MAT 4830 Mathematical Modeling 4.4 Matrix Models of Base Substitutions II .
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Transcript of MAT 4830 Mathematical Modeling 4.4 Matrix Models of Base Substitutions II .
Markov Models
Review of Eigenvalues and Eigenvectors An example a Markov model. Specific Markov models for base
substitution:• Jukes-Cantor Model
• Kimura Models (Read)
Recall
Characteristic polynomial of A
Eigenvalues of A
Eigenvectors of A
( ) det( )P A I
zeros of ( )P
( ) 0, 0A I x x
Recall
Use the transition matrix, we can estimate the base distribution vectors of descendent sequences by
An example of Markov model
, 1, 2,3,...kS k kp
1k kp Mp
Markov Models Assumption
What happens to the system over a given time step depends only on the state of the system and the transition matrix
Markov Models Assumption
What happens to the system over a given time step depends only on the state of the system and the transition matrix
In our case,
pk only depends on pk-1 and M 1k kp Mp
Markov Models Assumption
What happens to the system over a given time step depends only on the state of the system and the transition matrix
In our case,
pk only depends on pk-1 and M Mathematically, it implies
1k kp Mp
1 1 2 2 0 0
1 1
|
|
k k k k k k
k k k k
P S s S s S s S s
P S s S s
Markov Matrix
All entries are non-negative. column sum = 1.
| | | |
| | | ||
| | | |
| | | |
A A A G A C AT
G A G G G C G Ti j
C A C G C C C T
T A T G T C T T
P P P P
P P P PM P
P P P P
P P P P
Jukes-Cantor Model
Additional Assumptions
• All bases occurs with equal prob. in S0.
0
1 1 1 1
4 4 4 4
T
p
Jukes-Cantor Model
Additional Assumptions• Base substitutions from one to another are
equally likely.| | | |
| | | ||
| | | |
| | | |
| , for 3
A A A G A C AT
G A G G G C G Ti j
C A C G C C C T
T A T G T C T T
i j
P P P P
P P P PM P
P P P P
P P P P
P i j
constant
Jukes-Cantor Model
| | | |
| | | ||
| | | |
| | | |
| , for 3
A A A G A C AT
G A G G G C G Ti j
C A C G C C C T
T A T G T C T T
i j
P P P P
P P P PM P
P P P P
P P P P
P i j
constant
|
|
1 / 3 / 3 / 3
/ 3 1 / 3 / 3
/ 3 / 3 1 / 3
/ 3 / 3 / 3 1
, for 3
i j
i j
M P
P i j
constant
Jukes-Cantor Model
1 prob. of no base sub. in a site for 1 time step
prob. of having base sub. in a site for 1 time step
rate of base sub. sub. per site per time step
Observation #1
Mutation Rate
Mutation rates are difficult to find. Mutation rate may not be constant. If constant, there is said to be a
molecular clock More formally, a molecular clock
hypothesis states that mutations occur at a constant rate throughout the evolutionary tree.
Observation #2
|
0
1
1 / 3 / 3 / 3
/ 3 1 / 3 / 3
/ 3 / 3 1 / 3
/ 3 / 3 / 3 1
1 1 1 1
4 4 4 4
?
? for 1,2,3,...
i j
T
k
M P
p
p
p k
Observation #2
0
1
1
41 / 3 / 3 / 3 1
/ 3 1 / 3 / 3 4 ?/ 3 / 3 1 / 3 1
4/ 3 / 3 / 3 11
4
?
? for 1, 2,3,...k
Mp
p
p k
Observation #2
The proportion of the bases stay constant (equilibrium)
What is the relation between p0 and M?
Example 1
What proportion of the sites will have A in the ancestral sequence and a T in the descendent one time step later?
| 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
i jM P p
Example 2
What is the prob. that a base A in the ancestral seq. will have mutated to become a base T in the descendent seq. 100 time steps later?
