Stability, Complexity and Robustness in Population Dynamics

Abstract

The problem of stability in population dynamics concerns many domains of application in demography, biology, mechanics and mathematics. The problem is highly generic and independent of the population considered (human, animals, molecules,…). We give in this paper some examples of population dynamics concerning nucleic acids interacting through direct nucleic binding with small or cyclic RNAs acting on mRNAs or tRNAs as translation factors or through protein complexes expressed by genes and linked to DNA as transcription factors. The networks made of these interactions between nucleic acids (considered respectively as edges and nodes of their interaction graph) are complex, but exhibit simple emergent asymptotic behaviours, when time tends to infinity, called attractors. We show that the quantity called attractor entropy plays a crucial role in the study of the stability and robustness of such genetic networks.

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

“…By playing with the potential-Hamiltonian distribution [ 46 ], it is therefore possible, if the system is instantly perturbed, to increase the efficiency of the post-perturbation synchronization (due to the potential return to the limit-cycle) or to allow for desynchronizing (due to the Hamiltonian spiralization), which is necessary to avoid the perseveration in a synchronized state (observed in many neuro-degenerative pathologies).…”

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 .…”

“…By considering a Hopfield rule with all non-zero interaction weights w ij having the same absolute value c , we will study the robustness of the network in response to the variations of c , by proving the following Propositions [ 39 , 42 ]:…”

“…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.…”