Download Scoring the Alignment of Amino Acid Sequences

Scoring the Alignment
of Amino Acid Sequences
Constructing PAM and Blosum Matrices
Quotes from page 11 of our Lab Manual:
Proteins are huge molecules made up of large
numbers of amino acids.
The proteins are usually 100 to 500 amino acids
long
There are 20 different amino acids that make up
the proteins
Name
Abbr.
Linear structure formula
======================================================
Alanine
ala a
CH3-CH(NH2)-COOH
Arginine
arg r
HN=C(NH2)-NH-(CH2)3-CH(NH2)-COOH
Asparagine
asn n
H2N-CO-CH2-CH(NH2)-COOH
Aspartic acid asp d
HOOC-CH2-CH(NH2)-COOH
Cysteine
cys c
HS-CH2-CH(NH2)-COOH
Glutamine
gln q
H2N-CO-(CH2)2-CH(NH2)-COOH
Glutamic acid glu e
HOOC-(CH2)2-CH(NH2)-COOH
Glycine
gly g
NH2-CH2-COOH
Histidine
his h
NH-CH=N-CH=C-CH2-CH(NH2)-COOH
Isoleucine
ile i
CH3-CH2-CH(CH3)-CH(NH2)Leucine
leu l
(CH3)2-CH-CH2-CH(NH2)-COOH
Lysine
lys k
H2N-(CH2)4-CH(NH2)-COOH
Methionine
met m
CH3-S-(CH2)2-CH(NH2)-COOH
Phenylalanine phe f
Ph-CH2-CH(NH2)-COOH
Proline
pro p
NH-(CH2)3-CH-COOH
Serine
ser s
HO-CH2-CH(NH2)-COOH
Threonine
thr
t
CH3-CH(OH)-CH(NH2)-COOH
Tryptophan
trp w
Ph-NH-CH=C-CH2-CH(NH2)-COOH
Tyrosine
tyr
y
HO-p-Ph-CH2-CH(NH2)-COOH
Valine
val v
(CH3)2-CH-CH(NH2)-COOH
Constructing Probability Matrices
Using a Smaller Set of AA’s
Suppose we live in a world with only 3 amino acids:
Alanine
Leucine
Serine
Furthermore suppose:
Alanine
Leucine
with probability .2
Alanine
Serine
with probability .1
Leucine
Serine
with probability .3
We will assume that these probabilities are for changes that take place
during one time unit
We can summarize these observations using the language of probability
theory. We will use the notation (A|L, t) to mean: “A certain position in our
sequence initially contains Leucine and at time, t, it contains Alanine.”
Another way of saying this is, “After t time units the position contains
Alanine given that it initially contained Leucine.” , i.e. the vertical bar
means “given” So, Alanine given Leucine after t time units.
We then write:
Pr(A|A, 1) = .7
Pr(A|L, 1) = .2
Pr(A|S, 1) = .1
Pr(L|A, 1) = .2
Pr(L|L, 1) = .5
Pr(L|S, 1) = .3
Pr(S|A, 1) = .1
Pr(S|L, 1) = .3
Pr(S|S, 1) = .6
The above can be summarized in a table, called a matrix
1\2
A L S
A .7 .2 .1
L .2 .5 .3
S .1 .3 .6
.7 .2 .1
M  .2 .5 .3
.1 .3 .6
What about the probabilities two time units later? For example what is the
probability that a position that was originally Alanine is Alanine two time
units later?
This can happen in three ways:
A
A
A
L
A
S
A
In our original notation, we are saying:
(A|A, 2) = (A|A, 1)and(A|A, 1) or (L|A, 1)and(A|L, 1) or (S|A, 1)and(A|S, 1)
Thus, to compute the probability,
Pr(A|A,2) = Pr(A|A,1)Pr(A|A,1) + Pr(L|A,1)Pr(A|L,1) + Pr(S|A,1)Pr(A|S,1)
= .7*.7 + .2*.2 + .1*.1 = .49 + .04 +.01 = .54
We will work out the 8 other second time unit transition probabilities in class.
After we compute all 9 of the probabilities for the transitions after
2 time units we have the following table.
A
L
S
A
.54
.27
.19
L
.27
.38
.35
S
.19
.35
.46
This table required three multiplications and two additions to
compute the values placed in each of its nine cells. That is
there where 27 multiplications and 18 additions required to
produce the above table.
The Matrix Connection
Consider the matrix, M, that we constructed earlier when we made the
table of probabilities
.7 .2 .1


M  .2 .5 .3
.1 .3 .6
In matrix algebra, the product of two matrices is defined as follows:
To compute the product of two matrices A and B, the value placed
in row, i, and column, j, is obtained by multiplying each value in
row, i, of A by its corresponding element in column, j, of B and
summing the results.
Translation by way of an illustration to follow.
Let’s suppose we want to square M, i.e. multiply M by itself
.7 .2 .1 .7 .2 .1
.2 .5 .3  .2 .5 .3

 

