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 96

4.6 Perspective: The General Architecture of a Genetic
Regulatory Network and the Problem of Its Robustness

The general architecture of a genetic regulatory network is given like in Fig. 4.8 or
in Fig. 4.16 by a digraph framework in which the nodes sources are often micro-
RNAs exerting their partly unspecific basic inhibitory influence, and the nodes sink
are important genes controlling vital functions like the RAG (Recombinase
Activating Gene) responsible for the TCR building in the immune system
[cf. Fig. 4.16 and Demongeot and Waku ( 2012a ), Pasqual et al. ( 2002 ), Baum
et al. ( 2004 ), and Thuderoz et al. ( 2010 )].
The network robustness (or resilience) (Demongeot and Waku 2012a ;
Demongeot and Demetrius ( submitted ); Blanchini and Franco 2011 ; Lesne
2008 ; Gunawardena 2010 ; Waddington 1940 ;Thom 1972 ; Cinquin and
Demongeot 2002 ) can be defined as the capacity to return at its ordinary asymp-
totic dynamical behavior (called attractor) after endogenous or exogenous

perturbations affecting:

- The state of certain of its genes, e.g., in case of specific silencing by micro-RNAs
- The state of its boundary, notably the appearance of new regulations from a
mutation in the noncoding genome giving birth to new micro-RNAs
- Its architecture, by creating new links between proteins needing to take into
account nonlinear interactions (Demongeot and Sen´ 2011 ).

A study about the influence of the microRNAs on the robustness of a network
needs the exact counting of the dynamical attractors of this network, then to
know the reduction or on the contrary the amplification factor caused, for
example, by the circuit opening due to the effective inhibition of a gene (until
its possible knockout) by a microRNA, or by the appearance of new genes,
which corresponds to important architectural perturbations. A parameter devoted
to represent the robustness of a biological network is its evolutionary entropy
[cf. Mathematical Annex below and Demongeot et al. ( 2008b ), Demongeot and
Sen´ ( 2008 ), Demetrius ( 1983 , 1997) ,K¨hn ( 2010 ), Fogelman Souli´
et al. ( 1989 ), Cosnard and Goles ( 1977 ), and Demongeot et al. ( 2012 )], defined
as the Kolmogorov-Sina¨ entropy of the Markov process underlying the gene
states updating. The microRNAs can have a double opposite influence on this
parameter, causing its increase (and hence that of the network robustness) when
they create an unspecific inhibitory “noise”, dispatching into the gene state space
the probability to have more possible phenotypes, and they can also provoke a
decrease of the robustness, by opening circuits of the strongly connected
components of the network, hence by diminishing the number of positive
circuits, responsible of the attractor number of the whole network.
The example in Fig. 4.16 shows that we can count the attractor number of the
main subnetworks of a functional network like that organized around the gene
Engrailed. These attractor numbers are given in blue in Fig. 4.16 , as well as their
nature (steady state or limit cycle). The global attractor of the whole network is