  The proper positioning of the mitotic spindle is crucial for asymmetric cell
division and generating cell diversity during development. Proper position in
the single-cell embryo of Caenorhabditis elegans is achieved initially by the
migration and rotation of the pronuclear complex (PNC) and its two associated
centrosomal arrays of microtubules (MTs). We present here the first systematic
theoretical study of how these $O(1000)$ centrosomal microtubules (MTs)
interact through the immersing cytoplasm, the cell periphery and PNC, and with
each other, to achieve proper position. This study is made possible through our
development of a highly efficient and parallelized computational framework that
accounts explicitly for long-ranged hydrodynamic interactions (HIs) between the
MTs, while also capturing their flexibility, dynamic instability, and
interactions with molecular motors and boundaries. First, we show through
direct simulation that previous estimates of the PNC drag coefficient, based on
either ignoring or partially including HIs, lead to misprediction of the active
forces and time-scales of migration. We then directly study the dynamics of PNC
migration under various force-transduction models, including the pushing or
pulling of MTs at the cortex, and the pulling of MTs by cytoplasmically-bound
force generators. While achieving proper position and orientation on
physiologically reasonable time-scales does not uniquely choose a model, we
find that each model produces a different signature in its induced cytoplasmic
flow and MT conformations. We suggest then that cytoplasmic flows and MT
conformations can be used to differentiate between mechanisms and to determine
their contribution to the migration process.