.1 .3 .6 .1 .3 .6
To compute the value of the product matrix M2 in row, 2,
column, 3, we multiply each element in row 2 of the first matrix
by its corresponding element in row 3 of the second matrix and
sum the results:
.2*.1 + .5*.3 + .3*.6 = .02 +.15 + .18 = .35
But this is exactly how we calculated Pr(S|L, 2)! This
agreement between M2 and the table of transition probabilities
holds for each position.
It appears that Matrix Multiplication is exactly what we need to
generate the table of transition probabilities after t time units.
Thus, if we use the rules of matrix multiplication,
.7 .2 .1 .7 .2 .1 .54 .27 .19
M 2  .2 .5 .3  .2 .5 .3  .27 .38 .35
.1 .3 .6 .1 .3 .6 .19 .35 .46
Since the rules of matrix multiplication and those for
computing the transition probabilities are essentially the
same, we have a marriage made by the divine. So let’s use
them to our advantage.
The number, variety, and chemical properties of the
Amino Acids make the problem of scoring a pair of Amino
Acids a much more complicated problem than scoring a
pair of nucleotides.
In the late 1970’s Dayhoff, Schwartz, and Orcutt decided
to look at a database of similar proteins having common
ancestors and obtain substitution frequency data.
They looked at 71 groupings of protein data that differed
by no more than 15% of their residues, i.e. at least 85%
similar. They then built phylogenetic trees where each
transition from generation to generation has as few
changes as possible, given the data, in each ancestral
sequence.
From this a value is determined for the entry Aab in a
matrix giving the frequency data for each pairing.
Constructing a Parsimonious Phylogenetic Tree
(taken from page 40 of Krane & Raymer)
ACGCTAFKI
A -> G
GCGCTAFKI
A -> G
GCGCTGFKI
A -> L
GCGCTLFKI
I -> L
ACGCTAFKL
C -> S
ASGCTAFKL
G -> A
ACACTAFKL
Dayhoff and her team used sequences that were at least 85% similar
and calculated the frequency with which each protein was substituted
for each of the other proteins.
Dayhoff’s Data
NOTE: The diagonals are blank since only the changes are recorded. Also, the
upper triangular half of the matrix is not shown since it is assumed that the
changes α and α are symmetrical.
Calculating the Entry in The
Substitution Matrix
Let
P(b|a,t) = Probability that a is substituted for b in t time units
adjusted for divergence time (Dayhoff time unit)
Aab
c Aac
qaqb = Probability that a would randomly follow b
= (frequency if a)(freqency of b)
s(a,b | t) = an entry at position (a, b) or (b, a) in the scoring
matrix
Then,
 P(b | a, t ) 

s(a, b | t) = log 
qb


The Probabilities Found By Dayhoff
The entry in cell Mab is the probability that a would be followed by b in one
Dayhoff time unit multiplied by 100. Thus, for example, Alanine would be
followed by Proline 0.22% of the time.
Note: The previous matrix is NOT the scoring matrix. It is used
to derive the scoring matrix. Recall:
s(a, b | t) =
 P(b | a, t ) 

log 10 
qb


However, the probability matrix is the main tool for deriving a
sensible scoring matrix. To find the probability that amino acid
a will mutate be replaced by amino acid b at a time t time units
later, we need to calculate the a,b-th entry of the matrix Mt.
After calculating this entry, then we apply the “log-odds”
formula given above.
The reason that the logarithm is used in the scoring formula is
that it allows us, among other things, to add the scores of the
aligned residues when we compute the score for an overall
alignment of two sequences.
The matrix having scores found from the original probability matrix is
called a
1 PAM matrix
PAM stands for Point Accepted Mutation or Percent Accepted Mutation
Dayhoff’s term was Accepted Point Mutation, but PAM rolls off the
tongue easier than APM.
The 1 means that given the degree of similarity between the sequences
used to make up the matrix, the scores in this matrix are the frequencies
for one evolutionary time unit.
Scores representing longer times and are called PAMt matrices = Mt.
The most widely used matrix is PAM250 or the log-odds matrix based
on:
M250 = the 250th power of M.
This matrix shows the probability of change over a long period of time.
However, for closely related sequences, say mouse and rat MSH2, a
PAM10 matrix may be more appropriate
The PAM250 Matrix
We only show the top half because the bottom half is a reflection
of the top half, i.e. Sa,b = Sb,a
Discussion of PAM
• The 1 PAM matrix was derived by constructing
hypothetical phylogenetic trees relating sequences
in 71 families.
• The higher the power of the matrix, the more
evolutionary time units represented by the matrix.
• Criticism – raising M to high powers does not
capture the true difference between short time
substitutions and long time substitutions.
• Note short time substitutions are dominated by
amino acid substitutions that come from a single
base change in the codon triplets of an Amino Acid,
whereas the long time substitutions show all kinds
of codon changes
BLOSUM
(BLOck Substitution Matrix)
Matrices
The criticism given at the end of the last discussion
is that the large PAM matrices tend to minimize the
effects of short time substitutions such as
L<->I
L<->V and Y<->F
In 1991 – 1992 Henikoff and Henikoff used the
BLOCKS database at the Fred Hutchison Cancer
Research Center
This database contains blocks of multiple
alignments of more distantly related sequences
Such a database can be used to derive scores
more directly
Methodology
• Sequences from each block were clustered
• Two sequences were placed in the same cluster if
their percent differences were above some level, say
α%
• The frequency Aab is calculated from observing
residue a in one clustered alignment against residue b
in another clustered alignment.
• Corrections are made for clusters of differing sizes
Calculating the Matrix Entries
Let the following be determined from the observed
data:
qa = the fraction of pairings that include an a
pab = the fraction of parings of a and b
Then
qa



b
Aab
cd
Acd
and
The score is calculated as
p ab
q a qb

cd Acd
 p ab
s(a, b)  log 
 q a qb



These values are then scaled and rounded to make
calculations easier.
If we set the limit, α, to 62, the we have a BLOSUM62
Matrix
Most popular BLOSUM Matrices are BLOWSUM62 and
BLOWSUM50.
• BLOWSUM62 is used mainly for ungapped matching.
• BLOWSUM50 is used for alignments with gaps.
Note: the lower the number the longer the time span in
evolutionary units.
Differences Between PAM and BLOSUM
PAM assumes that substitutions probabilities for highly
related proteins can be extrapolated to the probabilities
for distantly related proteins.
BLOWSUM matrices are based on the observation of
more distantly related protein alignments.
NOTE: Both types of matrices use log-odds values in
their scoring systems.