a Geometry of a modelled cell, shown in cross-section with each paddle at a different rotation angle, θ. Solid lines represent the links within the paddles; dashed lines the links of the connecting backbone. Dots P1–4 are magnetic beads, dots B1–4 are non-magnetic beads and dot N shows the beginning of the next cell. Beads P1–4 are magnetized along their paddle orientation, as indicated by the arrows. b Modelled variation of x-axis flow with paddle 2 rotational stiffness, l2, at constant paddle 1 rotational stiffness l1 = 0.19 (this l1 reproduces the maximum rotation observed in the experimental axle of width w = 12 µm, see Supplementary Fig. 3). Data points highlight values of l2 when both paddles are identical (l2 = l1 = 0.19) and rotational stiffnesses corresponding to the mechanics of experimental axles w = 9 µm, 6 µm and 3 µm (l2 = 0.17, 0.15 and 0.13, respectively). c Illustration of dissimilar rotational stiffness generating non-reciprocal actuation, with left-hand paddles rotating by 60° (right-hand paddles by 45°) between time steps until rotated to near-180° and then returning to 0°. Time is represented by the arrow; the mirror symmetry in time of the driving field cycle is indicated by the solid line.