\chapter[Basics in Polymer Chemistry]{Basics in Polymer Chemistry}
\label{chap:basics-polymer-chemistry}

This chapter will introduce the basics of polymer
chemistry\index{polymer~chemistry}. The way this topic is going to be
covered is admittedly biased towards mass spectrometry and biological
polymers. Moreover, the aim of this chapter is to provide the reader
with the specialized words that will later be used to describe and
explain the (inner) workings of the \mXp\ program. This manual is not
a ``crash course'' in biochemistry.

\renewcommand{\sectitle}{Polymers? Where? Everywhere!}
\section*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{section}{\numberline{}\sectitle}

Indeed, polymers are everywhere. If you ask somebody to show you
something polymeric, he/she will point you at the first plastic object
in the vicinity. Right, plastic materials are made of hydrocarbon
polymers. We also have many different polymers in our body. Proteins
are polymers, complex sugars are polymers, DNA (the so-called
``molecule of heredity'' is a \emph{huge} polymer. There are polymers
in wine, in wood... Where? Everywhere!

\bigskip 

\noindent The \textsl{Oxford Advanced Learner's Dictionary of Current
  English} gives for \emph{polymer} the following definition:
\textit{natural or artificial compound made up of large molecules
  which are themselves made from combinations of small simple
  molecules}.

\bigskip 

\noindent A polymer is indeed made by covalently linking small simple
molecules together. These small simple molecules are called
\emph{monomer}s, and it is immediate that a \emph{polymer} is made of
a number of monomers. A general term to describe the process that
leads to the formation of a polymer is \emph{polymerization}. It
should be noted that there are many ways to polymerize monomers
together. For example, a polymer might be either linear or branched. A
polymer is linear if the monomers that are polymerized can be joined
at most two times. The first junction links the monomer to an
elongating polymer (thus making it the new end of the elongating
polymer which, by the way, is longer than before by one unit) and the
second junction links the new elongating polymer's end to another
monomer. This process goes on until the reaction is stopped, the point
at which the polymer reaches its \emph{finished
  state}\index{finished~state}. A branched polymer is a polymer in
which at least one monomer is able to contract more than two bonds. It
is thus clear that a single monomer linked three times to other
monomers will yield a ``T-structure'', which is nothing but a branched
structure.

In the following sections we'll describe a number of different kinds
of polymers. Each time, they will be described by initially detailing
the structure of their constitutive monomers; next the formation of
the polymer is described. At each step we shall try to set forth each
polymer characteristics in such a manner as to introduce the way \mXp\
``thinks polymers'' and to introduce specialized terminologies. Once
the basic chemistries (of the different polymers) have all been
described, we will enter a more complex subject that is of enormous
importance to the mass spectrometry specialist: polymer chain
disrupting chemistry. We shall see that this terminology actually
involves two kinds of chemistries: cleavage, on the one hand, and
fragmentation, on the other hand.

While \mXp\ is basically oriented to linear single-stranded polymer
chemistries, it can also be used to simulate highly complex polymer
chemistries. Biological polymers are the main focus of this manual,
however all the concepts described here may be applied with no
modification to synthetic polymer chemistries.
 

\renewcommand{\sectitle}{Various Biopolymer Structures}
\section*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{section}{\numberline{}\sectitle}

Biopolymers are amongst the most sophisticated and complex polymers on
earth and it certainly is not a mistake to take them as examples of
how monomers (be these complex or not) can assemble covalently into
life-enabling polymers. In this section we will visit three different
polymers encountered in the living world: proteins, nucleic acids and
polysaccharides. We shall be concerned with 1) the monomers'
structure, 2) the polymerization reaction and 3) the final end-capping
reaction responsible for putting the polymer in its finished state.

\renewcommand{\sectitle}{Proteins}
\subsection*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{subsection}{\numberline{}\sectitle}
\index{protein}

These biopolymers are made of amino acids. There are twenty major
amino acids in nature, and each protein is made of a number of these
amino acids\index{amino~acid}. The combinations are infinite,
providing enormous diversity of proteins to the living world.

A protein is a polar polymer: it has a left end and a right
end\index{protein!left/right ends}, and polymerization actually occurs
from left to right (from N-terminus to C-terminus, see
below). Figure~\ref{fig:peptbond-formation} shows that the chemical
reaction at the basis of protein synthesis is a
\emph{condensation}\index{condensation}. A protein is the result of
the condensation of amino acids with each other in an orderly polar
fashion. A protein has a left end, called \emph{N-terminus; amino
  terminal end} and a right end, called \emph{C-terminus; carboxyl
  terminal end}. The left end is an amino group ($\mathrm{_2HN--}$)
corresponding to the non-reacted amino group of the amino acid. Upon
condensation of a new amino acid onto the first one, the carboxyl
group of the first amino acid reacts with the amino group of the
second amino acid. A water molecule is released, and the formation of
an amide bond between the two amino acids yields a dipeptide.  The
right end of the dipeptide is a carboxyl group (--COOH) corresponding
to the un-reacted carboxyl group of the last amino acid to have
``polymerized in''.

