whereF:X!Yis an ill-posed operator between Hilbert spaces
X,Y. The inverse problem is to identify the model parameters
observed at various time under different experimental conditions
if only noisy datayδare given. Now denoting byxthe parameter
vector to be determined and withyδthe available noisy data the
inverse problem can be formulated with:(^) yδFðxÞ
2
Y!minx∈X ð^11 Þ
In recent years, many of the well-known methods for linear
ill-posed problems have been generalized to nonlinear operator
equations [15].
The iterative methods by Tikhonov regularization is obtained
by minimizing the Tikhonov functional
JαðxÞ¼
(^) yδFðxÞ
(^2) þα
(^) xx
2
xδα¼argmin
x
JαðxÞ
ð 12 Þ
The advantage of Tikhonov regularization is that convergence of
the method, i.e.xδ!x{forδ!0 and an appropriate parameter
choiceα¼α(δ) holds under weak assumptions to the operator.
However, the difficulties for Tikhonov regularization are a proper
choice of the regularization parameter and the computation of the
minimizer of the Tikhonov functional [15]. Other forms of regu-
larization are: Maximum entropy [15] and Bounded variation [16].
2.4 Sloppy Models
and Fisher Information
Matrix
Dynamic systems biology models involve many kinetic parameters,
the quantitative determination of which could be extracted from a
fit long before the experimental data constrained the parameters,
even to within orders of magnitude. This pattern was attributed to a
low sensitivity to model’s parameter also revealed by the fact that
sensitivity eigenvalues were roughly evenly spaced over many dec-
ades. Consequently, the model behavior depended effectively on
only a fewstiffparameter combinations.
Sloppiness is particularly relevant to biology, because the col-
lective behavior of most biological systems is much easier to mea-
sure in vivo than the values of individual parameters. Gutenkunst
[3], analyzing 17 system biology models drawn from the BioMo-
dels database [17], an online repository of models encoded in the
Systems Biology Markup Language, have shown that exist models
that are poorly constrained and/or ill-conditioned because it is
difficult to use experimental data to derive their parameters. These
models were called sloppy.
The change in model behavior as parametersyivaried from
their published valuesdi(SBML) [18] by the average squared
change in molecular species. This is accomplished by defining a
cost function that quantifies how different the model output is for a
Inverse Problems in Systems Biology 75