-\noindent$\sim_{d,T}$ is obviously an equivalence relation. When $d=1-\Delta$, where $\Delta$ is the similarity scoring function embedded into the emboss package (Needleman-Wunch released by EMBL), we will simply denote $\sim_{d,0.1}$ by $\sim$. The method starts by building an undirected graph based on
-the similarity rates $r_{ij}$ between sequences $g_{i}$ and $g_{j}$ (\emph{i.e.}, $r_{ij}=\Delta(g_{i},g_{j})$).
-In this latter, nodes are constituted by all the coding sequences of the set of genomes under consideration, and there is an edge between $g_{i}$ and $g_{j}$ if the
-similarity rate $r_{ij}$ is
-greater than the given similarity threshold. The Connected Components
-(CC) of the ``similarity'' graph are thus computed.
-This produces an equivalence
-relation between sequences in the same CC based on Definition~\ref{def1}.
-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 $\{g_{1}^G,...,g_{m_G}^G\}$ 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 $\sim$. In other words, $G$ is mapped into $\{\pi(g_{1}^G),...,\pi(g_{m_G}^G)\}$.
-Remark that a projected genome has no duplicated gene, as it is a set. The core genome (resp. the pan genome) of $G_{1}$ and $G_{2}$ is defined thus as the intersection (resp. as the union) of these 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. However such approach suffers from producing too small core genomes,
-for any chosen similarity threshold, compared to what is usually waited 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?
\ No newline at end of file
+\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 (Needleman-Wunch released by EMBL), we will simply denote $\sim_{d,0.1}$ by $\sim$.
+
+The method begins by building an undirected graph based on similarity
+rates $r_{ij}$ between DNA~sequences $g_{i}$ and $g_{j}$ (\emph{i.e.},
+$r_{ij}=\Delta\left(g_{i},g_{j}\right)$). In this latter graph, nodes
+are constituted by all the coding sequences of the set of genomes
+under consideration, and there is an edge between $g_{i}$ and $g_{j}$
+if the similarity rate $r_{ij}$ is greater than a given similarity
+threshold. The Connected Components (CC) of the ``similarity'' graph
+are thus computed.
+
+This process also results in an equivalence relation between sequences
+in the same CC based on Definition~\ref{def1}. 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 $\sim$. 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. However such approach suffers from producing
+too small core genomes, 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?
