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commit 5476053bc7b3307de76b87c70934b4ddb4965ca2 1 parent f30df95
@vreinharz vreinharz authored
Showing with 20 additions and 20 deletions.
  1. +20 −20 Recomb/methods_RECOMB.tex
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40 Recomb/methods_RECOMB.tex
@@ -33,14 +33,14 @@ \subsection{Definitions}
\subsection{Energy Model}
The energy we will be composed of two function, $\text{ES}^{\beta}_{ab\to a'b'}$ and
-$\text{EI}_{(i,j),ab}$. The former is equal to the
+$\text{EI}^{\Omega}_{(i,j),ab}$. The former is equal to the
stacking energy of the base pair with nucleotides $ab$ on top of the base pair with nucleotides
$a'b'$, as set in the NNDB~\cite{Turner2010}. If one of the base pair is not valid (i.e. not in
$\{\text{GU},\text{UG},\text{CG},\text{GC}, \text{AU or UA}\}$, the value is a parameter
$\beta \in [1,\infty]$. This allows
to completely forbid a sequence where a base pair is non valid, when $\beta = \infty$ or only
penalize it.
-$\text{EI}_{(i,j), a'b'}$ is the average of the sum of differences between the isostericity
+$\text{EI}^{\Omega}_{(i,j), a'b'}$ is the average of the sum of differences between the isostericity
of base pairs at positions $(i,j)$ in $\Omega$ and $s_is_j$, and the isostericity of base pairs
at positions $(i,j)$ in $\Omega$ and $ab$. If gives us an indication
if the base pair $ab$ is more isosteric to the set $\Omega$ than the one on the sequence
@@ -51,7 +51,7 @@ \subsection{Energy Model}
\subsection{Inside}
The \emph{Inside} function $\Z{i,j}{m}{a,b}$ is the partition function considering only the
-energy in subsequence $[i,j]$ over mutants of $s$ having exactly $m$ mutations between $[i,j]$ and who nucleotide at position $i-1$ is $a$ (resp. in position $j+1$ it is $b$).
+energy in subsequence $[i,j]$ over mutants of $s$ having exactly $m$ mutations between $[i,j]$ and whose nucleotide at position $i-1$ is $a$ (resp. in position $j+1$ it is $b$).
We define function $\Z{i,j}{m}{a,b}$ as a recurrence, and will use as initial conditions:
\[
@@ -74,13 +74,13 @@ \subsection{Inside}
\Z{i+1,j}{m-\Kron_{a',s_i}}{a',b} & \text{If }S_{i}=-1\\
\displaystyle
\sum_{\substack{a',b'\in \B^2,\\ \Kron_{a'b',s_is_j}\le m}}
- e^{\frac{-(\alpha \text{ES}^\beta_{a b \to a' b'}+(1-\alpha)\text{EI}_{(i,j),a'b'})}{RT}}
+ e^{\frac{-(\alpha \text{ES}^\beta_{a b \to a' b'}+(1-\alpha)\text{EI}^{\Omega}_{(i,j),a'b'})}{RT}}
\Z{i+1,j-1}{m-\Kron_{a'b',s_is_j}}{a',b'}&
\text{Elif }S_i=j \land S_{i-1}=j+1\\
