Nutations in Plant Shoots: Endogenous and Exogenous Factors in the Presence of Mechanical Deformations

Abstract

We present a three-dimensional morphoelastic rod model capable to describe the morphogenesis of growing plant shoots driven by differential growth. We discuss the evolution laws for endogenous oscillators, straightening mechanisms, and reorientations to directional cues, such as gravitropic reactions governed by the avalanche dynamics of statoliths. 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 an instability triggered by exogenous factors. When also oscillations due to endogenous cues are present, their weight relative to those associated with the 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 exogenous oscillations becomes dominant.

Nutations in growing plant shoots as a morphoelastic flutter instability

“…Let E 3 be the Euclidean space and choose a right-handed orthonormal basis {e 1 , e 2 , e 3 } such that the gravity vector is g := −e 2 . We follow Agostinelli et al [28] and model a growing slender organ as an unshearable and (elastically) inextensible rod of circular cross section. The current configuration of the rod has axis p(s, t) and orthonormal directors d j (s, t) for j = 1,2,3, where s ∈ [0, (t)] is the arc length coordinate.…”

“…At leading-order, differential growth, namely, a morphing mechanism consisting of a spatially non-homogeneous growth rate of the organ cross section, determines the evolution laws of the spontaneous flexural strains. In this respect, we consider several distinct contributions: endogenous oscillators, gravitropic responses (driven by the statoliths avalanche dynamics) and straightening mechanisms [28].…”

“…Following a common approach, tip growth can be modelled by prescribing the following coupled problems ∂s ∂S (S, t) = λ(S, t) with s(0, t) = 0, (3.1a) and 1 λ(S, t) ∂λ ∂t (S, t) =ε (S, t) with λ(S, 0) = 1, (3.1b) to be solved for S ∈ [0, 0 ] and t ≥ 0 [30]. Here a dot denotes the material time derivative so thaṫ ε is the strain rate [28], which is often referred to as Relative Elemental Growth Rate (REGR) in the plant science community. This is a growth quantifier that was first introduced by Erickson & Sax [32] in order to describe the primary growth of plant roots, and it can be experimentally measured by tracking material markers along the elongating organ [33][34][35][36][37][38].…”

“…For the numerical implementation of the morphoelastic rod model detailed in the previous sections, we extend the computational model presented in [28] to include a generic growth law characterized by a REGR of the form (3.5). At this point, we find it useful to summarize the full set of equations governing morphoelastic processes in elongating plant organs.…”

“…In fact, nutations in growing plant shoots may be described as spontaneous oscillations caused by a flutter-like instability (a Hopf-type bifurcation) in a growing elastic system subject to gravity loading while sensing and actively responding to external stimuli. We have discussed the merits of this hypothesis in previous studies [28,29] and, in this paper, we study in detail the impact of different growth models on the qualitative and quantitative features of these spontaneous oscillations. To this aim, we model the system using the theory of morphoelastic growing rods discussed in [30].…”