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\author{Steffen Neumann$^\ddagger$,
  Anton Pervukhin$^\P$, Sebastian B{\"o}cker$^\P$}
\begin{document}
\title{Mass decomposition with the Rdisop package}
\maketitle
\begin{center} $^\ddagger$Leibniz Institute of Plant Biochemistry,
Department of Stress and
Developmental Biology, {\tt sneumann@IPB-Halle.DE}\\
$^\P$Bioinformatics, Friedrich-Schiller-University Jena,
{\tt \{apervukh|boecker\}@minet.uni-jena.de}
 \end{center}

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\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

The BioConductor \Rpackage{Rdisop} package is designed to determine
the sum formula of metabolites solely from their exact mass and
isotope pattern as obtained from high resolution mass spectrometry
measurements.

It is designed with compatibility to the Bioconductor packages
\Rpackage{XCMS}, \Rpackage{MassSpecWavelet} and \Rpackage{rpubchem} in
mind.

\section{Decomposing isotope patterns}

After preprocessing, the output of a mass spectrometer is a list of
peaks which corresponds to the masses of the sample molecules and
their abundance, i.e., the amount of sample compounds with a certain
mass.  In fact, sum formulas of small molecules can be identified
using only accurate output masses.  However, even with very high mass
accuracy ($<$~1~ppm), many chemically possible formulas are found in
higher mass regions.  It has been shown that applying only this data
therefore does not suffice to identify a compound, and more
information, such as isotopic abundances, needs to be taken into
account. High resolution mass spectrometry allows us to obtain the
isotope pattern of sample molecule with outstanding accuracy.

\subsection{Chemical background}

Atoms are composed of electrons, protons, and neutrons. The number of
protons (the atomic number) is fixed and defines what element the atom
is. The number of neutrons, on the other hand, can vary: Atoms with
the same number of protons but different numbers of neutrons are
called \emph{isotopes} of the element.  Each of these isotopes occurs
in nature with a certain abundance. The \emph{nominal mass} of a
molecule is the sum of protons and neutrons of the constituting atoms.
The \emph{mass} of the molecule is the sum of masses of these atoms.
The \emph{monoisotopic} (\emph{nominal}) \emph{mass} of a molecule is
the sum of (nominal) masses of the constituting atoms where for every
element its lightest natural isotope is chosen. Clearly, nominal mass
and mass depend on the isotopes the molecule consists of, thus on the
\emph{isotope species} of the molecule.

No present-day analysis technique is capable of resolving isotope
species with identical nominal mass. Instead, these isotope species
appear as one single peak in the mass spectrometry output. For this
reason, we merge isotope species with identical nominal mass and refer
to the resulting distribution as the molecule's \emph{isotope pattern}.

\subsection{Identification schema}

Obtaining an accurate isotope pattern from a high resolution mass
spectrometer, we apply this information to identify the elemental
composition of the sample molecule. Our input is a list of masses with
normalized abundances that corresponds to the isotope pattern of the
sample molecule. We want to find that molecule's elemental composition
whose isotope pattern best matches the input.

Solving this task is divided into the following parts: First, all
elemental compositions are calculated that share some property, for
example monoisotopic mass, with the input spectrum.  Second, to remove
those compositions that do not exist in nature, chemical bonding rules
are applied, discarding formulas that have negative or non-integer
degree of unsaturation.  And third, for every remaining composition,
its theoretical isotope pattern is calculated and compared to the
measured isotope pattern.  Candidate patterns are ranked using
Bayesian Statistics, and the one with the highest score is chosen.

\section{Working with molecules and isotope peaklists}

The central object in Rdisop is the molecule, which is
a list containing the (sum-)formula, its isotope pattern, a score and other
information. Molecules can either be created explicitely through
\Rfunction{getMolecule()} or \Rfunction{initializeXXX()}, or through
\Rfunction{decomposeMass()} and \Rfunction{decomposeIsotopes()}. Most
functions operate only on a subset of the periodic system of elements
(PSE) given as ``elements'' argument.

\subsection{Handling of Molecules}

The \Rfunction{getMolecule} returns a list object containing the
information for a named single atom or a more complex molecule.

<<>>=
library(Rdisop)
molecule <- getMolecule("C2H5OH")
getFormula(molecule)
getMass(molecule)
@

Note that the formula is in a canonical form, and the mass
includes the decimals (the nominal mass for ethanol would be just 46).

