The mystery of plant nutations: Is mathematics of any help?

Plants do move (a lot!)

We often assume that plants are inanimate objects, just because they keep the same position for a long time. The truth is that human attention is typically focused on fast events — as they might be the most dangerous and threatening for our safety — so that we miss to detect many slow motions. Time-lapse photography allows us to admire an extraordinary variety of interesting movements proper to the plant kingdom, which fascinated generations of scientists since the pioneering work by Darwin [1].

Movements in plants are mainly classified in tropisms and nastic movements. Tropisms are responses to directional external cues — e.g., light (phototropism), gravity (gravitropism), touch (thigmotropism), etc — while nastic movements are motions independent of the stimulus direction — e.g., temperature (thermonasty), chemicals (chemonasty), touch (thigmonasty), etc.

Example of (negative) gravitropism: Plant shoots align their axis against the gravity vector.

In plants, active adaptations (biochemical processes) and passive reconfigurations (mechanical instabilities) cooperate to produce essential functions, such as reproduction, nutrition, and self-defense. Think about the explosive seed dispersal of some species, the snapping of Venus flytrap and the closing of Mimosa Pudica [2,3].

Examples of explosive seed dispersal.

Circumnutations: The question that puzzled scientists for more than a century

Circumnutations are another kind of motion that is widespread in plants. They are pendular, elliptical or circular oscillatory movements, such as the ones exhibited by the main stem of Arabidopsis thaliana (Fig. 1).

Fig. 1: Examples of tip trajectories from specimens of Arabidopsis thaliana (ecotype Col-0) grown at the SAMBA laboratory of SISSA: (a) Pendular oscillations, (b) elliptic and (c) circular patterns. Left: Stereo pair of images corresponding to the last instant of the tip trajectories. The superposed black dots are the tracked positions of the tip at time intervals of 1 minute. Right: Top view of the tip trajectories as reconstructed by matching corresponding points in the stereo pair of images. The coloured lines, from blue to red for increasing time, are obtained by moving averaging over ten detected positions, shown in black. From [13].

Circumnutations result from differential growth and reversible cell volume variations in opposite sides of the stem [4]. However, their regulatory mechanism remains unclear, with opinions split between three main hypotheses [5]. First, as already suggested by Darwin [1], circumnutations might be driven by endogenous oscillators, internally regulating differential growth. Second, they might be the byproduct of gravitropic and straightening (or autotropic) mechanisms: The deviation from the vertical line triggers a correcting movement that, due to a reaction time between perception and actuation, makes the plant overshoot giving rise to self-sustained oscillations. This is referred to as the overshooting hypothesis [6]. Third, the previous two mechanisms might be combined in a “two-oscillator” hypothesis in which endogenous prescriptions and delayed (gravitropic and autotropic) responses coexist [7].

Over the last few decades, new experiments on Earth and in Space fueled the debate about the role of gravity for the induction or continuation of circumnutations [8,9]. To date, the question remains unsolved even if

“[…] circumnutations are more readily presumed to be induced by both the
inner oscillator and gravity […]”, Stolarz [5].

Previous studies analysed the role of gravity as the trigger for gravitropic responses, but they neglected its effect on the mechanical (elastic) deformations of the plant organ. Let us investigate this question by means of a mathematical model!

Mathematical modeling: What do we learn?

We develop a model based on the theory of morphoelastic rods, which allows to describe growing slender structures (rods) by decoupling active growth processes and passive elastic reconfigurations [10]. We include endogenous oscillators, lignification, proprioception (i.e., a straightening mechanism), and gravitropic responses.

Fig. 2: Superposition of deformed shapes and respective directors. For supercritical lengths two types of nontrivial periodic solutions emerge in the absence of endogenous oscillations: (a) unstable pendular and (b) stable circular oscillatory patterns. From [13].

By means of a theoretical analysis and a computational study — based on this FEniCS implementation [11] — , we find that, in the absence of endogenous cues, spontaneous oscillations might arise as system instabilities (bifurcations) when a loading parameter exceeds a critical value, see Fig. 2 [12]. We refer to these oscillations as “flutter”, as they are reminiscent of dynamic instabilities exhibited by mechanical systems under nonconservative loads, or as “exogenous oscillations”, since endogenous cues need not to be included. This mechanism is robust with respect to changes in the underlying growth law [13].

Video 1: Computational results from the nonlinear rod model. In the beginning we observe only endogenous oscillations with gravitropic and autotropic responses. In the intermediate regime the oscillations due to the intrinsic oscillator and the mechanical flutter are comparable and give rise to trochoid-like patterns. In the end, the flutter instability dominates the movements. From [13].

When also oscillations of endogenous nature are present, their relative importance with respect to the ones associated with the exogenous oscillations varies in time and affects the resulting dynamics, as the shoot elongates [13]. The system gradually transitions from a dynamics mainly characterized by endogenous oscillations in the subcritical regime (i.e., when the shoot is short) to one in which flutter-induced oscillations dominate in the supercritical regime (i.e., when the shoot is long). Trochoid-like patterns — that are reminiscent of the trajectories observed by Schuster and Engelmann [15] in specimens of Arabidopsis thaliana — are visible in the intermediate regime of flutter initiation, see Video 1 and Fig. 3.

