On the Analysis of Optical Mapping Data - University of Wisconsin ...

14
Significance: An optimal alignment exists in any map comparison problem, irrespective
of any actual association. In order to minimize the potential effects of misaligned maps, it is
essential to limit alignments by some additional criterion. This is the problem of assessing
the significance of a given alignment. The significance problem in optical map alignment
is more difficult than in sequence alignment, because of a greater degree of noise and also
because of differences in the nature of the data. We find deficiencies in the current state of
the art, and in Chapter 3 we introduce and evaluate an alternative approach to measuring
the significance of optical map alignments. Here, we give a general overview of the mechanics
of map alignment.
Notation: We restrict our attention to pairwise alignments, i.e. those between two restriction
maps. Let x = (x 1 , . . .,x m ) and y = (y 1 , . . .,y n ) denote two restriction maps with
m and n fragments respectively. Let the corresponding representations in terms of cut sites
be S(x) = {s 0 < s 1 < · · · < s m } and S(y) = {t 0 < t 1 < · · · < t n }. An alignment between x
and y can be represented by an ordered set of index pairs
C = (( i 1
j1
)
,
( i2
j2
)
, . . .,
( ik
jk
))
indicating a correspondence between the cut sites s il and t jl for l = 1, . . .,k, where 0 < i 1 <
· · · < i k < m and 0 < j 1 < · · · < j k < n. To allow missing fragments in the alignment, this
last condition can be modified to allow successive indices to be equal, as long as successive
index pairs are not identical. For non-trivial alignments k ≥ 2, in which case the alignment
consists of k −1 aligned chunks. The l th chunk (l = 1, . . ., k −1) has lengths ˜x l = s il −s il−1 ,
and ỹ l = t jl − t jl−1 involving m l = i l − i l−1 and n l = j l − j l−1 fragments respectively in
the original maps x and y. To be used successfully in a dynamic programming algorithm, a
score function must be additive, in the sense that the score of a complete alignment must be
the sum of the scores for its component chunks.