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 widespread use of antibiotics is selecting for a variety of resistance mechanisms that seriously challenge our ability to treat bacterial infections. Resistant bacteria can be selected at the high concentrations of antibiotics used therapeutically, but what role the much lower antibiotic concentrations present in many environments plays in selection remains largely unclear. Here we show using highly sensitive competition experiments that selection of resistant bacteria occurs at extremely low antibiotic concentrations. Thus, for three clinically important antibiotics, drug concentrations up to several hundred-fold below the minimal inhibitory concentration of susceptible bacteria could enrich for resistant bacteria, even when present at a very low initial fraction. We also show that de novo mutants can be selected at sub-MIC concentrations of antibiotics, and we provide a mathematical model predicting how rapidly such mutants would take over in a susceptible population. These results add another dimension to the evolution of resistance and suggest that the low antibiotic concentrations found in many natural environments are important for enrichment and maintenance of resistance in bacterial populations.

Transcription factors (TFs) are proteins that regulate the expression of genes by binding sequence-specific sites on the chromosome. It has been proposed that to find these sites fast and accurately, TFs combine one-dimensional (1D) sliding on DNA with 3D diffusion in the cytoplasm. This facilitated diffusion mechanism has been demonstrated in vitro, but it has not been shown experimentally to be exploited in living cells. We have developed a single-molecule assay that allows us to investigate the sliding process in living bacteria. Here we show that the lac repressor slides 45 ± 10 base pairs on chromosomal DNA and that sliding can be obstructed by other DNA-bound proteins near the operator. Furthermore, the repressor frequently (>90%) slides over its natural lacO(1) operator several times before binding. This suggests a trade-off between rapid search on nonspecific sequences and fast binding at the specific sequence.

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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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Hydrolysis of vesicles of 1,2-dimyristoyl-sn-glycero-3-phosphomethanol (DMPM) by pig pancreatic phospholipase A2 (PLA2) occurs in a highly processive "scooting" mode, and the rate is comparable to or exceeds the rates observed with detergent-dispersed mixed micelles under optimal conditions. A complete kinetic description of the steady-state time course of the hydrolysis is developed. The analysis covers the whole Michaelis-Menten space: it emphasizes the key features of interfacial catalysis by a detailed theoretical analysis, describes the experimental protocols to determine the values of the kinetic and equilibrium constants for interfacial catalysis, and provides an interpretation of the effect of calcium, substrate, products, apparent activators, and competitive inhibitors on the reaction progress curve by a single set of rate and equilibrium parameters. In this paper, the integrated reaction progress curve was rigorously interpreted in terms of a minimal model involving the Michaelis-Menten reaction sequence in the interface: E* + S in equilibrium E*S----E*P in equilibrium E* + P, and most of the individual rate and equilibrium constants for the catalytic cycle were determined. This rigorous description of interfacial catalysis was made experimentally possible by examining the action of PLA2 in the scooting mode under conditions of at most one enzyme per vesicle, where it hydrolyzed all of the substrate in the outer monolayer of vesicles without leaving the surface. Other experimentally verified constraints for this analysis include the following: all enzyme was bound to vesicles; the integrity of vesicles was maintained during the course of hydrolysis; and the substrate, enzyme, and products did not exchange between vesicles nor did they exchange across the bilayer. The mechanistic significance of the rate constants is discussed in the accompanying papers.

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