Multiscale models in the biomechanics of plant growth

Multiscale Asymptotic Analysis Reveals How Cell Growth and Subcellular Compartments Affect Tissue-Scale Hormone Transport

Determining how cell-scale processes lead to tissue-scale patterns is key to understanding how hormones and morphogens are distributed within biological tissues and control developmental processes. In this article, we use multiscale asymptotic analysis to derive a continuum approximation for hormone transport in a long file of cells to determine how subcellular compartments and cell growth and division affect tissue-scale hormone transport. Focusing our study on plant tissues, we begin by presenting a discrete multicellular ODE model tracking the hormone concentration in each cell's cytoplasm, subcellular vacuole, and surrounding apoplast, represented by separate compartments in the cell-file geometry. We allow the cells to grow at a rate that can depend both on space and time, accounting for both cytoplasmic and vacuolar expansion. Multiscale asymptotic analysis enables us to systematically derive the corresponding continuum model, obtaining an effective reaction-advection-diffusion equation and revealing how the effective diffusivity, effective advective velocity, and the effective sink term depend on the parameters in the cell-scale model. The continuum approximation reveals how subcellular compartments, such as vacuoles, can act as storage vessels, that significantly alter the effective properties of hormone transport, such as the effective diffusivity and the induced effective velocity. Furthermore, we show how cell growth and spatial variance across cell lengths affect the effective diffusivity and the induced effective velocity, and how these affect the tissue-scale hormone distribution. In particular, we find that cell growth naturally induces an effective velocity in the direction of growth, whereas spatial variance across cell lengths induces effective velocity due to the presence of an extra compartment, such as the apoplast and the vacuole, and variations in the relative sizes between the compartments across the file of cells. It is revealed that hormone transport is faster across cells of decreasing lengths than cells with increasing lengths. We also investigate the effect of cell division on transport dynamics, assuming that each cell divides as soon as it doubles in size, and find that increasing the time between successive cell divisions decreases the growth rate, which enhances the effect of cell division in slowing hormone transport. Motivated by recent experimental discoveries, we discuss particular applications for transport of gibberellic acid (GA), an important growth hormone, within the Arabidopsis root. The model reveals precisely how membrane proteins that mediate facilitated GA transport affect the effective tissue-scale transport. However, the results are general enough to be relevant to other plant hormones, or other substances that are transported in a similar way in any type of cells.

RootSlice-A novel functional-structural model for root anatomical phenotypes

Root anatomy is an important determinant of root metabolic costs, soil exploration, and soil resource capture. Root anatomy varies substantially within and among plant species. RootSlice is a multicellular functional-structural model of root anatomy developed to facilitate the analysis and understanding of root anatomical phenotypes. RootSlice can capture phenotypically accurate root anatomy in three dimensions of different root classes and developmental zones, of both monocotyledonous and dicotyledonous species. Several case studies are presented illustrating the capabilities of the model. For maize nodal roots, the model illustrated the role of vacuole expansion in cell elongation; and confirmed the individual and synergistic role of increasing root cortical aerenchyma and reducing the number of cortical cell files in reducing root metabolic costs. Integration of RootSlice for different root zones as the temporal properties of the nodal roots in the whole-plant and soil model OpenSimRoot/maize enabled the multiscale evaluation of root anatomical phenotypes, highlighting the role of aerenchyma formation in enhancing the utility of cortical cell files for improving plant performance over varying soil nitrogen supply. Such integrative in silico approaches present avenues for exploring the fitness landscape of root anatomical phenotypes.

A coupled mechano-biochemical model for cell polarity guided anisotropic root growth

Plants develop new organs to adjust their bodies to dynamic changes in the environment. How independent organs achieve anisotropic shapes and polarities is poorly understood. To address this question, we constructed a mechano-biochemical model for Arabidopsis root meristem growth that integrates biologically plausible principles. Computer model simulations demonstrate how differential growth of neighboring tissues results in the initial symmetry-breaking leading to anisotropic root growth. Furthermore, the root growth feeds back on a polar transport network of the growth regulator auxin. Model, predictions are in close agreement with in vivo patterns of anisotropic growth, auxin distribution, and cell polarity, as well as several root phenotypes caused by chemical, mechanical, or genetic perturbations. Our study demonstrates that the combination of tissue mechanics and polar auxin transport organizes anisotropic root growth and cell polarities during organ outgrowth. Therefore, a mobile auxin signal transported through immobile cells drives polarity and growth mechanics to coordinate complex organ development.

