Class 5: Optimal pairwise alignment

Bioinformatics

Andrés Aravena

October 11, 2021

How does spell check work?

When you write a text in any good text editor
(or even in Microsoft Word®),
some words are automatically underlined with red

These are words that are not found in the dictionary

Sometimes the editor can suggest the correct word

How does the computer do that?

How do we find the suggestion?

One option is to define a rule, like a program, that will tell if a word is correct or not \[ isCorrect: Words ↦ \{True, False\} \] That is hard, in general

There is an easier way

Other option is to have a dictionary or database of know correct words, and a way to find the nearest word \[ Similar: (Words, Database) ↦ \{Words\in Database\} \]

What do we mean by similar?

When are two sequences similar?

Calculating a distance between strings

First idea:

  • We count the number of mismatches

  • Both strings must have the same length

      ASELLKYLTT
  ASELLKALTT

Here distance(ASELLKYLTT,ASELLKALTT) is 1

This is called Hamming distance

Hamming distance examples

How many substitutions we need to go from one sequence to the other

  • CAT and CAT have a Hamming distance of 0
  • CAT and BAT have a Hamming distance of 1
  • CAT and BAG have a Hamming distance of 2

Let’s make a Hamming distance calculator in Excel or Google Sheets

Let’s pause a minute and do this exercise

What if sequences have different length?

In that case we need to insert “gaps”

MOUSE
GROUSE

Placing two sequences face to face, inserting gaps if necessary is called Pairwise Alignment

There are many ways to insert gaps

The distance depends on the gaps positions

-MOUSE
GROUSE

Hamming Distance=2

MOUSE--
-GROUSE

Hamming Distance=7

So, which one is the distance?

In geometry we say that “the distance between a point X and a line L is the shortest length from X to any point in L”

Same idea

If there many “candidate distances”, we choose the smallest one

This is an important idea

Distance is the length of the shortest path

Now we also count gaps

Hamming distance counts substitutions between sequences

Now we counts substitutions, insertions and deletions

This is called Levenstein distance

Hamming versus Levenstein

Hamming distance counts substitutions

ABCDEFGHIJ
BCDEFGHIJA

Hamming distance=10

Levenstein counts substitutions, insertions and deletions

ABCDEFGHIJK-
-BCDEFGHIJKA

Levenstein distance=2

What is the best way to insert gaps?

If the sequence has m letters, there are 2m+1 ways to insert a single gap

CAT
CAT-
CA-T
CA-T-
C-AT
C-AT-
C-A-T
C-A-T-
-CAT
-CAT-
-CA-T
-CA-T-
-C-AT
-C-AT-
-C-A-T
-C-A-T-

And that is not even counting larger gaps

We could also do

C--A-----T

or

----CA-T--

or

-C--A--T--

There is a better way to find the best

We will see more details later

The idea is to draw a rectangle

  • One sequence in the rows
  • Another sequence in the columns
  • Mark the cells where row letter and column letter are equal

It looks like this

The goal is to move from one corner to the other

Jumping black to black is free

Horizontal and vertical moves are gaps

Exercise

Prepare a DotPlot in Excel or Google Sheets

We move from one corner to the other

White blocks add to the distance

Summary

  • Hamming distance
  • Levenstein distance
  • Dot plot