Figure 1.
Atomic Force Microscopy of Lac Repressor–DNA Complexes
(A) Schematic structures of biotin (bio)- and digoxigenin (dig)-labeled DNA constructs with one (O-539 and O-349) or two (O-153-O and O-158-O) ideal lac operator sequences (white bars) [82]. Numerals are DNA segment lengths (base pairs) measured from the center(s) of the operator(s).
(B) AFM image of molecules adsorbed to a mica surface (see Materials and Methods) from a mixture of repressor and di-operator DNA O-153-O (183-nm contour length). R, repressor; R-DNA, repressor–DNA complex. The color scale corresponds to a 3-nm range of heights.
(C) Scatter plot of the DNA arm contour lengths in repressor–DNA complexes. Measurements were made on images of complexes (n = 364) that had a single repressor molecule with two well-resolved protruding DNA arms; complexes with only one arm or with an arm that was too short for accurate measurement were excluded. In this graph, the data from each complex are plotted twice: the length of the longer arm (on the vertical axis) is plotted against the length of the shorter arm (horizontal axis); the same data are then replotted in a different color with the two axes interchanged. This produces a representation of the data that is symmetrical across a line (not shown) extending from the lower left to upper right corner. Data from both O-158-O (red and black) and O-153-O (green and blue) containing samples are included. The region between the dashed lines is that in which the sum of the arms lengths is 169 ± 16 nm, the image contour length mean ± 2 × S.D. measured for the uncomplexed DNAs. Green and red squares enclose the ranges of arm lengths predicted for operator-specific linear and looped repressor–DNA complexes, respectively.
(D and E) Left: projection images of two alternative three-dimensional structural models of looped complexes between Lac repressor (gray) and DNA (black) with two operators spaced approximately 50 nm apart. Right: the corresponding AFM images produced by numerical simulation of the AFM imaging process (see Materials and Methods).
(F) Representative images (120 × 120 nm) of looped complexes with O-158-O or O-153-O DNA.
Figure 2.
Dynamic Repressor-Mediated Loop Formation in Single DNA Molecules at Equilibrium
(A) Experimental design (not to scale). Left: a digoxigenin- and biotin-labeled DNA (black curve) containing two operator sites (white bars) is immobilized at low surface density on an anti-digoxigenin–coated coverslip; the distal end of the DNA is labeled with an avidin-conjugated 0.098-μm diameter polystyrene bead for visualization by light microscopy. Right: formation of a Lac repressor-mediated looped complex decreases the effective length of the DNA tether and thus reduces the observed Brownian motion of the tethered bead.
(B–D) Examples of Brownian motion recordings for single beads tethered by either O-539 (B), O-153-O (C), or O-158-O (D), in the presence of 5.4 nM Lac repressor. The extent of Brownian motion is expressed as the effective tether length, the length of DNA that gives the same amount of Brownian motion in the absence of repressor (see Materials and Methods).
(E–G) Effective tether length histograms (bin width 15 bp) from the individual records designated by arrows.
Figure 3.
State Lifetime Distributions (A–E) and the Mechanisms and Rate Constants Determined from the TPM Data (F and G)
For each state, N measured lifetimes were used to construct a histogram (circles) of bin width W, which is plotted here as a scaled probability density (see Materials and Methods); also shown (lines) are the theoretical distributions calculated from the mechanisms and rate constants.
(A) O-153-O unlooped state (N = 110, W = 15 s); (B) O-153-O looped state (N = 108, W = 18 s); (C) O-158-O unlooped state (N = 168, W = 15 s); (D) O-158-O long-tether loop (N = 245, W = 40 s); (E) O-158-O short-tether loop (N = 80, W = 30 s). (F and G) Proposed mechanisms for repressor-mediated looping of O-153-O (F) and O-158-O (G). O2 (lines), di-operator DNA molecule; R (squares), repressor tetramer. For simplicity, only one of the two identical reaction steps linking each of the two equivalent O2R linear species with the other states is shown; the rate constants are the microscopic rate constants for each of the individual reaction steps. Similarly, unlooping rate constants k-loop, k2, and k6 are for the dissociation of repressor–single operator interactions; there are two such interactions in a loop so that the overall unlooping rate constant is twice the value given. ka* is the pseudo first-order rate constant for repressor association with operator. Numbers in parentheses are the standard error (S.E.) of the final digit of the corresponding rate constant. Rate constants were determined from observed state lifetimes, equilibrium constants, and kinetic partition ratios as described in Materials and Methods. The O-158-O rate constant in brackets was not well determined by the experimental data and was instead assumed to be equal to the corresponding O-153-O rate constant (see Materials and Methods).