The bond formed by condensation of two amino acids is an amide
bond\index{protein!amide~bond}, also called---in protein chemistry---a
\emph{peptidic bond}. The elongation of the protein is a simple
repetition of the condensation reaction shown in
Figure~\ref{fig:peptbond-formation}, granted that the elongation
\emph{always} proceeds in the described direction (a new monomer
arrives to the right end of the elongating polymer, and elongation is
done from left to right).

\begin{figure}
  \begin{center}
    \includegraphics[width=0.4\textwidth]{figures/peptbond-formation.png}
  \end{center}
  \caption[Peptidic bond formation]{\textbf{Peptidic bond formation by
      condensation.} The left end monomer $\mathrm{R_1}$ is condensed
    to the right end monomer $\mathrm{R_2}$ to yield a peptidic bond.
    A water molecule is lost during the process.}
  \label{fig:peptbond-formation}
\end{figure}

Now we should point at a protein chemistry-specific terminology issue:
we have seen that a protein is a polymer made of a number of monomers,
called amino acids\index{amino~acid}. In protein chemistry, there is a
subtlety: once a monomer is polymerized into a protein it is no more
called a monomer, it is called a \emph{residue}\index{residue}. We may
say that a residue is an amino acid less a water molecule.

From what we have seen until now, we may define a protein this way:
---\textsl{``A protein is a chain of residues linked together in an
  orderly polar fashion, with the residues being numbered starting
  from 1 and ending at n, from the first residue on the left end to
  the last one on the right end''}. This definition is still partly
inexact, however.  Indeed, from what is shown in
Figure~\ref{fig:prot-polymer}, there is still a problem with the
extremities of the residual chain: what about the amino group on the
left end of a protein (the amino group sits right onto the first amino
acid of the protein), and what about the carboxyl group of the right
end of a protein (the carboxyl group sits right onto the last amino
acid of the protein)? Because these groups lie at the extremities of
the residual chain, they remained unreacted during the polymerization
process. But because we are simulating a residual chain using residues
and not amino-acids, we still need to put the residual chain in its
finished state: by \emph{capping} the left end with a proton
\emph{cap} (so as to complete the amino group) and the right end with
a hydroxyl cap (so as to complete the carboxyl
group)\index{protein!left/right~caps}. The capping of the residual
chain extremities ensures that the polymer is in its finished state,
and that it cannot be elongated anymore. The proton is the \emph{left
  cap} of the protein polymer and the hydroxyl is the \emph{right cap}
of the protein polymer.

\begin{figure}
  \begin{center}
    \includegraphics[width=0.4\textwidth]{figures/prot-polymer.png}
   \end{center}
   \caption[End capping chemistry of the protein polymer]{\textbf{End
       capping chemistry of the protein polymer.} A protein is made of
     a chain of residues and of two caps. The left cap is the
     N-terminal proton and the right cap is the C-terminal hydroxyl.
     Altogether, the residual chain (enclosed here in the blue
     polygon) and both the H and OH red-colored caps do form a
     complete protein polymer in its finished state.}
  \label{fig:prot-polymer}
\end{figure}

Now comes the question of unambiguously defining the structure of a
protein. It is commonly accepted that the simple ordered sequence of
each residue code in the protein, from left to right, constitutes an
unambiguous description of the protein's primary structure (that is
its sequence). Of course, proteins have three-dimensional structures,
but this is of no interest to a program like \mXp, which is aimed at
calculating masses of polymers. To enunciate unambiguously the
sequence of a protein, one would use a symbology like this:
\smallskip

\begin{mynoindent}
  {\footnotesize using the 3-letter code of the amino acids:}\\
  Ala Gly Trp Tyr Glu Gly Lys\\
  {\footnotesize or, using the 1-letter code of the amino acids:}\\
  A G W Y E G K\\
  Alanine is thus the residue 1 and Lysine is the last residue (n =
  7).
\end{mynoindent}


\renewcommand{\sectitle}{Nucleic Acids}
\subsection*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{subsection}{\numberline{}\sectitle}
\index{nucleic~acid}

These biopolymers are more complex than proteins, mainly because they
are composed of monomers \emph{(nucleotides)}\index{nucleotide} that
have three different chemical parts, and because those parts differ in
DNA and RNA. A nucleotide\index{nucleic~acid!left/right~ends} is the
nucleic acid's brick: \emph{a nucleotide consists of a nitrogenous
  base combined with a ribose/deoxyribose sugar and with a phosphate
  group}. There are two different kinds of nucleic acids:
deoxyribonucleic acid (DNA, the sugar is a deoxyribose) and
ribonucleic acid (RNA, the sugar is a ribose). DNA is most often found
in its double stranded form, while RNA is most often found in single
strand form.  There are four nitrogenous bases for each: Adenine,
Thymine, Guanine, Cytosine for DNA; in RNA only one of these bases
changes: Thymine is replaced by Uracile. As for proteins, nucleic
acids are polar polymers: the polymerization process is polar, from
left to right (sometimes left is up and right is down in certain
vertical representations found mainly in textbooks).

