Buckling and force propagation along intracellular microtubules

Motivated by recent experiments showing the compressive buckling of microtubules in cells, we study theoretically the mechanical response of and force propagation along elastic filaments embedded in a non-linear elastic medium. We find that embedded microtubules buckle when their compressive load exceeds a critical value fc, and that the resulting deformation is restricted to a penetration depth that depends on both the non-linear material properties of the surrounding cytoskeleton, as well as the direct coupling of the microtubule to the cytoskeleton. The deformation amplitude depends on the applied load f > fc as (f − fc). This work shows how the range of compressive force transmission by microtubules can be tens of microns and is governed by the mechanical coupling to the surrounding cytoskeleton. Copyright c © EPLA, 2008 The mechanical response of most eukaryotic cells depends on their cytoskeleton, a composite network of filamentous proteins [1]. Microtubules (MTs) are the stiffest of these cytoskeletal filaments, and they play an important role in organization of, and transport within the cell. Their mechanical rigidity allows them to support significant stresses in the cytoplasm. These stresses can be highly inhomogeneous, with compressive/tensile forces directed along stiff MTs, permitting directed force transmission and mechanical signaling over several microns within the cell. As with macroscopic elastic rods, however, even the comparatively rigid MTs cannot, on their own, withstand as large compressive loads as tensile loads. This is because of the classical Euler buckling instability limiting the compressive force to a maximum value, which actually vanishes for long rods. It was recently shown, however, that even long MTs can bear large compressive loads, as a result of their coupling to the surrounding elastic matrix of the cytoskeleton [2]. This composite aspect of the cytoskeleton has important consequences for cell mechanics and mechanotransduction [3–7] —the generation, transmission, and sensing of forces by the cell. Here, we develop a model for compressively loaded elastic filaments such as MTs embedded in an elastic (a)E-mail: [email protected] continuum. When their compressive load f exceeds a critical force fc, an oscillatory buckling of the filament is expected, with a wavelength depending on both the stiffness of the elastic filament and the shear modulus of the surrounding medium [2,8,9]. In the classical Euler buckling problem, and even in the presence of a (linear) elastic background, an elastic rod becomes unstable for f > fc. We include non-linear elastic properties expected for the cytoskeleton, and show that the system is stable to supercritical loads, with a buckling amplitude that increases above threshold as |f − fc|1/2. In addition, both the buckling amplitude and the compressive load decay away from the point of force application in a way that depends sensitively on the longitudinal mechanical coupling of the filament to its surroundings. This suggests that the range of force transmission in the cell can be effectively controlled by microtubule-associating proteins that couple MTs to the rest of the cytoskeleton. Surprisingly, the experiments in ref. [2] (see fig. 6 therein) also found evidence that, despite the existence of a threshold force for buckling, the deformation of the microtubule was attenuated, suggesting a spatially varying force. In studying the mechanical response of intracellular MTs, we extend the classical buckling theory of rods in an elastic medium [2,8,9] to take into account the mechanical coupling of both longitudinal and transverse deformations