Localized buckling of a microtubule surrounded by randomly distributed cross linkers.

Microtubules supported by surrounding cross linkers in eukaryotic cells can bear a much higher compressive force than free-standing microtubules. Different from some previous studies, which treated the surroundings as a continuum elastic foundation or elastic medium, the present paper develops a micromechanics numerical model to examine the role of randomly distributed discrete cross linkers in the buckling of compressed microtubules. First, the proposed numerical approach is validated by reproducing the uniform multiwave buckling mode predicted by the existing elastic-foundation model. For more realistic buckling of microtubules surrounded by randomly distributed cross linkers, the present numerical model predicts that the buckling mode is localized at one end in agreement with some known experimental observations. In particular, the critical force for localized buckling, predicted by the present model, is insensitive to microtubule length and can be about 1 order of magnitude lower than those given by the elastic-foundation model, which suggests that the elastic-foundation model may have overestimated the critical force for buckling of microtubules in vivo. In addition, unlike the elastic-foundation model, the present model can capture the effect of end conditions on the critical force and wavelength of localized buckling. Based on the known data of spacing and elastic constants of cross linkers available in literature, the critical force and wavelength of the localized buckling mode, predicted by the present model, are compared to some experimental data with reasonable agreement. Finally, two empirical formulas are proposed for the critical force and wavelength of the localized buckling of microtubules surrounded by cross linkers.