This manual is not to teach biochemistry, which is why the structure
of the monomers is not described in atomic detail. However, since it
is important to understand how the polymerization occurs,
Figure~\ref{fig:nucacbond-formation} represents the polymerization
reaction mechanism between a nucleotide and another one, to yield a
dinucleotide. That reaction is a \emph{trans-esterification}. A
nucleic acid has a left end---\emph{5' end; often this end is
  phosphorylated}---and a right end---\emph{3' end; hydroxyl end}. The
trans-esterification reaction is the attack of the phosphorus of the
new (deoxy)nucleotide triphosphate by the 3'OH of the right end of the
elongating nucleotidic chain. Upon
trans-esterification\index{trans-esterification}, an \emph{inorganic
  pyrophosphate} ($\mathrm{PP_i}$) is released, and the formation of a
phosphodiester bond\index{nucleic~acid!phosphodiester~bond} between the two
nucleotides yields a dinucleotide.  The elongation of the nucleic acid
polymer is a simple repetition of this esterification reaction so that
the chain growth is always in the 5'$\Longrightarrow$3' direction.
This is achieved in the living cells by what is called the
\emph{5'$\Longrightarrow$3' polymerase enzymatic activity}.

\begin{figure}
  \begin{center}
    \includegraphics[width=0.8\textwidth]{figures/nucacbond-formation.png}
   \end{center}
  \caption[Phosphodiester bond formation]{\textbf{Phosphodiester bond
      formation by esterification.} The arriving monomer (on the
    right) has its triphosphate on the 5' carbon of the sugar
    esterified by nucleophilic attack of the first phosphorus by the
    alcohol function beared by the 3' carbon of the (deoxy)ribose
    sugar ring of the left monomer. The bond that is formed is a
    phosphodiester bond, with release of a pyrophosphate group
    ($\mathrm{PP_i}$). Note that the sugar and nitrogenous bases are
    schematically represented in this figure.}
  \label{fig:nucacbond-formation}
\end{figure}


The conventional representation of a nucleic acid involves showing the
5' end on the left, and the 3' end on the right, horizontally.
Sometimes, to clearly indicate that the left end is phosphorylated,
while the right end is not, the ends are indicated as ``5'P'' and
``3'OH''. Figure~\ref{fig:nucac-polymer} shows a simple way to
formalize what a nucleic acid polymer is. The molecule represented on
the left is the ``monomer'' in the sense that the polymer is made of n
monomers. On the right side of that figure, the polymer made of n
monomers is shown as a residual chain (inside the blue polygon box)
that got capped with OH on its left end and H on its right end
(red-colored atoms)\index{nucleic~acid!left/right~caps}. Thus, in the
case of the nucleic acid polymers, the left cap is a hydroxyl and the
right cap is a proton.  This anecdotically happens to be the exact
converse of what was described earlier for proteins.

\begin{figure}
  \begin{center}
    \includegraphics[width=0.5\textwidth]{figures/nucac-polymer.png}
      \end{center}
  \caption[A nucleic acid is a capped nucleotide chain]{\textbf{End
      capping chemistry of the nucleic acid polymer.} A nucleic acid
    is made of a chain of nucleotides (left formula) and of two caps.
    The left cap is the hydroxyl group that belongs to the terminal
    phosphate of the 5' carbon of the sugar. The right cap is the
    proton that belongs to the hydroxyl group of the 3' carbon of the
    sugar ring (right formula). Altogether, a finished nucleic acid
    polymer is made of the nucleotidic chain (enclosed here in the
    blue polygon), made of the repetitive elements (one of which is
    shown on the left), and of the two caps (red-colored OH and H, out
    of the box on the right).}
  \label{fig:nucac-polymer}
\end{figure}

Now comes the question of unambiguously defining the structure of a
nucleic acid. It is commonly accepted that the listing of the named
nitrogenous bases in the nucleic acid---from left (5' end) to right
(3' end)---constitutes an unambiguous description of the nucleic acid
sequence. To enunciate the sequence of a gene, one would use a
symbology like this:
\smallskip

\begin{mynoindent}
  {\footnotesize for a DNA, using the 1-letter code of the nitrogenous
    bases:} A T G C A G T C\\
  {\footnotesize for an RNA, using the 1-letter code of the
    nitrogenous bases:} A U G C A G U C\\
  Adenine is thus the base 1 and Cytosine is the last base (n = 8).
\end{mynoindent}


\renewcommand{\sectitle}{Saccharides}
\subsection*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{subsection}{\numberline{}\sectitle}
\index{saccharide}

These biopolymers are certainly amongst the most complex ones in the
living world. This is mainly due to the fact that saccharides are
usually heavily modified in living cells with a huge variety of
chemical modifications. Furthermore, the ramifications in the polymer
structure are more often the normal situation than not. Interestingly,
these molecules are first thought of as the ``fuel'' for the cell,
which is certainly far from being total nonsense, but it is also
undoubtful that their structural role is extremely important (often in
combination with proteinaceous material). Another interesting aspect
of their ability to form complex structures is their use as ``key''
systems for identification processes: a number of complex sugars are
located on the cell walls and provide ``recognition patterns'' for the
other cells to deal with\dots

Nonetheless, the general picture is not that complex, if the way
monomers are polymerized together is the only concern (which is the
case in this manual). As far as we are concerned, in fact, the
polymerization mechanism is a simple condensation (much like what has
been described for proteins), yielding a sugar
bond\index{saccharide!sugar~bond}. Indeed, some people use the same
terminology: a monomeric sugar becomes a residue once polymerized in
the saccharidic chain. There are two main different kinds of sugars:
\emph{pentoses} (in $\mathrm{C_5}$) and \emph{hexoses} (in
$\mathrm{C_6}$); it should be noted, however, that there is a variety
of other common molecules, like \emph{sialic acids},
\emph{heptoses}\dots

Like already seen for proteins and nucleic acids, a saccharidic
polymer is polar: it has a left end and a right
end\index{saccharide!left/right~ends}.  The terminology regarding the
ends of a saccharidic polymer is rather unexpected at first sight: the
left end is said to be the \emph{non-reducing
  end}\index{non-reducing~end} while the right end is said to be the
\emph{reducing end}\index{saccharide!reducing~end}.  Historically this
was observed with monosaccharides (also called
\emph{monoses}\index{monose}), which reduced cupric
($\mathrm{Cu^{2+}}$) ions, thus getting oxydized themselves on the
carbonyl (when in the open ring aldehydic form).

