  We study theoretically the in vitro evolution of a DNA sequence by binding to
a transcription factor. Using a simple model of protein-DNA binding and
available binding constants for the Mnt protein, we perform large-scale,
realistic simulations of evolution starting from a single DNA sequence. We
identify different parameter regimes characterized by distinct evolutionary
behaviors. For each regime we find analytical estimates which agree well with
simulation results. For small population sizes, the DNA evolutional path is a
random walk on a smooth landscape. While for large population sizes, the
evolution dynamics can be well described by a mean-field theory. We also study
how the details of the DNA-protein interaction affect the evolution.
