Bistability and oscillations in cooperative microtubule and kinetochore dynamics in the mitotic spindle
Authors: Schwietert Felix , Kierfeld Jan 
Published: March 2019

In the mitotic spindle microtubules attach to kinetochores via catch bonds during metaphase. We investigate the cooperative stochastic microtubule dynamics in spindle models consisting of ensembles of parallel microtubules, which attach to a kinetochore via elastic linkers. We include the dynamic instability of microtubules and forces on microtubules and kinetochores from elastic linkers. We start with a one-sided model, where an external force acts on the kinetochore. A mean-field approach based on Fokker-Planck equations enables us to analytically solve the one-sided spindle model, which establishes a bistable force-velocity relation of the microtubule ensemble. All results are in agreement with stochastic simulations. We derive constraints on linker stiffness and microtubule number for bistability. The bistable force-velocity relation of the one-sided spindle model gives rise to oscillations in the two-sided model, which can explain stochastic chromosome oscillations in metaphase (directional instability). We also derive constraints on linker stiffness and microtubule number for metaphase chromosome oscillations. We can include poleward microtubule flux and polar ejection forces into the model and provide an explanation for the experimentally observed suppression of chromosome oscillations in cells with high poleward flux velocities. Chromosome oscillations persist in the presence of polar ejection forces, however, with a reduced amplitude and a phase shift between sister kinetochores. Moreover, polar ejection forces are necessary to align the chromosomes at the spindle equator and stabilize an alternating oscillation pattern of the two kinetochores. Finally, we modify the model such that microtubules can only exert tensile forces on the kinetochore resulting in a tug-of-war between the two microtubule ensembles. Then, induced microtubule catastrophes after reaching the kinetochore are necessary to stimulate oscillations.

Author summary

The mitotic spindle is responsible for proper separation of chromosomes during cell division. Microtubules are dynamic protein filaments that actively pull chromosomes apart during separation. Two ensembles of microtubules grow from the two spindle poles towards the chromosomes, attach on opposite sides, and pull chromosomes by depolymerization forces. In order to exert pulling forces, microtubules attach to chromosomes at protein complexes called kinetochores. Before the final separation, stochastic oscillations of chromosomes are observed, where the two opposing ensembles of microtubules move chromosome pairs back an forth in a tug-of-war.

Using a a combined computational and theoretical approach we quantitatively analyze the emerging chromosome dynamics starting from the stochastic growth dynamics of individual microtubules. Each of the opposing microtubule ensembles is a bistable system, and coupling two such systems in a tug-of-war results in stochastic oscillations. We can quantify constraints on the microtubule-kinetochore linker stiffness and the microtubule number both for bistability of the one-sided system and for oscillations in the full two-sided spindle system, which can rationalize several experimental observations. Our model can provide additional information on the microtubule-kinetochore linkers whose molecular nature is not completely known up to now.