Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 72

is satisfied by S if and only if every path starting from a state x , with x 1 =0, only
contains states y such that y 1 < 2. In other words, the formula means that when
the level of the first component is zero, then it will be always less than two. The
asynchronous state graph of Fig. 2.11 does not satisfy this property, because of the
presence of the path (0 , 0) → (1 , 0) → (2 , 0).
To summarize, the interest of CTL is twofold. Firstly, it allows to express, in
a simple way, rather complex dynamical properties on undeterministic transition
systems. Secondly, every CTL formula φ can be checked on a transition system
S in a polynomial time with respect to the size of φ and S . (More precisely, the
complexity of the verification is in

O ( | φ |·| S | ) where | φ | is the number of symbols

in φ seen as a string, and is the sum of the number of vertices and the number

of transitions of the state graph S .) Notice though, as discussed in Sect. 2.3.2.4 ,that
the number of state graphs grows exponentially with the number of components.

| S |

2.3.3 Success Story: Pseudomonas aeruginosa and Cystic

Fibrosis

The bacteria Pseudomonas aeruginosa [ 25 ] are commonly present in the environ-
ment and secrete mucus only in lungs affected by cystic fibrosis. As it increases
the respiratory deficiency of the patient, it is the major cause of mortality. Bacteria
isolated from cystic fibrosis lungs continue to grow in laboratory as mucous colonies
for numerous generations (mucoid phenotype). A majority of these bacteria present
a mutation. Does it mean that the mutation is the cause of the passage to the mucoid
state?
A majority of biologists tend to follow this hypothesis. However, the regulatory
network that controls the mucus production has been elucidated (Fig. 2.13 a) and the
regulatory graph contains two feedback circuits among which one is a positive one
(Fig. 2.13 b). This positive circuit makes possible a dynamic with two attractors that
would allow, from a biological point of view, an epigenetic change (stable change
of phenotype without mutation) from the non-mucoid state to the mucoid one.
From a biological point of view, it is very important to determine whether the
mucoidy can be induced by an epigenetic phenomenon or not. In such a case, the
elimination of the anti-AlgU gene ( via a mutation) could be favored later on because
an inhibitor complex is produced, which is toxic for the bacteria.
From a modeling point of view, and because the mathematical model of mucus
production system is not yet determined, this question becomes: Can we exhibit,
from the interaction graph of 2.13 , a dynamical model (an asynchronous state graph)
presenting at least two attractors, one in which mucus is regularly produced and one
in which mucus is not produced?
Assuming that AlgU activates the mucus production at its maximal level b AlgU ,
to state that a model which regularly produces mucus is equivalent to the fact that