Some further minor corrections

[chloroplast13.git] / classEquiv.tex

diff --git a/classEquiv.tex b/classEquiv.tex
index 00b227d0e30be5eff890d6dc5fe33a4f27ff5357..f3d3ed13e5f19a1161dcc58dc14fbbae6fa70bc5 100644 (file)
--- a/classEquiv.tex
+++ b/classEquiv.tex
@@ -1,7 +1,6 @@
-

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

-preliminary Definition:
+preliminary definition:

 
 \begin{definition}
 \label{def1}
 
 \begin{definition}
 \label{def1}


@@ -13,25 +12,31 @@ all   $x,y\in  A^{\ast}$,   we   will  say   that  $x\sim_{d,T}y$   if
 $d(x,y)\leqslant T$.
 \end{definition}
 
 $d(x,y)\leqslant T$.
 \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 (Needleman-Wunch released by EMBL), we will simply denote $\sim_{d,0.1}$ by $\sim$.
+%\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.

 
 

-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
+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
 (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
+$\pi$ 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.
 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.


@@ -42,8 +47,26 @@ 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
 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
+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
 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