Inference of Past Population Expansion from the Timing of Coalescence Events in a Gene Genealogy

We investigate the expected coalescent in populations growing exponentially. The distribution of expected times to coalescence events may show a linear relationship with a number of ancestral lineages, when the latter is subjected to the “epidemic transformation”. However, in a number of viral populations, upward curves are created when the epidemically transformed number of ancestral lineages is plotted against time. We consider possible causes of such upward curves. These include the possibility that a curved line is created through a transformation failure due to a sample size that is too large. We suggest a new formula for predicting such failure. The second cause is a population size increasing at an accelerating rate. However, the combination of recent coalescent events and an upward curve is created by an accelerating population increase only under restricted conditions. Specifically, such a pattern is expected only when, were population growth not to have accelerated, the transformation would have failed anyway. The third cause of nonlinearity arises in the estimated coalescent, as distinct from the real coalescent, if the mutation rate is small. However, coalescence times estimated from data typically give a straight line following epidemic transformation, but the rate of exponential increase, or r value, will be underestimated.