Example 2
What is the prob. that a base A in the ancestral seq. will have mutated to become a base T in the descendent seq. 100 time steps later?
1
100100 0
k kp Mp
p M p
Example 2100
100 0p M p
100100 0
p M p
1004,1 100 0( | )M "Must be" P S T S A
100100 0
100100 0
100100 0
100100 0
[ ]
( | )
[ ]
( | )
If M P S i S j
then p M p
If p M p
then M P S i S j
Example 2100
100 0p M p
100100 0
p M p
1004,1 100 0( | )M "Must be" P S T S A
100100 0
100100 0
100100 0
100100 0
[ ]
( | )
[ ]
( | )
If M P S i S j
then p M p
If p M p
then M P S i S j
For general n, can be prove by inductive arguments.
Example 2 (Book’s Solutions)
0t
tp M p
0
ttp M p
0( | ) ?tP S T S A
,
1, 2,3, 4
i
i
Find the eigenvalues and the
corresponding eigenvectors v
for i
Example 2 (Book’s Solutions)
0t
tp M p
0
ttp M p
,
1, 2,3, 4
i
i
Find the eigenvalues and the
corresponding eigenvectors v
for i
1
0
0
0
tM
Example 2 (Book’s Solutions)
0t
tp M p
,
1, 2,3, 4
i
i
Find the eigenvalues and the
corresponding eigenvectors v
for i
1
0
0
0
tM
1 2 3 4
1
0 1 1 1 1
0 4 4 4 4
0
v v v v
0
ttp M p
Example 2 (Book’s Solutions)
0t
tp M p
,
1, 2,3, 4
i
i
Find the eigenvalues and the
corresponding eigenvectors v
for i
1
0
0
0
tM
1 2 3 4
1
0 1 1 1 1
0 4 4 4 4
0
v v v v
0
ttp M p
Example 2 (Book’s Solutions)
0t
tp M p
1
0
0
0
tM
1 2 3 4
1 1 2 2 3 3 4 4
1
0 1 1 1 1
0 4 4 4 4
0
1 1 1 1
4 4 4 4
1 3 31
4 4 4
1 1 31
4 4 4
1 1 31
4 4 4
1 1 31
4 4 4
t t t t t
t t t t
t
t
t
t
M M v M v M v M v
v v v v
0
ttp M p
Example 2 (Book’s Solutions)
1 3 3 1 1 3 1 1 31 1 1
4 4 4 4 4 4 4 4 4
1 1 3 1 3 3 1 1 31 1 1
4 4 4 4 4 4 4 4 4
1 1 3 1 1 3 1 3 31 1 1
4 4 4 4 4 4 4 4 4
1 1 3 1 1 3 1 11 1 1
4 4 4 4 4 4 4 4
t t t
t t t
t
t t t
t t
M
1 1 31
4 4 4
1 1 31
4 4 4
1 1 31
4 4 4
3 1 3 31
4 4 4 4
t
t
t
t t
Our Solutions
1
21 1
1
Theorem:
Suppose is a symmetric matrix with eigenvalues and the
corresponding eigenvectors .
0
Let [ ] and
0
Then,
i
i
n
n
M
v
P v v v D
D P MP
Our Solutions
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 1 11 1 1
4 4 3 4 4 3 4 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
4 1 3 41
3 4 4 3
t
t
t
t t
Homework Problem 2
Although the Jukes-Cantor model assumes , a Jukes-Cantor transition matrix could describe mutations even a different . Write a Maple program to investigate the behavior of .
0 0.25 0.25 0.25 0.25T
p
kp
0p
Homework Problem 3 Read and understand the Kimura 2-
parameters model. Read the Maple Help to learn how to find
eigenvalues and eigenvectors. Suppose M is the transition matrix
corresponding to the Kimura 2-parameters model. Find a formula for Mt by doing experiments with Maple. Explain carefully your methodology and give evidences.