Shaping the Organ: A Biologist Guide to Quantitative Models of Plant Morphogenesis

Organ morphogenesis is the process of shape acquisition initiated with a small reservoir of undifferentiated cells. In plants, morphogenesis is a complex endeavor that comprises a large number of interacting elements, including mechanical stimuli, biochemical signaling, and genetic prerequisites. Because of the large body of data being produced by modern laboratories, solving this complexity requires the application of computational techniques and analyses. In the last two decades, computational models combined with wet-lab experiments have advanced our understanding of plant organ morphogenesis. Here, we provide a comprehensive review of the most important achievements in the field of computational plant morphodynamics. We present a brief history from the earliest attempts to describe plant forms using algorithmic pattern generation to the evolution of quantitative cell-based models fueled by increasing computational power. We then provide an overview of the most common types of "digital plant" paradigms, and demonstrate how models benefit from diverse techniques used to describe cell growth mechanics. Finally, we highlight the development of computational frameworks designed to resolve organ shape complexity through integration of mechanical, biochemical, and genetic cues into a quantitative standardized and user-friendly environment.

Mathematical Modelling of Residual-Stress Based Volumetric Growth in Soft Matter

Growth in nature is associated with the development of residual stresses and is in general heterogeneous and anisotropic at all scales. Residual stress in an unloaded configuration of a growing material provides direct evidence of the mechanical regulation of heterogeneity and anisotropy of growth. The present study explores a model of stress-mediated growth based on the unloaded configuration that considers either the residual stress or the deformation gradient relative to the unloaded configuration as a growth variable. This makes it possible to analyze stress-mediated growth without the need to invoke the existence of a fictitious stress-free grown configuration. Furthermore, applications based on the proposed theoretical framework relate directly to practical experimental scenarios involving the "opening-angle" in arteries as a measure of residual stress. An initial illustration of the theory is then provided by considering the growth of a spherically symmetric thick-walled shell subjected to the incompressibility constraint.

Lockhart with a twist: Modelling cellulose microfibril deposition and reorientation reveals twisting plant cell growth mechanisms

Plant morphology emerges from cellular growth and structure. The turgor-driven diffuse growth of a cell can be highly anisotropic: significant longitudinally and negligible radially. Such anisotropy is ensured by cellulose microfibrils (CMF) reinforcing the cell wall in the hoop direction. To maintain the cell's integrity during growth, new wall material including CMF must be continually deposited. We develop a mathematical model representing the cell as a cylindrical pressure vessel and the cell wall as a fibre-reinforced viscous sheet, explicitly including the mechano-sensitive angle of CMF deposition. The model incorporates interactions between turgor, external forces, CMF reorientation during wall extension, and matrix stiffening. Using the model, we reinterpret some recent experimental findings, and reexamine the popular hypothesis of CMF/microtubule alignment. We explore how the handedness of twisting cell growth depends on external torque and intrinsic wall properties, and find that cells twist left-handedly 'by default' in some suitable sense. Overall, this study provides a unified mechanical framework for understanding left- and right-handed twist-growth as seen in many plants.

The Structure of the Barley Husk Influences Its Resistance to Mechanical Stress

This paper explores the links between genotype, plant development, plant structure and plant material properties. The barley husk has two organs, the lemma and the palea, which protect the grain. When the husk is exposed to mechanical stress, such as during harvesting, it can be damaged or detached. This is known as grain skinning, which is detrimental to grain quality and has a significant economic impact on industry. This study focused on the lemma, the husk organ which is most susceptible to grain skinning. This study tested three hypotheses: (1) genotype and plant development determine lemma structure, (2) lemma structure influences the material properties of the lemma, and (3) the material properties of the lemma determine grain skinning risk. The effect of genotype was investigated by using plant material from four malting barley varieties: two with a high risk of grain skinning, two with a low risk. Plant material was assessed at two stages of plant development (anthesis, GS 65; grain filling, GS 77). Structure was assessed using light microscopy to measure three physiological features: thickness, vasculature and cell area. Material properties were approximated using a controlled impact assay and by analyzing fragmentation behavior. Genotype had a significant effect on lemma structure and material properties from anthesis. This indicates that differences between genotypes were established during floral development. The lemma was significantly thinner in high risk genotypes, compared to low risk genotypes. Consequently, in high risk genotypes, the lemma was significantly more likely to fragment. This indicates a relationship between reduced lemma thickness and increased fragmentation. Traditionally, a thin husk has been considered beneficial for malting quality, due to an association with malt extract. However, this study finds a thin lemma is less resistant to mechanical stress. This may explain the differences in grain skinning risk in the genotypes studied.

Multiscale integration of environmental stimuli in plant tropism produces complex behaviors

Plant tropism refers to the directed movement of an organ or organism in response to external stimuli. Typically, these stimuli induce hormone transport that triggers cell growth or deformation. In turn, these local cellular changes create mechanical forces on the plant tissue that are balanced by an overall deformation of the organ, hence changing its orientation with respect to the stimuli. This complex feedback mechanism takes place in a three-dimensional growing plant with varying stimuli depending on the environment. We model this multiscale process in filamentary organs for an arbitrary stimulus by explicitly linking hormone transport to local tissue deformation leading to the generation of mechanical forces and the deformation of the organ in three dimensions. We show, as examples, that the gravitropic, phototropic, nutational, and thigmotropic dynamic responses can be easily captured by this framework. Further, the integration of evolving stimuli and/or multiple contradictory stimuli can lead to complex behavior such as sun following, canopy escape, and plant twining.

