Biology Reference
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from the law of mass action, using Michaelis-Menten kinetics or justifies these
equations using power-law representations, such as S-systems, these models take
the form
d
dt x
state-transition
=
Vxtut
response map
yt
=
hxt
=
where x
x 1 x n , of the system and X is called
the state-space. One might consider V a mapping VX × U
X denotes the state, x
n , which in
the geometric setting of nonlinear systems theory becomes a vector field. The
mapping V defines for a particular state x and stimulus u = u 1 u m at time
t how the system changes, given initial condition xt 0 = x 0 . Mathematically,
these models are usually described in terms of finite-dimensional vector spaces.
The discussion above centred around the difference between analytical and
synthetic modelling. Formally, the main difference between these two approaches
is linked to the difference between the direct product and direct sum. Direct
products and sums differ for infinite indices. An element of the direct sum
is zero for all but a finite number of entries, while an element of the direct
product can have all nonzero entries. In other words, the direct sum consists of
the elements of the direct product which have only finitely many factors. For
finite-dimensional vector spaces, the direct sum is the same as the direct product.
Finite dimensional vector spaces are the basis for virtually all dynamic pathway
modelling projects in systems biology. In dynamic pathway modelling, we are
using (nonlinear) ordinary differential for the following reason:
In modelling cells, cell function or pathways, causation is the principal of expla-
nation of change; a relation not between things but changes of states. For anything
to be different (changed) to anything else, either space or time, or both have to be
supposed.
We noted above the central dogma of systems biology that systems dynamics
are at the root of cell function. Although the areas of genomics and bioinformatics
are identifying, cataloguing and characterising the components that make up a
cell, this static information will not be sufficient to understand cellular processes
such as growth, differentiation or apoptosis. The processes are not arbitrary
but coordinated, controlled and regulated. To control, regulate or coordinate
whatever, means to adapt, maintain or optimize; but this does mean that there
must exist a goal, objective or function. To induce a change towards this goal,
information must be fed back. Feedback implies a before and after, which is
why we have to understand cell function as a dynamic system (Wolkenhauer
et al., 2005b). Feedback loops are the basis for control and regulation in dynamic
systems. In regulation, the system tries to maintain a variable or its level;
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