Vertex-element models for anisotropic growth of elongated plant organs

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.

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.

Patterned proliferation orients tissue-wide stress to control root vascular symmetry in Arabidopsis

Symmetric tissue alignment is pivotal to the functions of plant vascular tissue, such as long-distance molecular transport and lateral organ formation. During the vascular development of the Arabidopsis roots, cytokinins initially determine cell-type boundaries among vascular stem cells and subsequently promote cell proliferation to establish vascular tissue symmetry. Although it is unknown whether and how the symmetry of initially defined boundaries is progressively refined under tissue growth in plants, such boundary shapes in animal tissues are regulated by cell fluidity, e.g., cell migration and intercalation, lacking in plant tissues. Here, we uncover that cell proliferation during vascular development produces anisotropic compressive stress, smoothing, and symmetrizing cell arrangement of the vascular-cell-type boundary. Mechanistically, the GATA transcription factor HANABA-TARANU cooperates with the type-B Arabidopsis response regulators to form an incoherent feedforward loop in cytokinin signaling. The incoherent feedforward loop fine-tunes the position and frequency of vascular cell proliferation, which in turn restricts the source of mechanical stress to the position distal and symmetric to the boundary. By combinatorial analyses of mechanical simulations and laser cell ablation, we show that the spatially constrained environment of vascular tissue efficiently entrains the stress orientation among the cells to produce a tissue-wide stress field. Together, our data indicate that the localized proliferation regulated by the cytokinin signaling circuit is decoded into a globally oriented mechanical stress to shape the vascular tissue symmetry, representing a reasonable mechanism controlling the boundary alignment and symmetry in tissue lacking cell fluidity.

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.

Complementary roles for auxin and auxin signalling revealed by reverse engineering lateral root stable prebranch site formation

Priming is the process through which periodic elevations in auxin signalling prepattern future sites for lateral root formation, called prebranch sites. Thus far, the extent to which elevations in auxin concentration and/or auxin signalling are required for priming and prebranch site formation has remained a matter of debate. Recently, we discovered a reflux-and-growth mechanism for priming generating periodic elevations in auxin concentration that subsequently dissipate. Here, we reverse engineer a mechanism for prebranch site formation that translates these transient elevations into a persistent increase in auxin signalling, resolving the prior debate into a two-step process of auxin concentration-mediated initial signal and auxin signalling capacity-mediated memorization. A crucial aspect of the prebranch site formation mechanism is its activation in response to time-integrated rather than instantaneous auxin signalling. The proposed mechanism is demonstrated to be consistent with prebranch site auxin signalling dynamics, lateral inhibition, and symmetry-breaking mechanisms and perturbations in auxin homeostasis.

Modeling Auxin Signaling in Roots: Auxin Computations

Auxin signaling and patterning is an inherently complex process, involving polarized auxin transport, metabolism, and signaling, its effect on developmental zones, as well as growth rates, and the feedback between all these different aspects. This complexity has led to an important role for computational modeling in unraveling the multifactorial roles of auxin in plant developmental and adaptive processes. Here we discuss the basic ingredients of auxin signaling and patterning models for root development as well as a series of key modeling studies in this area. These modeling studies have helped elucidate how plants use auxin signaling to compute the size of their root meristem, the direction in which to grow, and when and where to form lateral roots. Importantly, these models highlight how auxin, through patterning of and collaborating with other factors, can fulfill all these roles simultaneously.

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.

Feedback from Tissue Mechanics Self-Organizes Efficient Outgrowth of Plant Organ

Plant organ outgrowth superficially appears like the continuous mechanical deformation of a sheet of cells. Yet, how precisely cells as individual mechanical entities can act to morph a tissue reliably and efficiently into three dimensions during outgrowth is still puzzling, especially when cells are tightly connected as in plant tissue. In plants, the mechanics of cells within a tissue is particularly well-defined because individual cell growth is essentially the mechanical yielding of the cell wall in response to internal turgor pressure. Cell-wall stiffness is controlled by biological signaling, which is impacted by stresses, and hence, cell growth is observed to respond to mechanical stresses building up within a tissue. What is the role of the mechanical feedback during morphing of tissue in three dimensions? Here, we develop a three-dimensional vertex model to investigate tissue mechanics at the onset of organ outgrowth at the tip of a plant shoot. We find that organ height is primarily governed by the ratio of growth rates of faster-growing cells initiating the organ versus slower-growing cells surrounding them. Remarkably, the outgrowth rate is higher when cell growth responds to the tissue-wide mechanical stresses. Our quantitative analysis of simulation data shows that tissue mechanical feedback on cell growth can act via a twofold mechanism. First, the feedback guides patterns of cellular growth. Second, the feedback modifies the stress patterns on the cells, consequently amplifying and propagating growth anisotropies. This mechanism may allow plants to grow organs efficiently out of the meristem by reorganizing the cellular growth rather than inflating growth rates.

