Association kinetics with coupled three- and one-dimensional diffusion. Chain-length dependence of the association rate of specific DNA sites.

The effective target size of a specific binding site on a long polymeric chain can be considerably enhanced if the associating molecules slide along the chain in a one-dimensional diffusion. Such a mechanism has been proposed to explain the "faster than diffusion controlled" association of the lac repressor protein with the operator site on a long DNA chain. An important experimental tool for investigating the properties of rate-enhancing mechanisms of this type is now becoming available. In these experiments the DNA chain length is varied and the association rate of the investigated protein is measured. Previous attempts to describe theoretically how such one-dimensional sliding processes behave under those conditions were either oversimplified or connected with fundamental errors. We present here a theory for the association rate of proteins with a specific site on DNA molecules with variable lengths. We use a steady-state approach as do Schranner and Richter (Biophys. Chem. 8 (1978)47) but, in contrast, use boundary conditions which appear to be physically more reasonable.