Transcription factor (TF) proteins rapidly locate unique target sites on long genomic DNA molecules - and bind to them - during gene regulation. The search mechanism is known to involve a combination of three-dimensional diffusion through the bulk of the cell and one-dimensional sliding diffusion along the DNA. It is believed that the surprisingly high target binding rates of TF proteins relies on conformational fluctuations of the protein between a mobile state that is insensitive to the DNA sequence and an immobile state that is sequence-sensitive. Since TFs are not able to consume free energy during their search to obtain DNA sequence information, the Second Law of Thermodynamics must impose a strict limit on the efficiency of passive search mechanisms. In this article, we use a simple model for the protein conformational fluctuations to obtain the shortest binding time consistent with thermodynamics. The binding time is minimized if the spectrum of conformational fluctuations that take place during the search is impedance-matched to the large-scale conformational change that takes place at the target site. For parameter values appropriate for bacterial TF, this minimum binding time is within an order-of-magnitude of a limiting binding time corresponding to an idealized protein with instant target recognition. Numerical estimates suggest that typical bacteria operate in this regime of optimized conformational fluctuations.
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