Statistical Mechanics Provides Novel Insights into Microtubule Stability and Mechanism of Shrinkage

Abstract

Microtubules are nano-machines that grow and shrink stochastically, making use of the coupling between chemical kinetics and mechanics of its constituent protofilaments (PFs). We investigate the stability and shrinkage of microtubules taking into account inter-protofilament interactions and bending interactions of intrinsically curved PFs. Computing the free energy as a function of PF tip position, we show that the competition between curvature energy, inter-PF interaction energy and entropy leads to a rich landscape with a series of minima that repeat over a length-scale determined by the intrinsic curvature. Computing Langevin dynamics of the tip through the landscape and accounting for depolymerization, we calculate the average unzippering and shrinkage velocities of GDP protofilaments and compare them with the experimentally known results. Our analysis predicts that the strength of the inter-PF interaction ( $E_m^s$ ) has to be comparable to the strength of the curvature energy ( $E_m^b$ ) such that $E_m^s-E_m^b\approx 1k_{\text{B}}T$ , and questions the prevalent notion that unzippering results from the domination of bending energy of curved GDP PFs. Our work demonstrates how the shape of the free energy landscape is crucial in explaining the mechanism of MT shrinkage where the unzippered PFs will fluctuate in a set of partially peeled off states and subunit dissociation will reduce the length.

Author Summary

Microtubules are cylindrical machines inside biological cells, and are crucial for many functions such as chromosome segregation, intra-cellular transport, and cell motility. They are made of 13 elastic filaments (protofilaments) that can be either in a straight or in a curved conformation depending on the chemical state of the constituent tubulin molecules. The interplay between these two conformations help microtubules to display a fascinating phenomenon known as “dynamic instability,” in which the microtubule steadily self-assembles and catastrophically disassembles in a seemingly random process. During the disassembly, the protofilaments are known to curve out forming ram’s horn-like structures. Scientists have been trying to understand how the laws of mechanics and statistical thermodynamics determine the complex behavior of microtubules. In this paper, we investigate how the chemical bonding between protofilaments, bending elasticity of protofilaments, and thermal forces determine the speed of the disassembly of the microtubule cylinder. We show that the current notion of a bending-elasticity dominated disassembly will lead to a paradox; resolving the paradox, we argue that the disassembly is a result of an intricate molecular orchestration in which the thermal and curvature energies of protofilaments compete with inter-protofilament bonding energy leading to curving out and depolymerization.