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 99

P i;g β ðf

8 g 2f 0 ; 1 g; β 2 Ω; x i ð t þ 1 Þ¼ gx j ð t Þ¼βg Þ¼ exp ½ g ðΣ j 2Ni w i j β j θ i Þ= T = Z i ;

where Z i ¼½ , N i is the neighborhood of the gene i in

the getBren N , i.e., the set of genes j (including possibly i ) such that w ij

1 þ exp ½ðΣ j 2Ni w ij β j θ i Þ= T

0, and
P i;g β is the probability for the gene i of passing to the state g at time t +1, from
the state

6¼

at time t on N i . M denotes the transition matrix built from the P i;g β ,
s. M depends on the update mode chosen for changing the states of the getBren
automata. For the extreme values of the randomness parameter T , we have:

β

1. if T 0, the getBren becomes a deterministic threshold automata network and

the transition can be written as:

¼

x i ð t þ 1 Þ¼ h ðΣ j 2Ni w ij x j ð t Þθ i Þ;

where h is the Heaviside function: h ( y ) ¼ 1, if y > 0;

h ( y ) 0,

except for the case

¼ 0, if y <

Σ j 2Ni w ij x j ð t Þθ I ¼ 0, for which, if necessary, 1 and 0 are

both chosen with probability ½
2. When T tends to infinity, then P i;g β ¼

1

= 2 and each line of M is the uniform

distribution on .

If the mode of updating the states of the n genes is sequential, the invariant
measure

Ω

Ω
defined by (Demongeot and Waku 2012a ; Demongeot and Demetrius ( submit-
ted ); Demongeot et al. 2011b , 2008b , 2012 ; Demongeot and Sen´ 2008 , 2011 ;
Fogelman Souli ´ et al. 1989 ; Cosnard and Goles 1977 ):

μ expressing the asymptotic state of N is the Gibbs measure on

8 x 2 Ω; μ f ðÞ¼ x exp ððΣ i2x;

j2Ni w ij x i x j θ i Þ=

T Þ= Z ;

where Z .

We define the random energy U (Demongeot and Sen ´ 2008 , 2011 ;
Demongeot et al. 2008b ) and random frustration F (Demongeot and Waku
2012a ; Demongeot and Demetrius ( submitted )) of a getBren N by:

¼ Σ y 2Ω exp ððΣ j 2y;k2Nj w jk y j y k θ j Þ= T Þ

8 x 2 Ω; U ð x Þ¼Σ i ; j2f 1 ;ng α ij x i x j ¼ Q þ ð N Þ F ð x Þ;

where Q + ( N ) is the number of positive edges in the interaction graph G of the
network N and F ( x ) the global frustration of x , i.e., the number of pairs ( i , j ) where
the values of x i and x j are contradictory with the sign α ij of the interaction between

genes i and j : F

ð x Þ¼Σ i; j2f 1 ;ng F ij ð x Þ

, where F ij is the local frustration of the pair ( i , j )

defined by:

F ij ð x Þ¼ 1 ; if α ij ¼ 1 ; x j ¼ 1 and x i ¼ 0 ; or x j ¼ 0 and x i ¼ 1 ;

and if α ij ¼ 1 ; x j ¼ 1 and x i ¼ 1 ; or x j ¼ 0 and x i ¼ 0 ;