Water Affects Morphogenesis of Growing Aquatic Plant Leaves

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Water Affects Morphogenesis of Growing Aquatic Plant Leaves

Fan Xu, Chenbo Fu, and Yifan Yang

Phys. Rev. Lett. 124, 038003 – Published 24 January 2020

See Focus story: Explaining the Ruffles of Lotus Leaves

Abstract

Lotus leaves floating on water usually experience short-wavelength edge wrinkling that decays toward the center, while the leaves growing above water normally morph into a global bending cone shape with long rippled waves near the edge. Observations suggest that the underlying water (liquid substrate) significantly affects the morphogenesis of leaves. To understand the biophysical mechanism under such phenomena, we develop mathematical models that can effectively account for inhomogeneous differential growth of floating and freestanding leaves to quantitatively predict formation and evolution of their morphology. We find, both theoretically and experimentally, that the short-wavelength buckled configuration is energetically favorable for growing membranes lying on liquid, while the global buckling shape is more preferable for suspended ones. Other influencing factors such as the stem or vein, heterogeneity, and dimension are also investigated. Our results provide a fundamental insight into a variety of plant morphogenesis affected by water foundation and suggest that such surface instabilities can be harnessed for morphology control of biomimetic deployable structures using substrate or edge actuation.

Received 11 October 2019

DOI:https://doi.org/10.1103/PhysRevLett.124.038003

Physics Subject Headings (PhySH)

Applications of soft matter Biomechanics Elastic deformation Morphogenesis Pattern formation Surface instabilities

Plants

Physics of Living SystemsInterdisciplinary PhysicsPolymers & Soft Matter

Focus

Explaining the Ruffles of Lotus Leaves

Published 24 January 2020

A new theory accurately predicts a wide range of leaf shapes and explains the differences between dry lotus leaves and those that grow on water.

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Authors & Affiliations

Fan Xu ^*, Chenbo Fu, and Yifan Yang

Institute of Mechanics and Computational Engineering, Department of Aeronautics and Astronautics, Fudan University, Shanghai 200433, People’s Republic of China

^*Corresponding author. fanxu@fudan.edu.cn

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Issue

Vol. 124, Iss. 3 — 24 January 2020

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Images

Figure 1
Water effect on morphogenesis of diverse aquatic plant leaves. A floating lotus leaf (a) grows with short waves along the edge (the wavy edge is highlighted by red color in experiments), while suspended lotus leaves (b) and (c) morph into long-wavelength ripples (depending on the size of main stem [11]) near the margin. The more apparent global bending cone shape in natural lotus leaves might be caused by vein effect. With water foundation, the leaf of white water lily (d) remains flat, while for the suspended leaf (e) globally bends with long ripples near the edge. The leaf of Victoria water lily (f) where water substrate occupies about 80% of the entire leaf area (local effect of liquid foundation) grows into a bowllike shape with sharp edge bending. The blue background in (a), (d), and (f) represents water foundation. In theoretical calculations of (a)–(f), we took $R / h = 50$ and $K_{s} h / E = 1 / 1200$ . The growth strain exponentially attenuates from the edge to the center, satisfying $ϵ^{g} = ε^{0} \exp [- τ (R - r) / R]$ , in which $τ = 1$ and $ε^{0} \sim 10^{- 2}$ .
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Figure 2
The comparison of model, theory, and experiments for water-foundation leaves: (a) Critical wrinkling wave number $k_{c}$ of circular floating lotus leaves as a linear function of $\sqrt[4]{K_{s} R^{4} / E h^{3}}$ . Our model and theoretical predictions agree well with experiments. (b) Critical growth strain $ϵ_{c}^{g}$ of circular floating lotus leaves as a linear function of $k_{c}^{2} h^{2} / R^{2}$ . Fitting coefficients: $C_{1}$ and $C_{2}$ [11].
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Figure 3
Size effect ( $R / h$ ) on growing shapes of lotus (circular) and white water lily leaves (fan-shaped): (a),(b) water foundation (blue background), (c),(d) suspended (stem support). For circular leaves floating on water, larger dimensionless radius $R / h$ leads to more waves, i.e., larger wave number (the wavy edges are highlighted by red color in experiments), while for circular leaves constrained by the central main stem, the dimension has no apparent effect on the wrinkling wave number and leaf morphology. Apart from size effect, this difference implies the important role played by water foundation in the morphology of growing aquatic plants. As for fan-shaped leaves, size effect, however, remains insignificant with or without water substrate. Material parameter: $K_{s} h / E = 1 / 1200$ . Attenuation coefficient: $τ = 1$ . Growth strain: $ε^{0} \sim 10^{- 2}$ .
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Figure 4
Effect of growth-induced inhomogeneity on morphogenesis of diverse plant leaves. The upper line and lower line represent, respectively, our theoretical predictions of pattern selection without or by accounting for inhomogeneous growth-induced bending moment. The columns classify the leaves by different geometries: circular, fan-shaped, and rectangular. To distinguish between antisymmetric shape (c) and reflection-symmetric mode (f), the wrinkling edges are highlighted by red color in natural leaves and experiments. In theoretical calculations of (a)–(f), we took $R / h = 50$ for circular and fan-shaped leaves, while $L / W = 2$ and $W / h = 50$ for oblong leaves. Attenuation coefficient $τ = 1$ and growth strain at leaf margin $ε^{0} \sim 10^{- 3}$ are set. (g) Critical wrinkling wave number $k_{c}$ of suspended oblong leaves as a linear function of aspect ratio $L / W$ . Our model and theoretical predictions match well with experiments. (h) Critical growth strain $ϵ_{c}^{g}$ of suspended rectangular leaves as a linear function of $h^{2} / L^{2}$ . Fitting coefficients: $C_{3}$ and $C_{4}$ [11].
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