New Statistical Algorithms for the Analysis of Mass - FU Berlin, FB MI ...

52 CHAPTER 3. MATHEMATICAL MODELING AND ALGORITHMS
that a masterpeak is a set of peaks where each single peak stems from a distinct
spectrum and represents a molecule (or peptide) of a certain weight. If we
now find a masterpeak in two different groups at the same (m/z) position
we have most likely found the same molecule (peptide) in the two groups.
If these masterpeaks differ significantly in height we could differentiate by
them. However, if the two masterpeaks have similar height we call them noninformative
and want them to be tagged as such.
Following this, we first have to match masterpeaks of two groups occurring
at the same m/z position. Obviously, sometimes a masterpeak from group S1
cannot be matched with a masterpeak from group S2 because this molecule
(peptide) is just not present in S2. This process is called Masterpeak Alignment.
We have implemented two different approaches to solve this problem, a
naive version and an implementation as an Minimum Weight Maximum Cardinality
Bipartite Matching formulated as Linear Program and solved by the
Munkres (Hungarian) Algorithm which are compared in section 3.7.3.
Approach 1: Naive Solution
Algorithm 5 Masterpeak Assignment - Naive
Require: Lists of masterpeaks MP1, MP2 of groups 1 & 2 respectively
while MP1 has more elements do
MP 1cur ← next element from MP1
MP 2candidates : { s | m/z(s)−m/z(MP1)| ≤ 2}
if |MP 2candidates| = 0 then
insert (MP 1cur, ∅) into LMP P AIRS
else
for all p ∈ MP 2candidates do
insert (MP 1cur, p) into LMP P AIRS
mark p (in list MP2) as processed
end for
end if
for all p ∈ MP2 do
if p is not marked as processed then
insert (∅, p) into LMP P AIRS
end if
end for
end while
return MP PAIRS: tuples of aligned pairs of masterpeaks.
In this approach we simply check for masterpeaks in both groups that have
similar m/z values. An obvious problem here is that peaks can be assigned
more than once and no similarity measure is used to increase the quality of
the assignments.
Approach 2: Bipartite Graph Matching
In the second approach we re-formulate the problem as a Minimum Weight
Maximum Cardinality Bipartite Matching (assignment) problem. We are given
the two sets of masterpeaks (MP1, MP2) which can be seen as vertices of a
graph. This graph G = {V, E} is bipartite because the vertex set V is subdivided
into two sets.