MicroRNAs and Robustness in Biological Regulatory Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic, and Genetic

Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks

“…We define first the functions energy U , frustration F and dynamic entropy E of a genetic network N with n genes in interaction [ 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 ]. where x is a configuration of gene expression ( , if the gene i is expressed and , if not), denotes the set of all configurations of gene expression (i.e., the hypercube ) and is the sign of the interaction weight quantifying the influence the gene j exerts on the gene i : (resp.…”

“…If we consider the Hopfield rule with a constant absolute value for its non-zero interaction weights, we can study the robustness of the network with respect to the variations of w, by using the two following propositions [ 51 , 52 ]:…”

Entropy as a Robustness Marker in Genetic Regulatory Networks

“…We define first the functions energy U and frustration F for a genetic network N with n genes in interaction [ 1 , 2 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 78 ]: where x is a configuration of gene expression ( x i = 1, if the gene i is expressed, and x i = 0, if not), E denotes the set of all configurations of gene expression; that is, for a Boolean network, the hypercube {0,1} n , and α ij = sign( w ij ) is the sign of the interaction weight w ij , which quantifies the influence of the gene j on the gene i : α ij = −1 (with respect to +1), if j is an inhibitor (with respect to activator) of the expression of i , and α ij = 0, if j exerts no influence on i . Q + ( N ) is equal to the number of positive edges of the interaction graph G of the network N having n genes, whose incidence matrix is A = ( α ij ) i , j = 1, n .…”

“…The matrices involved in the network dynamics are constant or depend on time t [ 41 , 42 ]: Concerning W , the dependence is called the Hebbian dynamics: if the two vectors { x i ( s )} s < t and { x j ( s )} s < t have a correlation coefficient j ( t )≠0, then the dependence is expressed through the equation: w ij ( t + 1) = w ij ( t ) + h ij ( t ), with h > 0, corresponding to a reinforcement of the absolute value of interactions w ij ( t ) having succeeded to increase the x i( s )’s, if w ij ( t ) is positive, and conversely to decrease the x i (s )’s, if w ij ( t ) is negative. Concerning S, the updating can be state dependent or not.…”

“…In the present paper, we will focus on the study of robustness of genetic regulatory networks driven by Hopfield’ stochastic rule [ 35 ], by using the Kolmogorov-Sinaï entropy of the Markov process underlying the state transition dynamics [ 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 ]. In Section 2 , we define the concepts underlying the relationships between complexity, stability and robustness in the Markov framework of genetic threshold Boolean random regulatory networks.…”