Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. Models and theory

Genome regulatory proteins (e.g., repressors or polymerases) that function by binding to specific chromosomal target base pair sequences (e.g., operators or promoters) can appear to arrive at their targets at faster than diffusion-controlled rates. These proteins also exhibit appreciable affinity for nonspecific DNA, and thus this apparently facilitated binding rate must be interpreted in terms of a two-step binding mechanism. The first step involves free diffusion to any nonspecific binding site on the DNA, and the second step comprises a series of protein translocation events that are also driven by thermal fluctuations. Because of nonspecific binding, the search process in the second step is of reduced dimensionality (or volume); this results in an accelerated apparent rate of target location. In this paper we define four types of processes that may be involved in these protein translocation events between DNA sites. These are (i) "macroscopic" dissociation--reassociation processes within the domain of the DNA molecule, (ii) "microscopic" dissociation--reassociation events between closely spaced sites in the DNA molecule, (iii) "intersegment transfer" (via "ring-closure") processes between different segments of the DNA molecule, and (iv) "sliding" along the DNA molecule. We present mathematical and physical descriptions of each of these processes, and the consequences of each for the overall rate of target location are worked out as a function of both the nonspecific binding affinity between protein and DNA and the length of the DNA molecule containing the target sequence. The theory is developed in terms of the Escherichia coli lac repressor--operator interaction since data for testing these approaches are available for this system [Barkley, M. (1981) Biochemistry 20, 3833; Winter, R. B., & von Hippel, P. H. (1981) Biochemistry (second paper of three in this issue); Winter, R. B., Berg, O. G., & von Hippel, P. H. (1981) Biochemistry (third paper of three in this issue)]. However, we emphasize that this approach is general for the analysis of mechanisms of biological target location involving facilitated transfer processes via nonspecific binding to the general system of which the target forms a small part.