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# Mechanical Models of Plant Growth

Plants are the past, present, and future. They have supplied Earth’s ecosystem with its most fundamental ingredients for millions of years, and they comprise an indispensable source of food, medicine, construction materials, and so forth in modern human society. According to the Food and Agriculture Organization of the United Nations, agricultural productivity must increase by 60 percent to feed a projected population of 9.6 billion in 2050 — a monumental task for humanity against a backdrop of rapid climate change [2]. As applied mathematicians, we can and should contribute to a deeper understanding of the mechanics of plant growth in order to help improve crop resilience, protect biodiversity, and accelerate sustainable development.

One of the latest advances in plant growth modeling on the microscale is a theoretical framework that probes the mechanics of cell wall extension [1]. The plant cell wall is a complex material that responds viscoplastically to sufficiently high cytoplasmic turgor pressure, resulting in an irreversible deformation of the cell called “diffuse growth.” For nearly six decades, the paradigmatic mechanical theory of diffuse growth has been the Lockhart model [5]:

$\frac{1}{V}\frac{\textrm{d}V}{\textrm{d}t}=\Phi(P-Y) \quad \textrm{if} \quad P>Y.$

Using an enigmatic parameter known as extensibility $$(\Phi)$$, this model relates the cell’s volume $$(V)$$ over time $$(t)$$ to the turgor pressure $$(P)$$ and some yield threshold $$(Y)$$. In the simplest scenario, $$P$$, $$Y$$, and $$\Phi$$ are all constants, so the model predicts exponential expansion of cell volume.

Figure 1. The root of Arabidopsis thaliana, which illustrates the significant anisotropic growth of the cells. The meristem and elongation zone are both visible. Figure courtesy of [3].
In an archetypal plant organ like the root of Arabidopsis thaliana, newer cells that are created in the meristem will push an older cell along the elongation zone; the older cell undergoes 30-fold axial extension while retaining a roughly constant girth (see Figure 1). The Lockhart model does not explain this extreme anisotropy of diffuse growth. Therefore, the challenge for post-Lockhart modelers is to augment the Lockhart model to address the following two questions: Which of the cell’s mechanical properties constitute extensibility $$\Phi$$? And how can a Lockhart-type model explain anisotropic growth?

The latest theoretical framework addresses both of these questions by promoting the mechanical role of molecules called cellulose microfibrils (CMFs) to a higher importance than ever before. CMFs are inextensible, rod-like molecules that are embedded in a ground matrix within the cell wall, strengthening the wall much like a steel fiber mesh strengthens concrete (see Figure 2). Experiments indicate that hoop-like arrangements of CMFs in the side wall of a cylindrical cell prevent it from bulging radially when inflated from within, resulting in anisotropic axial extension [8]. The new theory describes the cell wall as a fiber-reinforced viscous fluid whose mechanical properties depend on CMF orientation and three viscosity-like parameters. When the two parameters that each control a specific property of the CMF mesh are sufficiently large, the model dictates that the cell’s radial expansion will be negligible as compared to its axial extension, thus corroborating observed cell behavior. The third viscosity-like parameter—which controls the ground matrix stiffness of the cell wall and is assumed to increase over time—represents a known biochemical phenomenon [6].

Figure 2. The walled cell as a pressurized axisymmetric structure and the wall as a fiber-reinforced viscous fluid. 2a. A thin-sheet geometry characterizes the permanently yielded wall, which is subjected to various external forces $$(P, P_\textrm{ext}, Q, \Sigma)$$. The cellulose microfibrils (partially illustrated by line segments in the wall) are a key component of the model. 2b. The cell wall. Figure 2a courtesy of [1], and Figure 2b courtesy of Wikimedia user LadyofHats.

One of the model’s key features is the dynamic reorientation of the CMFs, in which the underlying fluid flow advects and rotates the fibers. The CMF orientations throughout the spatial domain are therefore encapsulated by a normalized vector field that evolves over time, and the cell wall’s mechanical properties evolve accordingly. Moreover, the deposition of newly created material onto the inner surface of the cell wall is represented by a dynamical boundary condition that allows for the incorporation of any biologically realistic deposition mechanism into the model. When the system of equations is asymptotically reduced to leading order, the axial length $$(L)$$ of the cell obeys the following generalized Lockhart equation:

$\frac{1}{L}\frac{\textrm{d}L}{\textrm{d}t}=\frac{\Delta P}{\int^h_0[4\mu_0(t)+\mu_2\sin^4\phi(n,t)+4\mu_3\sin^2\phi(n,t)]\textrm{d}n}.$

On the right-hand side, $$\Delta P$$ is a generalized excess pressure, $$n$$ is the normal coordinate that varies across the uniformly constant cell wall thickness $$(h)$$, $$\mu_0(t)$$ is the ground matrix stiffness, $$\mu_2$$ and $$\mu_3$$ are the parameters that control viscous properties of the CMF mesh, and $$\phi(n,t)$$ is the latitude angle of the fiber orientation field. This system of equations is ultimately a Lockhart-type model that explains anisotropic diffuse growth, with a dynamical extensibility that is given by the multiplier of $$\Delta P$$ in the generalized Lockhart equation: the reciprocal of the spatial integral of an elegant combination of cell wall viscosities and CMF orientation.

