  The Drosophila melanogaster embryo, an elongated multi-nucleated cell, is a
classical model system for eukaryotic development and morphogenesis. Recent
work has shown that bulk cytoplasmic flows, driven by cortical contractions
along the walls of the embryo, enable the uniform spreading of nuclei along the
anterior-posterior (AP) axis necessary for proper embryonic development. Here
we propose two mathematical models to characterise cytoplasmic flows driven by
tangential cortical contractions in elongated cells. Assuming Newtonian fluid
flow at low Reynolds number in a spheroidal cell, we first compute the flow
field exactly, thereby bypassing the need for numerical computations. We then
apply our results to recent experiments on nuclear transport in cell cycles 4-6
of Drosophila embryo development. By fitting the cortical contractions in our
model to measurements, we reveal that experimental cortical flows enable
near-optimal axial spreading of nuclei. A second mathematical approach,
applicable to general elongated cell geometries, exploits a long-wavelength
approximation to produce an even simpler solution, with errors below 5%
compared to the full model. An application of this long-wavelength result to
transport leads to fully analytical solutions for the nuclear concentration
that capture the essential physics of the system, including optimal axial
spreading of nuclei.
