# McGill-CSB/RNApyro

less typos

 @@ -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=ki\\ \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