[Robustesse dans les réseaux de régulation biologique I : Approche mathématique]
De nombres indices ont été proposés pour quantifier la complexité des réseaux biologiques de régulation, comme le nombre de leurs composants, leur connectivité, ou le nombre des composantes fortement connexes de leur graphe dʼinteraction. Quant à la stabilité de ces réseaux biologiques, elle correspond à leur capacité à absorber les changements dynamiques ou paramétriques. La complexité est ici mesurée par lʼentropie évolutionnaire, qui décrit la manière dont la probabilité de présence asymptotique du système dynamique correspondant est distribuée dans lʼespace dʼétat, et la stabilité est caractérisée par la vitesse de retour à lʼéquilibre de cette distribution, après perturbation. Cet article montre les relations mathématiques existant entre entropie et vitesse de retour, de manière générale dans le cadre des chaînes de Markov.
Numerous indices of complexity are used in biological regulatory networks like the number of their components, their connectance (or connectivity), or the number of the strong connected components of their interaction graph. Concerning the stability of a biological network, it corresponds to its ability to recover from dynamical or parametric disturbance. Complexity is here quantified by the evolutionary entropy, which describes the way the asymptotic presence distribution of the corresponding dynamical system is spread over the state space and the stability (or robustness) is characterized by the rate at which the system returns to this equilibrium distribution after a perturbation. This article shows the mathematical relationships between entropy and stability rate in the general framework of a Markov chain.
Accepté le :
Publié le :
@article{CRMATH_2012__350_3-4_221_0,
author = {Demongeot, Jacques and Waku, Jules},
title = {Robustness in biological regulatory networks {I:} {Mathematical} approach},
journal = {Comptes Rendus. Math\'ematique},
pages = {221--224},
publisher = {Elsevier},
volume = {350},
number = {3-4},
year = {2012},
doi = {10.1016/j.crma.2012.01.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.01.003/}
}
TY - JOUR AU - Demongeot, Jacques AU - Waku, Jules TI - Robustness in biological regulatory networks I: Mathematical approach JO - Comptes Rendus. Mathématique PY - 2012 SP - 221 EP - 224 VL - 350 IS - 3-4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.01.003/ DO - 10.1016/j.crma.2012.01.003 LA - en ID - CRMATH_2012__350_3-4_221_0 ER -
%0 Journal Article %A Demongeot, Jacques %A Waku, Jules %T Robustness in biological regulatory networks I: Mathematical approach %J Comptes Rendus. Mathématique %D 2012 %P 221-224 %V 350 %N 3-4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.01.003/ %R 10.1016/j.crma.2012.01.003 %G en %F CRMATH_2012__350_3-4_221_0
Demongeot, Jacques; Waku, Jules. Robustness in biological regulatory networks I: Mathematical approach. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 221-224. doi : 10.1016/j.crma.2012.01.003. http://www.numdam.org/articles/10.1016/j.crma.2012.01.003/
[1] Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves, Philos. Trans. R. Soc. Lond. Ser. A, Volume 367 (2009), pp. 4829-4862
[2] Complexity and demographic stability in population models, Theor. Pop. Biol., Volume 65 (2004), pp. 211-225
[3] Darwinian fitness, Theor. Pop. Biol., Volume 72 (2007), pp. 323-345
[4] Natural selection and demographic drift: an empirical study of France (1860–1965), Population, Volume 2 (1989), pp. 231-248
[5] Asymptotic evaluation of certain Markov process expectations for large time. I, Comm. Pure Appl. Math., Volume 28 (1975), pp. 1-47
[6] Random Perturbations of Dynamical Systems, Springer, New York, 1984
[7] Entropy increase in dynamical systems, Israel J. Math., Volume 38 (1981), pp. 241-256
[8] A non-equilibrium entropy for dynamical systems, J. Stat. Phys., Volume 24 (1981), pp. 325-343
[9] Evaluation of distance measures between Gaussian mixture models of MFCCs, Proceedings ISMIR 2007, Austrian Computer Society, Wien, 2007, pp. 107-108
[10] Robustness: confronting lessons from physics and biology, Biol. Rev. Camb. Philos. Soc., Volume 83 (2008), pp. 509-532
[11] Entropy and convergence in dynamics and demography, J. Math. Biol., Volume 31 (1993), pp. 253-271
[12] On small random perturbations of dynamical systems, Russian Math. Surveys, Volume 25 (1970), pp. 1-55
Cité par Sources :
