Genetics 540 Models of DNA evolution, part 1

Joe Felsenstein Department of Genetics University of Washington joe@genetics Change of DNA sequence of a species ... is actually mutation, then substitution, in a population

T

TTTT C

C A u/3 G u/3 u/3 u/3 u/3 C T u/3

The Jukes-Cantor model (1969) the simplest symmetrical model of DNA evolution Transition probabilities under the Jukes-Cantor model • All sites change independently • All sites have the same stochastic process working at them • Make up a fictional kind of event, such that when it happens the site changes to one of the 4 bases chosen at random (equiprobably) 4 • Assertion: Having these events occur at rate 3u is the same as having the Jukes-Cantor model events occur at rate u • The probability of none of these fictional events happens in 4 time t is exp(−3ut) • No matter how many of these fictional events occur, provided it is not zero, the chance of ending up at a particular base is 1 4. • Putting all this together, the probability of changing to C, given the site is currently at A, in time t is

4

1 ¡ Prob (C|A, t) = 1 − e−3ut 4 while

4 4

1 ¡ Prob (A|A, t) = e−3t + 1 − e−3ut 4 or 4

1 ¡ Prob (A|A, t) = 1 + 3e−3ut 4 so that the total probability of change is

4

3 ¡ Prob (change|t) = 1 − e−3ut 4 1

0.75 Differences 0.49 per site

0 0 0.7945 Branch length

The fraction of sites differing after branches of different length, under the Jukes-Cantor model a A G

b b b b

C T a

Kimura’s (1980) K2P model of DNA change, which allows for different rates of transitions and transversions, A similar derivation can be done for the K2P model. It involves making two kinds of fictional events: • I. At rate α, if the site has a purine (A or G), choose one of the two purines at random and change to it. If the site has a pyrimidine (C or T), choose one of the pyrimidines at random and change to it. • II. At rate β, choose one of the 4 bases at random and change to it. By proper choice of α and β one can achieve the overall rate of change and Ts/Tn ratio R you want. For rate of change 1,

the transition probabilities (warning: terminological tangle).

1 1 1 R+2

Prob (transition|t) = 4 − 2 exp − R+1 t ¡

1 2 ¡

+4 exp −R+1t

1 1 2 ¡ Prob (transversion|t) = 2 − 2 exp −R+1t . (1) (the transversion probability is the sum of the probabilities of both kinds of transversions). 0.60 Total differences

0.50 Transitions 0.40

Differences 0.30

0.20 Transversions

0.10 R = 10 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (branch length)

Fractions of transitions and transversions expected in different amounts of branch length under the K2P model, for Ts/Tn = 10 0.70 Total differences 0.60

0.50 Transversions 0.40

Differences 0.30 Transitions 0.20

0.10 R = 2 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (branch length)

Fractions of transitions and transversions expected in different amounts of branch length under the K2P model, for Ts/Tn = 2 Other commonly used models include: Two models that specify the equilibrium base frequencies (you provide the frequencies πA, πC, πG, πT and they are set up to have an equilibrium which achieves them), and also let you control the transition/transversion ratio: The Hasegawa-Kishino-Yano (1985) model:

to : A G C T from : A − απG + βπG απC απT G απA + βπA − απC απT C απA απG − απT + βπT T απA απG απC + βπC −

My F84 model:

to : A G C T from : πG A − απG + β π απC απT π R G απ + β A − απ απ A πR C T βπT C απA απG − απT + π π Y T απ απ απ + β C − A G C πY where πR = πA + πG and πY = πC + πT (The equilibrium frequencies of purines and pyrimidines) Both of these models have formulas for the transition probabilities, and both are subcases of a slightly more general class of models, the Tamura-Nei model (1993). There is also the General Time-Reversible model (GTR), which maintains “detailed balance” so that the probability of starting at (say) A and ending at (say) T in evolution is the same as the probability of starting at T and ending at A:

to : A G C T from :

A − απG βπC γπT G απA − δπC πT C βπA δπG − υπT T γπA πG υπC −

And there is of course the general 12-parameter model which has arbitrary rates for each of the 12 possible changes (from each of the 4 nucleotides to each of the 3 others). Neither of these has formulas for the transition probabilities, but those can be done numerically. There are many other models, but these are the most widely- used ones. Here is a general scheme of which models are subcases of which other ones:

General 12−parameter model (12)

General time−reversible model (9)

Tamura−Nei (6)

HKY (5) F84 (5)

Kimura K2P (2)

Jukes−Cantor (1) α = 0.25 cv = 2

α = 11.1111 α = 1 cv = 0.3

frequency cv = 1

0 0.5 1 1.5 2 rate

Gamma distributions with mean 1 and different coefficients of variation (standard deviation / mean). α = 1/CV 2 is the “shape parameter” of the Gamma distribution References:

Barry, D., and J. A. Hartigan. 1987. Statistical analysis of hominoid molecular evolution. Statistical Science 2: 191-210.

Felsenstein, J. and G. A. Churchill. 1996. A hidden Markov model approach to variation among sites in rate of evolution Molecular Biology and Evolution 13: 93-104.

Hasegawa, M., H. Kishino, and T. Yano. 1985. Dating of the human-ape splitting by a molecular clock of mitochondrial DNA. Journal of Molecular Evolution 22: 160-174.

Jin, L. and M. Nei. 1990. Limitations of the evolutionary parsimony method of phylogenetic analysis. Molecular Biology and Evolution 7: 82-102.

Jukes, T. H. and C. Cantor. 1969. Evolution of protein molecules. pp. 21-132 in Mammalian Protein Metabolism, ed. M. N. Munro. Academic Press, New York.

Kimura, M. 1980. A simple model for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. Journal of Molecular Evolution 16: 111-120.

Lanave, C., G. Preparata, C. Saccone, and G. Serio. 1984. A new method for calculating evolutionary substitution rates. Journal of Molecular Evolution 20: 86-93. Lockhart, P. J., M. A. Steel, M. D. Hendy, and D. Penny. 1994. Recovering evolutionary trees under a more realistic model of sequence evolution. Molecular Biology and Evolution 11: 605- 612.

Olsen, G. J. 1987. Earliest phylogenetic branchings: comparing rRNA-based evolutionary trees inferred with various techniques. Cold Spring Harbor Symposia on Quantitative Biology 52: 825-837.

Tamura, K. and M. Nei. 1993. Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees. Molecular Biology and Evolution 10: 512-526.

Waddell, P. J. and M. A. Steel. 1997. General time-reversible distances with unequal rates across sites: mixing Γ and inverse Gaussian distributions with invariant sites. Molecular Phylogenies and Evolution 8: 398-414.

Yang, Z. 1994. Maximum likelihood phylogenetic estimation from DNA sequenc es with variable rates over sites: approximate methods. Journal of Molecular Evolution 39: 306-314.

Yang, Z. 1995. A space-time process model for the evolution of DNA sequen ces. Genetics 139: 993-1005.