FIg. 3: Superposition of deformed shapes and respective directors. Flutter was initiated in the clockwise direction by suitable initial perturbations and we obtained (a) epitrochoid-like and (b) hypotrochoid-like patterns for concordant and discordant endogenous oscillations, respectively. From [13].

Interestingly, experiments reveal a transition from small to large amplitudes (see Video 2), as Darwin already reported in [1]:

“[…] Again, climbing plants whilst young circumnutate in the ordinary manner, but as soon as the stem has grown to a certain height, which is different for different species, it elongates rapidly, and now the amplitude of the circumnutating movement is immensely increased, evidently to favour the stem catching hold of a support […]”.

Video 2: Tracking and reconstruction of a flower of the main stem in a sample of Arabidopsis thaliana (ecotype Col-0) grown at the SAMBA laboratory of SISSA. The points (black dots) were tracked on the stereo pair of images, which were calibrated to reconstruct the 3D position by exploiting the Computer Vision Toolbox in MATLAB R2019b. The coloured lines, from blue to red for increasing time, are obtained by moving averaging over ten detected positions. The red dot indicates the base of the plant. From [13].

In conclusion, we point out that this mathematical model is a simplification of the complex reality of plant circumnutations, and our findings require a thorough quantitative assessment in comparison with experiments. However, these results provide insight to reinterpret the two-oscillator hypothesis and shed light on the role of elasticity in the mystery of circumnutations.

References

[1] C. Darwin. The power of movement in plants. John Murray, London, 1880
[2] Y. Forterre, J.M. Skotheim, J. Dumais, and L. Mahadevan. How the Venus flytrap snaps. Nature, 433(7024):421–425, 2005
[3] H. Hofhuis, D. Moulton, T. Lessinnes, A.-L. Routier-Kierzkowska, R.J. Bomphrey, G. Mosca, H. Reinhardt, P. Sarchet, X. Gan, M. Tsiantis, Y. Ventikos, S. Walker, A. Goriely, R. Smith, and A. Hay. Morphomechanical innovation drives explosive seed dispersal. Cell, 166(1):222–233, 2016
[4] S. Mugnai, E. Azzarello, E. Masi, C. Pandolfi, and S. Mancuso. Nutation in Plants. In S. Mancuso, S.and Shabala, editor, Rhythms in Plants, pages 19–34. Springer, 2015
[5] Stolarz, M. Circumnutation as a visible plant action and reaction: physiological, cellular and molecular basis for circumnutations. Plant signaling & behavior 4, 380–387, 2009
[6] H. Gradmann. Die Fünfphasenbewegung der Ranken. Jahrbücher für wissenschaftliche Botanik, 61:169–204, 1922
[7] A. Johnsson, C. Jansen, W. Engelmann, and J. Schuster. Circumnutations without gravity: a two-oscillator model. Journal of Gravitational Physiology, 6(1):9–12, 1999
[8] J.Z. Kiss. Up, down, and all around: how plants sense and respond to environmental stimuli. Proceedings of the National Academy of Sciences of USA, 103(4):829–830, 2006
[9] A. Kobayashi, H.-J. Kim, Y. Tomita, Y. Miyazawa, N. Fujii, S. Yano, C. Yamazaki, M. Kamada, H. Kasahara, S. Miyabayashi, T. Shimazu, Y. Fusejima, and H. Takahashi. Circumnutational movement in rice coleoptiles involves the gravitropic response: analysis of an agravitropic mutant and space-grown seedlings. Physiologia Plantarum, 165:464–475, 2019
[10] A. Goriely. The mathematics and mechanics of biological growth, 45. Springer, 2017
[11] A. Logg, K.-A. Mardal, and G. Wells. Automated solution of differential equations by the finite element method: The FEniCS book, 84. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012.
[12] D. Agostinelli, A. Lucantonio, G. Noselli, and A. DeSimone. Nutations in growing plant shoots: The role of elastic deformations due to gravity loading. Journal of the Mechanics and Physics of Solids, 136:103702, 2020. The Davide Bigoni 60th Anniversary Issue
[13] D. Agostinelli, A. DeSimone, and G. Noselli. Nutations in plant shoots: Endogenous and exogenous factors in the presence of mechanical deformations. bioRxiv, 2020 (to appear in Frontiers in Plant Science — Plant Biophysics and Modeling, 2021)
[14] D. Agostinelli, G. Noselli, and A. DeSimone. Nutations in growing plant shoots as a morphoelastic flutter instability. Philosophical Transactions of the Royal Society A, 2021 (in press)
[15] J. Schuster and W. Engelmann. Circumnutations of Arabidopsis thaliana seedlings. Biological Rhythm Research, 28(4):422–440, 1997