Figure~\vref{fig:sacchbond-formation} shows the polymerization
reaction between a sugar and another one (2 glucose monomers,
actually), to yield a maltose disaccharide. The polymerization
mechanism is a simple condensation. The elongation of the saccharidic
polymer is a simple repetition of this condensation reaction so that
the chain growth is always in the same orientation, from the
non-reducing end to the reducing end. The conventional representation
of a polysaccharide involves showing the non-reducing end on the left,
and the reducing end on the right, horizontally.
Figure~\vref{fig:sacch-polymer} shows a simple way to formalize what a
saccharidic polymer is. The top formula is the representation of the
monomer. The bottom formula represents a polysaccharide, with the
repetitive elements boxed (there are n monomers polymerized). The
atoms shown in red (outside the boxed repetitive elements) are the
saccharidic polymer caps. Thus, we see clearly that in the case of
polysaccharides, the left cap is a proton and the right cap is a
hydroxyl\index{saccharide!left/right~caps}. This anecdotically happens
to be identical to proteins and the exact converse of what we
described previously for nucleic acids.



\begin{figure}
  \begin{center}
    \includegraphics[width=0.8\textwidth]{figures/sacchbond-formation.png}
  \end{center}
  \caption[Osidic bond formation]{\textbf{Osidic bond formation by
      condensation.} The two monomers are subject to condensation with
    loss of one molecule of water.}
  \label{fig:sacchbond-formation}
\end{figure}

\begin{figure}
  \begin{center}
    \includegraphics[width=0.4\textwidth]{figures/sacch-polymer.png}
  \end{center}
  \caption[A saccharidic polymer is a capped osidic residue
  chain]{\textbf{End capping chemistry of the polysaccharidic
      polymer.} A polysaccharide is made of a chain of osidic residues
    (blue-boxed formula) and of two caps (red-colored atoms). The left
    cap is the proton group that belongs to the non-reducing end of
    the polymer. The right cap is the hydroxyl group that belongs to
    the reducing end of the polymer.}
  \label{fig:sacch-polymer}
\end{figure}

Now comes the question of unambiguously defining the structure of a
saccharidic polymer. It is commonly accepted that the simple ordered
sequence of the named monoses in the saccharidic polymer, from left
(non-reducing end) to right (reducing end), constitutes an unambiguous
description of the glycan sequence. To enunciate the sequence of a
glycan, one would use a symbology like this:

\begin{mynoindent}
  {\footnotesize using a  3-letter code:}\\
  Ara Gal Xyl Glc Hep Man Fru\\
  Arabinose is thus the monose 1 and Fructose is the last monose (n =
  7).
\end{mynoindent}

\bigskip

\noindent Incidentally, this is where the ability of \mXp\ to handle monomer
codes of non-limited length comes in handy!

\renewcommand{\sectitle}{To Sum Up}
\section*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{section}{\numberline{}\sectitle}

We made a rapid overview of the three major polymers in the living
world. A great many other polymers exist around us.
Table~\vref{tab:three-biopolym-exples} tries to sum up all the
informations gathered so far. Note that the formul{\ae} given for the
monomers are the ``residual'' ones. For example, the formula of the
glycyl residue corresponds to the formula of the Glycine monomer less
one molecule of water. Many synthetic polymers are much simpler than
the ones we have rapidly reviewed, and it should be clear that, if
\mXp\ can deal with the complex biopolymers described so far, it
certainly will be very proficient with less complex synthetic
polymers. Describing the formation of polymers is one thing, but we
also have to describe how to disrupt polymers. This is what we shall
do in the next section.


\begin{table}
    \begin{small}
      \begin{tabular}{c|ccccc}\hline
        polymer     &   name & code  &    formula                &  left cap  & right cap \\ 
        \hline
        protein     &   &      &                           &      H         &           OH       \\
        & Glycine   &   G     & $\mathrm{C_2H_3O_1N_1}$   &                &                    \\
        & Alanine   &   A     & $\mathrm{C_3H_5O_1N_1}$   &                &                    \\
        & Tyrosine  &   T     & $\mathrm{C_9H_9O_2N_1}$   &                &                    \\
        nucleic acid&   &      &                           &      OH        &            H       \\
        & Adenine   &   A     & $\mathrm{C_{10}H_{12}O_5N_5P_1}$ &         &                    \\
        & Cytosine  &   C     & $\mathrm{C_9H_{12}O_6N_3P_1}$    &         &                    \\
        saccharide  &   &      &                           &      H         &            OH      \\
        & Arabinose &   Ara   & $\mathrm{C_5H_8O_4}$      &                &                    \\
        & Heptose   &   Hep   & $\mathrm{C_7H_{12}O_8}$   &                &                    \\
        \hline
        \multicolumn{6}{c}{Note: LC=left cap; RC= right cap}\\
        \hline
      \end{tabular}
      \caption[Comparison of three common biopolymers]{\textbf{Quick comparison of three biopolymers with examples of monomers}}\label{tab:three-biopolym-exples}
    \end{small}
\end{table}


