Biology Reference
In-Depth Information
Eqn 6 to define an observable as an operator that is a linear combination of
maps
j
=
j
j
j
(7)
where
j
is some real-valued parameter:
M
→
M
p
→
j
j
j
To relate an observable on M defined in an analysis of M with an observable
arising from a synthetic model, we can define an equivalence of the two,
p
=
j
j
j
q
j
j
j
j
p
=
(8)
only because of the involved linearity and the disjointness of summands O
j
,
which implied that the observables
ij
on O
j
, by definition of the synthetic
approach, are
unlinked
.
Note however that is an element of the set of all maps from M to
,
denoted HM
. There are, thus, many more observables than those that are
linear combinations of the kind Eqn 7. It would, therefore, be possible to find
an observable
in an analysis of M, where for any two q and q
in
the same equivalence class of O
j
/E
j
,
∈
HM
j
q
=
j
q
holds but where splits this equivalence class such that
q
q
=
In other words, although every synthetic model is an analytical model, there are
analytical encodings of M that are not synthetic models; that do not possess a
synthetic refinement.
6. DYNAMIC PATHWAY MODELLING
The vast majority of mathematical models and computational simulations in
systems biology are developed within the framework of time-invariant differen-
tial equations (rate-equations). Regardless of whether one derives the equations
