Algorithmic Approaches for Biological Data, …Algorithmic Approaches for Biological Data, Lecture...

Algorithmic Approaches for Biological Data, Lecture #20

Katherine St. John

City University of New YorkAmerican Museum of Natural History

20 April 2016

Outline

Aligning with Gaps and Substitution Matrices

Global versus Local Alignment

Searching Graphs: Breadth First & Depth First

K. St. John (CUNY & AMNH) Algorithms #20 20 April 2016 2 / 16

Outline

Pairwise Sequence Alignment

A G A G

0 -1 -2 -3 -4A -1 1G -2G -3

Pictorially:

As equations:

where:

A G A G

0 -1 -2 -3 -4A -1 1G -2G -3

Pictorially:

As equations:

where:

A G A G

0 -1 -2 -3 -4A -1 1G -2G -3

Pictorially:

As equations:

where:

The basic dynamic programming formatcan be adjusted for different gaps andsubstitutions models.

δ: the gap penalty

σ: scores matches/mismatches.

where:

δ: the gap penalty

where:

δ: the gap penalty

Gaps Are Treated Equally

A G A G

0 -1 -2 -3 -4

A -1 1

Commonly use affine gap penalty

function:

I h: penalty associated withopening a gap

I g : (smaller) penalty associatedwith extending the gap.

To implement this efficiently, use 2additional matrices that keeps track ofthe gaps (one for each sequence).

A G A G

0 -1 -2 -3 -4

A -1 1

function:

A G A G

0 -1 -2 -3 -4

A -1 1

function:

Affine Gap

Burr Settles, U Wisconsin, 2008

Using Substitution Matrices

A C G T

A 1 -1 -1 -1

C -1 1 -1 -1

G -1 -1 1 -1

T -1 -1 -1 1

Can view σ(i , j) as a substitution matrix.

Substitution matrices commonly used for proteinseqeunces.

PAM = Percent Accepted Mutation

I Dayhoff et al., 1978I Used for closely related protein sequencesI Based on global alignment

BLOSUM = Blocks Substitution Matrix

I Henikoff & Henikoff, 1992I Used for more divergent sequencesI Based on local alignment

A C G T

A 1 -1 -1 -1

C -1 1 -1 -1

G -1 -1 1 -1

T -1 -1 -1 1

A C G T

A 1 -1 -1 -1

C -1 1 -1 -1

G -1 -1 1 -1

T -1 -1 -1 1

A C G T

A 1 -1 -1 -1

C -1 1 -1 -1

G -1 -1 1 -1

T -1 -1 -1 1

A C G T

A 1 -1 -1 -1

C -1 1 -1 -1

G -1 -1 1 -1

T -1 -1 -1 1

A C G T

A 1 -1 -1 -1

C -1 1 -1 -1

G -1 -1 1 -1

T -1 -1 -1 1

Paul Reiners, IBM, 2008

Global: Needleman & Wunsch, 1970.

Local: Smith & Waterman, 1981.

Instead of looking for the global bestscore, look for the best score forsubsequences of the initial sequences.

Examples:

I finding motifs (conservedpatterns) across sequences,

I comparing sequences againstlonger sequences (e.g. blastsearch).

Global: Needleman & Wunsch, 1970.

Local: Smith & Waterman, 1981.

Instead of looking for the global bestscore, look for the best score forsubsequences of the initial sequences.

Examples:

I finding motifs (conservedpatterns) across sequences,

I comparing sequences againstlonger sequences (e.g. blastsearch).

Smith-Waterman Algorithm

The equation is slightly different:

s(i , j) = max

σ(i , j) + s(i − 1, j − 1)−δ + s(i , j − 1)−δ + s(i − 1, j)0

Initialize: first row and first column set to 0’s

Traceback: find maximum value of s(i , j) anywhere inthe the matrix, stop when we get to a cell with 0.

The equation is slightly different:

s(i , j) = max

σ(i , j) + s(i − 1, j − 1)−δ + s(i , j − 1)−δ + s(i − 1, j)0

Initialize: first row and first column set to 0’s

Traceback: find maximum value of s(i , j) anywhere inthe the matrix, stop when we get to a cell with 0.

In Pairs: Local Alignment

A A G A

Use σ from Monday, but δ = 2.

What are the best local alignments?

A A G A

0 0 0 0 0

A A G A

0 0 0 0 0

T 0 0 0 0 0

A 0 1 1 0 1

A 0 1 2 0 1

G 0 0 0 3 1

In Pairs: Searching Graphs

Bastert et al., 2002

Develop a strategy tovisit every node of thegraph(i.e. what datastructures areneeded?)

The bookkeeping isimportant.

Two common strategies:

I Breadth First Search (BFS): visit all theneighbors, then visit all the neighbors’neighbors, etc.

I Depth First Search (DFS): for eachneighbor, visit its’ neighbors, andcontinue as far down as possible.

Bookkeeping is important:

I Keep a “To Do” list (priority queue) ofnodes still to visit.

I Mark nodes as you visit them, so, youknow not to visit again.

Two common strategies:

I Breadth First Search (BFS): visit all theneighbors, then visit all the neighbors’neighbors, etc.

I Depth First Search (DFS): for eachneighbor, visit its’ neighbors, andcontinue as far down as possible.

Bookkeeping is important:

I Keep a “To Do” list (priority queue) ofnodes still to visit.

I Mark nodes as you visit them, so, youknow not to visit again.

Dynamic Programming: will do local &global alignments in lab today.

Algorithmic Approaches for Biological Data, …Algorithmic Approaches for Biological Data, Lecture...

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