Transcription factor specificity limits the number of DNA-binding motifs

We study the limits imposed by transcription factor specificity on the maximum number of binding motifs that can coexist in a gene regulatory network, using the SwissRegulon Fantom5 collection of 684 human transcription factor binding sites as a model. We describe transcription factor specificity using regular expressions and find that most human transcription factor binding site motifs are separated in sequence space by one to three motif-discriminating positions. We apply theorems based on the pigeonhole principle to calculate the maximum number of transcription factors that can coexist given this degree of specificity, which is in the order of ten thousand and would fully utilize the space of DNA subsequences. Taking into account an expanded DNA alphabet with modified bases can further raise this limit by several orders of magnitude, at a lower level of sequence space usage. Our results may guide the design of transcription factors at both the molecular and system scale.