\renewcommand{\sectitle}{Polymer Chain Disrupting Chemistry}
\section*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{section}{\numberline{}\sectitle}

\label{sect:pol-chain-disrupt-chem}

The ``polymer chain disrupting chemistry'' was mentioned earlier as a
complex subject that was of \emph{enormous} importance to the mass
spectrometrist. This is why that subject will be treated in a pretty
thorough manner. First of all it should be noted that a chemical
modification of a polymer does not necessarily involve the
perturbation of the chain structure of the polymer.  Here, however, we
are concerned specifically with a number of chemical modifications
that yield a polymer chain perturbation; \emph{cleavage} and
\emph{fragmentation}:
\smallskip

\begin{mynoindent}
  \textsc{A cleavage is a chemical process}\index{cleavage} by which a
  cleaving agent will act directly on the polymer chain making it fall
  into at least two separated pieces (the \emph{oligomers}). As a
  result of the cleavage reaction, groups originating in the cleaving
  molecule remain attached to the polymer at the precise cleavage
  location;

\smallskip

\textsc{A fragmentation is a chemical process}\index{fragmentation} by
which the polymer structure is disrupted into separated pieces (the
\emph{fragments}) mainly because of energy-dependent electron doublet
rearrangements leading to bond breakage.
\end{mynoindent}



\renewcommand{\sectitle}{Polymer Cleavage}
\subsection*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{subsection}{\numberline{}\sectitle}
\label{sect:polymer-cleavage}
\index{cleavage}

We said above that, upon cleavage of a polymer, the cleaving molecule
reacts with it, and by doing so directly or indirectly
``\emph{dissolves}'' an inter-monomer bond. A polymer cleavage always
occurs in such a way as to generate a set of \emph{true} polymers
(smaller in size than the parent polymer, evidently, which is why they
are called \emph{oligomers}). Indeed, let us take the example shown in
Figure~\ref{fig:prot-cleavage}, where a tripeptide (a very little
protein, containing a methionyl residue at position 2) is submitted
either to a water-mediated cleavage (hydrolysis, upper panel) or to a
cyanogen bromide-mediated cleavage (lower panel). The two cases
presented in this figure are similar in some respects and different in
others:

\begin{itemize}

\item In the first case the molecule that is responsible for the
  cleavage is water, while in the second case it is cyanogen bromide;

\item In both cases the bond that is cleaved is the inter-monomer bond
  (in protein chemistry this is a peptidic bond);

\item In both cases the Oligomer 2 has the same structure;

\item The structures of the Oligomer 1 species differ when produced
  using water or cyanogen bromide as the cleaving molecule.

\end{itemize}
%
%
The difference between hydrolysis and cyanogen bromide cleavage is in
the generation of the Oligomer 1 species: the cyanogen bromide
cleavage has a side effect of generating a homoserine as the right end
monomer of Oligomer 1, while hydrolysis generates a genuine methionine
monomer. This is because water reverses in a very symmetrical manner
what polymerization did (hydrolysis is the converse of condensation),
while cyanogen bromide\index{cyanogen~bromide} did some chemical
modification onto the generated Oligomer 1 species.

\begin{figure}
  \begin{center}
    \includegraphics[width=0.8\textwidth]{figures/prot-cleavage.png}
  \end{center}
  \caption[Protein cleavage by water and cyanogen
  bromide]{\textbf{Protein cleavage by water and cyanogen bromide.} A
    tripeptide is cleaved at position 1 either by hydrolysis (top) or
    by cyanogen bromide (bottom). Cyanogen bromide cleaves
    specifically on the right of a methionine monomer. Upon cleavage,
    the methionyl monomer gets converted into homoserine by the
    cyanogen bromide reagent.}
  \label{fig:prot-cleavage}
\end{figure}

Nonetheless, the reader might have noted that---interestingly---all
the four oligomers do effectively have their left cap (a proton) and
their right cap (the hydroxyl). This means that in both water- and
cyanogen bromide-mediated cleavages, all the generated oligomers are
indeed true polymers in the sense that: 1) they are a chain of
monomers (modified or not) and 2) they are correctly capped
(\textit{i.e.} they are polymers in their finished state). This is
important because it is the basis on which we shall make the
difference between a cleavage process and a fragmentation process.
Thus, the \mXp\ definition of an oligomer might be: \emph{an oligomer
  is a polymer (of at least one monomer) in its finished state that
  was generated upon cleavage of a longer polymer}.


When the polymer cleavage reaction precisely reverses the reaction
that was performed for the same polymer's synthesis, there is no
special difficulty. But when the cleavage reaction modifies the
substrate, then this should be carefully modelled. How? To answer this
question we might start by comparing the two different Oligomer 1
species that were yielded upon the water-mediated and the cyanogen
bromide-mediated cleavage reactions: ``the hydrolysis-generated
Oligomer 1 is equal to the cyanogen bromide-generated Oligomer 1 +S1
+C1 +H2 -O1''; this is a big difference! The observations we did so
far might be worded this way: \textsl{Whenever a protein undergoes a
  cyanogen bromide-mediated cleavage, the ``-C1H2S1+O1'' chemical
  reaction should be applied to the resulting oligomers \textit{if and
    only if} they have a methionine monomer at their right end}. In
\mXp's jargon, this logical condition is called a \emph{cleavage rule}
(described later; see page~\pageref{sect:cleavespecif}).

