Dependence of DNA length on binding affinity between TrpR and trpO of DNA

We scrutinize the length dependency of the binding affinity of bacterial repressor TrpR protein to trpO (specific site) on DNA. A footprinting experiment shows that the longer the DNA length, the larger the affinity of TrpR to the specific site on DNA. This effect termed "antenna effect" might be interpreted as follows: longer DNA provides higher probability for TrpR to access to the specific site aided by one-dimensional diffusion along the nonspecific sites of DNA. We show that, however, the antenna effect cannot be explained while detailed balance holds among three kinetic states, that is, free protein/DNA, nonspecific complexes, and specific complex. We propose a working hypothesis that slow degree(s) of freedom in the system switch(es) different potentials of mean force causing transitions among the three states. This results in a deviation from detailed balance on the switching timescale. We then derive a simple reaction diffusion/binding model that describes the antenna effect on TrpR binding to its target operator. Possible scenarios for such slow degree(s) of freedom in TrpR-DNA complex are addressed.