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In my book this is described as happening by kinetochores depolymerising from their attached microtubule spindle, then being dragged towards the spindle's centrosome by a dynein protein. I can understand this, but what it does not seem to answer is how different copies of chomatids beome attached to the correct spindles in the first place (i.e. spindles connected to opposing centrosomes). How does this work?

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It's a spatial constraint. As the DNA is replicated, the two resulting chromatids are kept stuck together by cohesin proteins. The DNA sequence that corresponds to the centromere then coalesces the kinetochore. It seems that the DNA-protein interactions at the centromeres creates a particular structure along the chromosome. Since the two chromatids have their kinetochores pointing in opposite directions, when a spindle microtubule attaches to the kinetochore, it is nearly impossible for another microtubule from the same spindle to attach to the opposing kinetochore.

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