Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 63

could be further analyzed using a constraint-based method [ 14 ]toevaluateand
improve the structure of the network under the parameter constraints on Table 2.1 .
This example illustrates the sequence of events delineated in the introduction:
the external or environmental signal is the presence ( u =0) or absence ( u =1)
of a nutrient source, which may trigger the activation of the transcription of some
genes, with production of the respective mRNA and proteins. Depending on the
external signal, the response of the system is different, leading to low (respectively,
high) expression of gene fis if nutrient is absent (respectively, present). Similar
conclusions hold for the remaining genes, and many of the predictions have been
experimentally observed.

2.3 Discrete Models of GRN

2.3.1 Challenges

In the previous section, the concentrations of molecular species are handled in
continuous frameworks, using differential equations and it is shown that some
regions (domains) of the space of state variables can be identified so that a more
abstract continuous modeling framework can be applied, namely the piecewise
affine systems. Going further into abstraction, many biological questions can be
answered by only looking at the successive boxes that the cells under study can
follow, forgetting the precise state in each box. Such models are called discrete
models, as the state of a variable at a given time can be described by an integer
value: the number of the interval containing the continuous state.
There are several motivations to consider qualitative models that forget the
precise continuous state into a box:

• Numerous biological questions are themselves of qualitative nature;
•The in vivo measurement capabilities offer a resolution that does not allow
to validate or refute a very precise value for the continuous parameters of a
differential equation;

• Discrete descriptions can be easily modeled and simulated by computers, even

when hundreds of variables are involved, allowing to experiment large genetic
networks in silico .

These biological considerations have motivated the discrete approach proposed by
Rene Thomas (presented in the next section).
Perhaps more importantly, discrete models can be studied using powerful
techniques from computer science:

• Combinatorial approaches , often based on graph theory, are able to establish

general laws about the link between the form of the interaction graph and the
dynamic behavior of the system (e.g. there are behaviors that are unreachable for
some interaction graphs);