In the mitotic spindle microtubules attach to kinetochores via catch bonds during metaphase, and microtubule depolymerization forces give rise to stochastic chromosome oscillations. 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. A one-sided model, where an external force acts on the kinetochore is solved analytically employing a mean-field approach based on Fokker–Planck equations. The solution establishes a bistable force–velocity relation of the microtubule ensemble 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 derive constraints on linker stiffness and microtubule number for metaphase chromosome oscillations. Including poleward microtubule flux into the model we can 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. The model can reproduce experimental results for kinetochore oscillations in PtK1 cells quantitatively.

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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 summaryThe 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 March 12, 2019 1/31 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 micro...

Regarding the experimental observation that microtubule catastrophe can be described as a multistep process, we extend the Dogterom-Leibler model for dynamic instability in order to discuss the effect that such a multistep catastrophe mechanism has on the distribution of microtubule lengths in the two regimes of bounded and unbounded growth. We show that in the former case, the steady state length distribution is non-exponential and has a lighter tail if multiple steps are required to undergo a catastrophe. If rescue events are possible, we detect a maximum in the distribution, i.e., the microtubule has a most probable length greater than zero. In the regime of unbounded growth, the length distribution converges to a Gaussian distribution whose variance decreases with the number of catastrophe steps. We extend our work by applying the multistep catastrophe model to microtubules that grow against an opposing force and to microtubules that are confined between two rigid walls. We determine critical forces below which the microtubule is in the bounded regime, and show that the multistep characteristics of the length distribution are largely lost if the growth of a microtubule in the unbounded regime is restricted by a rigid wall. All results are verified by stochastic simulations.

Regarding the experimental observation that microtubule catastrophe can be described as a multistep process, we extend the Dogterom--Leibler model for dynamic instability in order to discuss the effect that such a multistep catastrophe mechanism has on the distribution of microtubule lengths in the two regimes of bounded and unbounded growth. We show that in the former case, the steady state length distribution is non-exponential and has a lighter tail if multiple steps are required to undergo a catastrophe. If rescue events are possible, we detect a maximum in the distribution, i.e., the microtubule has a most probable length greater than zero. In the regime of unbounded growth, the length distribution converges to a Gaussian distribution whose variance decreases with the number of catastrophe steps. We extend our work by applying the multistep catastrophe model to microtubules that grow against an opposing force and to microtubules that are confined between two rigid walls. We determine critical forces below which the microtubule is in the bounded regime, and show that the multistep characteristics of the length distribution are largely lost if the growth of a microtubule in the unbounded regime is restricted by a rigid wall. All results are verified by stochastic simulations.