\color{red}
We still need to propose good methodology to predict good core genes that reflect natural biological relationships among species. Proposing new methods and compare them with previous ones can give us an indicator of which method can produce good functional genes. In this section, we will recall the definition with a fast revision of similarity based method (see \cite{Alkindy2014} for further details).
This method, considers annotated genomes from NCBI and DOGMA and uses a distance-based similarity measure on genes' coding sequences. Such an approach requires annotated genomes, like the ones provided by the NCBI website.

\subsubsection{Theoretical presentation}

We start by fast revision of first method by the following preliminary definition~\cite{acgs13:onp,Alkindy2014}.\color{black}

\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   function called
similarity measure   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 \emph{similarity measure} $d$.
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 say that two coding sequences 
$g_i$, $g_j$ are
equivalent with respect to the relation $\mathcal{R}$ if both $g_i$ and 
$g_j$  belong in the same 
connected component (CC) of this similarity graph, \textit{i.e.}, if there is a path between $g_i$ 
and $g_j$ in the 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  a ``gene'' 
in this article,   where   its  representatives
(DNA~sequences)  are the ``alleles''  of this  gene, such abuse of language
being proposed to set our ideas down.  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 finally 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
$\dot{x}$. This set will constitute the core genome of the whole species
under consideration. 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, first approach with NCBI annotations and Needleman-Wunsch similarity measure.}\label{Fig:sim:core:pan}
% \end{figure}
% 


% \begin{table}[h!]
% \centering
% \begin{tabular}{ccccccc}
% \hline
%           & \multicolumn{4}{c}{Method 1}                          & \multicolumn{2}{c}{Method 3}       \\ \hline
%           & \multicolumn{2}{c}{NCBI}  & \multicolumn{2}{c}{DOGMA} & \multicolumn{2}{c}{NCBI and DOGMA} \\ \hline
% Threshold & core       & pan          & core       & pan          & core            & pan              \\ \hline
% 50        & 1          & 163          & 1          & 118          & \textbf{5}      & \textbf{245}     \\
% 51        & 1          & 291          & 1          & 194          & -               & -                \\
% 52        & 1          & 412          & 1          & 258          & -               & -                \\
% 53        & 1          & 508          & 1          & 321          & -               & -                \\
% 54        & 4          & 617          & 2          & 372          & -               & -                \\
% 55        & \textbf{5} & \textbf{692} & 2          & 409          & -               & -                \\
% 56        & \textbf{5} & \textbf{761} & \textbf{3} & \textbf{445} & -               & -                \\
% 57        & 4          & 832          & \textbf{3} & \textbf{459} & -               & -                \\
% 58        & 4          & 905          & 2          & 477          & -               & -                \\
% 59        & 4          & 976          & 2          & 497          & -               & -                \\
% 60        & 2          & 1032         & 2          & 519          & 4               & 242              \\
% 61        & 2          & 1113         & 2          & 553          & -               & -                \\
% 62        & 2          & 1186         & 2          & 580          & -               & -                \\
% 63        & 2          & 1264         & 2          & 607          & -               & -                \\
% 64        & 2          & 1352         & 2          & 644          & -               & -                \\
% 65        & 1          & 1454         & 2          & 685          & -               & -                \\
% 66        & 1          & 1544         & 1          & 756          & -               & -                \\
% 67        & 0          & 1652         & 1          & 838          & -               & -                \\
% 68        & 0          & 1775         & 1          & 912          & -               & -                \\
% 69        & 0          & 1886         & 1          & 1007         & -               & -                \\
% 70        & 0          & 2000         & 1          & 1116         & 3               & 242              \\
% 80        & 0          & 3541         & 0          & 2730         & 1               & 242              \\
% 90        & 0          & 5703         & 0          & 5181         & 0               & 241              \\ \hline
% \end{tabular}
% \caption{Size of core and pan genomes w.r.t. the similarity threshold, first and third approaches.}
% \label{Fig:sim:core:pan}
% \end{table}
% 
\begin{table}[h!]
\centering
\begin{tabular}{cccccccc}
\hline
          & \multicolumn{4}{c}{Method 1}                          & \multicolumn{2}{c}{Method 2}       \\ 
\hline
          & \multicolumn{2}{c}{NCBI}  & \multicolumn{2}{c}{DOGMA} & \multicolumn{2}{c}{NCBI and DOGMA} \\ \hline
Threshold & core       & pan          & core       & pan          & core            & pan              \\ 
\hline
50        & 1          & 163          & 1          & 118          & \textbf{5}      & \textbf{245}     \\
51        & 1          & 291          & 1          & 194          & -               & -                \\
52        & 1          & 412          & 1          & 258          & -               & -                \\
53        & 1          & 508          & 1          & 321          & -               & -                \\
54        & 4          & 617          & 2          & 372          & -               & -                \\
55        & \textbf{5} & \textbf{692} & 2          & 409          & -               & -                \\
56        & \textbf{5} & \textbf{761} & \textbf{3} & \textbf{445} & -               & -                \\
57        & 4          & 832          & \textbf{3} & \textbf{459} & -               & -                \\
58        & 4          & 905          & 2          & 477          & -               & -                \\
59        & 4          & 976          & 2          & 497          & -               & -                \\
60        & 2          & 1032         & 2          & 519          & 4               & 242              \\
61        & 2          & 1113         & 2          & 553          & -               & -                \\
62        & 2          & 1186         & 2          & 580          & -               & -                \\
63        & 2          & 1264         & 2          & 607          & -               & -                \\
64        & 2          & 1352         & 2          & 644          & -               & -                \\
65        & 1          & 1454         & 2          & 685          & -               & -                \\
66        & 1          & 1544         & 1          & 756          & -               & -                \\
67        & 0          & 1652         & 1          & 838          & -               & -                \\
68        & 0          & 1775         & 1          & 912          & -               & -                \\
69        & 0          & 1886         & 1          & 1007         & -               & -                \\
70        & 0          & 2000         & 1          & 1116         & 3               & 242              \\
80        & 0          & 3541         & 0          & 2730         & 1               & 242              \\
90        & 0          & 5703         & 0          & 5181         & 0               &241              \\ 
\hline
\end{tabular}
\caption{Size of core and pan genomes w.r.t. the similarity threshold, first and second 
approache.}
\label{Fig:sim:core:pan}
\end{table}

