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.

The association and dissociation kinetics of the Escherichia coli lac repressor--operator (RO) complex have been examined as a function of monovalent ion concentration and operator-containing DNA fragment length in order to investigate the mechanisms used by repressor in locating (and dissociating from) the operator site. Association rate constants (ka) measured with an 80- or a 203-base-pair lac operator containing DNA fragment are 3--5-fold smaller than those determined with a 6700-base-pair operator fragment or with intact lambda plac5 DNA (50000 base pairs) at all salt concentrations tested. At salt concentrations less than approximately 0.1 M KCl, association rate constants to all operator-containing DNA fragments (except lambda plac5 DNA) are insensitive to variations in salt concentration, but the limiting low salt value of ka appears to depend upon operator-containing DNA length. The value of ka for lambda plac5 DNA decreases significantly from the approximately 0.1 M KCl maximum at low salt. Above approximately 0.1 M KCl, repressor--operator association rate constants for all operator-containing DNA substrates tested show a similar decrease with increasing salt concentration, which does not appear to depend upon the length of the DNA molecule (except for the very small DNA fragments). In contrast to the association reaction, kd, the dissociation rate constant, decreases linearly (on a log kd vs. log [KCl] plot) with decreasing salt concentration over virtually the entire salt concentration range studied (0.05--0.2 M KCl). These results are consistent with the explanation of the unusually fast association kinetics for this system in terms of a two-step model in which repressor initially diffuses to a nonoperator DNA binding site (forming an RD complex) and then rapidly "scans" (in a locally correlated fashion) adjacent sites until the operator is located or the repressor dissociates from the chain. Dissociation of the RO complex follows the same two-step process in reverse. Quantitative comparisons are made between these results and the theoretical predictions of the two facilitating translocation mechanisms (one-dimensional "sliding" along the DNA double helix and direct transfer between DNA segments) developed in the first paper of this series [Berg, O. G., Winter, R. B., & von Hippel, P. H. (1981) Biochemistry (first paper of three in this issue)]. We conclude that the experimental data for the "faster-than-diffusion-controlled" interaction of repressor and operator can be quantitatively modeled by a two-step process in which sliding is the dominant transfer mechanism. Molecular models of the initial nonspecific binding event (including "hopping") as well as sliding and interchain transfer are discussed, and the possible roles of facilitated translocation mechanisms of the diffusion-driven type in this and other in vitro and in vivo protein--nucleic acid interaction processes are considered.

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