Figure 4.
Lac Repressor and Proposed O-158-O Loop Geometries
(A) Crystal structure of Lac repressor complexed with two 21-bp symmetric operator DNAs (Protein Data Bank 1LBG; [20]). Each N-terminal headpiece of the tetrameric repressor is in contact with DNA (shown as thin wires). The C-terminal α-helical coiled-coil domains from each of the four subunits associate to form the stable 4-helix bundle tetramerization domain (TD). The center-to-center spacing of the two DNAs is approximately 7 nm.
(B and C) Proposed models and predicted effective tether lengths for long-tether (B) and short-tether (C) loops. Black curves and white bars represent flanking DNA and operator sites, respectively. Each of the four subunits of the repressor is shown in a different color. Structures are shown as if flat (for clarity), but actual loop structures are at least somewhat nonplanar. (B) Long-tether loop. The repressor has the crystallographic V-shaped conformation. (C) Short-tether loop. The repressor has opened up to a fully extended form by rotating the two dimers about the axis of the 4-helix bundle, which is roughly normal to the plane of the figure.
Figure 5.
Physical Models Corresponding to Possible In-Phase Loop Geometries
Photographs show loop models with the V-shaped repressor conformation (left column), and the corresponding configurations that result when the repressor V is opened (right column). In each pair, the structure on the right was produced from that on the left by rotating the two half-tetramers (represented by a blue or red paper clip) away from each other about the axis of the 4-helix bundle (represented by a silver bolt); the front dimer is rotated approximately 80° clockwise and the rear approximately 80° counterclockwise.
(A) Structures proposed (Figure 4) for the O-158-O long- (left) and short-tether loops (right). Left structure is similar to the “wrapping toward” model of Friedman et. al. [19], but with the DNA helix axes roughly perpendicular to the 4-helix bundle axis, as seen in the repressor–oligonucleotide co-crystal [20].
(B) “Wrapping away” loop (left) [19,35].
(C) Alternative wrapping toward loop [20,24,35].
(D) Alternative wrapping away loop.
The right-column structures (B) through (E) are predicted to be significantly less stable than that in (A) due to increased twisting strain (B), bending strain (C and D), or both (E). The paper strip representing the DNA is colored black on one side and white on the other to make any twist visible. The structures (A) through (E) are topologically equivalent to the P2, P1, P1, P2, and A2 configurations described by Swigon et al [28]. Loops with additional antiparallel configurations (A1 and A1*; not shown) are also possible, but all such antiparallel configurations produce highly strained opened structures similar to that shown for (E). In classifying the structures, we assume that the flanking DNA arms are constrained, either by attachment to surface and bead (as in the experiments reported here) or by incorporation into a larger DNA circle (as in a plasmid or E. coli chromosome). With constrained flanking DNA arms, the two P2 structures (A and D) are topologically distinct; they can only be interconverted by temporarily separating a DNA binding domain from its operator or by passing the loop segment through one of the arms. The same is true for the two P1 structures (B and C).
Figure 6.
Free-Energy Profile for the Looping and Unlooping Reactions
Free energies (G) of the stable species, and inferred G of transition states (‡) for looping/unlooping reactions, are shown with cartoon representations of the three-dimensional structures proposed (see Figure 4B and 4C, and text) for each stable species. The linear complexes with the repressor in the V” conformations (species 1 and 4) were chosen as reference states and assigned G = 0; the relative energies of other species were calculated from the rate constants (Figure 3). Transitions state energies are rough estimates based on the assumptions that all reactions are single-barrier processes with identical pre-exponential factors arbitrarily set at 6.1 × 1012 s−1. The conformational change of the repressor bound to linear DNA (species 4 ↔ 5) cannot be detected in the experiments; G of species 5 is instead estimated from the change solvent-exposed protein surface area (see Materials and Methods), and the transition state is not shown. The three-dimensional structure of species 2 is assumed to be similar to that of species 3, except that the repressor structure is distorted so as to minimize DNA twist (see text). Species 3 is shown twice so that all transitions in the cyclic O-158-O looping scheme (Figure 3G) can be shown. Free energies are expressed as multiples of the product of the Boltzmann constant (k) and the absolute temperature ( T ).