Both sides now: modeling motor regulation of microtubule length at both ends

Microtubules are dynamic biopolymers whose lengths are continuously regulated by the concerted actions of polymerization, depolymerization, and motor-protein activity. While numerous mathematical models have explored the regulation of filament length, most have been formulated in the context of growth and shrinking at a single tip of a microtubule, effectively ignoring the mechanistic description of complex phenomena such as treadmilling. Here, we develop a multiscale model for microtubule length regulation that explicitly couples the kinetics of two classes of kinesin molecular motors to filament dynamics at both microtubule tips. Motor densities along the filament are modeled using one-dimensional parabolic partial differential equations. The microtubule length evolves dynamically through a shrinkage term that depends on motor density and which closes the system. In the adiabatic regime, where motor kinetics are fast relative to length dynamics, we derive a reduced model amenable to analytic study and identify simple parameter relationships distinguishing growth, catastrophe, and treadmilling behavior. Numerical simulations of the full system reveal qualitatively distinct dynamical regimes and demonstrate how bidirectional motor transport modulates filament length distributions. We parameterize our model with both in vivo and in vitro data and thus lay the foundation for developing mathematical models yielding a better understanding of cytoskeleton dynamics in living cells.