Combined Sum of Squares Penalties for Molecular Divergence Time Estimation

Estimates of molecular divergence times when rates of evolution vary require the assumption of a model of rate change. Brownian motion is one such model, and since rates cannot become negative, a log Brownian model seems appropriate. Divergence time estimates can then be made using weighted least squares penalties. As sequences become long, this approach effectively becomes equivalent to penalized likelihood or Bayesian approaches. Different forms of the least squares penalty are considered to take into account correlation due to shared ancestors. It is shown that a scale parameter is also needed since the sum of squares changes with the scale of time. Errors or uncertainty on fossil calibrations, may be folded in with errors due to the stochastic nature of Brownian motion and ancestral polymorphism, giving a total sum of squares to be minimized. Applying these methods to placental mammal data the estimated age of the root decreases from 125 to about 94 mybp. However, multiple fossil calibration points and relative molecular divergence times inflate the sum of squares more than expected. If fossil data are also bootstrapped, then the confidence interval for the root of placental mammals varies widely from ~70 to 130 mybp. Such a wide interval suggests that more and better fossil calibration data is needed and/or better models of rate evolution are needed and/or better molecular data are needed. Until these issues are thoroughly investigated, it is premature to declare either the old molecular dates frequently obtained (e.g. > 110 mybp) or the lack of identified placental fossils in the Cretaceous, more indicative of when crown-group placental mammals evolved.