  We study competitive DNA sequence evolution directed by {\it in vitro}
protein binding. The steady-state dynamics of this process is well described by
a shape-preserving pulse which decelerates and eventually reaches equilibrium.
We explain this dynamical behavior within a continuum mean-field framework.
Analytical results obtained on the motion of the pulse agree with simulations.
Furthermore, finite population correction to the mean-field results are found
to be insignificant.
