Explain scoring matrices (BLOSUM, PAM) — Biology notes

WHY do we need a scoring matrix at all?

WHAT should the score represent? The right quantity is: is this pairing more likely because the two residues are truly homologous (related by descent), or is it just random coincidence? That is a likelihood ratio.

HOW to derive the log-odds score from first principles

Formula Deriving the log-odds score

Consider a column in a trusted alignment showing residues $a$ and $b$ paired.

Two competing hypotheses:

$H_1$ (homology): they are paired because they descend from a common ancestor. Probability of observing this pair = $q_{ab}$ (the observed joint frequency in trusted alignments).
$H_0$ (random): they landed together by chance. Probability = $p_a \, p_b$ , where $p_a$ is the background frequency of amino acid $a$ .

The odds that this pairing is real is $\text{odds} = \frac{q_{ab}}{p_a \, p_b}.$

Why take a log? Alignment scores must be additive across columns (total score = sum of column scores). But probabilities of independent columns multiply. Taking a log converts multiplication into addition: $\log\!\Big(\prod_i \tfrac{q_i}{p_i}\Big) = \sum_i \log\tfrac{q_i}{p_i}.$ So we define the substitution score $\boxed{\,s(a,b) = \frac{1}{\lambda}\,\log_2 \frac{q_{ab}}{p_a\, p_b}\,}$ Multiply by a scaling constant and round to integers (that's why BLOSUM entries are small whole numbers). BLOSUM uses base-2 logs (bits), scaled by $\lambda = 1/2$ (i.e. half-bits).

Why this step? Because the log-odds is the statistically optimal score for testing $H_1$ vs $H_0$ (a consequence of the Neyman–Pearson lemma). Sign meaning:

$s(a,b) > 0$ : substitution seen more than chance → conserved/favorable.
$s(a,b) < 0$ : seen less than chance → disruptive.
$s(a,b) = 0$ : exactly as expected by chance.

BLOSUM — BLOcks SUbstitution Matrix

HOW the number works (this trips everyone up):

PAM — Point Accepted Mutation

HOW higher-PAM matrices are made — extrapolation:

Build a $20\times20$ mutation probability matrix $M$ for 1 PAM directly from closely-related sequences.
To model more distant sequences, multiply the matrix by itself: $M^{(n)} = M^n \quad\Rightarrow\quad \text{PAM250} = M^{250}.$ This assumes mutations accumulate as a Markov process (each step independent of the past).
Convert $M^n$ to log-odds using the formula above.

Figure — Explain scoring matrices (BLOSUM, PAM)

Worked Examples

Active Recall

Recall What does a positive vs negative matrix entry mean?

Positive → substitution occurs more often than chance (conserved/favorable). Negative → less than chance (disruptive). Zero → as expected by chance.

Recall Which direction is "distant" for PAM and for BLOSUM?

PAM high = distant (more mutations). BLOSUM high = close (higher identity clustering). They run opposite.

Recall Why do we take the logarithm of the odds ratio?

So that per-column scores add up instead of multiplying — matching how alignment algorithms sum scores and how independent probabilities combine.

Recall Feynman: explain to a 12-year-old

Imagine trading cards. Some swaps are fair ("I'll give you a common card for a common card") and some are unfair. A scoring matrix is a cheat sheet telling you how fair each swap is. It looks at real card-trading history (real proteins evolving) and gives plus points to swaps that people actually make a lot (because those cards are basically interchangeable) and minus points to swaps almost nobody makes (because those cards are totally different). We add up the points to decide if two whole decks (proteins) are cousins.

Flashcards

What quantity does a scoring-matrix entry

s(a,b)

represent?

A scaled log-odds ratio:

\frac{1}{\lambda}\log_2 \frac{q_{ab}}{p_a p_b}

— how often pair (a,b) occurs in real alignments vs by chance.

Why is the log taken in the log-odds score?

To make per-column scores additive (logs turn the product of independent column probabilities into a sum).

What does the "62" in BLOSUM62 mean?

