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Fig. 5.6
Transformation of the circles by means of algebraic transformations
geometric concepts can be developed in an abstract setting. In general, a space where
points are identified by a number (the dimension of the space) of variables ranging
over the real numbers forms a (real)
vector space
. A more biologically oriented ex-
ample of vectors is constituted by a cell culture plate, divided into
k
compartments
where for each compartment different nutrients or drugs were placed. The number
of cells for each compartment provides a vector in
k
and the growth process pro-
vides a time series for each compartment. As another example, a DNA microarray is
a small chip where many thousand of spots are arranged in a matrix (any spot has a
row and a column position). Each spot contains DNA probes of the same type (sin-
gle strands of length around 20 bases), which are specific of genes, in such a way
that each column refers to one gene. When RNA coming from a cell is uniformly
distributed on the microarray, neglecting the details of the process, a quantity can be
assigned to each spot, which determines the overall expression of the genes associ-
ated to the columns for the cell under observation. In conclusion, a DNA microarray
yields a vector in
N
k
,if
k
are the columns in the chip (the vector of the overall ex-
pression relative to each of
k
genes). A population of objects of many different types,
that is a multiset over a given set, where an order is given, provides a vector formed
by the multiplicities of the different types of objects. This means that the state of
a biochemical system where some reactions happen is naturally represented by a
point of a suitable space, and the evolution of the system, starting from this state, is
represented by a motion of this point in this space.
R
