Week 5: Distance methods, DNA and protein models

Genome 570

February, 2010

Week 5: Distance methods, DNA and protein models – p.1/34

Kimura 2-parameter model

A G

C T

α

α

β β β β

Week 5: Distance methods, DNA and protein models – p.2/34

Parameters of K2P in terms of transition/transversion rati o

α = RR+1

β =(

12

)1

R+1

Week 5: Distance methods, DNA and protein models – p.3/34

Transition [sic] probabilities for K2P

Prob (transition|t) = 14− 1

2exp

(− 2R+1

R+1t)

+ 14exp

(− 2

R+1t)

Prob (transversion|t) = 12− 1

2exp

(− 2

R+1t).

Week 5: Distance methods, DNA and protein models – p.4/34

Transition and transversion when R = 10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Total differences

Transitions

Transversions

Time (branch length)

Diff

eren

ces

R = 10

Week 5: Distance methods, DNA and protein models – p.5/34

Transition and transversion when R = 2

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Transversions

Transitions

Total differences

Diff

eren

ces

Time (branch length)

R = 2

Week 5: Distance methods, DNA and protein models – p.6/34

ML estimates for the K2P model

t = −14ln

[(1 − 2Q)(1 − 2P − Q)2

]

R = − ln(1−2P−Q)− ln(1−2Q) − 1

2

Week 5: Distance methods, DNA and protein models – p.7/34

Likelihood for two species under the K2P model

L = Prob (data | t,R)

=(

14

)n(1 − P − Q)n−n1−n2 Pn1

(12Q

)n2

where n1 is the number of sites differing by transitions, and n2 is thenumber of sites differing by transversions. P and Q are the expectedfractions of transition and transversion differences, as given by theexpressions three screens above.

Week 5: Distance methods, DNA and protein models – p.8/34

The Tamura/Nei model, F84, and HKY

To : A G C TFrom :

A − αRπG/πR + βπG βπC βπT

G αRπA/πR + βπA − βπC βπT

C βπA βπG − αYπT/πY + βπT

T βπA βπG αYπC/πY + βπC −

For the F84 model, αR = αY

For the HKY model, αR/αY = πR/πY

Week 5: Distance methods, DNA and protein models – p.9/34

Transition/transversion ratio for the Tamura-Nei model

Ts = 2 αR πA πG / πR + 2 αY πC πT / πY

+ β ( πAπG + 2πC πT)

Tv = 2 β πR πY

To get Ts/Tv = R and Ts + Tv = 1,

β =1

2πRπY(1 + R)

αY =πRπYR − πAπG − πCπT

(1 + R) (πYπAπGρ + πRπCπT)

αR = ρ αY

Week 5: Distance methods, DNA and protein models – p.10/34

Using fictional events to mimic the Tamura-Nei model

We imagine two types of events:

Type I:If the existing base is a purine, draw a replacement from apurine pool with bases in relative proportions πA : πG . Thisevent has rate αR.If the existing base is a pyrimidine, draw a replacement from apyrimidine pool with bases in relative proportions πC : πT .This event has rate αY.

Type II: No matter what the existing base is, replace it by a basedrawn from a pool at the overall equilibrium frequencies:πA : πC : πG : πT. This event has rate β.

Week 5: Distance methods, DNA and protein models – p.11/34

Transition [sic] probabilities with the Tamura-Nei model

If the branch starts with a purine:No events exp(−(αR + β)t)

Some type I, no type II exp(−β t) (1 − exp(−αR t))

Some type II 1 − exp(−β t)

If the branch starts with a pyrimidine:No events exp(−(αY + β) t)

Some type I, no type II exp(−β t) (1 − exp(−αY t))

Some type II 1 − exp(−β t)

Week 5: Distance methods, DNA and protein models – p.12/34

A transition probability

So if we want to compute the probability of getting a G given that a branchstarts with an A, we add up

The probability of no events, times 0 (as you can’t get a G from an Awith no events)

The probability of “some type I, no type II” times πG/πY (as the lasttype I event puts in a G with probability equal to the fraction of G’sout of all purines).

The probability of “some type II” times πG (as if there is any type IIevent, we thereafter have a probability of G equal to its overallexpected frequency, and further type I events don’t change that).

Week 5: Distance methods, DNA and protein models – p.13/34

A transition probability

So that, for example

Prob (G|A, t) =

exp(−β t) (1 − exp(−αR t)) πG

πR

+ (1 − exp(−β t)) πG

Week 5: Distance methods, DNA and protein models – p.14/34

A more compact expression

More generally, we can use the Kronecker delta notation δij and the“Watson-Kronecker” notation εij to write

Prob (j | i, t) =

exp(−(αi + β)t) δij

+exp(−βt) (1 − exp(−αit))(

πjεijP

k εjkπk

)

+(1 − exp(−βt)) πj

where δij is 1 if the two bases i and j are different (0 otherwise),and εij is 1 if, of the two bases i and j, one is a purine and one is apyrimidine (0 otherwise).

