can also be used to model rate heterogeneity across sites (Felsenstein and Churchill 1996).
These allow rate correlations to be made between adjacent sites.
Invariant sitesIncorporation of invariant sites parameters into the substitution model decreases the variance
of estimated gamma values (Gu et al. 1995). In order to obtain the most unbiased gamma
estimate, invariant sites should be included, as this will provide most accurate branch length
estimates (Lockhart et al. 1994).
Secondary structureSecondary structure models attempt to account for evolutionary dependence between
nucleotide sites opposite one another in RNA stem regions (Muse 1995; Tillier and Collins
1995). The complicated secondary structure of RNA molecules can lead to compensatory
changes and thus a single substitution may result in two substitutions (Hickson et al. 1996)
widely separated in the sequences. The secondary structure of proteins can also influence
amino acid replacement rates (Thorne 2000). However, Goldman et al. (1998) showed that
solvent accessibility correlates more strongly with amino acid replacement than secondary
structure. They found that replacement rates at sites on the surface of globular proteins are
about twice the rates at sites that are less accessible to solvents. Models have been proposed
to account for separate amino acid replacement in differing structural environments (see
Thorne et al. 1996). Substitution matrices which take these into account provide more
realistic branch length estimates.
Methodology for optimizing parametersBayesian inferenceAn increasingly popular approach to determining divergence times involves the use of
likelihood models within a Bayesian framework (Huelsenbeck et al. 2001; Lewis 2001).
Bayesian approaches take prior parameter values as probability distributions, modify them
given the data available, and then calculate a posteriori distribution, reflecting the level of
uncertainty in the parameter after viewing the data. Bayesian approaches can produce
probabilities for hypotheses of interest. Likelihoods alone are harder to interpret in this
context. These represent the probability of the data given the hypothesis rather than the
(Bayesian) probability of the hypothesis given the data. An argument against the Bayesian
approach is that the choice of the prior distribution, to which the a posteriori distribution is
very sensitive, can be too subjective (details in Shoemaker et al. 1999). Furthermore,
implementing Bayesian methods can be very complicated, especially where the parameter
space is complex or the dimensionality high, or both.
58 RICHARD A.FORTEY ET AL.