MicroRNAs and Robustness in Biological Regulatory Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic, and Genetic - Systems Biology of Metabolic and Signaling Networks - page 98

Ben Amor et al. 2010a , b ), and the role of this inhibition could lead to a
restricted number of dynamical behaviors susceptible to be stored and evoked
inside the subnetworks of hippocampus. Many microRNAs are also controlling
morphogenesis of teeth (where miR-140, miR-31, miR-875-5p, and miR-141
were expressed mainly during tooth morphogenesis), feathers or hairs, with
miRs like miR-30 activating Wnt through WWP1 [cf. Fig. 4.16 and Michon
et al. ( 2012 ) and Michon et al. ( 2008 )] or miR-16 inhibited by Wnt/ß-catenin
signaling and inhibiting Cdk2 in the proliferation box [cf. Fig. 4.16 and
Takeshita et al. ( 2010 ) and Martello et al. ( 2007 )].
More generally, the passage from a random network structure to a small world
(Duchon et al. 2006 ; Demongeot et al. 2010b , 2011c ) during evolution, by increas-
ing the microRNAs number in case of networks controlling the same function (like
the cell cycle), could have contributed to increase network robustness. Eventually,
considering the role of a large inhibition by microRNAs in genetic networks
modeling the chromatin clock, could allow understanding the role of the
state-dependent expression schedule. These two last perspectives would lead to
further investigations and constitute challenges for future work.

Acknowledgments We are indebted to A. Doncescu, M. Noual, and S. Sen´ for many helpful
discussions. This work has been supported by the EC NoE VPH.

Mathematical Annex

In mathematical modeling, a real genetic regulatory network is called a genetic
threshold Boolean regulatory network (denoted in the following getBren). A
getBren N can be considered as a set of random automata, defined by:

1. Any random automaton i of the getBren N owns at time t a state x i ( t ) valued in
{0,1}, 0 (resp. 1) meaning that gene i is inactivated or in silence (resp. activated
or in expression). The global state of the getBren at time t , called configuration in
the sequel, is then defined by: x ( t ) {0,1} n

¼ ( x i ( t )) i 2 {1, n } 2Ω ¼

2. a getBren N of size n is a triplet ( W , Θ , P ), where:

- W is a matrix of order n , where the coefficient w ij 2 R represents the

interaction weight gene j has on gene i . Sign( W ) ¼ ( α ij ¼ sign( w ij )) is the

adjacency (or incidence) matrix of the interaction graph G .

Θ

is an activation threshold vector of dimension n , its component

θ i being the

-

activation threshold attributed to automaton i
- M : P(

2 n )isa
Markov transition matrix, built from local probability transitions P i giving the
new state of the gene i at time t +1 according to W ,

[0,1] mm (where P(

Ω ) ! Ω ) is the set of all subsets of Ω and m ¼

Θ , and configuration x ( t )of

N at time t such that: