Estimating the deviation from a molecular clock

We address the problem of estimating the degree to which the evolutionary history of a set of molecular sequences violates a strong molecular clock hypothesis.We quantify this deviation formally, by defining the “stretch” of a model tree with respect to the underlying ultrametric tree (indicated by time). We then define the “minimum stretch” of a dataset for a tree and show how this can be computed optimally in polynomial time. We also present a polynomial-time algorithm for computing a lower bound on the stretch of a given dataset for any tree. We then explore the performance of standard techniques in systematics for estimating the deviation of a dataset from a molecular clock. We show that standard methods, whether based upon maximum parsimony or maximum likelihood, can return infeasible values (i.e. values for the stretch which cannot be realized on a tree), and often under-estimate the true stretch. This suggests that current approximations of the degree to which data sets deviate from a molecular clock may significantly underestimate these deviations. We conclude with some suggestions for further research.