Mathematical principles and models of plant growth mechanics: from cell wall dynamics to tissue morphogenesis

Quantifying Plant Signaling Pathways by Integrating Luminescence-Based Biosensors and Mathematical Modeling

Plants have evolved intricate signaling pathways, which operate as networks governed by feedback to deal with stressors. Nevertheless, the sophisticated molecular mechanisms underlying these routes still need to be comprehended, and experimental validation poses significant challenges and expenses. Consequently, computational hypothesis evaluation gains prominence in understanding plant signaling dynamics. Biosensors are genetically modified to emit light when exposed to a particular hormone, such as abscisic acid (ABA), enabling quantification. We developed computational models to simulate the relationship between ABA concentrations and bioluminescent sensors utilizing the Hill equation and ordinary differential equations (ODEs), aiding better hypothesis development regarding plant signaling. Based on simulation results, the luminescence intensity was recorded for a concentration of 47.646 RLUs for 1.5 μmol, given the specified parameters and model assumptions. This method enhances our understanding of plant signaling pathways at the cellular level, offering significant benefits to the scientific community in a cost-effective manner. The alignment of these computational predictions with experimental results emphasizes the robustness of our approach, providing a cost-effective means to validate mathematical models empirically. The research intended to correlate the bioluminescence of biosensors with plant signaling and its mathematical models for quantified detection of specific plant hormone ABA.

Assessing the hydromechanical control of plant growth

Multicellular organisms grow and acquire their shapes through the differential expansion and deformation of their cells. Recent research has addressed the role of cell and tissue mechanical properties in these processes. In plants, it is believed that growth rate is a function of the mechanical stress exerted on the cell wall, the thin polymeric layer surrounding cells, involving an effective viscosity. Nevertheless, recent studies have questioned this view, suggesting that cell wall elasticity sets the growth rate or that uptake of water is limiting for plant growth. To assess these issues, we developed a microfluidic device to quantify the growth rates, elastic properties and hydraulic conductivity of individual Marchantia polymorpha plants in a controlled environment with a high throughput. We characterized the effect of osmotic treatment and abscisic acid on growth and hydromechanical properties. Overall, the instantaneous growth rate of individuals is correlated with both bulk elastic modulus and hydraulic conductivity. Our results are consistent with a framework in which the growth rate is determined primarily by the elasticity of the wall and its remodelling, and secondarily by hydraulic conductivity. Accordingly, the coupling between the chemistry of the cell wall and the hydromechanics of the cell appears as key to set growth patterns during morphogenesis.

A continuum mechanics model of the plant cell wall reveals interplay between enzyme action and cell wall structure

Plant cell growth is regulated through manipulation of the cell wall network, which consists of oriented cellulose microfibrils embedded within a ground matrix incorporating pectin and hemicellulose components. There remain many unknowns as to how this manipulation occurs. Experiments have shown that cellulose reorients in cell walls as the cell expands, while recent data suggest that growth is controlled by distinct collections of hemicellulose called biomechanical hotspots, which join the cellulose molecule together. The enzymes expansin and Cel12A have both been shown to induce growth of the cell wall; however, while Cel12A's wall-loosening action leads to a reduction in the cell wall strength, expansin's has been shown to increase the strength of the cell wall. In contrast, members of the XTH enzyme family hydrolyse hemicellulose but do not appear to cause wall creep. This experimentally observed behaviour still awaits a full explanation. We derive and analyse a mathematical model for the effective mechanical properties of the evolving cell wall network, incorporating cellulose microfibrils, which reorient with cell growth and are linked via biomechanical hotspots made up of regions of crosslinking hemicellulose. Assuming a visco-elastic response for the cell wall and using a continuum approach, we calculate the total stress resultant of the cell wall for a given overall growth rate. By changing appropriate parameters affecting breakage rate and viscous properties, we provide evidence for the biomechanical hotspot hypothesis and develop mechanistic understanding of the growth-inducing enzymes.

Nyctinastic movement in legumes: Developmental mechanisms, factors and biological significance

In legumes, a common phenomenon known as nyctinastic movement is observed. This movement involves the horizontal expansion of leaves during the day and relative vertical closure at night. Nyctinastic movement is driven by the pulvinus, which consists of flexor and extensor motor cells. The turgor pressure difference between these two cell types generates a driving force for the bending and deformation of the pulvinus. This review focuses on the developmental mechanisms of the pulvinus, the factors affecting nyctinastic movement, and the biological significance of this phenomenon in legumes, thus providing a reference for further research on nyctinastic movement.

