1. Models of Sequence Evolution

2. JC69: Jukes-Cantor (1969) Model
Assumes all bases interchange with equal probabilities, equal base frequencies

3. F81: Felsenstein (1981) Model
Assumes all bases interchange with equal probabilities, base frequencies may vary.

4. HKY85: Hasegawa, Kishino, Yano (1985) Model
Assumes unequal transition and transversion probabilities, base frequencies may vary.

5. REV: General Time Reversible Model
Assumes unequal probabilities for all substitution types, base frequencies may vary.

7. Model # par. ln Lik JC69 1 -1227.45 F81 4 -1187.54 K2P 2 -1210.49
Models applied to a dataset of 13 HIV pol sequences (273nt)
Model
# par.
ln Lik
JC69 1
F81 4
K2P 2
HKY85 5
REV 9

8. Other Forms of Rate Heterogeneity
Variation from gene to gene
Variation from site to site within a gene
Synonymous vs synonymous rates
Spatial rate heterogeneity
Variation from lineage to lineage
Correlations among sets of sites

9. Site-to-site rate heterogeneity

10. Gamma models of site-to-site heterogeneity
Hierarchical model:
Evolutionary rates at individual sites are drawn from a gamma distribution
Given the rate at a particular site, sequence evolution follows one of the previously discussed models
Original idea: Uzzel and Corbin (1971)
First likelihood treatment was by Yang (1993)

12. REV+G: General Time Reversible Model with Gamma Rate Heterogeneity (Yang 1993)

13. “All” we are doing here is integrating the likelihood function over all possible rate values, with those values being weighted according to probabilities assigned by the gamma distribution.

14. Calculating the integral in the continuous case is very expensive.
Yang (1994) suggested “discretizing” the gamma distribution, and using the discrete form of the likelihood function.
The cost of calculation increases only linearly with the number of rate categories, N.

17. 13 HIV pol sequences

18. 4 alpha-spectrin sequences

19. Correlations among sites
Codons
Dinucleotides (secondary structure)
General idea: move from 4-state nucleotide models to 16- or 64-state models.

20. MG94: Muse and Gaut (1994)

21. GY94:Goldman and Yang(1994)

22. This approach can be combined with any nucleotide model
Rate heterogeneity can be added in the same way as with nucleotide models
Account for correlations among nucleotide sites within codons
Avoids the problematic notion of “degeneracy classes”
Necessary for rigorous estimates of synonymous and nonsynonymous substitution rates.

23. Muse and Gaut (1994) model

24. Consider the result of allowing gamma variation in rates over codons:
While they are allowed to have different magnitudes, the two classes of rates have the same distribution.
Synonymous rates are likely to be less variable over sites than are nonsynonymous rates.

25. Site-to-site rate variation, transitions and transversions with independent distributionsC G T A C G T

28. Muse and Gaut (1994) modification

29. Each site in the sequence has a (random) synonymous rate, and a (random) nonsynonymous rate, drawn from some bivariate distribution,
The likelihood is again integrated with respect to f:

30. As before, the likelihood function is discretized for computational feasibility:
The discretization process can be tricky in general. In all that follows, we assume that are independent gamma random variables, which allows discretization of each axis separately.

34. Goals of molecular evolutionary analyses
Understand the structure of
Are parameter values (i.e., rates) equal among different branches?
Is the structure of (e.g., TS/TV ratio) the same for different branches?
Are the values of such parameters “related” among different genes?

35. Likelihood function for homologous DNA sequencesB G x
is the collection of all parameters affecting the evolution of
sequences A, B, and G.
is the collection of all data (sequences A, B, and G).

36. Are evolutionary rates the same along two lineages?
Relative Rate Tests
Are evolutionary rates the same along two lineages? B A G

37. Versions of the relative rate test
Distance based (Wu and Li 1985)
For 2 clades (Li and Bousquet 1992)
Likelihood ratio (Muse and Weir 1992)
Nonparametric (Tajima 1993)

38. Likelihood-based RR TestG
Maximize L assuming
Maximize L without constraints
LRT has a chi-squared distribution if rates are equal
Note: Use of outgroup insures that the unknown divergence time is irrelevant.

39. Distance-based RR TestsG a b g

40. Nonparametric RR Test A B G

41. ndhF
rbcL
Nonsyn
Syn

42. Locus A
Locus B
Locus effect
Lineage effect
Lineage X Locus effect

43. Relative Ratio Tests Are “branch lengths” proportional among loci?
Muse and Gaut 1997; Muse et al. 1997; Huelsenbeck et al. 1997; Yang 1995

44. Relative Ratio Test
The null hypothesis is that the relative proportions of branch
lengths are the same for all loci. The proportion need not be
known a priori. 1 2 3 4 1 2 3 4