Biology Reference
In-Depth Information
between 'analytical' and 'synthetic' modelling, which is at the root of Rosen's
distinction between organisms and mechanisms and his critique of the 'physical
approach' to model biological systems. Rosen's discussion takes place at a high
level of abstraction, occasionally leaving the reader to fill in details. In this
situation, a clear mathematical notation is important and typographical errors
can easily mislead the reader.
The outline for the remaining text is as follows: We introduce analytical
models and synthetic modelling in separate sections, before establishing a rela-
tionship between the two. Finally, we point out some of the conclusions Rosen
himself has drawn from the presented material and we give links to the literature
that take up some of the issues relevant to the present text.
3. ANALYTICAL MODELLING
We investigate a natural system by considering it an abstract set M of abstract
states p
∈
M. For the analysis of complex natural systems, and in particular
biological systems, Rosen argued that it is most important that we should not
make any assumptions about the structure of M. Observations or measurements
we make on M are encoded by observables
→
M
M
p
→
p
where we use square brackets to denote the range or set image values of a map.
In a complex system and with limited means for observation or measurement of
the system, it may happen that more than one p
M maps into the same value.
In other words, every map on M induces an equivalence relation
∈
E
pp
if and only if p
p
=
and hence equivalence classes for which elements in M are indistinguishable
w.r.t.
p
p
p
=
=
p
The set of equivalence classes on M forms a quotient set or partition, denoted
M/E
(Fig. 2). For complex systems, what we observe is not M directly but
a reduced state space M/E
. The image set M and the partition M/E
are
isomorphic, that is, there is a one-to-one mapping, which is denoted by
M
M/E
(1)
