Characterizing fitness mutations – the elusive raw material of evolution – from population samples taken in real time

We present a standard model of evolution by natural selection and mutation in integro-pde form. We show that Fourier transform of the model reduces to a variant of the transport equation and admits exact solution. We employ this solution, together with empirical characteristic functions, to estimate the distribution of fitness effects (DFE) of newly-arising mutations, given fitness measurements of random samples taken from evolving populations. We derive non-parametric methods to determine when the DFE undergoes major changes. We apply our methods to evolving populations of E coli.