Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 48

(PWA) systems. These models often assume an homogeneous distribution of the
molecules over a selected volume of space and describe, for instance, the dynamics
of the concentration of some protein in a population of cells.

2.2.1 Challenges

The choice of appropriate variables is one of the first steps in the construction
of a model for a biological network. The network is made of nodes (proteins,
or RNA) and the edges usually describe the fact that some biochemical species
acts positively or negatively on the variation with respect to time of some other
biochemical species. Each variable (node) will play a different role in the behavior
of the system, and have different degrees of relevance. Some variables can be
measured experimentally, and are thus easier to compare to the model. Other
variables may be easier to control from the exterior. Large systems of differential
equations will require the introduction of a large number of parameters which will
be unknown and should be estimated. In general, from a theoretical point of view,
large dimensional systems are difficult to analyze and can only be studied through
numerical simulations. Therefore, a fundamental step is the development of model
reduction methods to simplify large networks and obtain more tractable systems of
lower dimension, which are more easily studied in detail.
Two classical examples are the “positive” and “negative” feedback loops, formed
by variables (proteins, for instance) that influence one another in a closed circuit, or
loop. A circuit with two proteins that mutually repress or activate each other is a
positive loop; if one of the interactions is a repression and the other an activation,
then the circuit is a negative loop. Each of these two motifs appears frequently in
GRN, and has a well known dynamical behavior; they can be combined with other
motifs to represent the dynamics of complex regulatory networks. The negative loop
is a system that generates oscillatory behavior, while the positive loop generates one
or two stable steady states, and will be analyzed in detail in the next sections.

2.2.2 Mathematical Tools

This section quickly summarizes some basic mathematical results that will be useful
in the analysis of systems of ordinary differential equations. For further details see,
for instance [ 18 ].

2.2.2.1 Analysis of Two-Dimensional Systems

Consider a system with two variables, x =( x 1 ,x 2 ) t , where each x i represents the
concentration of some molecular species, and x evolves in the region of space where