Models of Sequence Evolution and Maximum Likelihood Trees
Maximum-likelihood tree inference uses explicit assumptions about nucleotide substitution and variation among sites. Comparing Jukes-Cantor, HKY and GTR shows how their parameter constraints differ. Model choice can affect fitted branch lengths and the preferred topology.
What a learner can do afterwards
- Compare Jukes-Cantor, HKY and the general time-reversible model by the parameters each adds
- Explain how a gamma distribution models rate variation among sites and describe possible biases from ignoring that variation
- Interpret the likelihood of a tree given an alignment and a stated model
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The previous topic used a substitution model to score sequence data. A nucleotide model specifies rates between A, C, G and T in a rate matrix. Jukes-Cantor, or JC, assumes equal equilibrium base frequencies and equal rates between distinct bases. It still allows repeated substitutions; an observed difference need not represent just one change.
JC sets base frequencies equal and treats every nucleotide pair alike. HKY allows unequal equilibrium frequencies and a parameter that changes transition exchangeabilities relative to transversion exchangeabilities. Transitions exchange A with G or C with T; the other pairs are transversions. HKY allows a transition preference but does not require one.
GTR allows unequal equilibrium base frequencies and six pair exchangeabilities: AC, AG, AT, CG, CT and GT. A rate toward a base also depends on that destination base's equilibrium frequency, so opposite directions can have different rates. Fixing the overall rate scale leaves five free relative exchangeability parameters, not six independent directional rates.
Sites can have different relative substitution rates. A gamma distribution models this variation; its shape parameter controls the spread. With the mean fixed, smaller positive shape values mean greater variation. This concerns alignment positions, not separate rates for species. The bars show illustrative site rates, not a fitted gamma distribution.
These example rates differ among sites. Their mean is 1.
Ignoring site-rate variation can bias branch lengths and sometimes topology. Suppose a grouping disappears when HKY with equal site rates is replaced by HKY with gamma rate variation on the same alignment. The grouping is sensitive to the rate assumption. Its disappearance does not prove the second tree true. High ordinary bootstrap support under the first procedure does not settle the disagreement.
Likelihood is the probability of the observed alignment under a tree, model and parameter settings. Sum over unobserved ancestral bases, weighting assignments by their model probabilities. Maximum likelihood fits branch lengths and other parameters for candidate topologies, or branching patterns, and seeks the largest likelihood. Model changes can alter the fitted lengths and preferred topology, but need not do so.
If one model is a constrained case of another, the models are nested. On the same alignment, the richer model can fit at least as well if optimization reaches the maximum. Extra parameters can improve fit without making the model a better choice. Criteria such as AIC balance fit against parameter count. Check model fit and whether relationships remain stable under plausible models; neither richness nor a high score proves a tree true.
In GTR, a directional rate is the pair's exchangeability times the destination base's equilibrium frequency, with a common factor setting the overall rate scale. To compare opposite directions, cancel their shared positive exchangeability and scale factor. The rate ratio equals the ratio of the destination frequencies. A shared exchangeability therefore need not give equal directional rates.
JC, HKY and GTR impose different rate and frequency constraints. Gamma models variation among sites. Likelihood fits the alignment under a model; compare fit, complexity and sensitivity.
Lesson, questions and study cards written for this dataset.
Readings and study cards written for this dataset. Linked videos: Khan Academy and the American Museum of Natural History. The Museum video is all rights reserved.
More to read
Counting model parameters
JC, HKY and GTR constrain nucleotide substitutions differently. JC sets the four equilibrium base frequencies equal and treats every distinct base pair alike. HKY permits unequal frequencies and a parameter distinguishing transition from transversion exchangeabilities. GTR permits a separate exchangeability for each unordered pair: AC, AG, AT, CG, CT and GT. A pair exchangeability is not a directional substitution rate. In GTR, the rate toward a destination base also depends on that base's equilibrium frequency. Opposite directions can therefore have different rates even though they share an exchangeability. Six exchangeabilities are listed before fixing the overall rate scale. Once that scale is fixed, five relative exchangeability parameters remain free. The four equilibrium frequencies sum to one, leaving three free frequency parameters. These constraints matter when comparing model complexity.
Testing dependence on model assumptions
Keep the alignment fixed when examining model sensitivity. Fit each candidate model and its tree parameters rather than comparing arbitrary parameter settings. If a simpler model is a constrained case of a richer one, the richer model can attain at least the simpler model's maximum likelihood. That statement assumes the optimization reaches the maximum. A higher likelihood alone does not settle model choice. Extra parameters provide more flexibility. A criterion such as AIC balances fit against parameter count, while model-fit checks examine whether assumptions suit the data. Site-rate variation is a separate modeling choice from the pair constraints distinguishing JC, HKY and GTR. To investigate it, compare the same substitution family with equal site rates and with gamma rate variation. Report which relationships change and which persist. A changed topology reveals sensitivity; it does not prove which topology is historically true.
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