Fig. 2.

Simulations reproduce waving and coiling. (A) Final configurations. Top view of final configurations of simulations for tilt angles $0^0\le \alpha \le {90}^0$, with $\beta =\gamma =0.1$. Pink and orange lines represent the direction of the vectors $\pm {\mathbf{d}}_1$ describing the material frame along the organ. The local planar angle $\theta (s,t)$ is the angle between the projection of the organ tangent on the tilted plane, and the projection of the direction of gravity ${\widehat{\mathbf{g}}}_{\Vert }$. (B) Snapshots of waving dynamics. Five snapshots during the developing waving pattern for a root at $\alpha ={40}^o$, with local gravisensing, $\beta =0.2$, and $\gamma =0.1$. Stages (i) and (ii) depict the first symmetry breaking due to an elastic instability that reduces bending energy by twisting sideways. Between stages (iii) and (v) left-handed twist is being accumulated in the mature zone as the root turns clock-wise (evident by following ${\mathbf{d}}_1$), in line with observed twisting cell files (26). (C–E): Properties of the growth zone during the development of a waving pattern, corresponding to the root in (B). Dashed lines represent the snapshots in (B), and time is shown in number of time-steps. (C) Planar tip angle ${\theta }^{\text{tip}}(t)$ (blue). For comparison, we show the behavior of a corresponding free organ (pink line), that is of a root growing without substrate but characterized by the same initial angle (with respect to gravity) of the waving root (blue line). Both profiles present oscillations with a similar period; however, in the absence of interactions with the plane, the oscillations decay in time. (D) Average elastic bending energy $E_b$ and twisting energy $E_t$ in the growth zone. (E) Total writhe Wr and twist Tw in the growth zone.