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MAT 4830Mathematical Modeling
4.5
Phylogenetic Distances I
http://myhome.spu.edu/lauw
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Phylogenetic: of or relating to the evolutionary development of organisms
Estimate the amount of total mutations (observed and hidden mutations).
Example from 4.1
S0 : Ancestral sequenceS1 : Descendant of S0S2 : Descendant of S1
S0 : ATGTCGCCTGATAATGCC
S1 : ATGCCGCTTGACAATGCC
S2 : ATGCCGCGTGATAATGCC
Example from 4.1
S0 : Ancestral sequenceS1 : Descendant of S0S2 : Descendant of S1
S0 : ATGTCGCCTGATAATGCC
S1 : ATGCCGCTTGACAATGCC
S2 : ATGCCGCGTGATAATGCC
Observed mutations: 2
Example from 4.1
S0 : Ancestral sequenceS1 : Descendant of S0S2 : Descendant of S1
S0 : ATGTCGCCTGATAATGCC
S1 : ATGCCGCTTGACAATGCC
S2 : ATGCCGCGTGATAATGCC
Actual mutations: 5
Example from 4.1
S0 : Ancestral sequenceS1 : Descendant of S0S2 : Descendant of S1
S0 : ATGTCGCCTGATAATGCC
S1 : ATGCCGCTTGACAATGCC
S2 : ATGCCGCGTGATAATGCC
Actual mutations: 5, (some are hidden mutations)
Distance of Two Sequences
We want to define the “distance” between two sequences.
It measures the average no. of mutations per site that occurred, including the hidden ones.
S0 : ATGTCGCCTGATAATGCC
S : ATGCCGCGTGATAATGCC
Distance of Two Sequences
Let d(S0,S) be the distance between sequences S0 and S. What properties it “should” have?
1.
2.
3.S0 : ATGTCGCCTGATAATGCC
S : ATGCCGCGTGATAATGCC
Jukes-Cantor Model
Assume α is small. Mutations per time step are “rare”.
0
1 / 3 / 3 / 3
/ 3 1 / 3 / 3 1 1 1 1( )
/ 3 / 3 1 / 3 4 4 4 4
/ 3 / 3 / 3 1
T
M p
Jukes-Cantor Model
q(t)=conditional prob. that the base at time t is the same as the base at time 0
( )q t
1 3 4 1 1 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 3 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3( )
1 1 4 1 1 4 1 3 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 1 4 11 1
4 4 3 4 4 3 4
t t t
t t t
t
t t t
t t
M
1 1 41
4 4 3
1 1 41
4 4 3
1 1 41
4 4 3
1 4 1 3 41 1
4 3 4 4 3
t
t
t
t t
Jukes-Cantor Model
q(t)=fraction of sites with no observed mutations
( )q t
1 3 4 1 1 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 3 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3( )
1 1 4 1 1 4 1 3 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 1 4 11 1
4 4 3 4 4 3 4
t t t
t t t
t
t t t
t t
M
1 1 41
4 4 3
1 1 41
4 4 3
1 1 41
4 4 3
1 4 1 3 41 1
4 3 4 4 3
t
t
t
t t
Jukes-Cantor Model
p(t)=1-q(t)=fractions of sites with observed mutations
( )q t
( ) 1 ( )p t q t
1 3 4 1 1 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 3 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3( )
1 1 4 1 1 4 1 3 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 1 4 11 1
4 4 3 4 4 3 4
t t t
t t t
t
t t t
t t
M
1 1 41
4 4 3
1 1 41
4 4 3
1 1 41
4 4 3
1 4 1 3 41 1
4 3 4 4 3
t
t
t
t t
Jukes-Cantor Model
p(t)=1-q(t)=fractions of sites with observed mutations
( )q t
( ) 1 ( )p t q t
1 3 4 1 1 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 3 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3( )
1 1 4 1 1 4 1 3 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 1 4 11 1
4 4 3 4 4 3 4
t t t
t t t
t
t t t
t t
M
1 1 41
4 4 3
1 1 41
4 4 3
1 1 41
4 4 3
1 4 1 3 41 1
4 3 4 4 3
t
t
t
t t
3 3 4( ) 1
4 4 3
t
p t
Jukes-Cantor Model
p can be estimated from the two sequences
( )q t
( ) 1 ( )p t q t
1 3 4 1 1 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 3 4 1 1 41 1 1
4 4 3 4 4 3 4 4 3( )
