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 100

F ij ( x ) 0, elsewhere.

Eventually, we define the random global dynamic frustration D by:

¼

Dx ðÞ¼Σ i ; j2f 1 ;ng D ij ð ð t x ð t ÞÞ;

where D ij is the local dynamic frustration of the pair ( i , j ) defined by:

D ij ð x ð t ÞÞ ¼ ; α ij ¼ ; x i ð t Þ 6¼ h ðΣ j 2Ni w ij x j ð t Þθ i Þ α ij ¼ ;

1 if 1 or 1

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

D ij ( x ( t )) 0, elsewhere.

Then we prove the following Proposition 1:

¼

Proposition 1. Let us consider the random energy U and the random frustration F
of a getBren N having a constant absolute value w for its interaction weights, null
threshold Θ , temperature T equal to 1 and being sequentially updated, then:

1. U(x) F(x), where Q + (N) is the number of positive

edges in the interaction graph G of the network
2. E μ ( U )

¼ Σ i, j2{1,n} α ij x i x j ¼ Q + (N)

¼ ∂ log Z/ w, where the free energy log Z is equal to the quantity log( Σ y2Ω

∂

exp( Σ j2y,k2y w ij y j y k )) and μ is the invariant Gibbs measure defined by: 8 x 2Ω ,

μ ({ x }) ¼ exp( Σ i2x,j2x w ij x i x j )/Z

3. Var μ U ¼ Var μ F ¼ ∂ E μ / log w, where E μ ¼Σ x2Ω μ ({ x })log( μ ({ x })) ¼

∂

log Z - wE(U) is the entropy of μ , maximal among entropies corresponding to all

probability distributions for the U's having the same given expectation

ν

E ν (U) ¼ E μ (U).

Proof. 1. It is easy to check that: U ( x ) F ( x ) ,

2. The expectation of U , denoted E μ ( U ), is given by:

¼ Q + ( N )

E μ ð U Þ¼Σ x2Ω Σ i2x; j2x α ij x i x j exp ðΣ i2x; j2x wx i x j Þ= Z ¼ @ log Z =@ w

3. Following Demongeot and Waku ( 2012a , b and submitted ), we have Var μ U ¼

Var μ F log w , and according to Demongeot et al. ( 2008b ),

Demongeot and Sen´ ( 2008 ), Demetrius ( 1983 , 1997 ), K¨hn ( 2010 ), and
Fogelman Souli´ et al. ( 1989 ), E μ

¼ ∂ E μ /

∂

is maximal among the proposed set of

entropies.

∎

Proposition 2. Let us consider the getBren N with T ¼ 0, sequentially or synchro-

nously updated, defined from a potential P defined by:

t xA k x t xWx

8 x 2 Ω; P ð x Þ¼Σ k ð Þ x k þ þ Θ x ;