akashiwo . a, Rotation rate, ω, of HA452 cells before the overturning treatment, as a function of the direction, θ, of the instantaneous swimming velocity, v, relative to the vertical. The rotation rate of the cells (n = 2,257) was quantified by tracking them in the time intervals 0–5 s (grey) and 5–15 s (magenta) directly after a single flip, and averaged over all the cells as a function of θ. The difference between the two curves denotes the presence of cells that reorient more rapidly and others that reorient more slowly. Dashed lines are sinusoidal fits to the experimental data, used to obtain the reorientation timescale B. Solid lines (colour-coded) denote the arithmetic mean over all cell trajectories. The shaded region denotes ± 1 s.e.m. The reorientation timescales obtained from these data are B = 7.2 s for the first 5 s and B = 12.2 s for the following 10 s, denoting a nearly twofold higher stability for cells that were observed reorienting in the first 5 s. b, Rotation rate, ω, of HA3107 cells before the overturning treatment, as a function of the swimming direction, θ. The rotation rate was quantified by tracking cells for 15 s directly after a single flip and averaged over all cells as a function of θ (n = 1,283). The dashed line is a sinusoidal fit to the data used to obtain B. The solid line denotes the arithmetic mean over all cell trajectories. The shaded region denotes ± 1 s.e.m. The reorientation timescale obtained for HA3017 from these data was B = 4.9 s. c, Distribution of swimming orientation of HA452 cells before the overturning treatment (same data as in a). The distribution was quantified by tracking cells in the time intervals 0–5 s (black), 5–10 s (green), 10–15 s (cyan), and 15–20 s (blue) directly after a single flip, and averaged over all cells as a function of θ. Note that after 15 s the distribution does not appreciably change. d, Time series of the vertical distribution of HA452 following a 100-flip treatment (period Q = 18 s). The cell distribution inside the chamber was tracked after the end of the overturning treatment, with time zero corresponding to the termination of the treatment (between 461 and 592 cells are included in each vertical profile). At t = 1 s (blue) the cell distribution is homogeneous because the cells have been continuously flipped for 30 min and 1 s is not long enough to allow cells to reach their equilibrium profile. To traverse the chamber (4 mm), it would take 80 s for cells swimming with a vertical speed of 50 μm s−1 (Extended Data Fig. 2). In fact, it takes 90 s (orange) to establish the bimodal distribution at equilibrium, corresponding to the population split induced by overturning. The population split is then maintained for at least 7 h (black). The upward bias shown in Figs 1 and 2 and in Extended Data Figs 1 and 3 is always computed 30 min after the overturning ceases. e, Effect of the torque generated by the offset LNb of the nucleus within the equatorial plane, obtained from the cell mechanics model, shown in terms of its effect on the dependence of the rotation rate on the body axis angle for the upward-swimming subpopulation HA452(↑). The dashed red line denotes the case without offset (LNb = 0), the purple and pink lines represent the cases in which the nucleus is offset by LNb = +0.25 μm and by LNb = −0.25 μm, respectively (the average offset measured experimentally; see Extended Data Fig. 4d), and the dark green and light green lines represent the cases in which the offset corresponds to mean + s.d. of the experimentally measured values, that is, LNb = +(0.25 + 0.26) = +0.51 μm and LNb = −(0.25 + 0.26) = −0.51 μm. Note that the overall upward stability of the cells remains unchanged when one accounts for the effect of LNb, since the stable points for all the cases (coloured dots) always occur for a swimming orientation θ that is smaller than ± π/2 (dashed vertical lines), which separates upward and downward swimming (θ = ±28° for LNb = ±0.25 μm; θ = ±35° for LNb = ±0.51 μm. Note that the results are symmetric around the vertical direction, θ = 0°). f, Stability analysis demonstrating that two assumptions made in our calculations have negligible consequences, in particular the assumptions that (1) the angle α between the body axis and the direction of motion is zero (compare orange and red lines), and (2) the drag on the fore–aft asymmetric upward swimmers can be approximated by the drag on a spheroid (compare red and pink lines). Shown is the rotation rate as a function of body axis angle for three cases: a spheroidal cell in which the major axis is aligned with the direction of motion (α = 0, orange), a spheroidal cell in which the misalignment between major axis and direction of motion is accounted for (α = 4°, red), and a fore–aft asymmetric cell in which the misalignment between major axis and direction of motion is accounted for (α = 4°, pink). Parameters were taken from Extended Data Table 2 (first row), for the upward-swimming cells. The fore–aft asymmetric case was simulated with Comsol Multiphysics. Note that the cell stability is the same in all three cases, as evidenced from the fact that the curve has a stable point at a swimming angle of θ = 0 and a negative minimum at θ = π/2, which together imply upward stability. Throughout our analyses, we have thus adopted the spheroidal approximation for the calculation of the drag, and taken into account the contribution to the cell stability by the angle α.