  Fluid mechanics plays a vital role in early vertebrate embryo development, an
example being the establishment of left-right asymmetry. Following the
dorsal-ventral and anterior-posterior axes, the left-right axis is the last to
be established; in several species it has been shown that an important process
involved with this is the production of a left-right asymmetric flow driven by
'whirling' cilia. It has previously been established in experimental and
mathematical models of the mouse ventral node that the combination of a
consistent rotational direction and posterior tilt creates left-right
asymmetric flow. The zebrafish organising structure, Kupffer's vesicle, has a
more complex internal arrangement of cilia than the mouse ventral node;
experimental studies show the flow exhibits an anticlockwise rotational motion
when viewing the embryo from the dorsal roof, looking in the ventral direction.
Reports of the arrangement and configuration of cilia suggest two possible
mechanisms for the generation of this flow from existing axis information: (1)
posterior tilt combined with increased cilia density on the dorsal roof, and
(2) dorsal tilt of 'equatorial' cilia. We develop a mathematical model of
symmetry breaking cilia-driven flow in Kupffer's vesicle using the regularized
Stokeslet boundary element method. Computations of the flow produced by tilted
whirling cilia in an enclosed domain suggest that a possible mechanism capable
of producing the flow field with qualitative and quantitative features closest
to those observed experimentally is a combination of posteriorly tilted roof
and floor cilia, and dorsally tilted equatorial cilia.
