scale dynamic biological models generally may have many
unknown, non-measurable parameters and their tuning may appear
unrealistic. Nevertheless if on one side some authors have shown,
among others we cite Paci et al. [27], that it is possible to explore
the biological networks by reverse engineering provided the analy-
sis of classifying the nodes in the network is defined as a whole, on
the other side, other authors, among whom we cite Villaverde et al.
[28], have compared different parameter estimations methods with
the aim to have a benchmark for optimal experimental design. Then
in this paper we have tried to define the minimal approach to
understand how a biological system can be studied without forcing,
with tuned parameters, the biological nature of process involved.Acknowledgements
The authors thank Giuseppe Macino and Lorenzo Farina for their
inspiring discussions on the whole analysis presented in this study.Appendix A: Notes
A.1 Deterministic
Solutions
This appendix can be an aide to characterize the system biology,
thanks to mathematical models which may consist of a system of
differential equations involving state variables and parameters. If
the variables of such system do not depend explicitly on time, the
system is said to be autonomous. Under certain assumptions
defined below such system may be considered as an autonomous
dynamical system. As time does not occur explicitly in equations,
solution of a system of differential equations may be projected in a
space called phase-space in which the behavior of the state variables
is described.
The plot of the solution in such space is called phase portrait.
The bifurcation of a system of differential equations, i.e., of an
autonomous dynamical system is concerned with changes in the
qualitative behavior of its phase portrait as parameters vary and
more precisely, when such a bifurcation parameter reaches a certain
value, called critical value. Thus, bifurcation theory is of great
importance in dynamical systems study because it indicates stability
changes, structural changes in a system, etc. So, plotting the solu-
tion of autonomous dynamical system according to the bifurcation
parameter leads to the construction of a bifurcation diagram. Such
diagram provides knowledge on the behavior of the solution: con-
stant, periodic, nonperiodic, or even chaotic. As there are many
kinds of behaviors of solutions there are many kinds of bifurcations.
The Hopf bifurcation corresponds to periodic solutions and period
doubling bifurcation, or period doubling cascade, which is one of
the routes to chaos for dynamical systems.Inverse Problems in Systems Biology 83