Armed with this new understanding of cell-level plant growth, my collaborators and I are working to unlock the secret mechanisms behind the emergence of curvature and torsion in slender plant organs. The cross section of a plant root (e.g., in Arabidopsis thaliana) comprises a network of cell wall segments that are under turgor pressure from the cytoplasmic compartments (see Figure 3). Heterogeneous cell wall properties in the cross section give rise to differential growth at the tissue level; the walls extend perpendicularly to the cross section, and stiffer walls extend more slowly than malleable ones. In a mechanical model of the organ, force and moment balances in the cell wall network couple macroscopic bending with the microscopic elongation of individual cells. Assuming the original Lockhart equation for cell growth (with each cell’s cross-sectional area held constant), studies have shown that the rate at which organ curvature evolves depends on the amount of heterogeneity in the cells’ extensibilities [4]. If every cell holds its extensibility constant, the organ’s central axis remains within a constant plane of curvature. A more sophisticated single-cell growth equation is necessary to allow the organ to bend out of plane. By employing the generalized Lockhart equation wherein each cell’s extensibility is non-constant, we hope to construct a model where the plane of curvature evolves over time — representing an organ that bends and twists in three-dimensional space.

Figure 3. The root cross section as a network of cell wall segments, and a mechanical model that features heterogeneous cell wall stiffness. 3a. Microscope image of a root cross section. 3b. Idealized cross section with a lateral gradient in cell wall stiffness $$(\mu)$$. Figure 3a courtesy of [4], and Figure 3b courtesy of Galane Luo.

Such a model, if validated by experimental data, would explain how curvature and torsion in slender plant organs emerge from the coordination of cell-level mechanical variations. This theory would significantly advance the current understanding of plant morphology and serve as a valuable analytical tool for a range of industries, from agriculture to biomimetic robotics. For example, imagine a self-growing robot that bends and twists by manipulating a few mechanical parameters of its unit cells, just like a plant root; this type of machine would be incredibly useful in terrain exploration or disaster rescue. These technologies are already in development, but they and countless other innovations will undoubtedly benefit from stronger theoretical frameworks [7]. Further progress on this front will rely upon the continued efforts of an interdisciplinary consortium of experts to explore the mechanical underpinnings of plant growth. Here, mathematical modelers everywhere can find an invigorating new purpose.

Galane J. Luo presented this research during a minisymposium presentation at the 2022 SIAM Conference on the Life Sciences, which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., last summer.

Acknowledgments: The author would like to thank his collaborators on the cell growth study, Rosemary Dyson (University of Birmingham) and Jeevanjyoti Chakraborty (Indian Institute of Technology Kharagpur); his former student, Zoe Godard (University of Birmingham), for her brilliant work on the emergent curvature model; and the Institute of Mathematics and its Applications for supporting his attendance at the 2022 SIAM Conference on the Life Sciences through a QJMAM Fund for Applied Mathematics.

References
[1] Chakraborty, J., Luo, J., & Dyson, R.J. (2021). Lockhart with a twist: Modelling cellulose microfibril deposition and reorientation reveals twisting plant cell growth mechanisms. J. Theor. Biol., 525, 110736.
[2] Da Silva, J.G. (2012). Feeding the world sustainably. UN chronicle (The future we want), 1-2(XLIX). Retrieved from https://www.un.org/en/chronicle/article/feeding-world-sustainably
[3] Dyson, R.J., & Jensen, O.E. (2010). A fibre-reinforced fluid model of anisotropic plant cell growth. J. Fluid Mechan., 655, 472-503.
[4] Dyson, R.J., Vizcay-Barrena, G., Band, L.R., Fernandes, A.N., French, A.P., Fozard, J.A., … Jensen, O.E. (2014). Mechanical modelling quantifies the functional importance of outer tissue layers during root elongation and bending. New Phytol., 202(4), 1212-1222.
[5] Lockhart, J.A. (1965). An analysis of irreversible plant cell elongation. J. Theor. Biol., 8(2), 264-275.
[6] Peaucelle, A., Wightman, R., & Höfte, H. (2015). The control of growth symmetry breaking in the Arabidopsis hypocotyl. Current Biol., 25(13), 1746-1752.
[7] Sadeghi, A., Mondini, A., & Mazzolai, B. (2017). Toward self-growing soft robots inspired by plant roots and based on additive manufacturing technologies. Soft Robot., 4(3), 211-223.
[8] Somerville, C., Bauer, S., Brininstool, G., Facette, M., Hamann, T., Milne, J., … Youngs, H. (2004). Toward a systems approach to understanding plant cell walls. Science, 306(5705), 2206-2211.

 Galane J. Luo earned a Ph.D. in mathematical sciences from Durham University in 2018. He has since worked as a Research Fellow and Teaching Fellow in the School of Mathematics at the University of Birmingham. His research interests are biomechanics, complex systems, and agent-based modeling in the social sciences.