Nutations in plant shoots: endogenous and exogenous factors in the presence of mechanical deformations

A bstract We present a three-dimensional morphoelastic rod model capable to describe the morphogenesis of growing plant shoots, as driven by differential growth at the tip. We discuss the evolution laws for endogenous oscillators, straightening mechanisms and reorientations to directional cues, such as phototropic responses to a far light source and gravitropic reactions governed by the statoliths avalanche dynamics. We use this model to investigate the role of elastic deflections due to gravity loading in circumnutating plant shoots. We show that, in the absence of endogenous cues, pendular and circular oscillations arise as a critical length is attained, thus suggesting the occurrence of a Hopf bifurcation reminiscent of flutter instabilities exhibited by structural systems under nonconservative loads. When also oscillations due to endogenous cues are present, their weight relative to those associated with the Hopf instability varies in time as the shoot length and other biomechanical properties change. Thanks to the simultaneous occurrence of these two oscillatory mechanisms, we are able to reproduce a variety of complex behaviors, including trochoid-like patterns, which evolve into circular orbits as the shoot length increases, and the amplitude of the flutter induced oscillations becomes dominant. Our findings suggest that the relative importance of the two mechanisms is an emergent property of the system that is affected by the amplitude of elastic deformations, and highlight the crucial role of elasticity in the analysis of circumnutations.