Week 5: Distance methods, DNA and protein models – p.15/34

Reversibility and the GTR model

πi Prob (j|i, t) = πj Prob (i|j, t)

The general time-reversible model:

To : A G C TFrom :

A − πG α πC β πT γG πA α − πC δ πT εC πA β πG δ − πT ηT πA γ πG ε πC η −

Week 5: Distance methods, DNA and protein models – p.16/34

The GTR model

A G

C T

βπβπ

γπ

γπαπ

απ

δπ

δπεπ

επ

ηπ

G

A

T

AA

C

G

C

T

G

T

ηπ C

Week 5: Distance methods, DNA and protein models – p.17/34

Standardizing the rates

2πA πG α + 2 πA πC β + 2 πA πT γ

+ 2 πG πC δ + 2 πG πT ε + 2 πC πT η = 1

Week 5: Distance methods, DNA and protein models – p.18/34

General Time Reversible models – inferenceA data example (simulated under a K2P model, true distance 0.2

transition/transversion ratio = 2

A G C T total

A 93 13 3 3 112

G 10 105 3 4 122

C 6 4 113 18 141

T 7 4 21 93 125

total 116 126 140 118 500

Week 5: Distance methods, DNA and protein models – p.19/34

Averaging across the diagonal ...

A G C T total

A 93 11.5 4.5 5 114

G 11.5 105 3.5 4 124

C 4.5 3.5 113 19.5 140.5T 5 4 19.5 93 121.5

total 114 124 140.5 121.5 500

Week 5: Distance methods, DNA and protein models – p.20/34

Dividing each column by its sum

(column, because Pij is to be the probability of change from j to i )

P =

0.815789 0.0927419 0.0320285 0.04115230.100877 0.846774 0.024911 0.03292180.0394737 0.0282258 0.80427 0.1604940.0438596 0.0322581 0.13879 0.765432

Week 5: Distance methods, DNA and protein models – p.21/34

Rate matrix from the matrix logarithm

If the rate matrix is A,

P = eA t

so that

At = log(P

)

=

−0.212413 0.110794 0.034160 0.0467260.120512 −0.174005 0.025043 0.0355540.0421002 0.028375 −0.236980 0.2055790.0498001 0.034837 0.177778 −0.287859

.

Week 5: Distance methods, DNA and protein models – p.22/34

Standardizing the rates

If we denote by D the diagonal matrix of observed base frequencies, andwe require that the rate of (potentially-observable) substitution is 1:

−trace(AD) = 1

We get:

t = −trace(AtD) = −trace(log(P)D

)

and that also gives us an estimate of the rate matrix:

A = log(P

) /− trace

(log(P)D

)

Week 5: Distance methods, DNA and protein models – p.23/34

The rate estimates

A =

−0.931124 0.485671 0.149741 0.2048260.528274 −0.762764 0.109776 0.1558520.184549 0.124383 −1.038820 0.9011680.218302 0.152710 0.779302 −1.261850

.

moderately close to the actual K2P rate matrix used in the simulationwhich was:

A =

−1 2/3 1/6 1/62/3 −1 1/6 1/61/6 1/6 −1 2/31/6 1/6 2/3 −1

(but if any of the eigenvalues of log(P) are negative, this doesn’t workand the divergence time is estimated to be infinite).

Week 5: Distance methods, DNA and protein models – p.24/34

The lattice of these modelsGeneral 12−parameter model (12)

General time−reversible model (9)

Kimura K2P (2)

Tamura−Nei (6)

HKY (5) F84 (5)

Jukes−Cantor (1)Week 5: Distance methods, DNA and protein models – p.25/34

Variation of rates of evolution across sites

L(t) =

sites∏

i=1

(∫∞

0

f(r) πniPmini

(r t) dr

)

Week 5: Distance methods, DNA and protein models – p.26/34

The Gamma distribution

f(r) =1

Γ(α) βαxα−1 e−

xβ

E[x] = α β

Var[x] = α β2

To get a mean of 1, set β = 1/α so that

f(r) =αα

Γ(α)rα−1 e−αr.

so that the squared coefficient of variation is 1/α.

