Microtubule buckling in an elastic matrix with quenched disorder

The intracellular elastic matrix has been recognized as an important factor to stabilize microtubules and increase their critical buckling force P in vivo. This phenomenon was qualitatively explained by the Winkler model, which investigated the buckling of a filament embedded in a homogeneous elastic medium. However, the assumption of homogeneity of the matrix in Winkler's, and other advanced models, is unrealistic inside cells, where the local environment is highly variable along the filament. Considering this to be a quenched-disorder system, we use a Poisson distribution for confinements and apply the replica technique combined with the Gaussian variational method to study the buckling of a long filament. The results show two types of filament bucklings: one corresponding to the first-order, and the other to a continuous second-order phase transition. The critical point, i.e., the switch from first- to second-order buckling transition, is induced by the increase in disorder strength. We also discover that this random disorder of the elastic environment destabilizes the filament by decreasing P from the Winkler result and the matrix with stronger mean elasticity has a stronger role of disorder (inhomogeneity). For microtubules in vivo, buckling follows the discontinuous first-order transition, with P reduced to the fraction between 0.9 and 0.75 of the Winkler prediction for the homogeneous elastic matrix. We also show that disorder can affect the force-displacement relationship at non-zero temperature, while at zero temperature this effect vanishes.