The role of topology and mechanics in uniaxially growing cell networks

In biological systems, the growth of cells, tissues and organs is influenced by mechanical cues. Locally, cell growth leads to a mechanically heterogeneous environment as cells pull and push their neighbours in a cell network. Despite this local heterogeneity, at the tissue level, the cell network is remarkably robust, as it is not easily perturbed by changes in the mechanical environment or the network connectivity. Through a network model, we relate global tissue structure (i.e. the cell network topology) and local growth mechanisms (growth laws) to the overall tissue response. Within this framework, we investigate the two main mechanical growth laws that have been proposed: stress-driven or strain-driven growth. We show that in order to create a robust and stable tissue environment, networks with predominantly series connections are naturally driven by stress-driven growth, whereas networks with predominantly parallel connections are associated with strain-driven growth.

Discrete mechanical growth model for plant tissue

We present a discrete mechanical model to study plant development. The method is built up of mass points, springs and hinges mimicking the plant cell wall’s microstructure. To model plastic growth the resting lengths of springs are adjusted; when springs exceed a threshold length, new mass points, springs and hinges, are added. We formulate a stiffness tensor for the springs and hinges as a function of the fourth rank tensor of elasticity and the geometry of the mesh. This allows us to approximate the material law as a generalized orthotropic Hooke’s law, and control material properties during growth. The material properties of the model are illustrated in numerical simulations for finite strain and plastic growth. To solve the equations of motion of mass points we assume elastostatics and use Verlet integration. The method is demonstrated in simulations when anisotropic growth causes emergent residual strain fields in cell walls and a bending of tissue. The method can be used in multilevel models to study plant development, for example by coupling it to models for cytoskeletal, hormonal and gene regulatory processes.

How water flow, geometry, and material properties drive plant movements

Plants are dynamic. They adjust their shape for feeding, defence, and reproduction. Such plant movements are critical for their survival. We present selected examples covering a range of movements from single cell to tissue level and over a range of time scales. We focus on reversible turgor-driven shape changes. Recent insights into the mechanisms of stomata, bladderwort, the waterwheel, and the Venus flytrap are presented. The underlying physical principles (turgor, osmosis, membrane permeability, wall stress, snap buckling, and elastic instability) are highlighted, and advances in our understanding of these processes are summarized.

Coupling water fluxes with cell wall mechanics in a multicellular model of plant development

The growth of plant organs is a complex process powered by osmosis that attracts water inside the cells; this influx induces simultaneously an elastic extension of the walls and pressure in the cells, called turgor pressure; above a threshold, the walls yield and the cells grow. Based on Lockhart's seminal work, various models of plant morphogenesis have been proposed, either for single cells, or focusing on the wall mechanical properties. However, the synergistic coupling of fluxes and wall mechanics has not yet been fully addressed in a multicellular model. This work lays the foundations of such a model, by simplifying as much as possible each process and putting emphasis on the coupling itself. Its emergent properties are rich and can help to understand plant morphogenesis. In particular, we show that the model can display a new type of lateral inhibitory mechanism that amplifies growth heterogeneities due e.g to cell wall loosening.

Mathematical models of lignin biosynthesis

BACKGROUND

Lignin is a natural polymer that is interwoven with cellulose and hemicellulose within plant cell walls. Due to this molecular arrangement, lignin is a major contributor to the recalcitrance of plant materials with respect to the extraction of sugars and their fermentation into ethanol, butanol, and other potential bioenergy crops. The lignin biosynthetic pathway is similar, but not identical in different plant species. It is in each case comprised of a moderate number of enzymatic steps, but its responses to manipulations, such as gene knock-downs, are complicated by the fact that several of the key enzymes are involved in several reaction steps. This feature poses a challenge to bioenergy production, as it renders it difficult to select the most promising combinations of genetic manipulations for the optimization of lignin composition and amount.

RESULTS

Here, we present several computational models than can aid in the analysis of data characterizing lignin biosynthesis. While minimizing technical details, we focus on the questions of what types of data are particularly useful for modeling and what genuine benefits the biofuel researcher may gain from the resulting models. We demonstrate our analysis with mathematical models for black cottonwood (Populus trichocarpa), alfalfa (Medicago truncatula), switchgrass (Panicum virgatum) and the grass Brachypodium distachyon.