Week 5: Distance methods, DNA and protein models – p.27/34

Gamma distributions

0.5 1 1.5 20rate

freq

uenc

yα

α = 0.25cv = 2

α = 1cv = 1

= 11.1111cv = 0.3

Week 5: Distance methods, DNA and protein models – p.28/34

Gamma rate variation in the Jukes-Cantor modelFor example, for the Jukes-Cantor distance, to get the fraction of sitesdifferent we do

DS =

∫∞

0

f(r)3

4

(1 − e−

43r ut

)dr

leading to the formula for D as a function of DS

D = −3

4α

[1 −

(1 −

4

3DS

)−1/α

]

Week 5: Distance methods, DNA and protein models – p.29/34

Gamma rate variation in other modelsFor many other distances such as the Tamura-Nei family, the transitionprobabilites are of the form

Pij(t) = Aij + Bij e−bt + Cij e−ct

and integrating termwise we can make use of the fact that

Er

[e−b r t

]=

(1 +

1

αb t

)−α

Week 5: Distance methods, DNA and protein models – p.30/34

Dayhoff’s PAM001 matrixA R N D C Q E G H I L K M F P S T Wala arg asn asp cys gln glu gly his ile leu lys met phe pro ser thr trp

A ala 9867 2 9 10 3 8 17 21 2 6 4 2 6 2 22 35 32 0R arg 1 9913 1 0 1 10 0 0 10 3 1 19 4 1 4 6 1 8N asn 4 1 9822 36 0 4 6 6 21 3 1 13 0 1 2 20 9 1D asp 6 0 42 9859 0 6 53 6 4 1 0 3 0 0 1 5 3 0C cys 1 1 0 0 9973 0 0 0 1 1 0 0 0 0 1 5 1 0Q gln 3 9 4 5 0 9876 27 1 23 1 3 6 4 0 6 2 2 0E glu 10 0 7 56 0 35 9865 4 2 3 1 4 1 0 3 4 2 0G gly 21 1 12 11 1 3 7 9935 1 0 1 2 1 1 3 21 3 0H his 1 8 18 3 1 20 1 0 9912 0 1 1 0 2 3 1 1 1I ile 2 2 3 1 2 1 2 0 0 9872 9 2 12 7 0 1 7 0L leu 3 1 3 0 0 6 1 1 4 22 9947 2 45 13 3 1 3 4K lys 2 37 25 6 0 12 7 2 2 4 1 9926 20 0 3 8 11 0M met 1 1 0 0 0 2 0 0 0 5 8 4 9874 1 0 1 2 0F phe 1 1 1 0 0 0 0 1 2 8 6 0 4 9946 0 2 1 3P pro 13 5 2 1 1 8 3 2 5 1 2 2 1 1 9926 12 4 0S ser 28 11 34 7 11 4 6 16 2 2 1 7 4 3 17 9840 38 5T thr 22 2 13 4 1 3 2 2 1 11 2 8 6 1 5 32 9871 0W trp 0 2 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 9976Y tyr 1 0 3 0 3 0 1 0 4 1 1 0 0 21 0 1 1 2V val 13 2 1 1 3 2 2 3 3 57 11 1 17 1 3 2 10 0

Week 5: Distance methods, DNA and protein models – p.31/34

The codon model

U C A G

U

C

A

G

phe

phe

leu

leu

leu

leu

leu

leu

ile

ile

ile

met

val

val

val

val

ser stop stop

U

C

C

U

U

C

A

G

A

G

A

G

U

C

A

G

UUU

UUC

UUA

UUG

CUU

CUC

CUA

CUG

AUU

AUC

AUA

AUG

GUU

GUC

GUA

GUG

UCA UAA UGA

changing, and to what

Probabilities of change vary dependingon whether amino acid is

Goldman & Yang, MBE 1994; Muse and Weir, MBE 1994

Week 5: Distance methods, DNA and protein models – p.32/34

A codon-based model of protein evolution

In each cell:Pij ij

(v) a

ijP is the probability of codon change

and is the probability that the change is accepted

where (v)

ij a

....

....

AAA AAC AAG AAT ACA ACC ACG ACT AGA AGC AGG

AAA

AAC

AAG

AAT

ACA

ACC

ACG

ACT

AGA

AGC

AGG

01010000

000

lys

lys

asn

asn

thr

thr

thr

thr

arg

ser

arg

observation:

Week 5: Distance methods, DNA and protein models – p.33/34

Considerations for a protein model

Making a model for protein evolution (a not-very-practical approach)

Use a good model of DNA evolution.

Use the appropriate genetic code.

When an amino acid changes, accept it with probability that declinesas the amino acids become more different.Fit this to empirical information on protein evolution.

Take into account variation of rate from site to site.Take into account correlation of rates in adjacent sites.

How about protein structure? Secondary structure? 3D structure?

(the first four steps are the “codon model” of Goldman and Yang, 1994and Muse and Gaut, 1994, both in Molecular Biology and Evolution. Thenext two are the rate variation machinery of Yang, 1995, 1996 andFelsenstein and Churchill, 1996).

Week 5: Distance methods, DNA and protein models – p.34/34