Well, all this sounds reasonable. But what about the ``normal'' case,
when the cleavage is done using water? Nothing special: the mass of
the oligomer is calculated by summing the mass of each monomer in the
oligomer (since the monomers are not modified, this is easily done)
and the masses corresponding to the left and right caps (these are
defined in the polymer chemistry definition; in our present case it
would be a proton on the left end, and a hydroxyl on the right end).
In this way, the oligomer complies with its definition, which states
that it is a faithful polymer made of monomers and that it is in its
finished state.

Yes, but then how will \mXp\ manage to calculate the mass of the
modified oligomer, like our Oligomer 1 in the case of the cyanogen
bromide-mediated cleavage?  Simple enough: in a first step it does
exactly the same way as for the unmodified oligomer. Next, each
oligomer is checked for presence or absence of a methionine residue on
its right end. If a methionine is found, the mass corresponding to the
``-C1H2S1+O1'' chemical reaction is applied. And that's it.

In the previous cyanogen bromide example, the logical condition was
involving the identity of the oligomers' right end monomer, but other
examples can involve not the right end monomer, but the left end
monomer, if some chemical modification was to occur to the monomer
sitting right of the cleavage location. In this case the user would
have to analyse the situation and provide \mXp\ with the proper
chemical reaction by stating something analog to: \textsl{\textit{if
    and only if} they have a Xyz monomer at their left end}. This
introduction to polymer cleavage abstraction should be enough to later
delve into the cleavage specification definition as \mXp\ conceives it
and that is thoroughly detailed at page~\pageref{sect:cleavespecif}.


\renewcommand{\sectitle}{Polymer Fragmentation}
\subsection*{\textcolor{sectioningcolor}{\sectitle}}
\addcontentsline{toc}{subsection}{\numberline{}\sectitle}
\label{sect:polymer-fragmentation}
\index{fragmentation}

In a fragmentation process, the bond that is broken is not necessarily
the inter-monomer bond. Indeed, fragmentations are oft-times high
energy chemical processes that can affect bonds that belong to the
monomers' internal structure. This is one of the reasons why
fragmentations do differ from cleavages: they are specific of the
polymer type in which they occur. Hydrolyzing a protein and an
oligosaccharide is just the same process, from a chemical point of
view. But fragmenting a protein or an oligosaccharide are truly
different processes because the way that the fragmentation happens in
the polymer sequence is so much dependent on the nature of each
monomer that makes it.

Another peculiarity of the fragmentations, compared with the cleavages
that were described above, is the fact that there is no cleaving
molecule starting the process. Instead, a fragmentation process is
often initiated by an intra molecular electron doublet rearragement
that propagates more or less in the polymer structure to eventually
break it. Fragmentations are mainly a gas phase process, not some
reaction that happens in solution as a result of putting in contact
the polymer and some reagent. It is precisely because no cleaving
molecule is involved in the fragmentation process that the fragments
are not necessarily capped like a normal polymer should be; and this
is another really important difference between cleavage and
fragmentation. The following examples should illustrate these
concepts: protein and nucleic acid fragmentation.


\subsubsection*{Protein Fragmentation}
\index{fragmentation!protein}

There is a pretty important number of different kinds of fragments
that can be generated upon fragmentation of peptides. We are going to
detail the most common ones; the user is invited to use the \mXp'
fragmentation-specification grammar to add less frequent (or newly
discovered) fragmentation types.

\begin{figure}
  \begin{center}
    \includegraphics[scale=0.2]{figures/prot-fragmentation.png}
  \end{center}
  \caption[Protein fragmentation]{\textbf{Protein fragmentation
      patterns most widely encountered.} An hexapeptide is fragmented
    in the seven most widely encountered manners, such as to generate
    a, b, c, x, y, z and immonium fragment ions. The figure
    illustrates the position of the cleavage for each kind of fragment
    (exemplified using the case of the smallest fragment possible) and
    the mass calculation method is described for each fragment kind;
    consider that each fragment bears only \emph{one positive}
    charge.}
  \label{fig:prot-fragmentation}
\end{figure}

As can be seen from Figure~\ref{fig:prot-fragmentation}, the
fragmentations do generate fragments of three categories: the ones
that include the left end of the precursor polymer (a, b, c), the ones
that include the right end of the precursor polymer (x, y, z), and
finally the special case in which the fragment is an \emph{internal
  fragment}, like the immonium ions. When looking at the
fragmentations described in the figure it becomes immediately clear
why a fragmentation cannot be mistaken for a cleavage: the ionization
of the fragment is not necessarily due to the captation of a proton by
the fragment. Furthermore, we can also see that a fragmentation is not
a cleavage because the fragment that is generated is \emph{absolutely}
not necessarily what we call a polymer, in the sense that the fragment
might not be capped the same way as the precursor polymer is (that is,
the fragment is not in its finished polymerizaton state).