1 1 4 1 1 4 1 3 41 1 1
4 4 3 4 4 3 4 4 3
1 1 4 1 1 4 11 1
4 4 3 4 4 3 4
t t t
t t t
t
t t t
t t
M
1 1 41
4 4 3
1 1 41
4 4 3
1 1 41
4 4 3
1 4 1 3 41 1
4 3 4 4 3
t
t
t
t t
3 3 4( ) 1
4 4 3
t
p t
Example from 4.1
S0 : ATGTCGCCTGATAATGCC
S1 : ATGCCGCTTGACAATGCC
S2 : ATGCCGCGTGATAATGCC
Observed mutations: 2
fractions of sites with observed mutations
2 0.11
18p
Jukes-Cantor Distance
Given p (and t), the J-C distance between two sequences S0 and S1 is defined as
0 1
3 4( , ) ln 1
4 3JCd S S p
0
1
: ATGTCGCCTGATAATGCC
: ATGCCGCGTGATAATGCC
S
S
Jukes-Cantor Distance
Given p (and t), the J-C distance between two sequences S0 and S1 is defined as
0 1
3 4( , ) ln 1
4 3JCd S S p
0
1
: ATGTCGCCTGATAATGCC
: ATGCCGCGTGATAATGCC
S
S
Jukes-Cantor Distance
rate of base sub. sub. per site per time step
t no. of time step
t total no. of sub. in t time steps sub. per site
Jukes-Cantor Distance
rate of base sub. sub. per site per time step
t no. of time step
t total no. of sub. in t time steps sub. per site
3 3 41
4 4 3
4 4ln 1 ln 1
3 3 when is small
44ln 1
33
t
p
p pt
Jukes-Cantor Distance
rate of base sub. sub. per site per time step
t no. of time step
t total no. of sub. in t time steps sub. per site
3 3 41
4 4 3
4 4ln 1 ln 1
3 3 when is small
44ln 1
33
t
p
p pt
3 4ln 1
4 3t p
Example from 4.3
Suppose a 40-base ancestral and descendent DNA sequences are
0
1
S : ACTTGTCGGATGATCAGCGGTCCATGCACCTGACAACGGT
S : ACATGTTGCTTGACGACAGGTCCATGCGCCTGAGAACGGC
1 0\
7 0 1 1 1 9 2 0
0 2 7 2
1 0 1 6
S S A G C T
A
G
C
T
Example from 4.3
Suppose a 40-base ancestral and descendent DNA sequences are
0
1
S : ACTTGTCGGATGATCAGCGGTCCATGCACCTGACAACGGT
S : ACATGTTGCTTGACGACAGGTCCATGCGCCTGAGAACGGC
1 0\
7 0 1 1 1 9 2 0
0 2 7 2
1 0 1 6
S S A G C T
A
G
C
T
110.275
403 4 11
ln 1 0.34264 3 40JC
p
d
0 1
3 4( , ) ln 1
4 3JCd S S p
Example from 4.3
0.275 observed sub. per site.
0.3426 sub. estimated per site.
0
1
S : ACTTGTCGGATGATCAGCGGTCCATGCACCTGACAACGGT
S : ACATGTTGCTTGACGACAGGTCCATGCGCCTGAGAACGGC
1 0\
7 0 1 1 1 9 2 0
0 2 7 2
1 0 1 6
S S A G C T
A
G
C
T
110.275
403 4 11
ln 1 0.34264 3 40JC
p
d
Example from 4.3
11 observed sub.
13.7 sub. estimated.
0
1
S : ACTTGTCGGATGATCAGCGGTCCATGCACCTGACAACGGT
S : ACATGTTGCTTGACGACAGGTCCATGCGCCTGAGAACGGC
1 0\
7 0 1 1 1 9 2 0
0 2 7 2
1 0 1 6
S S A G C T
A
G
C
T
110.275
403 4 11
ln 1 0.34264 3 40JC
p
d
Performance of JC distance (Homework Problem 4)
Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α.
Performance of JC distance (Homework Problem 4)
Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α.
Count the number of base substitutions occurred.
Performance of JC distance (Homework Problem 4)
Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α.
Count the number of base substitutions occurred.
Compute the Jukes-Cantor distance of the initial and finial sequence.
Performance of JC distance (Homework Problem 4)
Write a program to simulate of the mutations of a sequence for t time step using the Jukes-Cantor model with parameter α.
Count the number of base substitutions occurred.
Compute the Jukes-Cantor distance of the initial and finial sequence.
Compare the actual number of base substitutions and the estimation from the Jukes-Cantor distance.
Performance of JC distance (Homework Problem 4)
Maple: Strings Handling II
Concatenating two strings
Maple: Strings Handling II
However, no “re-assignment”.
Classwork
Work on HW #1, 2
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