1. Substitution Numbers and Scoring Matrices

2. Substitution Numbers
The number of observed substitutions K is an important quantity in molecular evolutionary analysis
A simple count may be misleading, so statistical models are developed to estimate the number of substitutions
Jukes-Cantor model
Kimura model
(both are for nucleotides, but the ideas can extend to amino acids)

3. Jukes-Cantor Model
Assumes that each nucleotide is equally likely to change into any other nucleotide with probability α per time step
What is the probability that if we start with C we end up with C after 2 time steps Pcc(2) =
C -> C -> C C -> A -> C C -> T -> C C-> G -> C α A T C G φ α G A α α α
φ = α α T C α

4. Jukes-Cantor Model
The entry M(a,b) in the matrix M1 represents the probability of substitution from nucleotide a to b in one time step
What is the matrix M2, i.e. whose entries M(a,b) represent the probability of substitution from a to b in two time steps
essentially what we did on prev. slide but for all pairs of bases
A->X->A A->X->T A->X->C A->X->G
T->X->A T->X->T T->X->C T->X->G
C->X->A C->X->T C->X->C C->X->G
G->X->A G->X->T G->X->C G->X->G A T C G φ α
C->X->C = α∙α + α∙α + φ∙φ + α∙α (prev. slide)
C->X->A = α∙φ + α∙α + φ∙α + α∙α
M1 =
φ = α

5. Jukes-Cantor Model
Turns out that Mn = (M1)n i.e. whose entries M(a,b) represent the probability of substitution from a to b in n time steps
In general under the J.C. model the probability that a site will contain a C after t time steps is given by:
Pc(t) = ¼ + (¾)e-4αt
This model can be used to derive an estimate of the number of substitutions that have occurred between the sequences
K = -¾ ln[ 1 – (4/3) p ]
p – the fraction of nucleotides that are considered mismatch

6. Kimura Model
Addresses the unrealistic assumption in J.C. model that all substitutions are equally likely
Two types of substitutions
transitions – purine<=>purine exchange or pyrimidine<=>pyrimidine
transversions – purine<=>pyrimidine exchange α A T C G φ β Α α G A β β β
φ = 1 – α – 2 β β T C α

7. Kimura Model
What is the probability that if we start with C we end up with C after 2 time steps Pcc(2) =
C -> C -> C C -> A -> C C -> T -> C C-> G -> C
In general under the Kimura model the probability that a site will contain a C after t time steps is given by:
Pc(t) = ¼ + (¼)e-4βt + (½)e-2(α+β)t
Estimated number of substitutions (TR – transitions, TV – transverions)
K = ½ ln[ 1 / (1 – 2*TR – TV)] + ¼ ln[ 1 / (1 – 2*TV)]

8. Scoring Matrices

9. Alignment Score
Alignment score attempts to measure likelihood of a common evolutionary ancestor
Two possible ways to explain a given pairwise alignment
random model – the alignment could be produced purely by chance
evolutionary model – there is high correlation between aligned pairs
Under random model
each position is independent of the others
probability of amino acid a occurring at each position is pa
Under non-random model
probability of amino acid a depends on matched residue b – qab

10. Substitution Matrices
Given a (non-gapped) pairwise alignment of sequences
A = a1 a2 a3 a4…an
B = b1 b2 b3 b4…bn
under non-random model probability of the alignment
Pnon-random = qa1b1qa2b2qa3b3qa4b4…qanbn
under random model probability of the alignment
Prandom = pa1pa2pa3pa4…pan pb1pb2pb3pb4…pbn =
pa1pb1pa2pb2pa3pb3qa4pb4…panpbn
Use ratio of probabilities (odds ratio) to compare the models
r = –––––––– r > 1, non-random more likely
Pnon-random
Prandom

11. Substitution Matrices
Ratio of probabilities (odds ratio)
r = –––––––– = ––––––––––––––––––––––––––––––
= ––––––––––––––––––––––––––––––
Typically the log-odds ratio is used
log(r) = log( –––––––––––––––––––––––––––––– )
= log(––––––)+log(––––––)+log(––––––) log(––––––)
Pnon-random
qa1b1qa2b2qa3b3qa4b4 …qanbn
Prandom
pa1pb1pa2pb2pa3pb3qa4pb4…panpbn
qa1b qa2b qa3b qa4b4 … qanbn
pa1pb1 pa2pb2 pa3pb3 qa4pb4 …panpbn
qa1b qa2b qa3b qa4b4 … qanbn
pa1pb1 pa2pb2 pa3pb3 qa4pb4 …panpbn
Entry (a1, b1) in the
substitution matrix
qa1b1
qa2b2
qa3b3
qanbn
pa1pb1
pa2pb2
pa3pb3
panpbn

