter Reprint, Die Grundlehren der mathematischen Wissenschaften. Berlin, Springer, 1974, p 636.
  13. Thomas R: Logical description, analysis, and feedback loops; in Nicolis G (ed): Aspects of Chemical Evolution. 17th Solvay Conference on Chemistry. Chichester, Wiley, 1980, pp 247–282.
  14. Kaufman M, Thomas R: Model analysis of the bases of multistationarity in the humoral immune response. J Theor Biol 1987;129:141–162.
  15. Campbell L.A: Rational Samuelson maps are univalent. J Pure Appl Algebra 1994;92/3:227–240.

Article / Publication Details

First-Page Preview
Abstract of Original Research Paper

Received: February 17, 2003
Accepted: June 30, 2003
Published online: March 10, 2004
Issue release date: 2003

Number of Print Pages: 11
Number of Figures: 0
Number of Tables: 0

ISSN: 1424-8492 (Print)
eISSN: 1424-8506 (Online)

For additional information: https://www.karger.com/CPU


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