X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chloroplast13.git/blobdiff_plain/f8689d1f199221556dc52e80c5237c45addb0218..1e61bcbdaedf0f72f05c3dd365fc11d3a2071330:/classEquiv.tex

diff --git a/classEquiv.tex b/classEquiv.tex
index 41abda4..829a8b4 100644
--- a/classEquiv.tex
+++ b/classEquiv.tex
@@ -1,28 +1,73 @@
-Identifying  core genes  is important  to understand  evolutionary and
-functional phylogenies. Therefore, in this work we present two methods
-to build a  genes content evolutionary tree. More  precisely, we focus
-on   the    following   questions   considering    a   collection   of
-99~chloroplasts  annotated from  NCBI \cite{Sayers01012011} and  Dogma
-\cite{RDogma} : how can we identify the best core genome and what
-is the evolutionary scenario of these chloroplasts.
-Two methods are considered here. The first one is based on NCBI annotation, it is explained below.
-We start by the following definition.
+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 $d(x,y)\leqslant T$. 
+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. 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 , 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 $g_i$ an,d 
+$g_j$  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 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=0.4]{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?
