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 70

Fig. 4.5 Left : Relative sizes of the five attraction basins of the attractors of the genetic network
controlling the flowering of Arabidopsis thaliana , in case of presence ( red ) or absence ( green )of
RGA regulation. Right : the network

A. Lotka (Lotka 1925 ) used the same system for representing the kinetics of
bimolecular chemical reactions:

d x = d t ¼ x ð a by Þ; d y = d t ¼ y ð cx d Þ

This system is purely Hamiltonian, by changing the variables X ¼ Log x and

Y ¼ Log y (population «affinities») with a Hamiltonian function H equal to:

ce X be Y

H ð X ; Y Þ¼ þ d X þ aY ;

the trajectories being exactly the contour lines of H . These trajectories are
Lyapunov, but not asymptotically, stable. Hence, they are very sensitive to the
initial conditions, which decide what will be the final shape of the temporal
evolution of the system (cf. Fig. 4.6 ).

4.3.8 Example of a Non-Robust (Due to a Sensitivity to the
Updating Mode) Genetic Network Controlling the Cell
Cycle

The core of the genetic network controlling the cell cycle in mammals (Kohn 1999 ),
the E2F box, has a strong connected component made of two intersecting positive
circuits, one of length 4 and another of length 3 (Demongeot et al. 2008a , 2009a ).
The number and nature of its attractors depend both on the updating mode and
on the state of a boundary node, the microRNA miRNA 159, which inhibits E2F.
Then the system is not robust, essentially due to the occurrence of one periodic