Sequence Analysis

CSC 487/687 Introduction to computing for Bioinformatics

Aligning Sequences

Sequences Representing proteins or nucleic acid (DNA/RNA)

molecules Order of amino acids (for proteins – nucleotides for

DNA/RNA) along one chain Sequence alignment

The identification of residue-residue correspondences Any assignment of correspondences that preserves the

order of residues within the sequences

Evolutionary Basis of Sequence Alignment Identity: Quantity that describes how much

two sequences are alike in the strictest terms. Similarity: Quantity that relates how much

two amino acid sequences are alike. Homology: a conclusion drawn from data

suggesting that two genes share a common

evolutionary history.

Evolutionary Basis of Sequence Alignment Homologous sequences

Related by evolution (common ancestors)

Alignment of homologous sequences Identifying relationship between the sequence

elements Match up characters coming from same

characters in ancestor

Alignment and Evolution Assume we know evolutionary history relating q and d:

The true alignment can be found using h as a template:h : GLVS Tq’: GLISVTd’: GIV--T

Alignment Evolution Given an alignment, several different evolutionary histories

may be (equally) plausible Example:

Alignment:q’: GLISVTd’: G-I-VT

One possible history: H*:GLIVT /\ ->S / \ L-> / \q:GLISVT d:GIVT

Global and Local Alignment Global

Assuming that the complete sequences are the results of evolution from the same ancestor sequence

Local Align segments of the sequences so that the segments are evolutionarily related

AncestorS1

S2

AncestorS1

S2

Pairwise sequence alignments Vs Multiple sequence alignments Pairwise sequence alignment: two sequence

Multiple sequence alignments: a mutual alignment of more than two sequences

The dotplot

The dotplot

Captures not only the overall similarity of two sequences, but also the complete set and relative quality of different possible alignments Diagonal ― Horizontal ― a gap is introduced in the sequence

indexing the rows Vertical ― a gap is introduced in the sequence

indexing the columns

Dotplots and alignments

A path through the dotplot is as an edit script; Each move performs an operation ― a

substitution, an insertion or a deletion. When the end of the path is reached, the

effect will change one sequence into the other.

Several different sequences of edit operations may convert one string to the other in the same number of steps.

Dotplots and alignments

Although a sequence of edit operations derived from an optimal alignment may correspond to an actual evolutionary pathway

Impossible to prove that it does. The larger the edit distance, the larger the

number of reasonable evolutionary pathways between two sequences.

Dotplots and alignments

The dotplots between pairs of proteins with increasingly more distant relationships.

The dotplot comparisons of the sulphydryl proteinase papain from papaya, with four homologues ― the close relative, kiwi fruit actinidin, the more distant relatives, human procathepsin L, human cathepsin B, and staphyloccus anueus.

Example

Example

Example

Example

Measures of sequence similarity Hamming distance ― the number of positions

with mismatching characters.

Edit distance ― the minimum number of “edit operations” required to change one string into the other.

What is an Alignment? A global alignment of two sequences A and B

contains all characters of A and B in the same order

one symbol from A can be aligned with one symbol from B a symbol can be aligned with a blank, written as ‘-’ two blanks cannot be aligned Every symbol from A and from B must be aligned

Example:A:INVEST, B:INTERESTIN--VEST INV--EST IN-V--ESTINTEREST INTEREST IN-TEREST

Computing Alignments

There exist a large number of alignments for a pair of sequences

In order to use a computer to do the alignment process in a meaningful way, we need Scoring scheme – mathematical way to

calculate goodness of candidate alignments Search method – algorithm able to identify

high scoring alignments

Choosing Scoring Scheme

Scoring scheme should be Simple – to allow for

efficient calculation and search for best alignment

Biologically meaningful (give score to biologically good alignments)

Simple Scoring Scheme Assign score to each column in the alignment Columns are of the following sorts:

Alignment score: sum of score over all columns

R: matrix giving score for all possible character pairs (e.g., all pairs of amino acid symbols)

Alignment Score – Example R identity matrix – identical characters

score1, unequal 0, g=1

ALIGN1:

V - E I T G E I S TP R E - T E R I - T0 -1 1 -1 1 0 0 1 -1 1 Score: 1

ALIGN2:

V E I T G E I S TP R E T - E R I T0 0 0 1 -1 1 0 0 1 Score: 2

Finding the Minimum Scoring Alignment

Large number of possible alignments – cannot generate all and score them to find the best

Task – alignA=a1a2...am and

B=b1b2...bn

Independence Between Sub-alignments Observations:

The score of the alignment up to and including character i from A and character j from B is independent of how the rest of the sequences are aligned

The best solution to (i,j) can be “locked”, its score recorded in Di,j

Dm,n is the score of the best global alignment

Amenable to dynamic Programming

Dynamic programming algorithm Individual edit operations include:

Substitution of bj for ai ― represented (ai, bj)

Deletion of ai from sequence A― represented (ai,)

Deletion of bj from sequence B― represented (,bj)

Dynamic programming algorithm A cost function d is defined on edit operations

d(ai, bj)=cost of a mutation in an alignment in which position i of sequence A corresponds to position j of sequence B

d(ai,) or d( bj) = cost of a deletion or insertion

The minimum weighted distance between sequences A and B as D(A,B)=min (d(x,y))

Three Alternative Alignment Ends The alignment between a1a2...ai and b1b2...bi

ends in one of three ways:

ai

-

a1..i-1

b1..j

ai

bj

a1..i-1

b1..j-1

-bj

a1..i

b1..j-1

To calculate Di,j we pick the one thatgives the lowest cost

Recurrence Relation

ai

-

a1..i-1

b1..j

ai

bj

a1..i-1

b1..j-1

-bj

a1..i

b1..j-1

min, jiD 1,1 jiD

1, jiD

jiD ,1jiD ,1

1,1 jiD

1, jiD

Assume that Di-1,j, Di-1,j-1, Di,j-1 have been calculated already

d(ai,)

d(ai,bj)

d(,bj)

Basis of Recursion

Align empty string to string of length i (resp. j) – can be done by aligning to i (resp. j) blanks:

i

kki

j

kkj

adD

bdD

00,

0,0

),(

),(

Calculating Score of Best Alignment Using Matrix

cost of bestalignment

H matrix

Time Complexity

Sequences of lengths n and m

Two sequences of length l

)(nmO

)( 2lO