Single-letter scores (example)

If we pair two letters \(x\) and \(y\), we need to know how they contribute to the score value. For example,

• If \(x = y\) the score increases by 1
• If \(x ≠ y\) the score decreases by 1
• If one of the letters is - (a gap), the score reduces by 2

In general, we have a substitution matrix \(C_{x, y}\) for each \(x\) and \(y\)

Single-letter scores (example)

In one equation

\[C_{x, y} = \begin{cases} +1 & \quad x=y\\ -1 & \quad x≠y\\ -2 & \quad x=\texttt{"-"} \\ -2 & \quad y=\texttt{"-"} \end{cases}\]

Scoring matrices

The scoring matrix (also known as a substitution matrix) has one row and one column for each letter in our alphabet

For example 4 rows and 4 columns for DNA sequences

In DNA it is common to have +1 for match and -2 for mismatch

Sometimes we can use (+1,-3), (+1,-1) or even (4,-5)

Amino-acid scoring matrices

For proteins the alphabet has 20 letters, so the scoring matrix is 20×20

The scores of these matrices are based on the relative frequency of each mutation in nature

Obviously, we only care about non-lethal mutations

More specifically, we care about the Point Mutations that are Accepted

These are called PAM matrices (Margaret Dayhoff, 1978)

PAM matrices

Based on 1572 mutations in 71 families of closely related proteins

Dayhoff only used alignments having at least 85% identity

She assumed that any aligned mismatches were the result of a single mutation event, rather than several at the same location

PAM matrices

PAM1 is the scoring matrix for proteins in a “short” period

With some easy mathematics we can calculate PAM2, PAM30, PAM250, etc

They represent mutations in 2, 30, and 250 short periods

BLOSUM matrices

Using local alignment we can identify conserved regions

In 1992 Steven Henikoff and Jorja Henikoff created new substitution matrices based on local alignment of blocks

BLOcks SUbstitution Matrix

Idea: each protein domain can evolve at different speeds

Affine gaps

Gap values must reflect how real insertions and deletions occur in nature

We observe that, once an indel event starts, it can easily grow

If the polymerase jumps, then it can jump a long distance

To represent this, we use affine gaps

Linear v/s affine

So far we considered only linear gaps

The penalty of \(n\) consecutive gaps is \(n\cdot G\)
(\(G\) is the gap penalty)

Now we consider affine gaps, where the first gap is expensive, but the consecutive are cheap

The penalty of \(n\) consecutive gaps is \(I + n\cdot E\)

\(I\) is the initial gap penalty, \(E\) is the gap extension penalty

Score can change

If mismatches and gaps have different cost, the score will change

Sometimes the optimal alignment changes

Therefore alignments are meaningless without knowing the scoring matrices

Later we will discuss how to choose the “best” scoring matrix for each case

Larger scores, less hits

A hit is a subject with score over a threshold

Larger score thresholds give less hits

We can estimate the number of hits in a given database, assuming randomness

That is called Expected value

Expected value as a threshold

In practice, we choose a small Expected value

(usually called E-value)

Something like 10^-5 or 10^-20

What we find is not random
and maybe it is biologically meaningful

E-value depends on the database

The formula for E-value depends on

• The substitution scoring matrix
• The query size
• The database size

Same alignments in different databases have different E-value
but the same score