We derive the shape equations in terms of Euler angles for a uniform elastic rod with isotropic bending rigidity and spontaneous curvature, and study within this model the elasticity and stability of a helical filament under uniaxial force and torque. We find that due to the special requirements on the boundary conditions, a static slightly distorted helix cannot exist in this system except in some special cases. We show analytically that the extension of a helix may undergo a one-step sharp transition. This agrees quantitatively with experimental observations for a stretched helix in a chemically-defined lipid concentrate (CDLC). We predict further that under twisting, the extension of a helix in CDLC may also exhibit similar behavior. We find that a negative twist tends to destabilize a helix.
