Transcription factor binding kinetics constrain noise suppression via negative feedback

Negative autoregulation, where a transcription factor regulates its own expression by preventing transcription, is commonly used to suppress fluctuations in gene expression. Recent single molecule in vivo imaging has shown that it takes significant time for a transcription factor molecule to bind its chromosomal binding site. Given the slow association kinetics, transcription factor mediated feedback cannot at the same time be fast and strong. Here we show that with a limited association rate follows an optimal transcription factor binding strength where noise is maximally suppressed. At the optimal binding strength the binding site is free a fixed fraction of the time independent of the transcription factor concentration. One consequence is that high-copy number transcription factors should bind weakly to their operators, which is observed for transcription factors in Escherichia coli. The results demonstrate that a binding site's strength may be uncorrelated to its functional importance.