Probalign

To score an alignment of two sequences two things are needed:

• a similarity function ${\displaystyle \sigma (x,y)}$ (e.g. PAM, BLOSUM,...)
• affine gap penalty: ${\displaystyle g(k)=\alpha +\beta k}$

The score ${\displaystyle S(a)}$ of an alignment a is defined as:

${\displaystyle S(a)=\sum _{x_{i}-y_{j}\in a}\sigma (x_{i},y_{j})+{\text{gap cost}}}$

Now the boltzmann weighted score of an alignment a is:

${\displaystyle e^{\frac {S(a)}{T}}=e^{\frac {\sum _{x_{i}-y_{j}\in a}\sigma (x_{i},y_{j})+{\text{gap cost}}}{T}}=\left(\prod _{x_{i}-y_{i}\in a}e^{\frac {\sum _{x_{i}-y_{j}\in a}\sigma (x_{i},y_{j})}{T}}\right)\cdot e^{\frac {gapcost}{T}}}$

Where ${\displaystyle T}$ is a scaling factor.

The probability of an alignment assuming boltzmann distribution is given by

${\displaystyle Pr[a|x,y]={\frac {e^{\frac {S(a)}{T}}}{Z}}}$

Where ${\displaystyle Z}$ is the partition function, i.e. the sum of the boltzmann weights of all alignments.

Let ${\displaystyle Z_{i,j}}$ denote the partition function of the prefixes ${\displaystyle x_{0},x_{1},...,x_{i}}$ and ${\displaystyle y_{0},y_{1},...,y_{j}}$. Three different cases are considered:

1. ${\displaystyle Z_{i,j}^{M}:}$ the partition function of all alignments of the two prefixes that end in a match.
2. ${\displaystyle Z_{i,j}^{I}:}$ the partition function of all alignments of the two prefixes that end in an insertion ${\displaystyle (-,y_{j})}$.
3. ${\displaystyle Z_{i,j}^{D}:}$ the partition function of all alignments of the two prefixes that end in a deletion ${\displaystyle (x_{i},-)}$.

Then we have: ${\displaystyle Z_{i,j}=Z_{i,j}^{M}+Z_{i,j}^{D}+Z_{i,j}^{I}}$

The matrixes are initialized as follows:

• ${\displaystyle Z_{0,j}^{M}=Z_{i,0}^{M}=0}$
• ${\displaystyle Z_{0,0}^{M}=1}$
• ${\displaystyle Z_{0,j}^{D}=0}$
• ${\displaystyle Z_{i,0}^{I}=0}$

The partition function for the alignments of two sequences ${\displaystyle x}$ and ${\displaystyle y}$ is given by ${\displaystyle Z_{|x|,|y|}}$, which can be recursively computed:

• ${\displaystyle Z_{i,j}^{M}=Z_{i-1,j-1}\cdot e^{\frac {\sigma (x_{i},y_{j})}{T}}}$
• ${\displaystyle Z_{i,j}^{D}=Z_{i-1,j}^{D}\cdot e^{\frac {\beta }{T}}+Z_{i-1,j}^{M}\cdot e^{\frac {g(1)}{T}}+Z_{i-1,j}^{I}\cdot e^{\frac {g(1)}{T}}}$
• ${\displaystyle Z_{i,j}^{I}}$ analogously

Finally the probability that positions ${\displaystyle x_{i}}$ and ${\displaystyle y_{j}}$ form a base pair is given by:

${\displaystyle P(x_{i}-y_{j}|x,y)={\frac {Z_{i-1,j-1}\cdot e^{\frac {\sigma (x_{i},y_{j})}{T}}\cdot Z'_{i',j'}}{Z_{|x|,|y|}}}}$

${\displaystyle Z',i',j'}$ are the respective values for the recalculated ${\displaystyle Z}$ with inversed base pair strings.