models of dna evolution

Models of DNA evolutionHow does DNA change, and how can we obtain distances?

The Jukes-Cantor model

Thomas H. Jukes (1906-1999)

King JL Jukes TH 1969. Non-DarwinianEvolution. Science 164: 788-798.

Charles R. Cantor (°1942)


A G

TC

u

u

u

u

u

u

in the JC model, each base in the sequence has an equal chance of changing, u, into one of the three other bases


A G

TC

u/3

u/3

u/3

u/3

u/3

u/3

fictionalising, each base has a chance of (4/3)u of changing to a base randomly drawn from all 4 possibilities


PrY=y = e-m my

y!

537 hits576 squaresaverage hit per square = 0.9323

probability of not being hit e-0.9323*0.93230

0!= = 0.3936

expected number of squares hit 226.74not at all

the probability of no event is given by the zero term of a Poisson distribution


PrY=y = e-m my

y!


probability being hit once e-0.9323*0.93231

1!= = 0.3670

expected number of squares hit 226.74 211.39 not at all 1x


PrY=y = e-m my

y!


probability of being hit twice e-0.9323*0.93232

2!= = 0.1711

expected number of squares hit 226.74 211.39 98.54not at all 1x 2x


PrY=y = e-m my

y!


probability of being hit four times e-0.9323*0.93234

4!= = 0.012

expected number of squares hit 226.74 211.39 98.54 30.62 7.13 1.6not at all 1x 2x 3x 4x 5+

observed number of squares hit 229 211 93 35 7 1


PrY=y = e-m my

y!

u/3

A G

TC

u/3

u/3

u/3

u/3

u/3

probability of no event = e-(4/3)ut

probability of ≥1 event = 1 - e-(4/3)ut

probability of C at the end of a branch that started with A = (¼)(1 - e-(4/3)ut)

probability that a site is differentat two ends of a branch = (¾)(1 - e-(4/3)ut)


branch length (ut)

0 1 2 3

diffe

renc

es p

er si

te

0.0

0.2

0.4

0.6

0.8

y = (¾)(1 - e-(4/3)ut)

the expected difference per site between two sequences increases with branch length but reaches a plateau at 0.75

The Jukes-Cantor model not using the J&C correction will distort the tree

A D

B C A B C D

A 0 0.57698 0.59858 0.70439

B 0.57698 0 0.24726 0.59858

C 0.59858 0.24726 0 0.57698

D 0.70439 0.59858 0.57698 0

the real tree expected uncorrected sequence differences

A D

BC

least squares tree


A G

TC

u/3

u/3

u/3

u/3

u/3

u/3

the J&C model assumes no difference in substitution rates between transversions and transitions

Kimura’s two-parameter model

A G

TC

a

b

a

b

b

b

R = number of transitionsnumber of transversions

= a2b

the Kimura model allows a difference in substitution rate between transversions and transitions


Prob (transition|t) = ¼ - ½ e + ¼ e - 2R+1

R+1 t 2R+1 t-

probability that a transition will occur in a time interval t

R = a2b


Prob (transition|t) = ¼ - ½ e + ¼ e - 2R+1

R+1 t 2R+1 t-

probability that any tranversion will occur in a time interval t

Prob (transversion|t) = ½ - ½ e

2R+1 t


transversions

transitions

total

R=10

Time (branch length)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Diffe

renc

es

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(50% different)

Time (branch length)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Diffe

renc

es

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


transversions

transitions

total

R=2

(50% different)

Tamura-Nei models

P(event type I)purine>purinepyrim>pyrim

P(event type II) random base A G C T

purine

A aR b -aRpG/pR

+ bpGbpC bpT

G aR baRpA/pR

+ bpA- bpC bpT

pyrimidine

C aY b bpA bpG -aYpT/pY +

bpT

T aY b bpA bpGaYpC/pY +

bpC-

pA,G,C,T: relative proportion of A,G,C,T in the poolpR= pA+ pG

pY = pC+ pT

the T&N models allow asymmetric base frequencies

The general time-reversible model (GTR)

A G C T

A - apG bpC gpT

G apA - dpC epT

C bpA dpG - hYpT

T gpA epG hpC -

The general 12-parameter model

A G C T

A - apG bpC gpT

G dpA - epC fpT

C gpA hpG - iYpT

T jpA kpG lpC -

models of dna evolution

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