+\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?
