Fast estimation of the difference between two PAM/JTT evolutionary distances in triplets of homologous sequences

Abstract

Background: The estimation of the difference between two evolutionary distances within a triplet of homologs is a common operation that is used for example to determine which of two sequences is closer to a third one. The most accurate method is currently maximum likelihood over the entire triplet. However, this approach is relatively time consuming.

Results: We show that an alternative estimator, based on pairwise estimates and therefore much faster to compute, has almost the same statistical power as the maximum likelihood estimator. We also provide a numerical approximation for its variance, which could otherwise only be estimated through an expensive re-sampling approach such as bootstrapping. An extensive simulation demonstrates that the approximation delivers precise confidence intervals. To illustrate the possible applications of these results, we show how they improve the detection of asymmetric evolution, and the identification of the closest relative to a given sequence in a group of homologs.

Conclusion: The results presented in this paper constitute a basis for large-scale protein cross-comparisons of pairwise evolutionary distances.

Figure 1

Unrooted tree topology of all triplets of homologs. Sequences X, Y and Z originating from O. The problem addressed here is the estimation of the difference Δ = d_XY - d_XZ = d_OY - d_OZ

Figure 2

Scatter plots comparing the variance estimators. The upper-left plot shows the strong agreement between σ²($\widehat{\Delta }$_triplet) and our approximation σ²($\widehat{\Delta }$_pairwise). From the upper-right and the lower-left plots, it can be seen that both have similar correlation with ${\sigma }_{bootstrap}^2$($\widehat{\Delta }$_pairwise). Finally, the lower-right plot confirms that variance estimation under the assumption of independence can yield a large overestimation of the correct variance.

Figure 3

Detection of asymmetric evolution. Detection of Asymmetric Evolution. Comparison between the results of Kellis et al. and the three variants of closer, with k = 1.96. The circles separate cases of significant asymmetry (inside) from insignificant asymmetry (outside). For instance, there were 92 cases where all three variants of closer reported significant asymmetry, while the method of Kellis et al. did not detect significant asymmetry.

Figure 4

Tree randomly generated for closest homolog simulation. Example of a random tree (see text for description of the procedure) used to compare the different methods to infer the closest homolog to each leaf. Distances indicated are in PAM units.

Figure 5

Identification of the closest homolog. Identification of the closest homolog: comparison between methods using alignment score (1), distance with assumption of independence (2) and distance using our variance approximation (3), on simulated data.