CONCLUSIONS

Despite commonality in pathway structure, different plant species show different regulatory features and distinct spatial and topological characteristics. The putative lignin biosynthes pathway is not able to explain the plant specific laboratory data, and the necessity of plant specific modeling should be heeded.

Plant Science View on Biohybrid Development

Biohybrid consists of a living organism or cell and at least one engineered component. Designing robot-plant biohybrids is a great challenge: it requires interdisciplinary reconsideration of capabilities intimate specific to the biology of plants. Envisioned advances should improve agricultural/horticultural/social practice and could open new directions in utilization of plants by humans. Proper biohybrid cooperation depends upon effective communication. During evolution, plants developed many ways to communicate with each other, with animals, and with microorganisms. The most notable examples are: the use of phytohormones, rapid long-distance signaling, gravity, and light perception. These processes can now be intentionally re-shaped to establish plant-robot communication. In this article, we focus on plants physiological and molecular processes that could be used in bio-hybrids. We show phototropism and biomechanics as promising ways of effective communication, resulting in an alteration in plant architecture, and discuss the specifics of plants anatomy, physiology and development with regards to the bio-hybrids. Moreover, we discuss ways how robots could influence plants growth and development and present aims, ideas, and realized projects of plant-robot biohybrids.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

The Impact of Microfibril Orientations on the Biomechanics of Plant Cell Walls and Tissues

The microscopic structure and anisotropy of plant cell walls greatly influence the mechanical properties, morphogenesis, and growth of plant cells and tissues. The microscopic structure and properties of cell walls are determined by the orientation and mechanical properties of the cellulose microfibrils and the mechanical properties of the cell wall matrix. Viewing the shape of a plant cell as a square prism with the axis aligning with the primary direction of expansion and growth, the orientation of the microfibrils within the side walls, i.e. the parts of the cell walls on the sides of the cells, is known. However, not much is known about their orientation at the upper and lower ends of the cell. Here we investigate the impact of the orientation of cellulose microfibrils within the upper and lower parts of the plant cell walls by solving the equations of linear elasticity numerically. Three different scenarios for the orientation of the microfibrils are considered. We also distinguish between the microstructure in the side walls given by microfibrils perpendicular to the main direction of the expansion and the situation where the microfibrils are rotated through the wall thickness. The macroscopic elastic properties of the cell wall are obtained using homogenization theory from the microscopic description of the elastic properties of the cell wall microfibrils and wall matrix. It is found that the orientation of the microfibrils in the upper and lower parts of the cell walls affects the expansion of the cell in the lateral directions and is particularly important in the case of forces acting on plant cell walls and tissues.

Hierarchies of plant stiffness

Plants must meet mechanical as well as physiological and reproductive requirements for survival. Management of internal and external stresses is achieved through their unique hierarchical architecture. Stiffness is determined by a combination of morphological (geometrical) and compositional variables that vary across multiple length scales ranging from the whole plant to organ, tissue, cell and cell wall levels. These parameters include, among others, organ diameter, tissue organization, cell size, density and turgor pressure, and the thickness and composition of cell walls. These structural parameters and their consequences on plant stiffness are reviewed in the context of work on stems of the genetic reference plant Arabidopsis thaliana (Arabidopsis), and the suitability of Arabidopsis as a model system for consistent investigation of factors controlling plant stiffness is put forward. Moving beyond Arabidopsis, the presence of morphological parameters causing stiffness gradients across length-scales leads to beneficial emergent properties such as increased load-bearing capacity and reversible actuation. Tailoring of plant stiffness for old and new purposes in agriculture and forestry can be achieved through bioengineering based on the knowledge of the morphological and compositional parameters of plant stiffness in combination with gene identification through the use of genetics.

Making microscopy count: quantitative light microscopy of dynamic processes in living plants

Cell theory has officially reached 350 years of age as the first use of the word 'cell' in a biological context can be traced to a description of plant material by Robert Hooke in his historic publication 'Micrographia: or some physiological definitions of minute bodies'. The 2015 Royal Microscopical Society Botanical Microscopy meeting was a celebration of the streams of investigation initiated by Hooke to understand at the subcellular scale how plant cell function and form arises. Much of the work presented, and Honorary Fellowships awarded, reflected the advanced application of bioimaging informatics to extract quantitative data from micrographs that reveal dynamic molecular processes driving cell growth and physiology. The field has progressed from collecting many pixels in multiple modes to associating these measurements with objects or features that are meaningful biologically. The additional complexity involves object identification that draws on a different type of expertise from computer science and statistics that is often impenetrable to biologists. There are many useful tools and approaches being developed, but we now need more interdisciplinary exchange to use them effectively. In this review we show how this quiet revolution has provided tools available to any personal computer user. We also discuss the oft-neglected issue of quantifying algorithm robustness and the exciting possibilities offered through the integration of physiological information generated by biosensors with object detection and tracking.