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 66

Fig. 4.2 Left : Immunetwork upstream the gene GATA3 regulating the T helper cell maturation
(Demongeot et al. 2011b ). Right : levels of the Hamiltonian discrete kinetic energy for the positive
circuit of length 5 containing the gene GATA3 ( in red on the left )

4.3.3 Example of a Mixed Potential-Hamiltonian Metabolic
Network in Physiology

The van der Pol system has been used since about 80 years (van der Pol and van der
Mark 1928 ; http://www.sciencesunivnantesfr_sites_genevieve_tulloue_index_
fichiers_animflash.html ) for representing the activity of cardiac cells (Fig. 4.3 ,
bottom) and the differential equations representing its dynamics are defined by:

y

x 2 d x = d t ¼ y ; d y = d t ¼ x þ μ 1

If μ ¼ 0, these equations are those of the simple pendulum and if - x and

x 2 ) are replaced by polynomials in x of high order, they become Li´nard
systems (Demongeot et al. 2007a , b ; Glade et al. 2007 ). It is possible to obtain
a potential-Hamiltonian decomposition: d x /d t

(1

¼ ∂ P / x + H / y ,d y /d t ¼

∂ ∂ ∂

x 2 /4 + y 2 /4)/2. This
decomposition is not unique and allows obtaining an approximation for
the equation of

( x 2 + y 2 )/2 ∂ P / y ∂ H / x , with H ( x , y ) ¼ μ xy (1

∂ ∂

its limit cycle in the form: H ( x , y ) ¼ c (Demongeot and

Fran¸oise 2006 ).
Figure 4.3 (top) shows a dynamical system in which H and P have a revolution
symmetry and share contour lines of the corresponding surfaces especially a limit
cycle in the state space E ( x 1 0 x 2 ): the system can be assimilated, when the state

has a norm sufficiently big, to the motion of the projection (in green) of a ball (in red)
descending along the potential surface P until its minimal set, which is the contour
line (in green) of a Hamiltonian surface (in red). Figure 4.3 (middle) shows the limit
cycle for different values of the anharmonic parameter

¼

of the van der Pol equation.
Figure 4.3 (bottom) shows that the rhythm of the potential of an isolated cardiac cell
fits with the solution of a van der Pol equation, justifying the use of this equation to
represent the whole heart kinetics (Van der Pol and van der Mark 1928 ; http://www.
sciencesunivnantesfr_sites_genevieve_tulloue_index_fichiers_animflash.html ) .

μ