Bistability and oscillations in cooperative microtubule and kinetochore dynamics in the mitotic spindle

Abstract

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.

Modeling reveals cortical dynein-dependent fluctuations in bipolar spindle length

“…Although mathematical models have extensively examined MT attachments to chromosomes and how these attachments drive chromosome movement and alignment during mitosis (16,110,111), we omit chromosomes and chromosomederived forces in our model, as end-on MT attachments to kinetochores are dispensable for bipolar spindle formation (33,34). However, our data suggest that kinetochore-microtubule interactions may reinforce spindle stability.…”

Dynamics and length distributions of microtubules with a multistep catastrophe mechanism

“…Finally, the oscillatory behavior of multistep MTs that follows from the deterministic growth durations may also support the characteristic chromosome oscillations occurring during metaphase [27]. This influence may be examined theoretically by means of mitotic spindle models that reproduce chromosome oscillations and already include dynamic instability of individual MTs [28][29][30][31]. Such models could be easily extended by multistep catastrophes.…”

Modeling reveals cortical dynein-dependent fluctuations in bipolar spindle length

“…While mathematical models have extensively examined MT attachments to chromosomes and how these attachments drive chromosome movement and alignment during mitosis (16,110,111), we omit chromosomes and chromosome-derived forces in our model as end-on MT attachments to kinetochores are dispensable for bipolar spindle formation (33,34). However, our data suggest that kinetochore-microtubule interactions may reinforce spindle stability.…”