Twist and Stretch of Helices Explained via the Kirchhoff-Love Rod Model of Elastic Filaments

In various single-molecule experiments, a chiral polymer, such as DNA, is simultaneously pulled and twisted. We address an elementary but fundamental question raised by various authors: does the molecule overwind or unwind under tension? We show that within the context of the classic Kirchhoff-Love rod model of elastic filaments, both behaviors are possible, depending on the precise constitutive relations of the polymer. More generally, our analysis provides an effective linear response theory for helical structures that relates axial force and axial torque to axial translation and rotation.

Figure 1

A wrench applied to a helical spring gives rise to another helical equilibrium. Two simple questions are to determine (a) the sign of the increment of the rotation angle $\theta$ when an axial tension is applied with no torque and (b) whether a spring extends or contracts as a result of an axial torque loading with no applied force.

Figure 2

Left: For the particular constitutive relations (7), the set of $M=0$ zero axial torque helical equilibria [corresponding to Fig. 1a] lie on an ellipse in the curvature-torsion plane determined by the value of $\Gamma$. Close to the unstressed helix with $N=0$ (point $\mathbf{a}$), $N>0$ equilibria correspond to $|\tau |>|\widehat{\tau }|$ (the case of $\tau >0$ is shown). Initial overwinding in extension occurs for $\Gamma <1$, which is when the section of the ellipse outside the circle centered at the origin and passing through $\mathbf{a}$ has $|\tau |>|\widehat{\tau }|$. The set of $N=0$ zero force equilibria [corresponding to Fig. 1b] is also shown. Depending on $\Gamma$, the helix either lengthens when wound ($\Gamma <1$) or shortens ($\Gamma >1$). Right: For $M=0$, the relative coiling angle as a function of the axial force. For $\Gamma <1$ and $N>0$, the helix first overwinds, then unwinds.

Figure 3

Left: The set of equilibria for a helix under pure axial force ($M=0$) is the intersection of the hyperboloid $\mathcal{H}$ (yellow region) and the ellipsoid $\mathcal{E}$ (red region). The set of equilibria without winding $\theta =0$ is the intersection of the hyperboloid $\mathcal{H}$ (yellow region) and the sphere $\mathcal{S}$ (blue region). Right: For the particular case shown, the helix initially unwinds when pulled since the tangent ${\mathbf{t}}_{\mathcal{E}}$ to the curve of equilibria ${\mathsf{u}}_N$ (red curve) oriented toward $N>0$ lies within the sphere $\mathcal{S}$, whose intersection with $\mathcal{H}$ is shown as the blue curve.