Elasticity and stability of a helical filament

We derive the general shape equations in terms of Euler angles for a uniform elastic rod with spontaneous torsion and curvatures and subjected to external force and torque. Our results based on an analytic formalism show that the extension of a helical rod may undergo a one-step discontinuous transition with increasing stretching force. This agrees quantitatively with experimental observations for a helix in a chemically defined lipid concentrate. The larger the twisting rigidity, the larger the jump in the extension. The effect of torque on the jump is, however, dependent on the value of the spontaneous torsion. In contrast, increasing the spontaneous torsion encourages the continuous variation of the extension. An "over-collapse" behavior is observed for the rod with asymmetric bending rigidity, and an intrinsic asymmetric elasticity under twisting force is found.