Dynamic Programming & Smith-Waterman algorithm - …...The algorithm - an example complexity...

Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

References

pgflastimage

Dynamic Programming & Smith-Waterman

algorithmSeminar: Classical Papers in Bioinformatics

Yvonne Herrmann

May 3rd, 2010

Yvonne Herrmann Dynamic Programming & Smith-Waterman algorithm

Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

References

pgflastimage Overview

1 Dynamic Programming

2 Sequence comparison

3 Smith-Waterman algorithm


Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

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Smith-Waterman

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pgflastimageDynamic ProgrammingIntroduction

Definition

Dynamic Programming is a method of solving problemsby breaking them down into simpler steps

problem need to contain overlapping subproblems andshould have an optimal substructure

method is used for mathematical optimization andcomputer programming


Dynamic

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Smith-Waterman

algorith

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Dynamic

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Divide&Conquer

Divide&Conquer is used when all subproblems areindependent.

calculate partitions and combine the solutions to solvethe entire problem.

vs.

Dynamic Programming

Dynamic Programming is used when subproblems aredependent

there are no partitions, since the subproblems overlap.


Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

References


Definition

Dynamic Programming is a method of solving problemsby breaking them down into simpler steps

problem need to contain overlapping subproblems andshould have an optimal substructure

method is used for mathematical optimization andcomputer programming


Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

References

pgflastimageDynamic ProgrammingThe Principle of Optimality

The Principle of Optimality

”An optimal policy has the property that

whatever the initial state and initial decision are,

the remaining decisions must constitute an optimal

policy with regard to the state resulting from the

first decision.” a

aBellman, R.E. 1957. Dynamic Programming, Chap.III.3., Princeton

University Press


Dynamic

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pgflastimageDynamic ProgrammingThe Principle of Optimality - Example

shortest path

shortest way by car to get from Bielefeld to Cologne

have to pass through Hamm(Westf) and Dortmund

shortest route from Hamm(Westf) to Cologne, needs togo through Dortmund

⇛ The second problem is inside the first one.


Dynamic

Programming &

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Dynamic

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pgflastimageDynamic ProgrammingAlgorithms

Dynamic Programming is used by...

Floyd-Warshall algorithm (shortest path algorithm)

Needleman-Wunsch algorithm

Smith-Waterman algorithm

Bellman-Ford algorithm, etc.


Dynamic

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Dynamic

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2 Sequence comparison



Dynamic

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pgflastimageSequence comparisonIntentions

Why compare sequences?

Quantify the similarity or dissimilarity between two or moresequences and find out where they are similar or different.


Dynamic

Programming &

Smith-Waterman

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Overview

Dynamic

Programming

Sequence

comparison

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Alignments

Smith-Waterman

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pgflastimageSequence comparisonWhy compare sequences?

The analysis of this can help to determint:

if genes from two different organism are related

if similar nucleotide sequences lead to similar proteinstructures

which species is likely more related to another one

what kind of development happened in the evolution?(Mutations, insertions and deletions of gens or morespecific in the aminoacid sequence itself)


Dynamic

Programming &

Smith-Waterman

algorith

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Dynamic

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Sequence

comparison

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pgflastimageAlignmentsHow to compare sequences?

sequence alignment

Method of arranging the sequences of DNA, RNA oraminoacids of proteins to find regions of similarity whichmight be a consequence of functional, structural orevolutionary relationships between the sequences.


Dynamic

Programming &

Smith-Waterman

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Dynamic

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Sequence

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Smith-Waterman

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Conditions a alignment has to fulfill

all symbols have to be in the same order they appear inthe given sequences

a symbol can be aligned with a blank (’-’)

two blanks cannot be aligned


Dynamic

Programming &

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Dynamic

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Sequence

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Example

sequence s and t are given:s: A C T G A A C T Gt: A T G G A C C T Ga possible alignment is:A C T - G A - A C T GA - T G G A C - C T G


Dynamic

Programming &

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Dynamic

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pgflastimageLocal vs. global alignmentWhat’s the difference?

global alignment

The sequences must be aligned from start to end.

local alignment

Local alignments identify regions of high similarity withinsequences.


Dynamic

Programming &

Smith-Waterman

algorith

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Dynamic

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comparison

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pgflastimageLocal vs. global alignmentWhat’s the difference?

global alignment

The sequences must be aligned from start to end.

local alignment

Local alignments identify regions of high similarity withinsequences which are often widely different overall.

Smith-Waterman algorithm calculates the optimal localalignment!


Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Intentions

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Smith-Waterman

algorithm

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2 Sequence comparisonIntentionsAlignments



Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

History

Goal of the algorithm

The algorithm

The algorithm - anexample

complexity analysis

Disadvantages

Applications

References

pgflastimageSmith-Waterman algorithmA little history

algorithm was proposed in 1981 by Temple F. Smithand Michael S. Waterman

algorithm uses dynamic programming and is a variationof the Needleman-Wunsch algorithm


Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

History


The algorithm


complexity analysis

Disadvantages

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pgflastimageSmith-Waterman algorithmWhat’s the goal of this algorithm?

