Sequence Alignments with Indels

Evolution produces insertions and deletions (indels)– In addition to substitutions

Good example:

MHHNALQRRTVWVNAY MHHNALQRRTVWVNAYMHHALQRRTVWVNAY- MHH-ALQRRTVWVNAYBlosum Score = 2 (end = -6) Score = 79 (gap = -6)

An alignment must have equal length aligned sequences– So, we must add gaps at the start and the ends

Combinatorially difficult problem to find best indel solution

Gap

So far we ignored gaps A gap corresponds to an insertion or a deletion of a

residue A conventional wisdom dictates that the penalty for

a gap must be several times greater than the penalty for a mutation. That is because a gap/extra residue– Interrupts the entire polymer chain– In DNA shifts the reading frame

Gap Penalties

Gaps are penalised– Write wx to indicate the penalty for a gap of length x– For example, each gap scores -6, so wx = -6*x

One common scheme is– Score -12 for opening a gap– And -2 for every subsequent gap– i.e., wx = -12 - 2*(x-1)

Start and end gap penalties often set to zero– But this can leave a doubt

About evolutionary conclusions

Dot Matrix Representations (Dotplots)To help visualise best alignments

Plot where each pair is the same, then draw best line

M N A L S Q L N

N

A

L

M

S

Q

N

H

M N A L S Q L N

N

A

L

M

S

Q

N

H

Getting Alignments from Dotplot Paths

M N A L S Q L N

N

A

L

M

S

Q

N

H

Indicates that M matches with a gap

Indicates that L matches with a gap

Stage 1:– Align middle– Use triangles

To indicate gaps

NAL-SQLN NALMSQ-N Stage 2:

– Sort the ends out

MNAL-SQLN- -NALMSQ-NH

Dotplots for Real Proteins

Need a way to automatically find the best path(s)

Dynamic Programming Approach

BLAST is quick– But not guaranteed to find best alignment– Gapped blast has indels, but no guarantee…

Dynamic Programming:– Also known as: Needleman-Wunsch Algorithm

Can use it to draw the Dotplot paths– From that we can get the alignment

Mathematically guaranteed– To find the best scoring alignment– Given a substitution scheme (scoring scheme, e.g., BLOSUM)– And given a gap penalty

The Needleman-Wunsch algorithm

A smart way to reduce the massive number of possibilities that need to be considered, yet still guarantees that the best solution will be found (Saul Needleman and Christian Wunsch, 1970).

The basic idea is to build up the best alignment by using optimal alignments of smaller subsequences.

The Needleman-Wunsch algorithm is an example of dynamic programming, a discipline invented by Richard Bellman (an American mathematician) in 1953!

Dynamic Programming

A divide-and-conquer strategy:– Break the problem into smaller subproblems.– Solve the smaller problems optimally.– Use the sub-problem solutions to construct an optimal

solution for the original problem. Dynamic programming can be applied only to problems

exhibiting the properties of overlapping subproblems. Examples include

– Trevelling salesman problem– Finding the best chess move

Overview of Needleman-Wunsch

Four Stages1. Initialise a matrix for the sequences

2. Fill in the entries of that matrix (call these Si,j) At the same time drawing arrows in the matrix

3. Use the arrows to find the best scoring path(s)

4. Interpret the paths as alignments as before

Illustrate with: MNALQM & NALMSQA

Stage 1Initialising the Matrix

Draw the grid

Put in increasing gap penalties Then put in BLOSUM scores

Stage 2Putting Scores and Arrows in

Put the score in Draw the arrow

Mathematically, we are calculating:

Where: – Si,j is the matrix entry at (i,j) [the one we want to fill in]

Si-1,j-1 is above and to the left of this

– s(ai,bj) is the BLOSUM score for the i-th residue from the horizontal sequence and j-th residue from the vertical sequance (i.e., just the scores we have written in brackets)

This diagram might help:

Fill in the next row and column

A Close up View

Continue filling in the Si,j entries

Stage 3Finding the best path

Scores Si,j in the matrix – Are the BLOSUM scores for alignments

However!– We must take into account final gap penalties

Look down the final column and along the final row– Find the highest scoring number– Remembering to take off the gap penalty the correct

number of times

Finding the best path

So, the best path is:

Stage 4: Generating the Alignment Firstly, draw the Dotplot

Secondly, Generate the Alignment

Using the technique previously mentioned– This path gives us an alignment with three gaps

M N A L - - Q M - N A L M S Q AS = -6 6 4 4 -6 -6 5 -1 = 0

Should check that you get the same score– As on the diagram

Other Alignments

MNALQ-M- MNALQM--

-NALMSQA (score=-4) -NALMSQA (score=-5)

Smith - Waterman Alterations

To make the algorithm find best local alignments Adjustments only to the scoring scheme for Si,j:

– The scoring scheme must include: Some negative scores for mismatches

– When Si,j becomes negative, set it to zero So local paths are not penalised for earlier bad routes

To find best local alignment– Find highest scoring matrix position (anywhere)– And work backwards until a zero is reached

Local and Global Alignments

Needleman & Wunsch best global alignments

Smith & Watermanbest local alignments

For illustration purposes only– Calculations done slightly differently (don’t worry)