Microtubule elasticity: Connecting all-atom simulations with continuum mechanics

Abstract

The mechanical properties of microtubules have been extensively studied using a wide range of biophysical techniques, seeking to understand the mechanics of these cylindrical polymers. Here we develop a method for connecting all-atom molecular dynamics simulations with continuum mechanics and show how this can be applied to understand microtubule mechanics. Our coarse-graining technique applied to the microscopic simulation system yields consistent predictions for the Young’s modulus and persistence length of microtubules, while clearly demonstrating how binding of the drug Taxol decreases the stiffness of microtubules. The techniques we develop should be widely applicable to other macromolecular systems. Microtubules are essential players in a wide range of cellular functions. The protein tubulin polymerizes to form these filaments resulting in a hollow cylinder with a diameter of about 25 nm and lengths that range from tens of nanometers to tens of microns. Microtubules are the most rigid structures within the cell and their mechanical stability is particularly important for the role that microtubules play in cell division, cellular transport, and their basic ability to help provide cell shape. The mechanical properties of purified microtubules have been studied extensively using many biophysical techniques, including atomic force microscopy, thermal bending, and single molecule techniques [1–11]. Since microtubules are dynamic polymers and have intrinsic instability, many of these experimental studies have been performed on Taxol-stabilized microtubules, although the mechanical consequences of the drug Taxol for microtubule mechanics remains unclear. The majority of studies have found that the addition of Taxol results in ‘‘softer’’ microtubules with a shorter persistence length and lower Young’s modulus [1–5], while one published study had found the opposite result [6]. Recently, even the standard elastic model of microtubules characterized by a single elastic bending constant has been called into question by reports of length-dependent stiffness [12]. Thus, in spite of decades of experimental study, many basic questions and puzzles remain concerning the mechanics of even individual microtubules in vitro, let alone in vivo. Here, we combine all-atom simulations and continuum elastic modeling in order to determine both the elastic parameters of persistence length and Young’s modulus, as well as to test the applicability of simple, continuum shell elasticity for microtubules. Based on a molecular dynamics simulation of a small section of six tubulin monomers [13], we show that the observed fluctuations of this section are consistent with linear elastic constants for both bending and stretching along the microtubule axis. From these measurements, we determine the effective elastic properties for the microtubule wall, treated as an elastic sheet. With the addition of Taxol, we find increased flexibility and reduced bending and stretching moduli. We then examine the shear strain fluctuations of the tubulin sheet and find evidence for enhanced shear compliance, relative to the predictions of homogenous and isotropic elasticity. While anomalously large shear compliance has been suggested recently in connection with lengthdependent bending stiffness of microtubules [12], we find a much smaller degree of anisotropy than previously suggested. The shear elasticity we find is closer to recent experiments [9].