Without further arguments only the elements C, H, N, O, P and S are
available. For metabolomics research, these are the most relevant ones.
A different subset of the PSE can be returned and passed to
the functions, but keep in mind that a larger set of elements yields a
(much) larger result set when decomposing masses later.

<<>>=
essentialElements <- initializeCHNOPSMgKCaFe()
chlorophyll <- getMolecule("C55H72MgN4O5H", z=1, 
  elements=essentialElements)
isotopes <- getIsotope(chlorophyll, seq(1,4))
isotopes
@

<<isotopes, include = FALSE, fig = TRUE, eps = FALSE, width = 7, height = 7>>=
plot(t(isotopes), type="h", xlab="m/z", ylab="Intensity")
@

In this case we have created a complex molecule with a charge (z=~+1)
containing a metal ion and check its first four isotope peaks. For a
visual inspection the isotope pattern can be plotted, see
figure~\ref{fig:Rdisop-isotopes}.

\begin{figure}
\begin{center}
\includegraphics{Rdisop-isotopes}
\end{center}
\caption{Isotope pattern for a protonated chlorophyll ion, which could
  be observed on a high-resolution mass spectrometer in positive
  mode.\label{fig:Rdisop-isotopes}}
\end{figure}
\subsection{decomposeMass and decomposeIsotopes}

The \Rfunction{decomposeMass} returns a list molecules which have a
given exact mass (within an error window in ppm):

<<>>=
molecules <- decomposeMass(46.042, ppm=20)
molecules
@

This call produces a list of potential molecules (with a single
element in this case). The larger the masses, the allowed ppm
deviation and the allowed elements list, the larger the result list
will grow. For each hypothesis there is its formula and weight and
score. The parity, validity (using the nitrogen rule) and double bond
equivalents (DBE) are simple, yet commonly used hints for the
plausibility of a solution and can be used for filtering the results
list. For an amino acid this simple method guesses already eight
hypotheses:

<<>>=
length(decomposeMass(147.053))
@

On modern mass spectrometers a full isotope pattern can be obtained
for a molecule, and the masses and intensities improve the accuracy
of the sum formula prediction. Accessor functions return only subsets
of the molecule data structure:

<<>>=
# glutamic acid (C5H9NO4)
masses <- c(147.053, 148.056)
intensities <- c(93, 5.8)

molecules <- decomposeIsotopes(masses, intensities)
cbind(getFormula(molecules), getScore(molecules), getValid(molecules))
@

The first ranked solution already has a score close to one, and if
using an N-rule filter, only one solution would remain. These cases
are not removed by default, because a few compound classes do not obey
the N-rule, which after all is just a simple heuristic.

I the masses were obtained by an LC-ESI-MS, it is likely that the
measured mass signal actually resembles an adduct ion, such as
[M+H]$^+$.  The sum formula obtained through
\Rfunction{decomposeIsotopes} will have one H too much, and will not
be found in PubChem or other libraries, unless the adduct has been
removed:

<<>>=
querymolecule <- subMolecules("C5H10NO4", "H")
getFormula(querymolecule)
@

Similarly, if during ionisation an in-source fragmentation occurred,
the lost fragment can be added before querying using
\Rfunction{addMolecules}.

\subsection{Interaction with other BioConductor packages}

This section will give some suggestions how the Rdisop functionality
can be combined with other BioConductor packages.

Usually the masses and intensities will be obtained from a
high-resolution mass spectrometer such as an FTICR-MS or QTOF-MS.
BioConductor currently has two packages dealing with peak picking on
raw machine data, \Rpackage{MassSpecWavelet} and \Rpackage{XCMS}. The
latter contains a wrapper for MassSpecWavelet, so we need to deal with
XCMS peak lists only. The ESI package\footnote{not part of
  BioConductor, see \url{http://msbi.ipb-halle.de/}} can extract a set
of isotope clusters from peak lists.

After Rdisop has created a set of candidate molecular formulae, the
open-access compound databases PubChem or ChEBI can be queried whether
any information about this compound exists. Nota bene: a hit or
non-hit does not indicate a correct or incorrect formula, but merely
helps in further verification or structure elucidation steps.
For other cheminformatics functionality in BioConductor see e.g.
\Rpackage{RCDK}.

\section*{Acknowledgments}
AP supported by Deutsche Forschungsgemeinschaft (BO 1910/1),
additional programming by Marcel Martin, whom we thank for his
unfailing support, and by Marco Kortkamp.

\end{document}