\subsubsection{Case study}% using NCBI annotations}

Let us now consider the 99 chloroplastic genomes introduced earlier.
We will use in this case study either the coding sequences 
downloaded from NCBI website or the sequences predicted by DOGMA. 
DOGMA, which stands for \textit{Dual  Organellar GenoMe Annotator}, has
already been evoked in this article. This is a
 tool  developed in 2004 at  University of Texas    for annotating plant
chloroplast and  animal mitochondrial genomes.  This tool  
translates  a  genome  in all  six  reading frames  and then
queries  its own     amino     acid     sequence     database     using
Blast  (blastx~\cite{altschul1990basic})  with  various ad hoc
parameters. The choice of DOGMA is natural, as this annotation tool is
reputed and specific to chloroplasts. 

Each genome is thus constituted
by a list of coding sequences. In this illustration study,
we have evaluated the similarity between two sequences by
using a global alignment. More precisely, the measure $d$ 
introduced above is the similarity score provided after
a Needleman-Wunch global alignment, as obtained by running
the \emph{needle} command from the \emph{emboss} package
released by EMBL~\cite{Rice2000}. Parameters of the \emph{needle}
command are the default ones: 10.0 for gap open penalty and 0.5
for  gap extension.

The number of genes in the core genome and in the pan genome, 
according to this first method using data and measure described above
have been computed using the supercomputer facilities of the M\'esocentre
de calcul de Franche-Comt\'e. Obtained results are
represented in  Table~\ref{Fig:sim:core:pan} with respect to various
threshold values on Needleman-Wunsch similarity scores. 
Remark that when the threshold is large, 
we obtain more connected components, but with small sizes (a large number
of genes, with a few numbers of alleles for each of them). In other words,
%the number 
%of alleles of one gene is naturally small if the threshold is large.
%Similarly, 
when the threshold is large, the 
pan genome is large too. %However due to the construction method of the
%core genome,  this set of genes has few elements in such a  situation.  
No matter the chosen annotation tool, this first approach  suffers from producing
too small core genomes,  for any chosen similarity threshold, compared
to   what  is   usually   expected  by   biologists.
%  regarding   these
%chloroplasts and their commonly accepted evolutionary scenario. 
For NCBI, it is certainly due to a wrong determination of start and stop
codons in some annotated genomes, due to a large variety of annotation 
tools used during genomes submission on the NCBI server, some of them
being old or deficient: such truncated 
genes will not produce a large similarity score with their orthologous genes
present in other genomes. The case of DOGMA is more 
difficult to explain as, according to our experiments and to the
state of the art, this gene prediction tool produces normally good
results in average. The best explanation of such an under-performance
is that a few genomes are very specific and far from the remainder ones, in terms
of gene contents, which leads to a small number of genes in the global core
genome. However this first approach cannot help us to determine
which genomes must be removed from our set of data. To do so, we
need to introduce a second approach based on gene names: from the
problematic gene names, we will be able to trace back to the
problematic genomes.

% 
% This first illustration emphasizes the importance to deal with
% well-predicted coding sequences: gene prediction must be
% achieved with the same tool on each genome, and this tool must
% be well chosen among the state of the art ones. As we will use
% good annotation tools, a natural idea is to take advantage to
% the whole produced annotations (not only the predicted coding
% sequence, but its name and location too for instance). The implementation
% of this idea is the principle of the approaches for finding core
% and pan genomes detailed below.
% % We are  then left with the following  questions: how can
% we improve the confidence put in the produced core? That is, how to obtain
% such a core genome without considering coding sequences provided by the NCBI?
