  Evolution of gene regulation is crucial for our understanding of the
phenotypic differences between species, populations and individuals.
Sequence-specific binding of transcription factors to the regulatory regions on
the DNA is a key regulatory mechanism that determines gene expression and hence
heritable phenotypic variation. We use a biophysical model for directional
selection on gene expression to estimate the rates of gain and loss of
transcription factor binding sites (TFBS) in finite populations under both
point and insertion/deletion mutations. Our results show that these rates are
typically slow for a single TFBS in an isolated DNA region, unless the
selection is extremely strong. These rates decrease drastically with increasing
TFBS length or increasingly specific protein-DNA interactions, making the
evolution of sites longer than ~10 bp unlikely on typical eukaryotic speciation
timescales. Similarly, evolution converges to the stationary distribution of
binding sequences very slowly, making the equilibrium assumption questionable.
The availability of longer regulatory sequences in which multiple binding sites
can evolve simultaneously, the presence of "pre-sites" or partially decayed old
sites in the initial sequence, and biophysical cooperativity between
transcription factors, can all facilitate gain of TFBS and reconcile
theoretical calculations with timescales inferred from comparative genetics.
