Figure 3. J factor and ΔG_loop values versus DNA length or LacR interior angle for four classes of loop conformations.

(A) Comparison of J factors and ΔG_loop 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 ΔG_loop for the extended (SL) and LB conformations. Protein-flexibility parameters are given in parentheses as (σ_θ^PP = σ_φ^PP = σ_τ^PP, σ_θ^DP = σ_θ^PD = σ_φ^DP = σ_φ^PD = σ_τ^PD = σ_τ^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 ΔG_loop 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 (∼1.5k_B T) between the asymptotic ΔG_loop 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.