Biology Reference
In-Depth Information
regulation is thus understood as a form of disturbance rejection, leading to
robustness of the system against external perturbations. Control on the other
hand, is related to sensitivity, the systems response to stimuli in an efficient way
(Wolkenhauer et al., 2005a).
Differential equations are
the
natural language of change but with ordinary
differential equations we 'only' capture time but not space. Apart from inter-
actions through dynamics, spatial organisation is the second key aspect to bio-
logical systems. Partial differential equations are an option if spatial processes,
like diffusion, cannot be ignored. The theory of partial differential equations is,
however, rather complex and in many situations it will not be easy to iden-
tify parameter values from experimental data. Similarly, the translocation of
proteins, for example nucleo-cytoplasmic shuttling in cell signalling, leads to
transport delays or dead-times that have a significant influence on the dynam-
ics of a system. This effect is widely ignored in systems biology, because the
theory of delay differential equations is nontrivial. Processes would therefore
be more accurately described by partial differential equations or delay dif-
ference equations. Infinite-dimensional systems theory is concerned with the
formulation of such systems in state-space form, analogous to those for those
described by ordinary differential equations. In this setting, one introduces a
suitable infinite-dimensional state space and suitable operators instead of the
usual matrices.
The differences of different approaches, the assumptions implied and
consequences to prediction is therefore particularly important in systems biology.
7. ALL MODELS ARE WRONG, SOME ARE USEFUL
We presented a concise summary of Rosen's conceptual framework on the basis
of which he discusses the difference between organisms and mechanisms, and
which is also at the root of his critique of conventional mathematical mod-
elling of cellular systems. In this setting, the synthetic approach of mathemat-
ical modelling as tightly associated with the direct sum, while the notion of
an analytical model is tied to the notion of a direct product. In the context
of category theory, these two concepts are dual and only in special circum-
stances (which Rosen argues involves linearity and finiteness) the two can be
equivalent.
The most important aspect of the conceptual framework presented here is
that it provides a means to discuss the modelling process itself, formally.
Starting with basic considerations about the way we observe a natural sys-
tem, we are able to define categories of models. It is then possible to relate
models and investigate abstractions not just of natural systems but also of
models.