Deciphering the role of mechanosensitive channels in plant root biology: perception, signaling, and adaptive responses

MAIN CONCLUSION

Mechanosensitive channels are integral membrane proteins that rapidly translate extrinsic or intrinsic mechanical tensions into biological responses. They can serve as potential candidates for developing smart-resilient crops with efficient root systems. Mechanosensitive (MS) calcium channels are molecular switches for mechanoperception and signal transduction in all living organisms. Although tremendous progress has been made in understanding mechanoperception and signal transduction in bacteria and animals, this remains largely unknown in plants. However, identification and validation of MS channels such as Mid1-complementing activity channels (MCAs), mechanosensitive-like channels (MSLs), and Piezo channels (PIEZO) has been the most significant discovery in plant mechanobiology, providing novel insights into plant mechanoperception. This review summarizes recent advances in root mechanobiology, focusing on MS channels and their related signaling players, such as calcium ions (Ca2+), reactive oxygen species (ROS), and phytohormones. Despite significant advances in understanding the role of Ca2+ signaling in root biology, little is known about the involvement of MS channel-driven Ca2+ and ROS signaling. Additionally, the hotspots connecting the upstream and downstream signaling of MS channels remain unclear. In light of this, we discuss the present knowledge of MS channels in root biology and their role in root developmental and adaptive traits. We also provide a model highlighting upstream (cell wall sensors) and downstream signaling players, viz., Ca2+, ROS, and hormones, connected with MS channels. Furthermore, we highlighted the importance of emerging signaling molecules, such as nitric oxide (NO), hydrogen sulfide (H2S), and neurotransmitters (NTs), and their association with root mechanoperception. Finally, we conclude with future directions and knowledge gaps that warrant further research to decipher the complexity of root mechanosensing.

Simulation of microtubule-cytoplasm interaction revealed the importance of fluid dynamics in determining the organization of microtubules

Although microtubules in plant cells have been extensively studied, the mechanisms that regulate the spatial organization of microtubules are poorly understood. We hypothesize that the interaction between microtubules and cytoplasmic flow plays an important role in the assembly and orientation of microtubules. To test this hypothesis, we developed a new computational modeling framework for microtubules based on theory and methods from the fluid-structure interaction. We employed the immersed boundary method to track the movement of microtubules in cytoplasmic flow. We also incorporated details of the encounter dynamics when two microtubules collide with each other. We verified our computational model through several numerical tests before applying it to the simulation of the microtubule-cytoplasm interaction in a growing plant cell. Our computational investigation demonstrated that microtubules are primarily oriented in the direction orthogonal to the axis of cell elongation. We validated the simulation results through a comparison with the measurement from laboratory experiments. We found that our computational model, with further calibration, was capable of generating microtubule orientation patterns that were qualitatively and quantitatively consistent with the experimental results. The computational model proposed in this study can be naturally extended to many other cellular systems that involve the interaction between microstructures and the intracellular fluid.

Pulvinar slits: Cellulose-deficient and de-methyl-esterified pectin-rich structures in a legume motor cell

The cortical motor cells (CMCs) in a legume pulvinus execute the reversible deformation in leaf movement that is driven by changes in turgor pressure. In contrast to the underlying osmotic regulation property, the cell wall structure of CMCs that contributes to the movement has yet to be characterized in detail. Here, we report that the cell wall of CMCs has circumferential slits with low levels of cellulose deposition, which are widely conserved among legume species. This structure is unique and distinct from that of any other primary cell walls reported so far; thus, we named them "pulvinar slits." Notably, we predominantly detected de-methyl-esterified homogalacturonan inside pulvinar slits, with a low deposition of highly methyl-esterified homogalacturonan, as with cellulose. In addition, Fourier transform infrared spectroscopy analysis indicated that the cell wall composition of pulvini is different from that of other axial organs, such as petioles or stems. Moreover, monosaccharide analysis showed that pulvini are pectin-rich organs like developing stems and that the amount of galacturonic acid in pulvini is greater than in developing stems. Computer modeling suggested that pulvinar slits facilitate anisotropic extension in the direction perpendicular to the slits in the presence of turgor pressure. When tissue slices of CMCs were transferred to different extracellular osmotic conditions, pulvinar slits altered their opening width, indicating their deformability. In this study, we thus characterized a distinctive cell wall structure of CMCs, adding to our knowledge of repetitive and reversible organ deformation as well as the structural diversity and function of the plant cell wall.