12. Substitution Matrices
Provide the “likelihood” that two amino acids (nucleotides) will occur as aligned pair
Common substitution matrices for protein alignment
PAM family – derived from alignments of high sequence identity
(Dayhoff, Schwartz, and Orcutt. “A model of evolutionary change in proteins. In Atlas of
Protein Sequence and Structure Volume : )
BLOSUM family – derived from alignments of low sequence identity
(Henikoff and Henikoff. “Amino acid substitution matrices from protein blocks” Proc. Natl. Acad. Sci (22): 10915–10919.)
BLOSUM62
A R N D C Q E G H I L K M F P S T W Y V B Z X * A R N D C Q E G H I L K M F P S T W Y V B Z X *

13. BLOSUM Matrices
Based on ungapped multiple local alignments of conserved regions of proteins with low sequence identity
These alignments are used to derive qab pa pb which give the substitution score for amino acids a and b
score(a, b) = log(––––––)
Procedure
obtain known ungapped multiple local alignments
split into clusters, so that every pair in a cluster has ≥ C% identity
for each pair of amino acids a and b calculate
qab = frequency of a,b pair / total # pairs
(sequences within a cluster are given weight 1 / size_of_cluster)
qab
papb

14. BLOSUM Matrices
Calculating qQN for BLOSUM62 – within a cluster for each sequence there is one with (≥ 62% identity)
ATCKQ
ATCRN
ASCKN
SSCRN
SDCEQ
SECEN
TECRQ
7 clusters, 21 pairs of clusters
5*21 = 105 total # of aligned pairs
QN matched in 12 pairs of clusters
qQN = frequency of QN pair / total # aligned pairs
= 12 / 105 = 0.114

15. BLOSUM Matrices
Calculating qQN for BLOSUM50 – within a cluster for each sequence there is one with (≥ 50% identity)
ATCKQ
ATCRN
ASCKN
SSCRN
SDCEQ
SECEN
TECRQ
3 clusters, 3 pairs of clusters
5 bases * 3 clusters = 15 total # of aligned pairs
QN match frequency (between clusters):
top, mid:
top, bot:
mid, bot:
total:
qQN = frequency of QN pair / total # aligned pairs
= 14/8 / 15 =

16. BLOSUM Matrices
Calculating qQN for BLOSUM50 – within a cluster for each sequence there is one with (≥ 50% identity)
ATCKQ
ATCRN
ASCKN
SSCRN
SDCEQ
SECEN
TECRQ
3 clusters, 3 pairs of clusters
5 bases * 3 clusters = 15 total # of aligned pairs
QN match frequency (between clusters):
top, mid: ¼*½ + ¾*½
top, bot: ¾*1
mid, bot: ½*1
total: 1/8+3/8+3/4+1/2 = 14/8
qQN = frequency of QN pair / total # aligned pairs
= 14/8 / 15 =

17. BLOSUM Matrices
So far calculated qabN (i.e. probability that a and b will be paired up under non-random model)
To compute the substitution score need to know pa and pb (i.e. probability that a and b occur by chance)
pa = qaa + ½ Σa≠bqab ≈ fraction of all amino acids that are type a
The entry computed in the substitution matrix is:
qab
score(a, b) = log(––––––)
papb

18. PAM Matrices
Based on ungapped multiple local alignments of conserved regions of proteins with high sequence identity (> 85%)
Uses phylogenetic trees to compute the entries in the substitution matrix
Procedure
build a phylogenetic tree for sequence of high identity
compute relative mutability, ma, of each amino acid
(frequency of a substitutions in the phylogenetic tree)
compute Fab (number of substitutions of a with b)
compute Mab (mutation probability that a will be replaced by b)
Mab = mb Fab / ΣcFcb
compute entry in scoring matrix
score(a, b) = log(Mab / frequency of a)

19. PAM Matrices Constructing a PAM matrix ACGCTAFKI GCGCTAFKI ACGCTAFKL
GCGCTGFKI
GCGCTLFKI
ASGCTAFKL
ACACTAFKL
ACGCTAFKI
GCGCTAFKI ACGCTAFKL
GCGCTGFKI GCGCTLFKI ASGCTAFKL ACACTAFKL
A->G
I->L
A->G
A->L
C->S
G->A
Compute score(G, A) – need mA, FGA, ΣcFcA
ma = 4 / 2*6
FGA = 3
Σ FcA = 4
Mab = mA FGA / ΣcFcA
score(G, A) = log(MGA/ frequency_of_G) = log(MGA/ (10/63))