Three-dimensional flow in Kupffer's Vesicle

Abstract

Whilst many vertebrates appear externally left-right symmetric, the arrangement of internal organs is asymmetric. In zebrafish, the breaking of left-right symmetry is organised by Kupffer's Vesicle (KV): an approximately spherical, fluid-filled structure that begins to form in the embryo 10 hours post fertilisation. A crucial component of zebrafish symmetry breaking is the establishment of a cilia-driven fluid flow within KV. However, it is still unclear (a) how dorsal, ventral and equatorial cilia contribute to the global vortical flow, and (b) if this flow breaks left-right symmetry through mechanical transduction or morphogen transport. Fully answering these questions requires knowledge of the three-dimensional flow patterns within KV, which have not been quantified in previous work. In this study, we calculate and analyse the three-dimensional flow in KV. We consider flow from both individual and groups of cilia, and (a) find anticlockwise flow can arise purely from excess of cilia on the dorsal roof over the ventral floor, showing how this vortical flow is stabilised by dorsal tilt of equatorial cilia, and (b) show that anterior clustering of dorsal cilia leads to around 40 % faster flow in the anterior over the posterior corner. We argue that these flow features are supportive of symmetry breaking through mechano-sensory cilia, and suggest a novel experiment to test this hypothesis. From our new understanding of the flow, we propose a further experiment to reverse the flow within KV to potentially induce situs inversus.

Keywords: Cilia; Kupffer’s Vesicle; Symmetry-breaking flow; Zebrafish embryo.

Fig. 1

a Schematic of a zebrafish embryo, showing the approximate location of KV squeezed between the posterior and the yolk sac. b Cilia populate the inner surface of KV randomly, with a greater number on the dorsal roof and the highest density in the anterior-dorsal corner (Kreiling et al. , redrawn). These cilia drive an anticlockwise vortical flow in the coronal midplane (midway between the dorsal and ventral poles)

Fig. 2

Nodal cilia. a A ciliary beat-pattern described by Eq. (7) with semicone angle $\mathit{\psi }={30}^{\circ }$ and tilt angle $\mathit{\theta }={30}^{\circ }$, demonstrating clockwise rotation when the cilium is viewed from tip to base. b Schematic demonstrating the tilt and semicone angles $\mathit{\theta },\mathit{\psi }$ respectively, showing the beat conical envelope and the measurement of the arclength s

Fig. 3

Flows from a single rotlet located at the anterior equator, viewed from the right side of KV. a An untilted rotlet, showing vortical flow throughout KV. b A ${90}^{\circ }$ tilt is applied to the rotlet, showing an anticlockwise vortex when viewed from the dorsal pole. By linearity of the Stokes flow equations, the flow arising from equatorial cilia tilted by ${30}^{\circ }$ is a linear combination of these two flows. See the supplementary material for the videos corresponding to these plots

Fig. 4

Flows from a ring of equatorial cilia viewed from the right side of KV, with the same format as Fig. 3. a Flow from an untilted ring of 10 equatorial cilia, showing an anticlockwise vortex in the ventral hemisphere (bottom) and a clockwise vortex in the dorsal hemisphere (top). b After these equatorial cilia are tilted ${30}^{\circ }$ to the dorsal pole, flow is an anticlockwise vortex throughout. See the supplementary material for the videos corresponding to these plots

Fig. 5

Box plots of velocity magnitude sampled at 1500 random points in the anterior third and posterior third of the coronal midplane with $r\le 17.5\mathrm{\mu }\mathrm{m}$, i.e. at least half a length from any equatorial cilia. a The velocity for three separate random placements of cilia following the Kreiling distribution (Kreiling et al. 2007), showing consistently higher velocity in the anterior. b The velocity for three further random placements of cilia without anterior clustering, i.e. with 80 % of cilia on the dorsal roof and 20 % on the ventral floor, showing equal velocity in the anterior and posterior

Fig. 6

Flow speed in the coronal midplane for $r\le 15$ for a a natural distribution (dist 1), showing faster flow in the anterior (right) and a displaced vortex centre point, and b an unclustered distribution (dist 6), showing no significant anterior–posterior speed asymmetry and a vortex centre point located at the origin. A displaced centre point was observed experimentally by Supatto et al. (2008)

Fig. 7

Flows from a natural, random distribution of cilia viewed from the right side of KV. a Untilted cilia following the distribution of Kreiling et al. (2007), showing an anticlockwise vortical flow about the anterior dorsal corner and b the same placement of cilia tilted to the dorsal pole, flattening this vortex. See the supplementary material for the videos corresponding to these plots

Fig. 8

Coronal midplane flows from distributions of 40 cilia for which only cilia in the bottom ventral third are motile. a Flow in natural distribution 2, showing slight directional motion and b flow in an unclustered, random distribution such as those created by Wang et al. (2012) showing clockwise vortical flows in contrast with the naturally-occurring anticlockwise vortex (Fig. 6a)

Fig. 9

Comparing time averaged flow (a) of a $5\mathrm{\mu }\mathrm{m}$ with $\mathit{\psi }={30}^{\circ },\mathit{\theta }={30}^{\circ }$ beating at 30 Hz with the equivalent rotlet (b), in the plane $z=7.5\mathrm{\mu }\mathrm{m}$. Distances are shown in microns, and the colormaps in figures a–c are the speed of the flow in microns per second. Visual comparison of a and b show remarkable similarity between the two solutions, with c showing at worst a 10 % error in the speed calculated by the rotlet model, and very good qualitative agreement in flow direction and magnitude. Evaluation of the results in a took approximately 2 h on a laptop computer, whereas b took approximately 10 s. d Shows the envelope of the cilium, the location of the equivalent rotlet, and the location of the plane in which the velocity is evaluated

Fig. 10

Time averaged flow of the same cilium (a) and equivalent rotlet (b) as in Fig. 9 in the plane $y=-2.5\mathrm{\mu }\mathrm{m}$ half a length away from the cilium. Again a and b are remarkably similar, with c showing at worst a 10 % error in the speed calculated by the rotlet model, and very good qualitative agreement in flow direction and magnitude