Smith-Waterman algorithm calculates the localalignment of two given sequences

used to identify similar DNA, RNA and proteinsegments

alignments of any possible length starting and ending atany position in the two sequences are compared toobtain the optimal local alignment


Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

History


The algorithm


complexity analysis

Disadvantages

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pgflastimageSmith-Waterman algorithmWhat’s the goal of this algorithm?

it guarantees to find the optimal local alignmentconsidering the given scoring system.

scoring system includes a substitution matrix and agap-scoring scheme.

scores consider matches, mismatches, substitutions orinsertions/deletions

main difference to the Needleman-Wunsch algorithm is:negative scores are set to zero


Dynamic

Programming &

Smith-Waterman

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comparison

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The algorithm


complexity analysis

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pgflastimageSmith-Waterman algorithmThe algorithm

Starting conditions

two molecular sequences A=a1a2...an and B=b1b2...bm.

scoring theme

course of events

first: setting up matrix HHk0 = H0l = 0 (for 0 ≤ k ≤ n and 0 ≤ l ≤ m)

next: calculate score for each cell

last: backtrace the path to obtain optimal alignment


Dynamic

Programming &

Smith-Waterman

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The algorithm


complexity analysis

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pgflastimageSmith-Waterman algorithmThe algorithm

How to calculate the score for each cell?

Individual pair-wise comparisons between the characters as:

Hi j = max

Hi−1,j−1 +s(ai ,bj),

maxk{ Hi−k ,j - Wk},

maxl{ Hi ,j−l - Wl},

0.

k = deletion of length kl = deletion of length lWk and Wl is the gap − cost function


Dynamic

Programming &

Smith-Waterman

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Dynamic

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comparison

Smith-Waterman

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The algorithm


complexity analysis

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pgflastimageSmith-Waterman algorithmDefintion

backtracing

During the filling of matrix H you have to use backpointersto reconstruct from which cell you came.Then when you found the highest score in the matrix H youcan backtrace the path and obtain the optimal alignment.

caption of backpointers:

← Deletion

↑ Insertion

տ Substitution


Dynamic

Programming &

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comparison

Smith-Waterman

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The algorithm


complexity analysis

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pgflastimageSmith-Waterman algorithmSmith-Waterman - Example

Example

sequence A and B are given:

A: A G C T T and B: A G A C T

scoring theme:match = +1

mismatch = −1

3

Wk = 1 +1

3∗ k


Dynamic

Programming &

Smith-Waterman

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Dynamic

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The algorithm


complexity analysis

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Example

sequence A and B are given:A: A G C T T and B: A G A C T

Figure: Filled matrix H


Dynamic

Programming &

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The algorithm


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Example

optimal local alignment:A G A CTA G - CT

Figure: Filled matrix H and backtracing path


Dynamic

Programming &

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Dynamic

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comparison

Smith-Waterman

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The algorithm


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pgflastimageSmith-Waterman algorithmSmith-Waterman - Example 2

best optimal local alignment can be anywhere in thesequences→ Find highest score in matrix H as backtracing start point

Figure: Example from the original paper


Dynamic

Programming &

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Dynamic

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Sequence

comparison

Smith-Waterman

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The algorithm


complexity analysis

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pgflastimageSmith-Waterman algorithmSmith-Waterman - Example 2

optimal local alignment:G C A U U GG C - U C G

Figure: Example from the original paper


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The algorithm


complexity analysis

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pgflastimageSmith-Waterman algorithmComplexity of the algorithm

Complexity of the algorithm

running-time: O(nm)

algorithm is exact, but very time consuming.FASTA is an heuristic approximation and mostly usedtoday.

need of space: O(nm)


Dynamic

Programming &

Smith-Waterman

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Dynamic

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Sequence

comparison

Smith-Waterman

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The algorithm


complexity analysis

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pgflastimageSmith-Waterman algorithmDisadvantages

time and space cost are very high

finds the alignment with maximal score, but not withmaximal percent of matches

algorithm makes ’mosaics’ of well-conserved fragmentswith connections by poorly-conserved fragmentssolution: length-normalized local alignment

→ obtains the region with maximum degree of similarity


Dynamic

Programming &

Smith-Waterman

algorith

Overview

Dynamic

Programming

Sequence

comparison

Smith-Waterman

algorithm

History


The algorithm


complexity analysis

Disadvantages

Applications

References

pgflastimageSmith-Waterman algorithmApplications

JAligner

SSEARCH (in FASTA package)

Live-Demo of the Smith-Waterman algorithm:http://baba.sourceforge.net/


Dynamic

Programming &

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pgflastimage Bibliography

[1] Alison Cawsey, Dynamic Programming,http://www.macs.hw.ac.uk/alison/ds98/node122.html,1998

[2] Temple F. Smith and Michael S. Waterman,Identification of Common Molecular Subsequences,J. Mol. Biol., 147(1):195-197, March 1981

[3] Script: Sequence Analysis I+II, Lecture notes Faculty ofTechnology, Bielefeld University,Winter 2008/09 and Summer 2009

[4] Norman Casagrande, Basic-Algorithms of BioinformaticsApplet,http://baba.sourceforge.net/, 2003

[5] University of Southern California, University Professor,http://www.cmb.usc.edu/people/msw/Waterman.html,2005


Dynamic

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Dynamic Programming & Smith-Waterman algorithm - …...The algorithm - an example complexity...

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