Determining evolutionary distances from highly diverged nucleic acid sequences: operator metrics

Operator metrics are explicitly designed to measure evolutionary distances from nucleic acid sequences when substitution rates differ greatly among the organisms being compared, or when substitutions have been extensive. Unlike lengths calculated by the distance matrix and parsimony methods, in which substitutions in one branch of a tree can alter the measured length of another branch, lengths determined by operator metrics are not affected by substitutions outside the branch. In the method, lengths (operator metrics) corresponding to each of the branches of an unrooted tree are calculated. The metric length of a branch reconstructs the number of (transversion) differences between sequences at a tip and a node (or between nodes) of a tree. The theory is general and is fundamentally independent of differences in substitution rates among the organisms being compared. Mathematically, the independence has been obtained because the metrics are eigenvectors of fundamental equations which describe the evolution of all unrooted trees. Even under conditions when both the distance matrix method or a simple parismony length method are shown to indicate lengths that are an order of magnitude too large or too small, the operator metrics are accurate. Examples, using data calculated with evolutionary rates and branchings designed to confuse the measurement of branch lengths and to camouflage the topology of the true tree, demonstrate the validity of operator metrics. The method is robust. Operator metric distances are easy to calculate, can be extended to any number of taxa, and provide a statistical estimate of their variances. The utility of the method is demonstrated by using it to analyze the origins and evolution of chloroplasts, mitochondria, and eubacteria.