Lévy–Prokhorov metric: Difference between revisions

Revision as of 20:22, 6 May 2013

ProbCons is an open source probabilistic consistency-based multiple alignment of amino acid sequences. It is an efficient protein multiple sequence alignment program, which has demonstrated a statistically significant improvement in accuracy compared to several leading alignment tools.^[1]

Algorithm

The following describes the basic outline of the ProbCons algorithm. ^[2]

Step 1: Reliability of an alignment edge

For every pair of sequences compute the probability that letters $x_{i}$ and $y_{i}$ are paired in $a^{*}$ an alignment that is generated by the model.

$\begin{aligned} P (x_{i} \sim y_{i} | x, y) & \overset{d e f}{=} P r [x_{i} \sim y_{i} in some a | x, y] \\ = \sum_{alignment a with x_{i} - y_{i}} P r [a | x, y] \\ = \sum_{alignment a} 1 {x_{i} - y_{i} \in a} P r [a | x, y] \end{aligned}$

(Where $1 {x_{i} \sim y_{i} \in a}$ is equal to 1 if $x_{i}$ and $y_{i}$ are in the alignment and 0 otherwise.)

Step 2: Maximum expected accuracy

The accuracy of an alignment $a^{*}$ with respect to another alignment $a$ is defined as the number of common aligned pairs divided by the length of the shorter sequence.

Calculate expected accuracy of each sequence:

$\begin{aligned} E_{P r [a | x, y]} (a c c (a^{*}, a)) & = \sum_{a} P r [a | x, y] a c c (a^{*}, a) \\ = \frac{1}{m i n (| x |, | y |)} \cdot \sum_{a} 1 {x_{i} \sim y_{i} \in a} P r [a | x, y] \\ = \frac{1}{m i n (| x |, | y |)} \cdot \sum_{x_{i} - y_{i}} P (x_{i} \sim y_{j} | x, y) \end{aligned}$

This yields a maximum expected accuracy (MEA) alignment:

$E (x, y) = \arg \max_{a^{*}} E_{P r [a | x, y]} (a c c (a^{*}, a))$

Step 3: Probabilistic Consistency Transformation

All pairs of sequences x,y from the set of all sequences $𝒮$ are now re-estimated using all intermediate sequences z:

$P^{'} (x_{i} - y_{i} | x, y) = \frac{1}{| 𝒮 |} \sum_{z} \sum_{1 \leq k \leq | z |} P (x_{i} \sim z_{i} | x, z) \cdot P (z_{i} \sim y_{i} | z, y)$

This step can be iterated.

Step 4: Computation of guide tree

Construct a guide tree by hierarchical clustering using MEA score as sequence similarity score. Cluster similarity is defined using weighted average over pairwise sequence similarity.

Step 5: Compute MSA

Finally compute the MSA using progressive alignment or iterative alignment.

References

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External links

Template:Official website

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↑ Lecture "Bioinformatics II" at University of Freiburg [1]

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[2] Lecture "Bioinformatics II" at University of Freiburg [1]

[1]

[2]

