Torque and dynamics of linking number relaxation in stretched supercoiled DNA

In micromechanical studies of DNA, plectonemically supercoiled domains are often used as sources of constant torque. These torques are not easily measured and are instead usually estimated. Here, coexisting extended and supercoiled DNA domains are analyzed, and closed-form expressions for the dependence of extension and torque on force and linking number are presented. When there are coexisting domains of plectonemic and extended DNA, the torque depends only on force, with no dependence on linking number. However, torque depends on force in a manner more complex than a simple power law, involving the free energy of the extended and plectonemic DNA. A simple strategy is described for measurement of the free energies of both extended and plectonemic DNA without reference to specific microscopic polymer models. Applications of the theory to analysis of relaxation of supercoiling by enzymes which permit friction-controlled rotational relaxation of linking number is also presented. Such enzymes must display a breaking of symmetry between relaxations driven by equal magnitude but opposite direction torques.

Figure 1

Free energies of extended [dot-dashed curve, $\mathcal{S}\left(\sigma \right)$] and plectonemic supercoil [dashed curve, $\mathcal{P}\left(\sigma \right)$] DNA states as a function of linking number $\sigma$. For $\sigma <{\sigma }_{\mathrm{s}}$, the $\mathcal{S}$ state is lower in free energy than either $\mathcal{P}$ or any mixture of the two. Similarly, for $\sigma >{\sigma }_{\mathrm{p}}$, pure $\mathcal{P}$ is the lowest free energy configuration. However, for $\sigma$ between ${\sigma }_{\mathrm{s}}$ and ${\sigma }_{\mathrm{p}}$ the tangent construction shown (solid line segment between tangent points indicated by stars), representing coexisting domains of $\mathcal{S}\left({\sigma }_{\mathrm{s}}\right)$ and $\mathcal{P}\left({\sigma }_{\mathrm{p}}\right)$, is the lowest free energy state. Note that the gap between the two states at $\sigma =0$ is the free energy difference between random coil DNA [$\mathcal{P}\left(0\right)$] and stretched unsupercoiled DNA [$\mathcal{S}\left(0\right)$]; this difference grows with applied force.

Figure 2

Extension versus linking number, for forces 0.25 (lowest curve), 0.5, 1.0, and $2.0\mathrm{pN}$ (highest curve), for positive linking number change $(\sigma >0)$. As the force is increased, the extension increases, and the effect of torsional stress (linking number) is reduced. Parameter values are $C=95\mathrm{nm}$, $A=50\mathrm{nm}$, $P=24\mathrm{nm}$. The parabolic peak of each extension curve occurs when the DNA is pure extended state; extended and plectonemic DNA are in coexistence on the linear part of each extension curve. The beginning of the linear segments indicates ${\sigma }_{\mathrm{s}}$, and their $\sigma$ intercepts indicate ${\sigma }_{\mathrm{p}}$.

Figure 3

Torque vs force curve for coexisting state densities corresponding to fixed $\sigma =+0.03$, 0.04, 0.05, and 0.06, with other parameters as in Fig. 2. For forces below the lower critical forces $f_{\mathrm{p}}$ where the extended DNA state disappears (horizontal line segments at left of graph), torque is constant. Above the upper critical force $f_{\mathrm{s}}$ where the plectonemic DNA disappears, the torque slowly approaches its maximum value $(c∕{\omega }_0)\sigma$ (nearly flat regions to right). Between $f_{\mathrm{p}}$ and $f_{\mathrm{s}}$, the torque follows the same curve for each $\sigma$ value (concave part of curve). Dashed curve shows $(2k_{B}TAf{)}^{1∕2}$ for comparison.

