Download Document

Distances
A natural or ideal measure of distance
between two sequences should have an
evolutionary meaning.
One such measure may be the number
of nucleotide substitutions that have
accumulated in the two sequences
since they have diverged from each
other.
To derive a measure of
distance, we need to make
several simplifying
assumptions regarding the
probability of substitution of
a nucleotide by another.
Jukes & Cantor
one-parameter
model
Assumption:
• Substitutions occur with equal probabilities
among the four nucleotide types.
Kimura’s
two-parameter
model
Assumptions:
• The rate of transitional
substitution at each nucleotide
site is  per unit time.
• The rate of each type of
transversional substitution is 
per unit time.
NUMBER OF NUCLEOTIDE
SUBSTITUTIONS BETWEEN
TWO DNA SEQUENCES
After two nucleotide sequences diverge from
each other, each of them will start accumulating
nucleotide substitutions.
If two sequences of length N differ from each
other at n sites, then the proportion of
differences, n/N, is referred to as the degree of
divergence or Hamming distance.
Degrees of divergence are usually expressed
as percentages (n/N  100%).
The observed
number of
differences is
likely to be
smaller than the
actual number of
substitutions due
to multiple hits at
the same site.
13 mutations
=
3 differences
Number of
substitutions between
two noncoding
sequences
The one-parameter model
In this model, it is sufficient to
consider only I(t), which is the
probability that the nucleotide at a
given site at time t is the same in both
sequences.
where p is the observed proportion of
different nucleotides between the two
sequences.
p p
V (K) 
2
 4 
L1 p
 3 
2
L = number of sites compared
in the ungapped alignment
between the two sequences.
The two-parameter model
The differences between two
sequences are classified into
transitions and transversions.
P = proportion of transitional differences
Q = proportion of transversional
differences
ATCGG
ACCCG
Q = 0.2
P = 0.2
2
2
2


1  
1
1
1
P
Q
Q



V(K)  P
 Q

 


2  4P  2Q 2  4Q  1  2P  Q 2  4P  2Q 2  4Q  
L 1  2P  Q 


Numerical example (2P-model)
-Substitution schemes
with more than two
parameters.
- Parameter-free
substitution schemes.
Number of
substitutions between
two protein-coding
genes
Number of synonymous substitutions
Number of synonymous sites
Number of nonsynonymous substitutions
Number of nonsynonymous sites
Difficulties with
denominator:
1. The classification of a site
changes with time: For example, the
third position of CGG (Arg) is
synonymous. However, if the first
position changes to T, then the third
position of the resulting codon, TGG
(Trp), becomes nonsynonymous.
T
Trp
Nonsynonymous
Difficulties with denominator:
2. Many sites are neither completely
synonymous nor completely
nonsynonymous. For example, a
transition in the third position of GAT
(Asp) will be synonymous, while a
transversion to either GAG or GAA
will alter the amino acid.
Difficulties with
nominator:
1. The classification of the
change depends on the
order in which the
substitutions had occurred.
Difficulties with
nominator:
2. Transitions occur with different
frequencies than transversions.
3. The type of substitution depends
on the mutation. Transitions result
more frequently in synonymous
substitutions than transversions.
Miyata & Yasunaga (1980)
and
Nei & Gojobori (1986)
method
Step 1: Classify Nucleotides into non-degenerate, twofold and fourfold
degenerate sites
U
U
C
A
G
UUU
Phe
UUC
UUA
Leu
UUG
CUU
CUC
Leu
CUA
CUG
AUU
AUC
Ile
AUA
AUG Met
GUU
GUC
Val
GUA
GUG
C
UCU
UCC
UCA
UCG
CCU
CCC
CCA
CCG
ACU
ACC
ACA
ACG
GCU
GCC
GCA
GCG
A
Ser
Pro
Thr
Ala
UAU
UAC
UAA
UAG
CAU
CAC
CAA
CAG
AAU
AAC
AAA
AAG
GAU
GAC
GAA
GAG
G
Tyr
Stop
Stop
His
Gln
Asn
Lys
Asp
Glu
UGU
UGC
UGA
UGG
CGU
CGC
CGA
CGG
AGU
AGC
AGA
AGG
GGU
GGC
GGA
GGG
Cys
Stop
Trp
Arg
Ser
Arg
Gly
U
C
A
G
U
C
A
G
U
C
A
G
U
C
A
G
L0
L2
L4
L A L A
2
2
4
4
KS 
 B4
L L
2
4

2
2
2b Q a P  c (1 Q )
L V(A )  L V(A )
2
2
4
4
4 4 4 4 4
4
V(K ) 
 V(B ) 
S
4
L L
(L  L )2
2
4
2
4

L B L B
0
0
2
2
K A  A0 
L L
0
2


2
2
2b Q a P  c (1  Q )
L V(B )  L V(B )
0
2
2 
0 0 0 0 0
0
V(K )  V(A )  0
A
0
L L
(L  L )2
0
2
0
2
Number of Amino-Acid Replacements
between Two Proteins
• The observed proportion of different
amino acids between the two
sequences (p) is
p = n /L
• n = number of amino acid differences
between the two sequences
• L = length of the aligned sequences.
Number of Amino-Acid Replacements
between Two Proteins
The Poisson model is used to convert p into the number
of amino replacements between two sequences (d ):
d = - ln(1 – p)
The variance of d is estimated as
V(d) = p/L (1 – p)
How do you detect adaptive
evolution at the genetic level?
Theoretical Expectations
Deleterious mutations
Neutral mutations
Advantageous mutations
Overdominant mutations
48
49
50
51
52
53