A Continuum Model for Cytoplasmic Streaming in the Drosophila melanogaster Embryo

Cytoplasmic streaming is observed within developing embryos such as those of the Drosophila melanogaster and is hypothesized to be driven by the collective cooperation between microtubule and molecular-motor couplings. Specifically in the Drosophila melanogaster, this process is described by Dutta et al. using a mathematical coupling between the forced Stokes equation for the cytoplasm and slender body theory for the microtubules [Nature Physics 20:666-674, 2024]. They show that beyond a threshold microtubule areal density and motor force the flow spontaneously develops into regimes with fluid velocities sufficient for proper intracellular transport. To rationalize this boundary between distinct states of motion, we employ a continuum approach on the microtubule scale. We approximate discrete microtubules at fixed intervals as a continuum with variable packing density. Under the continuum model, we perform a linear stability analysis to investigate the emergence of distinct streaming regimes with respect to the parameter space. Doing so we gain insight into the physics driving cytoplasmic streaming and investigate the asymptotic boundary which arises between the stationary and dynamic regimes.