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+'''ProbCons''' is an open source probabilistic consistency-based multiple alignment of [[amino acid]] sequences. It is an efficient protein [[multiple sequence alignment]] program, which has demonstrated a statistically significant improvement in accuracy compared to several leading alignment tools.<ref>{{cite journal |doi=10.1101/gr.2821705 |author=Do CB, Mahabhashyam MSP, Brudno M, Batzoglou S |year=2005 |title=PROBCONS: Probabilistic Consistency-based Multiple Sequence Alignment |journal=Genome Research |volume=15 |issue=2 |pages=330–340 |pmid=15687296 |pmc=546535}}</ref>
+==Algorithm==
+The following describes the basic outline of the ProbCons algorithm. <ref>Lecture "Bioinformatics II" at University of Freiburg [http://www.bioinf.uni-freiburg.de//Lehre/Courses/2011_WS/V_BioinfoII/slides_probcons.pdf]</ref>
+===Step 1: Reliability of an alignment edge===
+For every pair of sequences compute the probability that letters <math>x_i</math> and <math>y_i</math> are paired in <math>a^*</math> an alignment that is generated by the model.
+<math>\begin{align}
+P(x_i \sim y_i|x,y) & \stackrel{def}{=} Pr[x_i \sim y_i \text{ in some a }|x,y] \\
+& = \sum_{\text{alignment a with }x_i - y_i} Pr[a|x,y]\\
+& = \sum_{\text{alignment a}} \mathbf{1}\{x_i - y_i \in a\} Pr[a|x,y]
+\end{align}</math>
+(Where <math>\mathbf{1}\{x_i \sim y_i \in a\}</math> is equal to 1 if <math>x_i</math> and <math>y_i</math> are in the alignment and 0 otherwise.)
+===Step 2: Maximum expected accuracy===
+The accuracy of an alignment <math>a^*</math> with respect to another alignment <math>a</math> is defined as the number of common aligned pairs divided by the length of the shorter sequence.
+Calculate expected accuracy of each sequence:
+<math>\begin{align}
+E_{Pr[a|x,y]}(acc(a^*,a)) & = \sum_{a}Pr[a|x,y]acc(a^*,a) \\
+& = \frac{1}{min(|x|,|y|)} \cdot \sum_{a}\mathbf{1}\{x_i \sim y_i \in a\} Pr[a|x,y]\\
+& = \frac{1}{min(|x|,|y|)} \cdot \sum_{x_i - y_i} P(x_i \sim y_j|x,y)
+\end{align}</math>
+This yields a maximum expected accuracy (MEA) alignment:
+<math>
+E(x,y) = \arg\max_{a^*} \; E_{Pr[a|x,y]}(acc(a^*,a))
+</math>
+===Step 3: Probabilistic Consistency Transformation===
+All pairs of sequences x,y from the set of all sequences <math>\mathcal{S}</math> are now re-estimated using all intermediate sequences z:
+<math>
+P'(x_i - y_i|x,y) = \frac{1}{|\mathcal{S}|} \sum_{z} \sum_{1 \leq k \leq |z|} P(x_i \sim z_i|x,z) \cdot P(z_i \sim y_i|z,y)
+</math>
+This step can be iterated.
+===Step 4: Computation of guide tree===
+Construct a guide tree by hierarchical clustering using MEA score as sequence similarity score. Cluster similarity is defined using weighted average over pairwise sequence similarity.
+===Step 5: Compute MSA===
+Finally compute the MSA using progressive alignment or iterative alignment.
+== See also ==
+* [[Sequence alignment software]]
+* [[Clustal]]
+* [[MUSCLE (alignment software)|MUSCLE]]
+* [[AMAP]]
+* [[T-Coffee]]
+* [[Probalign]]
+* [[ProbConsRNA]] &mdash; for [[nucleotide]] sequences
+==References==
+{{Reflist}}
+==External links==
+*{{Official website|http://probcons.stanford.edu/}}
+[[Category:Bioinformatics]]
+[[Category:Computational phylogenetics]]

Lévy–Prokhorov metric: Difference between revisions

Revision as of 20:22, 6 May 2013

Contents

Algorithm

Step 1: Reliability of an alignment edge

Step 2: Maximum expected accuracy

Step 3: Probabilistic Consistency Transformation

Step 4: Computation of guide tree

Step 5: Compute MSA

See also

References

External links

Navigation menu

Lévy–Prokhorov metric: Difference between revisions

Revision as of 20:22, 6 May 2013

Algorithm

Step 1: Reliability of an alignment edge

Step 2: Maximum expected accuracy

Step 3: Probabilistic Consistency Transformation

Step 4: Computation of guide tree

Step 5: Compute MSA

See also

References

External links

Navigation menu

Search