Systems Biology (Methods in Molecular Biology)

3.2 Identifiability In general a mathematical model helps to better understand the
biological phenomenon studied. It enables experiments to be spe-
cifically designed to make predictions of certain characteristics of
the system that can then be experimentally verified. It summarizes
the current body of knowledge in a format that can be easily
communicated. Then the model will be conditioned by elements
which may have an impact on the questions addressed by the users.
The mathematical assumptions are defined from the network
architecture and from the modelling framework like deterministic
or stochastic laws, partial differential equations, etc.
A crucial step is that to define a number of unknown
non-measurable parameters that can be determined by means of
experimental data fitting, the so-called identification. Raue et al. [9]
report several methods to be used for identifying parameters.
Among others the DAISY approach basic idea is that of manipulat-
ing algebraic differential equations as polynomials depending also
on derivatives of the variable. This algorithm permits to eliminate
the non-observed state variables from the system of equations and
to find the input–output relation of the system [23]. The EAR
approach developed by Fraunhofer Chalmers [7, 8] is based on a
method for local algebraic observability [13]. PL approach checks
for non-identifiability by posing a parameter estimation problem
using real or simulated data. The central idea is that
non-identifiability manifests as a flat manifold in the parameter
space of the estimation problem, e.g. the likelihood function.
Here we use the Sedoglavic’s approach, modified by Fraunho-
fer Chalmers as previously mentioned, applied to the Tyson’s
model. When the same system has the same input but one less
output, i.e.n1 states are measured, for examplepM, results
reported in Table2 show that thek4,k 4 prime, andk 5 notPare
unidentifiable. On the contrary all data are uniquely identifiable.

0.0012

1.25

CP

Y

1.20

1.15

1.10

1.05

1.00

0.08

YP

0.06

0.04

0.02

0.35

0.30

0.25

0.20

0.15

0.10

0.05

C2

pM

20 40 60 80 100

Time

20 40 60 80 100

Time

20 40 60 80 100

Time 20 40 60 80 100Time

20 40 60 80 100

Time

20 40 60 80 100

Time

0.0011

0.0010

0.035

M

0.030

0.025

0.020

0.015

0.010

0.005

0.00008

0.00007

0.00006

0.00005

Fig. 1Dynamical behavior of the components ofcdc2-cyclinmodel

Inverse Problems in Systems Biology 79