Figure 1.
Structures of the lac-repressor tetramer complexed with DNA.
(A) Crystallographic structure of the “v”-shaped LacR complexed with the symmetric operator sequence GAATTGTGAGCGCTCACAATT (PDF accession number 1LBG) [34], shown along the z axis. The α-carbon trace of each lac monomer is rendered in a separate color; DNA segments are shown in a space-filling representation. The three repressor domains are indicated: H, headpiece; C, core; T, tetramerization. The x axis is an approximate two-fold or dyad axis in the structure. (B) View of the complex shown in (A) along the x axis. Helical axes of the bound DNA segments project slightly (35°) out of the plane of the “v” structure, implying that a small degree of DNA writhe may be induced by LacR-mediated looping. (C) A hypothetical structure of LacR in its extended conformation. This model was generated from the “v”-shaped structure shown in (A) by increasing the interior angle from 60° to ∼180°. The three semi-flexible joints modeling the elastic properties of the tetramer are indicated by vertical arrows. Note that an increase in the length of the LacR major axis from 20 bp to 25 bp occurs when the tetramer isomerizes from the “v-shaped” to the extended structure. (D) Simplified elastic model for LacR and a simulated 137-bp DNA loop mediated by the extended LacR structure. DNA base pairs are represented by rectangular slabs (red). Two sets of coordinate axes (green) represent the local coordinate frames embedded in the protein subunits (gold) that mediate DNA looping. The coupling of protein and DNA geometry is characterized by tilt, roll, and twist values for the DNA-protein, protein-protein, and protein-DNA interfaces. Three of these variables are shown here: the DNA-protein roll angle, φDP ; the protein-protein twist angle, τPP ; and the protein-DNA roll angle, φPD (see Materials and Methods for details).
Figure 2.
Looped DNA conformations mediated by the v-shaped LacR tetramer structure for 179- and 163-bp DNAs.
Three classes of loop conformations: (A) “wrap toward” (“WT”), (B), (C) “wrap away” (“WA”), and (D), (E) “loop beside” (“LB”), are shown along with their respective J factors and ΔGloop values. At most two alternative loop topoisomers are found for small DNA loops such as those considered here; these correspond to underwound (−) and overwound (+) DNA conformations. The DNA-loop sizes 179 and 163 bp correspond to maxima and minima, respectively, in the DNA-length dependence of J for the “WA” or “WT” conformations (see Figure 3A). The 179-bp “WT” conformation shown in (A) is dominated by one planar topoisomer because of its in-phase LacR binding sites at this DNA length; however, there are two alternative 179-bp “WA” conformations, which are shown in (B). The two “WA” topoisomers for 163-bp DNA are essentially isoenergetic and shown in (C); note that the loop crossings differ in topological sign. In contrast, the “LB” conformations (D) and (E) have minimum and maximum J values at 179 and 163 bp, respectively. Only the 179-bp (−) “LB” topoisomer shown in (D) is populated to any appreciable extent at thermal equilibrium, whereas both (−) and (+) forms have similar free energies in the case of 163-bp “LB” loops.
Figure 3.
J factor and ΔGloop values versus DNA length or LacR interior angle for four classes of loop conformations.
(A) Comparison of J factors and ΔGloop for the three classes of loop conformations mediated by the v-shaped lac repressor. Protein assemblies are taken to be rigid (i.e., DNA-protein, protein-protein, and protein-DNA flexibility parameters were all set to 0). (B) Length dependence of J factors and ΔGloop for the extended (SL) and LB conformations. Protein-flexibility parameters are given in parentheses as (σθ PP = σφ PP = στ PP , σθ DP = σθ PD = σφ DP = σφ PD = στ PD = στ DP ). Taken together with (A), the J-factor length dependence shows that the extended LacR conformation dominates all of the v-shaped forms for loops smaller than 180 bp. (C) The dependence of J and ΔGloop values on the interior angle between LacR domains is shown for three classes of 153-bp loops as the repressor structure opens from the v-shape (60°) to an extended form (180°). Protein assemblies were taken to be rigid, as in (A). WA and WT loops become degenerate at large angles, which can be seen from the identical J factors attained with the extended form of LacR. A small difference (∼1.5kB T) between the asymptotic ΔGloop value for WA and WT conformations in (C) and the corresponding value on the SL curve in (B) is due to differences in protein flexibility and tetramer dimensions (dimer major-axis length of 25 bp in (B) versus 20 bp in (C)). Because of broken symmetry, LB loops adopt a highly strained conformation as the interior angle approaches 180°. For comparison, projections of 3-d conformations for LB loops with interior-angle values of 60° and 180° are shown as insets. Gaps in the curves indicate that no stable mechanical-equilibrium conformations were found for “LB” loops when the interior angle was between 146° and 156°, nor for “WT” loops having interior-angle values less than 98°. This behavior is characteristic of abrupt transitions between mechanical minima, usually when a loop bifurcates to either an over-twisted or under-twisted conformation. Without a stable mechanical-equilibrium conformation, the perturbation method employed in our statistical-mechanical theory cannot be applied.
Figure 4.
Chemical equilibria among LacR-operator association states assuming a single looped conformation for the LacR-DNA complex.
(A) A LacR tetramer can bind to two cognate sites with different affinities, for which K1 and K2 are apparent dissociation constants. Assuming site “1” is the primary site near the promoter of the lac operon, its occupancy by LacR prevents RNA polymerase from binding to the promoter, blocking transcription of genes under its control. (B) Coupled equilbria involving different LacR-DNA complexes. Here Kc (1) and Kc (2) denote the unimolecular association constants associated with DNA looping and are related to K1, K2 , λ, and J by Equation 9 (see Materials and Methods).
Figure 5.
Best fit of the thermodynamic model to in-vivo repression data.
Best-fit values of the adjustable parameters for all data sets are given in Table 1. (A) In-vivo repression data from Figure 3A of Müller et al. [5] and optimized fit to Equation 1. (B) In-vivo repression data of Becker et al. [19] for wild-type (“WT”) and HU-deletion (ΔHU) E. coli strains along with their respective optimized fits.
Figure 6.
J factor versus operator spacing under different intracellular conditions.
The J values corresponding to wild-type, WT, or HU-deletion, ΔHU, E. coli strains were calculated using the respective best-fit parameters given in Table 1. J factors corresponding to LacR-mediated loops having normal DNA elasticity were computed using canonical parameters for DNA persistence length (150 bp) and torsional rigidity (2.4 10−19 erg cm), and a LacR flexibility parameter identical to that for the wild-type strain (19°). Calculations of the J factor for the case of a rigid extended LacR conformation used the same parameters as for the wild-type strain except for a protein-flexibility parameter set equal to 2.0°.
Table 1.
Best-fit Values of Adjustable Parameters for LacR-loop-mediated Repression Data.
Table 2.
Effect of operator quality on DNA looping and gene repression at approximately constant helical phasing.
Figure 7.
Predicted repressor occupancy of the primary operator and loop yield as a function of operator spacing corresponding to the data of Becker et al. [19].
Upper panel: the proportion of the primary operator site (O2) bound by LacR for wild-type and HU-deletion strains computed from their respective best-fit parameters using formula 1−Eloop (see Equation 13). The occupancy of the primary operator in the absence of the looping contribution is shown for reference (dashed line). Lower panel: the loop yield computed using Equation 10 for both wild-type and HU-deletion strains. J values used in these calculations are those given in Figure 6.