  Recent studies of in vitro evolution of DNA via protein binding indicate that
the evolution behavior is qualitatively different in different parameter
regimes. I here present a general theory that is valid for a wide range of
parameters, and which reproduces and extends previous results. Specifically,
the mean-field theory of a general translation-invariant model can be reduced
to the basic diffusion equation with a dynamic boundary condition. The simple
analytical form yields both quantitatively accurate predictions and valuable
insight into the principles involved. In particular, I introduce a cutoff
criterion for finite populations that illustrates both of these qualities.
