Covariance of maximum likelihood evolutionary distances between sequences aligned pairwise

Background: The estimation of a distance between two biological sequences is a fundamental process in molecular evolution. It is usually performed by maximum likelihood ( ML) on characters aligned either pairwise or jointly in a multiple sequence alignment ( MSA). Estimators for the covariance of pairs from an MSA are known, but we are not aware of any solution for cases of pairs aligned independently. In large-scale analyses, it may be too costly to compute MSAs every time distances must be compared, and therefore a covariance estimator for distances estimated from pairs aligned independently is desirable. Knowledge of covariances improves any process that compares or combines distances, such as in generalized least-squares phylogenetic tree building, orthology inference, or lateral gene transfer detection. Results: In this paper, we introduce an estimator for the covariance of distances from sequences aligned pairwise. Its performance is analyzed through extensive Monte Carlo simulations, and compared to the well-known variance estimator of ML distances. Our covariance estimator can be used together with the ML variance estimator to form covariance matrices. Conclusion: The estimator performs similarly to the ML variance estimator. In particular, it shows no sign of bias when sequence divergence is below 150 PAM units ( i. e. above similar to 29% expected sequence identity). Above that distance, the covariances tend to be underestimated, but then ML variances are also underestimated.