The clustering threshold: sequences ≥62% identical within a block are merged/counted as one before tallying substitutions.

Higher BLOSUM number is for which sequences?

More closely related (higher identity) sequences.

What does 1 PAM unit mean?

An evolutionary distance where 1 accepted point mutation has occurred per 100 residues (1% change).

How are high-PAM matrices generated?

By multiplying the 1-PAM mutation matrix by itself: PAM250 = M^250 (Markov extrapolation).

Higher PAM number is for which sequences?

More distantly related (more accumulated mutations) sequences.

BLOSUM vs PAM: how do their numbers relate directionally?

Opposite — high BLOSUM = close; high PAM = distant. (e.g. BLOSUM62 ≈ PAM160–200 behavior.)

Why does W↔W score +11 but L↔L only +4 in BLOSUM62?

Trp is rare, so a Trp match is statistically surprising (high info), giving a larger log-odds; Leu is common, so its match is less informative.

Which matrix for a ~25% identity distant homolog search?

BLOSUM45 or PAM250.

Which statistical principle makes log-odds the optimal score?

Neyman–Pearson lemma (likelihood-ratio test for homology H1 vs random H0).

What data source is BLOSUM built from?

The BLOCKS database of ungapped, conserved local protein alignments (observed directly).

What model is PAM built from?

A Markov evolutionary model of point mutations fitted to closely related proteins (Dayhoff).

Mnemonic

Connections

Sequence Alignment — scoring matrices feed into Needleman-Wunsch Algorithm & Smith-Waterman Algorithm.
BLAST — default matrix is BLOSUM62; choice affects sensitivity.
Log-odds and Likelihood Ratios — statistical backbone (Neyman–Pearson).
Markov Chains — PAM extrapolation via matrix powers $M^n$ .
Amino Acid Properties — hydrophobicity/charge explain why certain substitutions score high.
Gap Penalties — the other half of an alignment score.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab hum do proteins ko align karte hain, toh har position pe do amino acids match ho rahe hote hain. Lekin sab substitutions barabar nahi hote — Leucine ko Isoleucine se badalna almost harmless hai (dono hydrophobic hain), par Leucine ko charged Aspartate se badalna protein ko kharab kar sakta hai. Isliye humein ek scoring matrix chahiye jo har pair ko ek number de: positive agar wo swap evolution mein chance se zyada dikhta hai, negative agar kam dikhta hai. Yeh number nikalta hai log-odds se: $\log_2(q_{ab}/(p_a p_b))$ — matlab "observed frequency divided by random frequency", phir log le lo taaki columns ke scores add ho sakein (kyunki independent probabilities multiply hoti hain, aur log unhe addition mein badal deta hai).

BLOSUM aur PAM do families hain. BLOSUM real conserved alignments (BLOCKS database) se seedha count karke banti hai. Yaad rakho: BLOSUM ka number = clustering threshold, aur HIGH BLOSUM = closely related sequences ke liye (BLOSUM80 close, BLOSUM45 distant). PAM ek evolutionary Markov model se banti hai — 1 PAM matlab 100 residues mein 1 accepted mutation. Distant sequences ke liye matrix ko khud se multiply karte hain (PAM250 = $M^{250}$ ), isliye HIGH PAM = distant.

Sabse important trap: PAM aur BLOSUM ke numbers ulti direction mein chalte hain! High PAM = door ke rishtedar, High BLOSUM = kareebi rishtedar. Exam mein yahi confuse karte hain students. Mnemonic yaad rakho: "BLOSUM Big = Buddies, PAM Plenty = Parted."

Kyun important hai? Kyunki BLAST search karte waqt sahi matrix choose karna padta hai. Agar aap distant homolog dhoond rahe ho (~25% identity), toh BLOSUM45/PAM250 use karo — warna hit miss ho jayega. Yeh matrix hi decide karti hai ki algorithm asli homology dhoondh payega ya nahi.

Test yourself — Bioinformatics & Computational Biology

Connections