Figure 3.1

Figure 3.1

(A) Overpressure distribution in various 2-D central canals of length 300 µm and diameter d = 10 µm. The overpressure represents the excess of pressure with respect to the inlet of the CC (at the junction with the brain ventricle). In the case of a homogeneous cilia distribution (red curve), modeled as a constant force f_v = αμf/h, the pressure increases linearly in the caudal direction. In the case of a loose distribution of cilia, represented by an alternation of active regions with the aforementioned force fv and passive regions without any bulk force, the pressure follows a piecewise linear increase, whose frequency is controlled by the distance between active and passive regions: w = d for the blue curve and w = d = 2 for the green curve. The cilia activity is thus directly correlated to the excess of pressure in the central canal, as dramatically put in evidence by the magenta flat curve in the absence of any cilia activity. This excess of pressure in the central canal induced by the cilia activity could help maintaining the lumen of the channel open, explaining why the central canal is found to be collapsed in cilia mutants (Figure 4C). (B) In the main manuscript, we modeled the contribution of the cilia as a constant bulk force f_v = αμf/h in the ventral region occupied by the cilia. In a prior publication, Siyahhan et al. (2014) modeled it as a force linearly increasing with the distance to the wall (also located in the ciliary region). We show in this figure that this choice (blue triangles) does not substantially alter the velocity profile predicted by our model (red squares), which simply turns into a 3rd degree polynomial in the ventral half of the channel. The y-position for vanishing local velocity is slightly shifted towards the dorsal wall. (C) Our model still holds for variations of the relative thickness of the ciliary region and the dorsal region. Following the derivation detailed in the manuscript, the velocity profiles can be obtained for arbitrary values of h, at fixed t = 10 µm. The green curve corresponds to the zebrafish CC features, and the red curves corresponds to very thin cilia with respect to the diameter of the tube. This situation is representative to that encountered by Shields et al. (2010) with artificial magnetic cilia. They decided to model the effect of cilia as a moving wall entraining fluid. While this moving wall modeling correctly accounted for their observations in these high length ratio conditions, we point out that it would not successfully predict the flow in more common length ratios as it fully neglects the flow in the ciliary region. Conversely, our model accounts for the flow in the whole channel, which renders it very versatile to different geometries. (D) In Figure 3C, we have shown by numerical simulations that the alternation of active and passive regions can give birth to vorticity in the flow profile. Here, we show that recirculation regions can also be obtained from the succession of neighboring cilia with different beating frequencies. And the larger the difference in frequency, the larger the vortices. The black lines correspond to the trajectory of the flow, and the colormap in the channel corresponds to the rostro-caudal velocity. The colored rectangles under the ventral wall indicate the frequency for each spot in the simulation for the bulk force f_v = αμf/h. This choice is consistent with the in vivo measurements detailed in the Figure 2B of the main manuscript.