Transcription factor binding sites vary in their specificity, both within and
between species. Binding specificity has a strong impact on the evolution of
gene expression, because it determines how easily regulatory interactions are
gained and lost. Nevertheless, we have a relatively poor understanding of what
evolutionary forces determine the specificity of binding sites. Here we address
this question by studying regulatory modules composed of multiple binding
sites. Using a population-genetic model, we show that more complex regulatory
modules, composed of a greater number of binding sites, must employ binding
sites that are individually less specific, compared to less complex regulatory
modules. This effect is extremely general, and it hold regardless of the
regulatory logic of a module. We attribute this phenomenon to the inability of
stabilising selection to maintain highly specific sites in large regulatory
modules. Our analysis helps to explain broad empirical trends in the yeast
regulatory network: those genes with a greater number of transcriptional
regulators feature by less specific binding sites, and there is less variance
in their specificity, compared to genes with fewer regulators. Likewise, our
results also help to explain the well-known trend towards lower specificity in
the transcription factor binding sites of higher eukaryotes, which perform
complex regulatory tasks, compared to prokaryotes