Systems Biology (Methods in Molecular Biology)

biology is a bifurcation phenomenon of the model equations.

These bifurcations are tuned by the conservation law constants of

the equations, resulting from the catalytic role of genes.

2.2 Identifiability,
Observability,
and Sloppiness

In several applications, models’ parameters of dynamic systems are

not directly measurable but only indirectly accessible through

inputs and measurements outputs. Furthermore signals are time

varying and are subjected to some applied perturbations. Then a

fundamental question to be answered before some methods are

selected is if the model structure in question is identifiable. Struc-

tural identifiability is a model property that ensures that parameters

can be globally or locally determined from knowledge of the inpu-

t–output behavior of the system. Sedoglavic [13, 14] presents a

probabilistic semi-numerical algorithm for testing the local struc-

tural identifiability of a model. A-priori non-identifiability may be

caused by over-parameterization of the model that includes its

observation function; while a-posteriori non-identifiability is gen-

erally due to lack of information on the available data. Sloppy

models are, however, often unidentifiable, i.e., characterized by

many parameters that are poorly constrained by experimental

data. In principle, however, these two concepts, identifiability and

sloppiness, are distinct.

In general we can consider a state variable with time invariant

parameters defined by the following algebraic system

P x_ðtÞ¼fðxðtÞ,uðtÞ,θÞ, xð 0 Þ¼x^0 ðθÞ

yðtÞ¼gðxðtÞ,uðtÞ,θÞ



ð 6 Þ

where xðtÞ∈Rn,uðtÞ∈Rm,θðtÞ∈Rd,yðtÞ∈Rp andf and gare

rational functions ofx,u,θ. Higher-order derivatives of the output

with respect to timeyνcan be obtained by repeated use of the chain

rule and replacingx_ using the system dynamics (also known as

extended Lie-derivative alongf)

y ¼ g

y_ ¼

∂g

∂x

fþ

∂g

∂u

u_¼Lfg

€y ¼LfðℒfgÞ¼L^2 fg

⋮

yν ¼Lνfg

withν¼n+d1 and whereLf¼

Xn

i¼ 1

fi∂∂xiþ

X^1

i¼ 0

uiþ^1 ∂u∂ðiÞis the

formal Lie derivation.

The output derivatives may be expressed in terms of the state

and parameters and the inputs and its derivatives asY¼YðxθÞthat

can be uniquely solved forxandθif the Jacobian

Inverse Problems in Systems Biology 73