Mechanical property of the helical configuration for a twisted intrinsically straight biopolymer

We explore the effects of two typical torques on the mechanical property of the helical configuration for an intrinsically straight filament or biopolymer either in three-dimensional space or on a cylinder. One torque is parallel to the direction of a uniaxial applied force, and is coupled to the cross section of the filament. We obtain some algebraic equations for the helical configuration and find that the boundary conditions are crucial. In three-dimensional space, we show that the extension is always a monotonic function of the applied force. On the other hand, for a filament confined on a cylinder, the twisting rigidity and torque coupled to the cross section are irrelevant in forming a helix if the filament is isotropic and under free boundary condition. However, the twisting rigidity and the torque coupled to the cross section become crucial when the Euler angle at two ends of the filament are fixed. Particularly, the extension of a helix can subject to a first-order transition so that in such a condition a biopolymer can act as a switch or sensor in some biological processes. We also present several phase diagrams to provide the conditions to form a helix.