Biology Reference
In-Depth Information
Fig. 18.1 An analogy between the optical spectra and “ribonic spectra.” The prism and optical
spectra have been reproduced from
http://wimminwiselpts.wordpress.com
can be represented as a set of n points in an N-dimensional mathematical space (to be
called the “ribonic space” or “RNA concentration space”), where N is either the time
points of measurements or the number of different samples analyzed. The ribonic
space is depicted in the top figure in the second column of Table
18.1
, where the
undulating membrane indicates a principal manifold onto which the nearest points
are projected by the ViDaExpert program (Sect.
2.8.1
). The ViDaExpert software is
based on several well-established mathematical and computational frameworks.
Each of the n point in the N-dimensional concentration space represents a
ribon
with N coordinate values which is the main reason for referring to the mathematical
space as the
ribonic space
. In the case of a
t-ribon
(e.g., see the upper figure in the
first column of Table
18.1
), its coordinate values are the levels (or copy numbers) of
an RNA measured at N different time points. In the case of an s-ribon (e.g., see the
lower figure in the first column of Table
18.1
), its coordinate values represent the
levels or copy numbers of an RNA measured in N different samples.
The raw data of microarray measurements can also be represented as a “distance
matrix” (see the table in the second column of Table
18.1
) where the Euclidean
distances between all possible pairs of the points (i.e., ribons) in the N-dimensional
concentration space have been calculated based on the Pythagorean formula. It
should be noted that
All the information contained in the raw RNA data measured with microarrays is encoded
in the distance matrix which is symmetric with respect to its diagonal because the distance
between a and b is the same as that between b and a. (18.2)
The distance matrix defined in Statement 18.2 may be referred to as the “ribonic
matrix,” since the first row and the first column of the matrix are composed of the
names of the ribons measured by microarrays.
It is possible to analyze the raw data, that is, both t- and s-ribons, in the forms of
ribonic matrices, without relying on any specialized analytical soft wares such as
hierarchical clustering or ViDaExpert. The top two figures in the third column of
Table
18.1
represent significant results of analyzing ribonic matrices without
utilizing any computational softwares (Ji et al. 2009b, c). The third figure in the
