Natural tristability of a confined helical filament with anisotropic bending rigidities

We find that when \(c_0 R\sim 0.5\) $c_{0}R\sim 0.5$ and \(\tau _0R<0.11<c_0R\) ${\tau }_{0}R<0.11<c_{0}R$ , confining a helical filament with anisotropic bending rigidities within a cylindrical tube of radius R can create a natural tristable status which is consisted of two low-pitch helices and one high-pitch helix, where a helical filament is referred to as a filament that has both an intrinsic curvature ( \(c_0\) $c_0$ ) and an intrinsic twist rate ( \(\tau _0\) ${\tau }_0$ ). The formation of the tristable status also requires that the filament has a significant difference between two bending rigidities and a large twisting rigidity. The relative heights of two low-pitch helices in a tristable status are close to zero, and the smaller the intrinsic twisting angle, the smaller the difference between these two heights. Moreover, at a large intrinsic twisting angle, two low-pitch helices display a large energy difference, and the energy difference increases with decreasing \(\tau _0\) ${\tau }_0$ . Meanwhile, the relative height of the high-pitch helix is always close to that of a straight line. Finally, at some specific intrinsic parameters, the tristable status can include an isoenergic status with two helices of the same energy but distinct pitches.