The two observations above should make clear to the reader that
calculating masses for fragments is a more difficult process than what
was described above for the oligomers. Indeed, while it was simple to
calculate the mass of an oligomer (by simply adding the masses of its
constitutive monomer units, plus the left and right caps, plus
ionization), here there is no chemical formalism generally applicable
to all the fragment types. This is why the specification of the
fragmentation is left to the user's responsibility.

By looking at Figure~\ref{fig:prot-fragmentation}, the reader should
have noticed that the fragment naming scheme takes into consideration
the fact that the fragment bears the left or the right end of the
precursor polymer (or none, also). Indeed, the numbering of fragments
holding the left end of the precursor polymer sequence begins at the
left end, and for fragments that hold the right end, at the right end.
Thus the third fragment of series \emph{a}---\emph{a3}---would involve
monomers [1$\rightarrow$3]; and the third fragment of series
\emph{y}---\emph{y3}---would involve monomers [6$\rightarrow$4] (in
the figure, these left-to-right and right-to-left directions are
symbolized using arrows). Therefore, it should appear to the reader
how important---when specifying a fragmentation---it is to clearly
indicate from which end of the precursor polymer the fragment is
generated (in \mXp's jargon this is ``LE'' for left end, ``RE'' for
right end and ``NE'' for no end). \mXp\ knows what action it should
take when it encounters one of these three specifications; for
example, if a ``LE'' specification is found for a given fragmentation
specification, \mXp\ adds to the fragment's mass the mass
corresponding to the left cap of the precursor polymer.


\paragraph{\emph{a} fragment series} If we take the \emph{a} fragment
series, the Figure~\ref{fig:prot-fragmentation} indicates that the
fragments include the left end and that their last monomer lacks its
carbonyl group (see, on top of Figure~\ref{fig:prot-fragmentation},
that the \emph{a1} arrow goes between the C$\alpha$H and the CO of
monomer 1?).  So we would say that each fragment of the \emph{a}
series should be challenged with the following chemical treatments: 1)
addition of the mass corresponding to the left cap (proton), 2)
removal of the mass corresponding to the lacking CO group. This way we
have the mass of fragment \emph{a1}. If we were interested in the
fragment \emph{a4} we would have summed the masses of monomers 1 to 4,
added the mass of the left cap, and finally removed the mass of a CO.
The mass calculation is thus mathematically expressed \[a_i = LC +
\sum_{1}^{i} M_i - CO\]

\paragraph{\emph{b} fragment series} Similarly, the mass calculation
is mathematically expressed \[b_i = LC + \sum_{1}^{i} M_i\]

\paragraph{\emph{c} fragment series} The mass calculation is
mathematically expressed \[c_i = LC + \sum_{1}^{i} M_i + NH_3\]

\paragraph{\emph{x} fragment series} For this series of fragments we
do not add the left cap anymore, but replace it with the right cap,
since the fragments hold the right end of the precursor polymer. Note
also that the numbering of the monomers using the variable \emph{i} in
the following mathematical expressions goes from right to left
(contrary to what happened for the \emph{a, b, c} fragment series. All
the fragments that hold the precursor polymer right end are numbered
this way, so this applies to fragments \emph{x, y, z}. The mass
calculation is mathematically expressed \[x_i = RC + \sum_{1}^{i} M_i
+ CO\]

\paragraph{\emph{y} fragment series} The calculation is mathematically
expressed \[y_i = RC + \sum_{1}^{i} M_i + H_2\]

\paragraph{\emph{z} fragment series} In low energy CID, the \emph{z}
fragments are expressed this way: \[z_i = RC + \sum_{1}^{i} M_i - NH\]
which is equivalent to \emph{y-$NH_3$}; in high energy CID an
additional proton is often measured: \[z_i = RC + \sum_{1}^{i} M_i -
NH + H\]

\paragraph{\emph{immonium} fragment series} These fragments are
internal fragments in the sense that they do not hold neither of the
two precursor polymer's ends. \mXp\ understands that the user is
speaking of this kind of fragment when the ``from which end'' piece of
data --in the fragmentation specification-- states ``NE'' instead of
``LE'' or ``RE'' (see page~\pageref{sect:fragspecif}). The mass
calculation for these fragments does not take into account the
monomers surrounding the one for which the calculation is done. The
mass for an immonium ion --at position \emph{i} in the precursor
polymer-- will be the mass of the monomer at position \emph{i}, less
the mass of a CO, plus the mass of a proton. The mass calculation for
these special internal fragments is expressed \[imm_i = M_i + H - CO\]


\subsubsection*{Nucleic Acid Fragmentation}
\index{fragmentation!nucleic~acid}

The fragmentations that can be obtained with nucleic acids are
numerous and it is more complicated than with proteins to describe
them fully.  The main reason for this is that there are a big number
of fragmentation combinations because of the loss of nitrogenous bases
from the skeleton. The mechanisms by which this loss happens are
fairly complex, and I am not going to detail any of them.  Figure
~\vref{fig:dna-fragmentation} shows the most common fragmentations
(without taking into consideration the potential loss of bases). An
example of fragment is given for each fragment series (pretty the same
way as we did before for proteins). Note that the fragment
representations are aimed at helping the reader to figure out what the
product ion is, not taking into account where the negative charge lies
on the fragment, since this charge can float around at every
de-protonatable group. All the fragments shown bear one and one only
negative charge.

