  Flows over remarkably long distances are crucial to the functioning of many
organisms, across all kingdoms of life. Coordinated flows are fundamental to
power deformations, required for migration or development, or to spread
resources and signals. A ubiquitous mechanism to generate flows, particularly
prominent in animals and amoeba, is acto-myosin cortex driven mechanical
deformations that pump the fluid enclosed by the cortex. Yet, it is unclear how
cortex dynamics can self-organize to give rise to coordinated flows across the
largely varying scales of biological systems. Here, we develop a
mechanochemical model of acto-myosin cortex mechanics coupled to a
contraction-triggering, soluble chemical. The chemical itself is advected with
the flows generated by the cortex driven deformations of the tubular-shaped
cell. The theoretical model predicts a dynamic instability giving rise to
stable patterns of cortex contraction waves and oscillatory flows.
Surprisingly, simulated patterns extend beyond the intrinsic length scale of
the dynamic instability - scaling with system size instead. Patterns appear
randomly but can be robustly generated in a growing system or by
flow-generating boundary conditions. We identify oscillatory flows as the key
for the scaling of contraction waves with system size. Our work shows the
importance of active flows in biophysical models of patterning, not only as a
regulating input or an emergent output, but rather as a full part of a
self-organized machinery. Contractions and fluid flows are observed in all
kinds of organisms, so this concept is likely to be relevant for a broad class
of systems.