Particle-based model shows complex rearrangement of tissue mechanical properties are needed for roots to grow in hard soil

When exposed to increased mechanical resistance from the soil, plant roots display non-linear growth responses that cannot be solely explained by mechanical principles. Here, we aim to investigate how changes in tissue mechanical properties are biologically regulated in response to soil strength. A particle-based model was developed to solve root-soil mechanical interactions at the cellular scale, and a detailed numerical study explored factors that affect root responses to soil resistance. Results showed how softening of root tissues at the tip may contribute to root responses to soil impedance, a mechanism likely linked to soil cavity expansion. The model also predicted the shortening and decreased anisotropy of the zone where growth occurs, which may improve the mechanical stability of the root against axial forces. The study demonstrates the potential of advanced modeling tools to help identify traits that confer plant resistance to abiotic stress.

Building an extensible cell wall

This article recounts, from my perspective of four decades in this field, evolving paradigms of primary cell wall structure and the mechanism of surface enlargement of growing cell walls. Updates of the structures, physical interactions, and roles of cellulose, xyloglucan, and pectins are presented. This leads to an example of how a conceptual depiction of wall structure can be translated into an explicit quantitative model based on molecular dynamics methods. Comparison of the model's mechanical behavior with experimental results provides insights into the molecular basis of complex mechanical behaviors of primary cell wall and uncovers the dominant role of cellulose-cellulose interactions in forming a strong yet extensible network.

In vitro experiments and kinetic models of Arabidopsis pollen hydration mechanics show that MSL8 is not a simple tension-gated osmoregulator

Pollen, a neighbor-less cell containing the male gametes, undergoes mechanical challenges during plant sexual reproduction, including desiccation and rehydration. It was previously shown that the pollen-specific mechanosensitive ion channel MscS-like (MSL)8 is essential for pollen survival during hydration and proposed that it functions as a tension-gated osmoregulator. Here, we test this hypothesis with a combination of mathematical modeling and laboratory experiments. Time-lapse imaging revealed that wild-type pollen grains swell, and then they stabilize in volume rapidly during hydration. msl8 mutant pollen grains, however, continue to expand and eventually burst. We found that a mathematical model, wherein MSL8 acts as a simple-tension-gated osmoregulator, does not replicate this behavior. A better fit was obtained from variations of the model, wherein MSL8 inactivates independent of its membrane tension gating threshold or MSL8 strengthens the cell wall without osmotic regulation. Experimental and computational testing of several perturbations, including hydration in an osmolyte-rich solution, hyper-desiccation of the grains, and MSL8-YFP overexpression, indicated that the cell wall strengthening model best simulated experimental responses. Finally, the expression of a nonconducting MSL8 variant did not complement the msl8 overexpansion phenotype. These data indicate that contrary to our hypothesis and to the current understanding of MS ion channel function in bacteria, MSL8 does not act as a simple membrane tension-gated osmoregulator. Instead, they support a model wherein ion flux through MSL8 is required to alter pollen cell wall properties. These results demonstrate the utility of pollen as a cellular scale model system and illustrate how mathematical models can correct intuitive hypotheses.

Multiscale Mechanical Performance of Wood: From Nano- to Macro-Scale across Structure Hierarchy and Size Effects

This review describes methods and results of studying the mechanical properties of wood at all scales: from nano- to macro-scale. The connection between the mechanical properties of material and its structure at all these levels is explored. It is shown that the existing size effects in the mechanical properties of wood, in a range of the characteristic sizes of the structure of about six orders of magnitude, correspond to the empirical Hall-Petch relation. This “law” was revealed more than 60 years ago in metals and alloys and later in other materials. The nature, as well as the particular type of the size dependences in different classes of materials can vary, but the general trend, “the smaller the stronger”, remains true both for wood and for other cellulose-containing materials. The possible mechanisms of the size effects in wood are being discussed. The correlations between the mechanical and thermophysical properties of wood are described. Several examples are used to demonstrate the possibility to forecast the macromechanical properties of wood by means of contactless thermographic express methods based on measuring temperature diffusivity. The research technique for dendrochronological and dendroclimatological studies by means of the analysis of microhardness and Young’s modulus radial dependences in annual growth rings is described.

