Torsion of the central pair microtubules in eukaryotic flagella due to bending-driven lateral buckling

Inspired by recent interest in torsion of the central pair microtubules in eukaryotic flagella, a novel thin-walled elastic beam model is suggested to study critical condition under which uniform bending of a flagellum will cause lateral/torsional buckling of the central pair. The model is directed to the central pair itself and the role of all surrounding cross-linkings inside the flagellum is modeled as an equivalent surrounding elastic medium. The model predicts that bending-driven torsion of the central pair does occur when the radius of curvature of the bent flagellum reduces to a moderate critical value typically of tens of microns. In particular, this critical value is almost independent of the flagellum length, and more sensitive to the parameters defining the surrounding elastic medium than the shear modulus of microtubules. The predicted wavelengths of the torsional buckling mode are insensitive to the flagellum length and comparable to some known related experimental data. These results indicate that torsion of the central pair microtubules in flagella is inevitable as a result of bending-driven lateral buckling. This offers an entirely new insight into the ongoing research on the mechanism of the central pair torsion.