The reader might have noticed at the bottom of Figure
~\vref{fig:dna-fragmentation} that a provision is made in the case the
fragmented molecular species are not 5' end-phosphorylated but 5'
end-hydroxylated. Indeed, the canonical monomer is such that, upon
polymerization and left capping, the 5' end is phosphorylated.
However, oft-times the oligonucleotides are synthesized chemically
without the 5' end phosphate group, thus ending in hydroxyl. This
special case should be accounted for by applying to all the fragments
that bear the left end of the precursor polymer the following chemical
reaction: $\mathrm{-HPO_3}$. This chemical reaction should be applied
\emph{in addition} to the chemical reaction that yields the fragment
\emph{per se}.

\begin{figure}
  \begin{center}
    \includegraphics[scale=0.2]{figures/dna-fragmentation.png}
  \end{center}
  \caption[DNA fragmentation]{\textbf{DNA fragmentation patterns most
      widely encountered.} A short DNA sequence is fragmented in the
    eight most widely encountered manners, such as to generate a, b,
    c, d, w, x, y, z fragment ions. The figure illustrates the
    position of the cleavage for each kind of fragment (exemplified
    using the case of the smallest fragment possible). and the mass
    calculation method is described for each fragment kind;
    considering that each fragment is protonated only once (+1).}
  \label{fig:dna-fragmentation}
\end{figure}

Exactly as done earlier for the protein fragments, the mathematical
expressions used to calculate the mass of different series of nucleic
acid fragments are provided; in these calculations it is assumed that
the left end of the precursor polymer is phosphorylated (5'P) and the
reader should bear in mind that this precise phosphate might itself be
expelled by the fragmentation. The fragment naming schemed detailed
earlier for proteins applies to nucleic acids in the very same manner.

\paragraph{\emph{a} fragment series} 
These fragments most often appear with base loss. \[a_i = LC +
\sum_{1}^{i} M_i - O\]

\paragraph{\emph{b} fragment series}
\[b_i = LC + \sum_{1}^{i} M_i\]

\paragraph{\emph{c} fragment series}
\[c_i = LC + \sum_{1}^{i} M_i - HPO_2\]

\paragraph{\emph{d} fragment series}
\[d_i = LC + \sum_{1}^{i} M_i - HPO_3\]

\paragraph{\emph{w} fragment series}
\[w_i = RC + \sum_{1}^{i} M_i + O\]

\paragraph{\emph{x} fragment series}
\[x_i = RC + \sum_{1}^{i} M_i\]

\paragraph{\emph{y} fragment series}
\[y_i = RC + \sum_{1}^{i} M_i - HPO_2\]

\paragraph{\emph{z} fragment series}
\[z_i = RC + \sum_{1}^{i} M_i - HPO_3\]

There are also a variety of fragments for which a base is lost.


\subsubsection*{More Complex Patterns Of Fragmentation}


Before finishing with fragmentations, it is necessary to describe a
powerful feature of the fragmentation specification grammar available
in \mXp. This feature was required for the fragmentation of
oligosaccharides and also sometimes for proteins. When the
fragmentation (the bond breakage reaction itself) occurs at the level
of certain monomers, it might be necessary to be able to specify some
particular chemistry that would arise on the monomer in question.

We have seen in the cleavage documentation that, upon cleavage of a
protein sequence with cyanogen bromide, for example, a particular
chemical reaction had to be applied to the oligomers that were
generated with a methionine monomer as their right end monomer. Well,
in a fragmentation specification it is possible to apply comparable
chemical reactions but in a more thorough manner. Indeed, while in the
cleavage it was possible to say something like ``\textsl{apply a given
  chemical reaction to the oligomer if the right end monomer is
  Xyz''}, in the fragmentation the logical condition can be bound not
only to the identity of the currently fragmented monomer, but also
(optionally) to the identity of the previous and/or next monomer in
the precursor polymer sequence. For example: ---\textsl{``Apply a
  given chemical reaction if fragmentation occurs at the level of
  ``Xyz'' monomer only if it is preceded by a ``Yxz'' monomer and
  followed by a ``Zyx'' monomer''}.

These logical conditions are called \emph{fragmentation rules}. A
\emph{fragmentation specification} can hold as many rules as
necessary. All of this is described in great detail at
page~\pageref{sect:fragspecif}.

\subsubsection*{To Sum Up}


To sum up all what we have seen so far with polymer chain disrupting
chemistries:

\begin{itemize}
\item A polymer sequence gets cleaved into oligomers when a chemical
  reaction occurs in it at the level of one or more inter-monomer
  bond(s); monomer-specific chemical reactions can be modelled into
  the cleavage specification using at most one leftrighrule;
\item A polymer sequence gets fragmented into fragments when a bond
  breakage occurs, without the help of any exterior molecule, at any
  level of the polymer structure, with no limitation to the
  inter-monomer bond; monomer-specific chemical reactions can be
  modelled into the fragmentation specification using any number of
  fragrules;
\item Oligomers are automatically capped---\emph{on both ends}---using
  the rules described in the precursor polymer's definition;
\item Fragments are capped automatically only---\emph{on the end they
    hold, if any}---using the rules described in the precursor
  polymer's definition;
\item Oligomers are automatically ionized (if required by the user)
  using the rules described in the precursor polymer's definition;
\item Fragments are never ionized automatically; ionization (gain/loss
  of a charged group) is necessarily integrated in the fragmentation
  specification.
\end{itemize}


\cleardoublepage


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