Information theoretic interpretation of PAM

matricesSorin Istrail and Derek Aguiar

Local alignments• The preferred method to compute

regions of local similarity for two sequences of amino acids is to consider the entire length of the sequence and optimize a similarity matrix.• PAM and BLOSUM both a number of

different matrices constructed to model similarity between amino acid sequences at different evolutionary distances.• Here, we follow Altschul c99c to

investigate PAM matrices from an information theoretic perspective.

Caveats and assumptions• The following theory only applies to

locally aligned segments that lack gaps.• Why is this assumption easier to tolerate

in local alignment vs. global alignment?• Why is this assumption still restrictive for

local alignments?

Notation and definitions• Amino acids: ai

• Substitution score of aligned amino acids ai and aj: sij

• A Maximal Segment Pair (MSP) is a pair of equal length segments from two amino acid sequences that, when aligned, have maximum score.

Random model• For any two amino acid sequences, there

exists at least one MSP.• It is convenient to compute what MSP scores

look like for random sequences to serve as a basis for comparison.• We will consider a very simple model

• Each amino acid ai appears randomly with probability pi reflecting actual frequencies of amino acid sequences

• What could be a more biologically accurate (yet mathematically less feasible) method for generating amino acid sequences?

More assumptions...• There is at least one positive score in the

substitution matrix• Why is this reasonable? (think about what

the optimal alignment would be)• The expected score for the matrix is

negative

𝜆• Previous theory defines a key parameter

• But what is ?• Consider the case of multiplying a

similarity matrix by some constant c, what happens to an alignment?

𝜆𝑐∗

A C G T -A c -c -c -c -cC -c c -c -c -cG -c -c c -c -cT -c -c -c c -c- -c -c -c -c -

A C G T -A 1 -1 -1 -1 -1C -1 1 -1 -1 -1G -1 -1 1 -1 -1T -1 -1 -1 1 -1- -1 -1 -1 -1 -

¿MSP score = 8 MSP score = c*8...AGCGCTAC... ...AGCGCTAC......AGCGCTAC... ...AGCGCTAC...

The MSP score changes but the MSP and alignment does not.

How does this affect ?

𝜆• To preserve after scalar multiplication of

the similarity matrix, we can simply set • So, one may view as a scaling parameter

for a similarity matrix.

Random Model• Given two random sequences how many

MSPs with score at least S can we expect by chance?• where N is the product of the sequences’

lengths and K is a calculable parameter.• This equation is related to the limiting

probability distribution for where M(N) is the MSP score.• Theorem 1 (single sequence, Karlin and

Altshul 1990):

Random Model• Which leads to a Poisson approximation

for the number of MSP with scores exceeding with parameter • So, the probability of finding m or more

distinct segments with score greater than or equal to S is approximated by which we can take m=1 for finding a single segment (and yield the Theorem of the previous slide)

Random Model• An extension of Theorem 1 to two

sequences of length m and n yields an important result:• where * denotes an estimated parameter

and M is the MSP score• Which leads to the result in Altshul 1991• The number of MSPs with score at least S is

well approximated by where N is the product of the sequences’ lengths and K and are calculable parameters.

Substitution matrices• Substitution matrices store scores which

encode the target frequencies of amino acids in a true alignment and the background amino acid probabilities• The scores, where are the target

frequencies, and are the background amino acid probabilities, and is a scaling factor.• This ratio compares the probability of an

alternate hypothesis (target frequencies) to the probability of the null (product of frequencies)

Local alignment and information theory• Because scaling the substitution matrix changes

but not the target frequencies, we have the freedom of adjusting • Let’s set • Then, we can set the number of MSPs with score

at least S to p and solve for S

• An alignment is significant when and K is typically near 0.1 thus the score needed to distinguish an MSP from chance is approximated by the number of bits needed to represent the MSP

Relative entropy and substitution matrices• So, what substitution matrices are the

most appropriate for the comparison of two particular sequences?• To answer this question, consider the

average score per residue pair in an alignment

• H is exactly the notion of relative entropy of the target and background probability distributions

Relative entropy • Relative entropy (KL divergence) is a

measure of how closely related two probability distributions are• Given two probability distributions Q and

P, relative entropy can be informally stated in several different manners• The amount of additional bits required to

code samples from P when using Q• The amount of information lost when Q is

used and P is the true distribution of the data

Relative entropy and substitution matrices• But how does this relate to substitution

matrices?• Well, if the target and background

frequency distributions are closely related, then the relative entropy is low and it is very difficult to distinguish between the target and background frequencies. We would therefore require a much longer alignment.• On the other hand, if the target and

background frequency distributions are very different, the relative entropy is high and we’re able to compute much shorter alignments.

Example 1 – cystic fibrosis• Variants in a transport protein have been

associated with cystic fibrosis• A search of this gene in the PIR protein

sequence database yields the table on the following slide

Example 1 – cystic fibrosis

Altshul, S.F. (c99c) “Amino Acid Substitution Matrices from an Information Theoretic Perspective”, Journal of Molecular Biology, 2c9:555-565

Example 1 – cystic fibrosis• Of note, the best PAM-250 score is not

higher than the highest score of a random alignment given the background frequencies.• On the other hand, PAM-120 gives

alignments in the same region with scores higher than the highest chance alignment• Why do you think PAM-120 a better fit

here?

References• Explains the connection between information

theory and substitution matrices• Altshul, S.F. (c99c) “Amino Acid Substitution Matrices

from an Information Theoretic Perspective”, Journal of Molecular Biology, 2c9:555-565

• Provides much of the theory for the above article• Karlin, S. Dembo, A. Kawabata, T. “Statistical

Composition of High-Scoring Segments from Molecular Sequences.” The Annals of Statistics 18 (1990), (2), 571--581.

• Karlin, S. and Altschul SF. “Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes” PNAS 1990 87 (6) 2264-2268