Energetics at the DNA supercoiling transition

Abstract

Twisting a DNA molecule held under constant tension is accompanied by a transition from a linear to a plectonemic DNA configuration, in which part of the applied twist is absorbed in a superhelical structure. Recent experiments revealed the occurrence of an abrupt extension change at the onset of this transition. To elucidate its origin we study this abrupt DNA shortening using magnetic tweezers. We find that it strongly depends on the length of the DNA molecule and the ionic strength of the solution. This behavior can be well understood in the framework of a model in which the energy per writhe for the initial plectonemic loop is larger than for subsequent turns of the superhelix. By quantitative data analysis, relevant plectoneme energies and other parameters were extracted, providing good agreement with a simple theory. As a direct confirmation of the initial-loop model, we find that for a kinked DNA molecule the abrupt extension change occurs at significantly lower twist than the subsequent superhelix formation. This should allow pinning of the plectoneme position within supercoiled DNA if a kinked substrate is used, and enable the detection of enzymes and proteins which, themselves, bend or kink DNA.

Figure 1

Buckling transition measured for different ionic strengths and DNA lengths. (A) An ∼1.9-kbp DNA molecule held at a constant force of 3.0 pN is continuously twisted with a frequency of 0.5 Hz and its length is recorded simultaneously. Curves are shown for Na⁺ concentrations of 20 mM, 60 mM, and 320 mM (red, green, and black line). The cartoons illustrate the experimental configuration and the transition from linear to superhelical DNA (red line). The magnetic bead (gray sphere) and the pair of magnets (blue squares) are also shown. (Inset) Enlarged view on the supercoiling curves of the main figure at the buckling transition. (B) Supercoiling curve for a 10.9-kbp DNA molecule twisted with 1 Hz at 3.0 pN force in buffer containing 320 mM Na⁺. (Inset) Enlarged view on the supercoiling curves of the main figure at the buckling transition. Data were taken at 300 Hz and smoothed to 20 Hz.

Figure 2

Equilibrium occupancy of pre- and postbuckling state. (A) 1.9 kbp and (B) 10.9 kbp DNA molecules were held at a constant force of 3.0 pN in buffer containing 320 mM Na⁺. Time traces were recorded at different amounts of added turns N in the vicinity of the buckling transition. Data were taken at 300 Hz. Normalized length histograms are shown on the right. The DNA was observed to rapidly fluctuate between two distinct states, the pre- and the postbuckling state. Dotted lines are centered on the peaks of the uppermost histogram and indicate the shift of the states throughout the transition. (C) Occupancy of the postbuckling state as function of added turns for both DNA lengths (red dots). Experimental conditions are as in panel A. Solid lines are fits to the data according to Eq. 5, where ΔN_b was taken to be free (black line, ΔN_b of best fit shown in graph) or fixed to ΔN_b = 1 (gray). (D) Illustration representing the behavior of DNA at the buckling transition, which can involve the formation of a structure comprising >1 turn of writhe.

Figure 3

Kinetics of the buckling transition. Data shown is for 4.0 pN and 320 mM Na⁺. (A) Distributions of the residence times for the pre- and postbuckling state close to the buckling equilibrium (9.8 turns) for the 1.9 kbp DNA molecule. The solid lines represent a single exponential function with the mean residence time as characteristic decay time. (B) Mean residence times of the prebuckling (red solid circles) and the postbuckling states (blue solid circles) as function of added turns for the 1.9 kbp and 10.9 kbp DNA molecule. Solid lines are exponential fits to the data according to Eq. 7. The resulting distances to the transition state $\Delta N_{\text{b}}^{\text{pre}}$ and $\Delta N_{\text{b}}^{\text{post}}$ are given in the figure. (C) Schematic drawing of the hypothetical energy landscape for plectoneme formation. Independent of the actual landscape for the initial loop (dark blue straight line and light blue line with transition state), the supercoiling energy attains E₁ after one turn and increases with E₂ for each subsequent turn. (Inset) Illustration of the plectoneme formation energies E₁ and E₂.

Figure 4

Measured supercoiling curves and predictions from the initial-loop model. (A and B) Supercoiling curves for a 1.9 kbp DNA molecule at 3.0 pN in a buffer containing 320 mM and 60 mM Na⁺ (as indicated). Data were taken at 300 Hz (dark shading) and smoothed to 20 Hz (light shading). The solid red line is the prediction from the initial-loop model according to Eq. 15 using estimates for the plectoneme formation energies E₁ and E₂ (Eqs. 16 and 17). (C) Salt dependence of supercoiling curves and (D) torque development as predicted by the initial-loop model. The shaded area between the torque overshoot and the postbuckling torque under the black curve corresponds to the difference between the initial loop energy E₁ and superhelix formation energy E₂.

Figure 5

Force dependence of the buckling transition at 320 mM Na⁺ for the 1.9 kbp (black) and the 10.9 kbp DNA molecule (blue). Experimental values are shown as solid circles, predictions from the end-loop model as solid lines. In the case of an overlap, only the black curve is depicted. (A) Jump size at buckling equilibrium. (B) Position of the buckling equilibrium. To highlight the DNA length dependence, the right axis, corresponding to the 10.9 kbp DNA molecule, was scaled linearly with DNA length compared to the left axis corresponding to the 1.9 kbp DNA molecule. (C) Change of twist transferred into writhe during buckling $\Delta N_{\text{b}}^{\text{p}}$ at buckling equilibrium and corresponding torque change $\Delta {\Gamma }_{\text{b}}=2\pi C_{\text{s}}/L\times \Delta N_{\text{b}}^{\text{p}}$. The expected scaling of $\Delta N_{\text{b}}^{\text{p}}$ with the square root of the DNA length is highlighted by scaling the axes accordingly (Eq. 13). (D) (Top) Superhelix-formation energy E₂ and inferred postbuckling torque. (Bottom) Energy difference between the initial-loop formation energy E₁ and E₂. Error bars represent the statistical error of the data.

Figure 6

Salt dependence of the buckling transition parameters at 3.5 pN for the 1.9 kbp DNA molecule. Experimental values are shown as circles. Predictions from the initial-loop model are shown as black lines. (A) Jump size at buckling equilibrium. (B) Position of the buckling equilibrium. (C) Change of twist transferred into writhe during buckling $\Delta N_{\text{b}}^{\text{p}}$ at the buckling equilibrium and corresponding torque change ΔΓ_b. (D) Superhelix-formation energy E₂, inferred postbuckling torque (solid circles and solid line), and initial-loop formation energy E₁ (open circles and dashed line). (Inset) Energy difference between E₁ and E₂.

Figure 7

(A) Supercoiling curves for a kinked (lower) and a straight, i.e., unkinked (upper) DNA molecule taken at 3.5 pN and 320 mM Na⁺. Data are acquired at 300 Hz (dark shaded lines) and filtered to 20 Hz (light shaded lines). Solid red and black lines are calculated according to Eq. 15 with the initial-loop length reduction ΔL₁, the initial-loop energy E₁, and the superhelix energy E₂ taken from fits to the data. The illustration shows the structure of the DNA kink. A 20-bp hairpin was introduced into the substrate, leading to the formation of a small three-arm junction. Neighboring arms should join at an angle of ∼120°. (Inset) Histogram of the DNA length at 7.0 turns for the kinked molecule. (B) Torque development for both molecules as predicted by the model. The shaded area enclosed by the two curves corresponds to the total difference between the initial-loop energies E₁ for the straight and the kinked DNA molecule.