Figure 4

Comparison of the theoretical torque formula of this paper [solid curve shows Eq. (17); horizontal line segments to left show pure-plectonemic-state torques; to right pure-extended-state torques] with results of MC calculations (symbols) following the method of Ref. 24 for an effective DNA thickness appropriate to describe $200\mathrm{mM}$ univalent salt solution. MC data are shown for $\sigma =0.01$ $(+)$, 0.02 $(\times )$, 0.03 $(◻)$, 0.04 $(엯)$ and 0.06 $(▵)$; successively higher pure-state torques are computed for the same linking numbers. Both analytical and MC calculations use $A=50\mathrm{nm}$ and $C=95\mathrm{nm}$; the analytical calculation uses $P=24\mathrm{nm}$ determined from the MC calculation (see text). Dashed curve shows $(2k_{B}TAf{)}^{1∕2}$ for comparison.

Figure 5

Free energies of extended (dot-dashed curve), plectonemic (short-dashed curve, minimum at $\sigma =0$), and denatured (long-dashed curve, minimum at $\sigma =-1$) DNA states, for force of $5\mathrm{pN}$. Solid straight line segments terminating in stars indicate coexistence regions between extended and plectonemic states. Solid straight line segments terminating in circles indicate coexistence region between extended and underwound denatured states. For negative $\sigma$ the lowest-energy states are combinations of extended and denatured DNA; the extended-denatured DNA coexistence line is below that for extended-plectonemic states for $\sigma <0$. Because the lowest-energy twist state of denatured DNA has ${\sigma }_0=-1$, this $\sigma <0$ coexistence occurs between extended DNA of $\sigma \approx -0.02$ and denatured DNA of $\sigma \approx -2$. For $\sigma >0$ the first coexistence that occurs is between extended and plectonemic DNA; for sufficiently large $\sigma$ a second plectonemic-denatured coexistence occurs but is not shown on this graph.

Figure 6

Extension versus linking number $\sigma$ for DNA at constant forces. (a) Results of theory for forces of 0.5 (lowest curve), 1, 2, 5, and $10\mathrm{pN}$ (highest curve), using $A=50\mathrm{nm}$, $C=95\mathrm{nm}$, $P=24\mathrm{nm}$, and denaturation parameters as described in the text. For $0.5\mathrm{pN}$ the extension is symmetric under sign change of $\sigma$, due to coexistence occurring between extended and plectonemically supercoiled DNA states, which have this symmetry. However, for forces of 1, 2, and $5\mathrm{pN}$, coexistence occurs between extended and plectonemic DNA for $\sigma >0$, but between extended and denatured DNA for $\sigma <0$, exhibiting a strong breaking of $\sigma \to -\sigma$ symmetry. Finally for $10\mathrm{pN}$, denaturation occurs for both positive and negative $\sigma$. (b) Comparison of theoretical results (solid curves) to experimental data of Charvin for $48\mathrm{kb}$ $\lambda$-DNA in $10\mathrm{mM}$ phosphate buffer (symbols) taken from Ref. 23. Forces used for experiment and corresponding theoretical curves are 0.25 $(엯)$, 0.44 $(◻)$, 0.74 $(◇)$, 1.31 $(+)$, and $2.95\mathrm{pN}$ $(\times )$. Parameters as in (a), except for $P=28\mathrm{nm}$ which is appropriate for the $10\mathrm{mM}$ ${\mathrm{Na}}^{+}$ buffer.

Figure 7

Force-torque phase diagram showing transitions between extended double helix (B), plectonemically supercoiled B-DNA (scB), underwound denatured DNA $(\mathrm{D}-)$, and overwound denatured DNA $(\mathrm{D}+)$. Note that the denaturation transitions occur at torques of ${\tau }_{\mathrm{d},-}\approx -10\mathrm{pN}\mathrm{nm}$ and ${\tau }_{\mathrm{d},+}\approx +40\mathrm{pN}\mathrm{nm}$. The small triangular region below $0.6\mathrm{pN}$ and between torques of $-5$ and $-10\mathrm{pN}\mathrm{nm}$ is plectonemically supercoiled B-DNA with negative linking number. The denatured states correspond to the “Z”/L and scP states observed experimentally.