Broad Specific Xyloglucan:Xyloglucosyl Transferases Are Formidable Players in the Re-Modelling of Plant Cell Wall Structures

Plant xyloglucan:xyloglucosyl transferases, known as xyloglucan endo-transglycosylases (XETs) are the key players that underlie plant cell wall dynamics and mechanics. These fundamental roles are central for the assembly and modifications of cell walls during embryogenesis, vegetative and reproductive growth, and adaptations to living environments under biotic and abiotic (environmental) stresses. XET enzymes (EC 2.4.1.207) have the β-sandwich architecture and the β-jelly-roll topology, and are classified in the glycoside hydrolase family 16 based on their evolutionary history. XET enzymes catalyse transglycosylation reactions with xyloglucan (XG)-derived and other than XG-derived donors and acceptors, and this poly-specificity originates from the structural plasticity and evolutionary diversification that has evolved through expansion and duplication. In phyletic groups, XETs form the gene families that are differentially expressed in organs and tissues in time- and space-dependent manners, and in response to environmental conditions. Here, we examine higher plant XET enzymes and dissect how their exclusively carbohydrate-linked transglycosylation catalytic function inter-connects complex plant cell wall components. Further, we discuss progress in technologies that advance the knowledge of plant cell walls and how this knowledge defines the roles of XETs. We construe that the broad specificity of the plant XETs underscores their roles in continuous cell wall restructuring and re-modelling.

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.

Molecular insights into the complex mechanics of plant epidermal cell walls

Plants have evolved complex nanofibril-based cell walls to meet diverse biological and physical constraints. How strength and extensibility emerge from the nanoscale-to-mesoscale organization of growing cell walls has long been unresolved. We sought to clarify the mechanical roles of cellulose and matrix polysaccharides by developing a coarse-grained model based on polymer physics that recapitulates aspects of assembly and tensile mechanics of epidermal cell walls. Simple noncovalent binding interactions in the model generate bundled cellulose networks resembling that of primary cell walls and possessing stress-dependent elasticity, stiffening, and plasticity beyond a yield threshold. Plasticity originates from fibril-fibril sliding in aligned cellulose networks. This physical model provides quantitative insight into fundamental questions of plant mechanobiology and reveals design principles of biomaterials that combine stiffness with yielding and extensibility.

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.

Pectin methylesterase selectively softens the onion epidermal wall yet reduces acid-induced creep

De-esterification of homogalacturonan (HG) is thought to stiffen pectin gels and primary cell walls by increasing calcium cross-linking between HG chains. Contrary to this idea, recent studies found that HG de-esterification correlated with reduced stiffness of living tissues, measured by surface indentation. The physical basis of such apparent wall softening is unclear, but possibly involves complex biological responses to HG modification. To assess the direct physical consequences of HG de-esterification on wall mechanics without such complications, we treated isolated onion (Allium cepa) epidermal walls with pectin methylesterase (PME) and assessed wall biomechanics with indentation and tensile tests. In nanoindentation assays, PME action softened the wall (reduced the indentation modulus). In tensile force/extension assays, PME increased plasticity, but not elasticity. These softening effects are attributed, at least in part, to increased electrostatic repulsion and swelling of the wall after PME treatment. Despite softening and swelling upon HG de-esterification, PME treatment alone failed to induce cell wall creep. Instead, acid-induced creep, mediated by endogenous α-expansin, was reduced. We conclude that HG de-esterification physically softens the onion wall, yet reduces expansin-mediated wall extensibility.

Methods to quantify primary plant cell wall mechanics

The primary plant cell wall is a dynamically regulated composite material of multiple biopolymers that forms a scaffold enclosing the plant cells. The mechanochemical make-up of this polymer network regulates growth, morphogenesis, and stability at the cell and tissue scales. To understand the dynamics of cell wall mechanics, and how it correlates with cellular activities, several experimental frameworks have been deployed in recent years to quantify the mechanical properties of plant cells and tissues. Here we critically review the application of biomechanical tool sets pertinent to plant cell mechanics and outline some of their findings, relevance, and limitations. We also discuss methods that are less explored but hold great potential for the field, including multiscale in silico mechanical modeling that will enable a unified understanding of the mechanical behavior across the scales. Our overview reveals significant differences between the results of different mechanical testing techniques on plant material. Specifically, indentation techniques seem to consistently report lower values compared with tensile tests. Such differences may in part be due to inherent differences among the technical approaches and consequently the wall properties that they measure, and partly due to differences between experimental conditions.