finish section four

[chloroplast13.git] / classEquiv.tex

The first method, described below, considers NCBI annotations and uses
a distance-based similarity measure. We start with the following
preliminary definition:

\begin{definition}
\label{def1}
Let $A=\{A,T,C,G\}$  be the nucleotides alphabet, and  $A^\ast$ be the
set  of finite  words on  $A$  (\emph{i.e.}, of  DNA sequences).   Let
$d:A^{\ast}\times   A^{\ast}\rightarrow[0,1]$   be   a   distance   on
$A^{\ast}$. Consider a given value $T\in[0,1]$ called a threshold. For
all   $x,y\in  A^{\ast}$,   we   will  say   that  $x\sim_{d,T}y$   if
$d(x,y)\leqslant T$.
\end{definition}

%\noindent $\sim_{d,T}$ is obviously an equivalence relation and when $d=1-\Delta$, where $\Delta$ is the similarity scoring function embedded into the emboss package , we will simply denote $\sim_{d,0.1}$ by $\sim$.

Let be given a \emph{similarity} threshold $T$  and a distance $d$
(Needleman-Wunch released by EMBL for instance).
The method begins by building  an undirected graph 
between all the DNA~sequences $g$ of the set  of genomes as follows:
there is  an edge between $g_{i}$ and $g_{j}$
if  $g_i \sim_{d,T} g_j$ is established.
This graph is further denoted as the ``similarity'' graph.

We thus consider that the pair of two coding sequences 
$(g_i,g_j)$ belongs in the relation $\mathcal{R}$ if both $g_i$ and 
$g_j$  belong in the same 
connected component (CC), \textit{i.e.} if there is a path between $g_i$ 
and $g_j$ in the similarity graph. It is not hard to see that this relation is an
equivalence relation whereas $\sim$ is not.


Any class for this relation   is  called   ``gene'' 
here,   where   its  representatives
(DNA~sequences)  are the ``alleles''  of this  gene.  Thus  this first
method   produces   for   each    genome   $G$,   which   is   a   set
$\left\{g_{1}^G,...,g_{m_G}^G\right\}$    of   $m_{G}$    DNA   coding
sequences, the  projection of each sequence according  to $\pi$, where
$\pi$ maps each sequence into its gene (class) according to $\mathcal{R}$. In
other     words,      a     genome     $G$      is     mapped     into
$\left\{\pi(g_{1}^G),...,\pi(g_{m_G}^G)\right\}$.    Note    that    a
projected genome has no duplicated gene since it is a set.

Consequently, the core  genome (resp.  the pan genome)  of two genomes
$G_{1}$  and $G_{2}$  is defined  as  the intersection  (resp. as  the
union) of their projected  genomes.  We then consider the intersection
of  all the  projected genomes,  which  is the  set of  all the  genes
$\dot{x}$  such  that   each  genome  has  at  least   one  allele  in
$\dot{x}$. The  pan genome is computed  similarly as the  union of all
the projected  genomes. 

\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{stats.png}
\end{center}
\caption{Size of core and pan genomes w.r.t. the similarity threshold}\label{Fig:sim:core:pan}
\end{figure}

The number of genes in the core genome and in the pan genome are 
represented in  Figure~\ref{Fig:sim:core:pan} with respect to the 
threshold value. 
First of all, the higher is the threshold, 
the smaller the connected components are. In other words, the number 
of alleles of one gene is small if the threshold is high.
When the threshold is high, the number of genes and the size of 
pan genome is high too. However due to the construction method of the
core genome,  this set of genes has few elements in such a  situation.  
This approach even suffers from producing
too small core genomes (of size 0 or 1),  for any chosen similarity threshold, compared
to   what  is   usually   expected  by   biologists  regarding   these
chloroplasts. We are  then left with the following  questions: how can
we improve the confidence put in  the produced core? Can we thus guess
the evolution scenario of these genomes?