Information Technology Reference
In-Depth Information
5.6. Conclusions
We have presented here analysis of Boolean dynamics on both a simplied model of
signalling networks and a real genetic regulatory network model (in our particular
case the GRN underlying cell-fate determination during early stages of ower devel-
opment of Arabidopsis thaliana. In both cases we have observed through computer
simulations that dierent rules, noise intensities, and topologies can give rise to
very dierent complex dynamical behaviors. When the complexity of the dynami-
cal behavior is quantied in terms of the spectrum of the correlation function of the
variable describing the state of the system, we can say that the observed behaviors
range from white noise to Brownian noise. These are two limiting cases very well
known in linear dynamical systems. But complex non-linear systems also show, for
intermediate noise levels, a very interesting 1=f behavior.
We have explored if such behavior is also observed in biological systems. In
particular, we have observed that the correlations of the uctuations are related to
the rate of visitation to the dierent deterministic attractors that the static network
and rules generate. Robustness and exibility are necessary in biological systems,
and thus the observed behavior may be precisely interpreted as the ability of the
system to visit all the available attractors. Therefore, relating such compromise be-
tween robustness and exibility to the complexity of the time signal is a remarkable
nding. In this context, it is remarkable that the wild type experimentally grounded
GRN studied, also showed a similar compromised behavior. Such behavior is lost
when the probability of violating the documented rules is increased. The prelimi-
nary analysis provided here concerning the Arabidopsis thaliana GRN in response
to noise, adds up to previous works showing that this, as well as other biological
GRNs, show a critical behavior at the brisk between ordered and chaotic dynamics.
References
Albert, R. and Barabasi, A.-L. (2002). Statistical mechanics of complex networks, Rev.
Mod. Phys. 74, pp. 47{96.
Albert, R. and Othmer, H. G. (2003). The topology of the regulatory interactions predicts
the expression pattern of the segment polarity genes in drosophila melanogaster,
Journal of Theoretical Biology 223, pp. 1{18.
Aldana, M., Balleza, E., Kauman, S. and Resendiz, O. (2007). Robustness and evolvabil-
ity in genetic regulatory networks, J. Theor. Bio. 245, pp. 433{448.
Aldana, M., Coppersmith, S. and Kadano, L. P. (2003). Boolean dynamics with random
coupling, in E. Kaplan, J. E. Marsden and K. R. Sreenivasan (eds.), Perspectives and
Problems in Nonlinear Science. A celebratory volume in honor of Lawrence Sirovich,
Springer Applied Mathematical Sciences Series (Springer, Berlin), pp. 23{90.
Alvarez-Buylla, E., Benitez, M., Chaos, A., Cortes-Poza, Y., Espinosa-Soto, C., Beau-
Lotto, R., Malkin, D., Escalera-Santos, G. and Padilla-Longoria, P. (2008). Flo-
ral morphogenesis: Stochastic explorations of a gene network epigenetic landscape,
PLoS ONE (in press).
Amaral, L. A. N., Daz-Guilera, A., Moreira, A. A., Goldberger, A. L. and Lipsitz, L. A.
