To test for functional constraint (case 1), I used equations 3-5 above to compute E[ΔS] = -1.61 and V[ΔS] = 2.77 for the Bcd matrix, using as the null hypothesis an HKY model with parameters (kappa = 2.26 and total evolutionary distance = 0.36 subs) estimated from an alignment of the entire regulatory region (see Methods).
Underestimation of branch length was correlated with underestimation of substitution model parameters, calculated as kappa (the transition/transversion ratio scaled by base frequencies) (Pearson's correlation coefficient = 0.343; P < 0.0001).
Furthermore, an increase in the parameter (kappa) (e.g., from (kappa =4) to (kappa =8)) leads to an enhancement of (V_{text {g}}) as (krightarrow 0).
Each weight represents a free model parameter.
On the other hand, the parameter kappa did not differ from zero (0.00) for FD.
Estimating model parameter differences.
To characterize the directional covariance between codon bias and intron content across species, we included three scaling parameters into the model B. These scaling parameters (kappa, lambda, and delta) were estimated for both the traits simultaneously.
When model parameters were not fixed, the median estimate of kappa was correct for 4-taxon datasets for branch lengths ≤ 1.2 substitutions/site, but overestimated (kappa = 4.8) for branch lengths of 1.4, with a wide range of estimates across 1 kb simulated datasets.
The constant variance random walk model has only one parameter, alpha, which describes the instantaneous variance of evolution [ 3, 65]; this model represents the default model with all branch length scaling parameters, kappa, lambda and delta, set as 1 [ 3, 65].
For this dataset, JMODELTEST selected the HKY+G substitution model with a transition-transversion-parameter kappa of 80.0366 and a gamma shape parameter alpha of 0.0740.
