Modeling Biochemical Pathways - Mathematical Approaches to Polymer Sequence Analysis and Related Problems - page 124

= individuals is obtained.

This mechanism determines that parents survive only for one generation.
Stochastic GO approaches can provide good solutions in modest computation
time, are quite simple to implement and do not require a transformation of the orig-
inal problem [ 18 ]. However, large instances of the NLP-ODE problem cannot be
solved in a reasonable time, even considering stochastic approaches. It is therefore
important to consider strategies to face the problem of parameter estimation of
systems biology ODE models on high performance computing and distributed plat-
forms [ 19 ].

offspring on average so that a new population of

6.7 Conclusions

Living organisms are complex and must be studied considering both reductionistic
and holistic approaches. At the molecular level, biological processes arise from the
interactions among a number of molecular entities. Importantly, sequence analysis
methods predict properties of both macromolecules and intermolecular interactions.
These data are useful to construct molecular circuits that can be studied using math-
ematical modeling and computer-based numerical simulations.
The definition of a systems biology model begins with the assembly of the
wiring diagram, which essentially capture the system structure in terms of molecular
entities and interactions among entities. Graphically, the wiring diagram should be
drawn according to the specifications of the SBGN. Formally, the structure of a
molecular circuit can be represented by a graph and by the stoichiometric matrix.
The analysis of the structure leads to two important results: the conservation rela-
tions, indicating the linear combinations of molecular entities the concentration of
which do not vary during the system evolution; the flux relations at steady state,
defining the biochemical processes that have the same intensity when the system is
at steady state.
The time evolution of a well-stirred biochemical system is defined by the CME;
unfortunately both analytical and numerical solutions are available only in a few
cases. The Gillespies's algorithm provides exact numerical simulations of the
stochastic process defined by the CME, but it is computationally intensive as the
system size increases. The

leaping method is an approximation that reduces
the SSA computational cost. As the number of molecules included in the system
increases, the leaping method can be approximated by the chemical Langevin

equation, that is a stochastic and continuous approach. In the end, for even larger
number of molecules, the reaction rate equations can be derived from the chemical
Langevin equation, excluding the stochastic drift term. The reaction rate equations
models (ODE models) are the most used and a lot of theories and computational
methods are available for their analysis (e.g., steady-state analysis, bifurcation
analysis, sensitivity analysis).
Once both, the structure and the kinetic formulation of the model are defined,
a set of values for the model parameters must be found. This task is not easy due