Probalign

Probalign is a sequence alignment tool that calculates a maximum expected accuracy alignment using partition function posterior probabilities. Base pair probabilities are estimated using an estimate similar to Boltzmann distribution. The partition function is calculated using a dynamic programming approach.

Algorithm
The following describes the algorithm used by probalign to determine the base pair probabilities.

Alignment score
To score an alignment of two sequences two things are needed: The score $$S(a)$$ of an alignment a is defined as:
 * a similarity function $$\sigma(x,y)$$ (e.g. PAM, BLOSUM,...)
 * affine gap penalty: $$ g(k) = \alpha + \beta k$$

$$ S(a) = \sum_{x_i-y_j \in a} \sigma(x_i,y_j) + \text{gap cost}$$

Now the boltzmann weighted score of an alignment a is:

$$ e^{\frac{S(a)}{T}} = e^{\frac{\sum_{x_i-y_j \in a} \sigma(x_i,y_j) + \text{gap cost}}{T}} = \left( \prod_{x_i - y_i \in a} e^{\frac{\sigma(x_i,y_j)}{T}} \right) \cdot e^{\frac{gapcost}{T}}$$

Where $$T$$ is a scaling factor.

The probability of an alignment assuming boltzmann distribution is given by

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

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

Dynamic programming
Let $$Z_{i,j}$$ denote the partition function of the prefixes $$x_0,x_1,...,x_i$$ and $$y_0,y_1,...,y_j$$. Three different cases are considered: Then we have: $$Z_{i,j} = Z^{M}_{i,j} + Z^{D}_{i,j} + Z^{I}_{i,j}$$
 * 1) $$Z^{M}_{i,j}:$$ the partition function of all alignments of the two prefixes that end in a match.
 * 2) $$Z^{I}_{i,j}:$$ the partition function of all alignments of the two prefixes that end in an insertion $$(-,y_j)$$.
 * 3) $$Z^{D}_{i,j}:$$ the partition function of all alignments of the two prefixes that end in a deletion $$(x_i,-)$$.

Initialization
The matrixes are initialized as follows:
 * $$Z^{M}_{0,j} = Z^{M}_{i,0} = 0$$
 * $$Z^{M}_{0,0} = 1$$
 * $$Z^{D}_{0,j} = 0$$
 * $$Z^{I}_{i,0} = 0$$

Recursion
The partition function for the alignments of two sequences $$x$$ and $$y$$ is given by $$Z_{|x|,|y|}$$, which can be recursively computed:
 * $$Z^{M}_{i,j} = Z_{i-1,j-1} \cdot e^{\frac{\sigma(x_i,y_j)}{T}}$$
 * $$Z^{D}_{i,j} = Z^{D}_{i-1,j} \cdot e^{\frac{\beta}{T}} + Z^{M}_{i-1,j} \cdot e^{\frac{g(1)}{T}} + Z^{I}_{i-1,j} \cdot e^{\frac{g(1)}{T}}$$
 * $$Z^{I}_{i,j}$$ analogously

Base pair probability
Finally the probability that positions $$x_i$$ and $$y_j$$ form a base pair is given by:

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

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