Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 68

Rene Thomas highlighted the predominant (dynamical) role of positive and
negative circuits by stating the following two rules [ 39 ]: (1) A necessary condition
for the presence of several stable states is the presence of a positive circuit in G .(2)
A necessary condition for the presence of sustained oscillations is the presence of a
negative circuit in G .
These rules are “transversal” to the considered modeling framework in the sense
that they have been proved for differential models [ 12 , 23 , 26 , 29 , 36 - 38 ], Boolean
models [ 4 , 5 , 30 ] and discrete models [ 32 , 33 ]. The obvious interest of these two rules
is that they relate the rather simple information contained in the interaction graph
G of a network to its much more complex dynamical behaviors. In addition, multi-
stationarity and sustained oscillations are, from a biological point of view, important
dynamical properties, respectively associated with differentiation processes and
homeostasis phenomena.
Inside Thomas' logical method, Thomas' rules take the following form:

Theorem 2 ([ 32 , 33 ]).

1. If G has no positive circuit, then for all bounds b , logical thresholds t , and logical
parameters K , the resulting asynchronous state graph has at most one attractor.
2. If G has no negative circuit, then for all bounds b , logical thresholds t , and
logical parameters K , the resulting asynchronous state graph has no cyclic
attractor.

In fact, if G has no positive (resp. negative) circuit, then every associated
asynchronous state graph contains a unique attractor (resp. at least one stable state).
These are simple consequences of the above theorem and the basic observation,
already mentioned, that a state graph has always at least one attractor.
Most often, real interaction graphs contains positive and negative circuits, so that
the previous theorem cannot be applied to obtain information on the dynamics of
the system. However, the following theorem, which extends the first point of the
previous theorem, can always be used (in the worst case, take I = { 1 ,...,n } ).

Theorem 3 ([ 4 , 31 ]). If I is a set of vertices such that every positive circuit of G has
at least one vertex in I , then the asynchronous state graph resulting from the bounds

b , logical thresholds t and logical parameters K contains at most i∈I ( b i +1)
attractors.

This theorem shows that the number of attractors is small when positive circuits
are highly connected. The number of positive circuits is not the relevant parameter:
if there is one million of positive circuits, but if all these circuits contain a vertex i
with b i =1, then there are at most two attractors. Note also that the upper bound is
tight in some cases. For instance, if G consists in a single vertex (vertex 1) with a
positive arc 1 → 1,andif b 1 = t 11 = K 1 ,{ 1 } =1 >K 1 ,∅ =0, then the resulting
asynchronous state graph has 2= b 1 +1attractors (that are stable states).