under very specific conditions. A good example of an inverse prob-
lem is the derivation of the structure of a molecule from the X-ray
diffraction pattern of a crystal. The universe of potential models for
any complex system like the function of a cell has very large dimen-
sions and, in the absence of any theory of the system, there is no
guide to constrain the choice of model.”
Then every systems biology project essentially results in a
model that tries to solve the problem of divining reality from
experimental data. However, a model is not reality, it is an imperfect
picture of reality constructed from bits and pieces of data. In
addition, data in biological measurements are often noisy with
large error and incomplete. Then systems biology may fall in the
inverse problems that Brenner points out.
This means, by one side, that models derived from systems
biology might be useful, and often this is a sufficient requirement
for using them, despite they might likely leave out some important
feature of the system. By the other side one of the major challenges
in inverse problems is to find a minimal set of parameters that can
describe the system under examination or to extract from the
models the information included in the system. Ideally those para-
meters should be sensitive to variation so that one constrains the
parameter space describing the given system.
However since the objective is to describe the interactions of
distinct molecular entities (for example, proteins, transcripts, or
regulatory sites), which give rise to particular cellular behaviors,
the current models consist of sets of linear or nonlinear ordinary
differential equations involving a high number of states (e.g., con-
centrations or amounts of the components of the network) and a
large number of parameters describing the reaction kinetics.
Unfortunately, in most cases the parameters introduced into
the set of equations are completely unknown and/or only rough
estimates of their values are available. Therefore, their values are
usually estimated from time-series experimental data. The so-called
parameter estimation problem is then formulated as an optimiza-
tion problem where the objective is to find the parameter set so as
to minimize a given cost function that relates model predictions
and experimental data, e.g. the least squares function or a similar
cost index. Furthermore since parameter estimation problems in
dynamic models of biochemical systems are characterized by lim-
ited observability, large number of parameters and a limited
amount of noisy data, the solution of the problem is in general
challenging and, even when using robust and efficient optimization
methods, computationally expensive.
A particularly promising example is the use ofsloppy models
developed by Sethna and collaborators [2] in which parameter
combinations rather than individual parameters are varied and
those combinations which are most tightly constrained are then
picked as the right ones. Then model building cycle requires the70 Rodolfo Guzzi et al.