\displaystyle
\sum_{\substack{a',b'\in \B^2,\\ \Kron_{a'b',s_is_k}\le m}}
\sum_{m'=0}^{m-\Kron_{a'b',s_is_k}}
- e^{\frac{-(1-\alpha)\text{EI}_{(i,k),a'b'}}{RT}}
+ e^{\frac{-(1-\alpha)\text{EI}^{\Omega}_{(i,k),a'b'}}{RT}}
\Z{i+1,k-1}{m-\Kron_{a'b',s_is_k}-m'}{a',b'}
\Z{k+1,j}{m'}{b',b} & \text{Elif }S_i=k \land i < k \leq j\\
0 &\text{Else}
@@ -133,20 +133,20 @@ \subsection{Outside}
\text{Elif }S_i=-1 \\
\displaystyle
\sum_{\substack{a'b'\in \B^2,\\ \Kron_{a'b',s_is_j}\le m}}
- e^{\frac{-(\alpha \text{ES}^\beta_{a b \to a' b'}+(1-\alpha)\text{EI}_{(i,j),a'b'})}{RT}}
+ e^{\frac{-(\alpha \text{ES}^\beta_{a b \to a' b'}+(1-\alpha)\text{EI}^{\Omega}_{(i,j),a'b'})}{RT}}
\Y{i-1,j+1}{m- \Kron_{a'b',s_is_j}}{a',b'} &
\text{Elif }S_{i}=j \land S_{i+1}=j-1\\
\displaystyle
\sum_{\substack{a'b'\in \B^2,\\ \Kron_{a'b',s_is_k}\le m}}
\sum_{m'=0}^{m-\Kron_{a'b',s_is_k}}
- e^{\frac{-(1-\alpha)\text{EI}_{(i,k),a'b'}}{RT}}
+ e^{\frac{-(1-\alpha)\text{EI}^{\Omega}_{(i,k),a'b'}}{RT}}
\Y{i-1,k+1}{m- \Kron_{a'b',s_is_k} - m'}{a',b'}
\Z{j,k-1}{m'}{b,b'} &
\text{Elif }S_{i}=k \geq j\\
\displaystyle
\sum_{\substack{a'b'\in \B^2,\\ \Kron_{a'b',s_ks_i}\le m}}
\sum_{m'=0}^{m-\Kron_{a'b',s_ks_i}}
- e^{\frac{-(1-\alpha)\text{EI}_{(k,i),a'b'}}{RT}}
+ e^{\frac{-(1-\alpha)\text{EI}^{\Omega}_{(k,i),a'b'}}{RT}}
\Y{k-1,j}{m- \Kron_{a'b',s_ks_i} - m'}{a',b}
\Z{k+1,i-1}{m'}{a',b'} &
\text{Elif }-1 < S_{i}=k < i\\
@@ -207,33 +207,33 @@ \section{Inside-Outside}
}
&\text{If }S_i = -1\\
\displaystyle
- \frac{
+ \frac{
\displaystyle
\sum_{m=0}^{M}
- \sum_{\substack{b\in Bases\\\Kron_{bx,s_ks_i\leq m}}}
- \sum_{m'=0}^{m-\Kron_{bx,s_ks_i}}
- \Y{k-1,i+1}{m-\Kron_{bx,s_ks_i-m'}}{b,x}
- \Z{k+1,i-1}{m'}{b,x}
+ \sum_{\substack{b\in Bases\\\Kron_{xb,s_is_k\leq m}}}
+ \sum_{m'=0}^{m-\Kron_{xb,s_is_k}}
+ \Y{i-1,k+1}{m-\Kron_{xb,s_is_k-m'}}{x,b}
+ \Z{i+1,k-1}{m'}{x,b}
}{
\displaystyle
\sum_{m=0}^{M}
\Z{0,n-1}{m}{X,X}
}
- &\text{If }S_i=k<i\\
+ &\text{If }S_i=k>i\\
\displaystyle
- \frac{
+ \frac{
\displaystyle
\sum_{m=0}^{M}
- \sum_{\substack{b\in Bases\\\Kron_{xb,s_is_k\leq m}}}
- \sum_{m'=0}^{m-\Kron_{xb,s_is_k}}
- \Y{i-1,k+1}{m-\Kron_{xb,s_is_k-m'}}{x,b}
- \Z{i+1,k-1}{m'}{x,b}
+ \sum_{\substack{b\in Bases\\\Kron_{bx,s_ks_i\leq m}}}
+ \sum_{m'=0}^{m-\Kron_{bx,s_ks_i}}
+ \Y{k-1,i+1}{m-\Kron_{bx,s_ks_i-m'}}{b,x}
+ \Z{k+1,i-1}{m'}{b,x}
}{
\displaystyle
\sum_{m=0}^{M}
\Z{0,n-1}{m}{X,X}
}
- &\text{If }S[i]=k>i
+ &\text{If }S_i=k<i
\end{array}\right.
$$
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