Elastic and damping forces generated by confined arrays of dynamic microtubules

In addition to serving as structural elements and as tracks for motor proteins, microtubules use chemical energy derived from the hydrolysis of GTP to generate forces when growing and shrinking. These forces are used to push or pull on organelles such as chromosomes and the mitotic spindle. If an array of microtubules grows out from a nucleation site and is confined by the periphery of the cell, pushing and pulling forces can give rise to interesting collective phenomena. In this paper, I show that pushing forces center the array provided that the microtubules are dynamic in the sense that they switch from pushing to shrinking after reaching the periphery. Microtubule dynamics of free ends is neither necessary nor sufficient for centering. Buckling can augment the centering force. For small displacements and velocities, the array can be modeled very simply as a damped spring. The dynamic stiffness of the array is orders of magnitude smaller than its static stiffness, and the relaxation time is on the order of the time that it takes for a microtubule to grow from the center to the periphery. Replacement of a dynamic polymer array with an equivalent mechanical circuit provides a bridge between molecular and cellular mechanics.