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 63

have: d x ( a , t )/d t ¼r P ¼ grad P ¼ ∂ P / x , where ∂ P / x is the vector

∂ ∂

( , x n ).

The system is called dissipative, because the potential P decreases along trajectories
until attractors which are located on the minima of P .

∂ P / x 1 , , ∂ P / x n )andthestate x is a vector of dimension n : x ¼ ( x 1 ,

∂ ... ∂ ...

4.2.8 Definition of a Hamiltonian dynamics

A dynamical system has a Hamiltonian dynamics if the velocity along its
trajectories is tangent to the contour lines projected on E from the surface represen-
tative of an energy function H defined on E :d x ( a , t )/d t ¼ tang H . If the dimension of

the system is 2, then the vector Tang H is equal to ( x 1 ). The system

is said conservative, because the energy function H is constant along a trajectory.

H / x 2 , ∂ H /

∂ ∂ ∂

4.2.9 Definition of a Mixed Potential-Hamiltonian Dynamics

A dynamical system has a mixed potential-Hamiltonian dynamics if the velocity
along its trajectories can be decomposed into two parts, one potential and one
Hamiltonian: d x ( a , t )/d t grad P + tang H . If the set of minima of P is a contour

line of the surface H on E , then its connected components are attractors of the
system.

¼

4.2.10 Definition of a Principal Potential Part Dynamics

A mixed potential-Hamiltonian system has a principal potential part dynamics, if
the ratio

between the norms of the potential part and the
Hamiltonian one tends to 0 when t tends to infinity.

k tangH k = k grad P k

4.3 Examples of Robust and Non-Robust Regulatory
Networks

We will give as examples in the following some toy models coming from regulatory
networks studied more in details in (Demongeot et al. 2000 ; Thellier et al. 2004 ;
Cinquin and Demongeot 2005 ; Forest and Demongeot 2006 ; Forest et al. 2006 ;
Jolliot and Prochiantz 2004 ; Demongeot et al. 2007a , b , 2008a , 2009a , 2010a ,
2011a , b ; Glade et al. 2007 ; Demongeot and Fran¸oise 2006 ; Ben Amor et al. 2008 ;