DLITE Uses Cell-Cell Interface Movement to Better Infer Cell-Cell Tensions

Cell shapes and connectivities evolve over time as the colony changes shape or embryos develop. Shapes of intercellular interfaces are closely coupled with the forces resulting from actomyosin interactions, membrane tension, or cell-cell adhesions. Although it is possible to computationally infer cell-cell forces from a mechanical model of collective cell behavior, doing so for temporally evolving forces in a manner robust to digitization difficulties is challenging. Here, we introduce a method for dynamic local intercellular tension estimation (DLITE) that infers such evolution in temporal force with less sensitivity to digitization ambiguities or errors. This method builds upon previous work on single time points (cellular force-inference toolkit). We validate our method using synthetic geometries. DLITE's inferred cell colony tension evolutions correlate better with ground truth for these synthetic geometries as compared to tension values inferred from methods that consider each time point in isolation. We introduce cell connectivity errors, angle estimate errors, connection mislocalization, and connection topological changes to synthetic data and show that DLITE has reduced sensitivity to these conditions. Finally, we apply DLITE to time series of human-induced pluripotent stem cell colonies with endogenously expressed GFP-tagged zonulae occludentes-1. We show that DLITE offers improved stability in the inference of cell-cell tensions and supports a correlation between the dynamics of cell-cell forces and colony rearrangement.

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.

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

Plant growth research produces a catalogue of complex open questions. We argue that plant growth is a highly mechanical process, and that mathematics gives an underlying framework with which to probe its fundamental unrevealed mechanisms. This review serves to illustrate the biological insights afforded by mathematical modelling and demonstrate the breadth of mathematically rich problems available within plant sciences, thereby promoting a mutual appreciation across the disciplines. On the one hand, we explain the general mathematical principles behind mechanical growth models; on the other, we describe how modelling addresses specific problems in microscale cell wall mechanics, tip growth, morphogenesis, and stress feedback. We conclude by identifying possible future directions for both biologists and mathematicians, including as yet unanswered questions within various topics, stressing that interdisciplinary collaboration is vital for tackling the challenge of understanding plant growth mechanics.

Formal description of plant morphogenesis

Plant morphogenesis may be characterized by complex feedback mechanisms between signals specifying growth and by the growth of the plant body itself. Comprehension of such feedback mechanisms is an ongoing research task and can be aided with formal descriptions of morphogenesis. In this review, we present a number of established mathematical paradigms that are useful to the formal representation of plant shape, and of biomechanical and biochemical signaling. Specifically, we discuss work from a range of research areas including plant biology, material sciences, fluid dynamics, and computer graphics. Treating plants as organized systems of information processing allows us to compare these different mathematical methods in terms of their expressive power of biological hypotheses. This is an attempt to bring together a large number of computational modeling concepts and make them accessible to the analytical as well as empirical student of plant morphogenesis.

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.

Cell-Based Model of the Generation and Maintenance of the Shape and Structure of the Multilayered Shoot Apical Meristem of Arabidopsis thaliana

One of the central problems in animal and plant developmental biology is deciphering how chemical and mechanical signals interact within a tissue to produce organs of defined size, shape, and function. Cell walls in plants impose a unique constraint on cell expansion since cells are under turgor pressure and do not move relative to one another. Cell wall extensibility and constantly changing distribution of stress on the wall are mechanical properties that vary between individual cells and contribute to rates of expansion and orientation of cell division. How exactly cell wall mechanical properties influence cell behavior is still largely unknown. To address this problem, a novel, subcellular element computational model of growth of stem cells within the multilayered shoot apical meristem (SAM) of Arabidopsis thaliana is developed and calibrated using experimental data. Novel features of the model include separate, detailed descriptions of cell wall extensibility and mechanical stiffness, deformation of the middle lamella, and increase in cytoplasmic pressure generating internal turgor pressure. The model is used to test novel hypothesized mechanisms of formation of the shape and structure of the growing, multilayered SAM based on WUS concentration of individual cells controlling cell growth rates and layer-dependent anisotropic mechanical properties of subcellular components of individual cells determining anisotropic cell expansion directions. Model simulations also provide a detailed prediction of distribution of stresses in the growing tissue which can be tested in future experiments.

Regulation of plant cell wall stiffness by mechanical stress: a mesoscale physical model

A crucial question in developmental biology is how cell growth is coordinated in living tissue to generate complex and reproducible shapes. We address this issue here in plants, where stiff extracellular walls prevent cell migration and morphogenesis mostly results from growth driven by turgor pressure. How cells grow in response to pressure partly depends on the mechanical properties of their walls, which are generally heterogeneous, anisotropic and dynamic. The active control of these properties is therefore a cornerstone of plant morphogenesis. Here, we focus on wall stiffness, which is under the control of both molecular and mechanical signaling. Indeed, in plant tissues, the balance between turgor and cell wall elasticity generates a tissue-wide stress field. Within cells, mechano-sensitive structures, such as cortical microtubules, adapt their behavior accordingly and locally influence cell wall remodeling dynamics. To fully apprehend the properties of this feedback loop, modeling approaches are indispensable. To that end, several modeling tools in the form of virtual tissues have been developed. However, these models often relate mechanical stress and cell wall stiffness in relatively abstract manners, where the molecular specificities of the various actors are not fully captured. In this paper, we propose to refine this approach by including parsimonious biochemical and biomechanical properties of the main molecular actors involved. Through a coarse-grained approach and through finite element simulations, we study the role of stress-sensing microtubules on organ-scale mechanics.

Compression simulations of plant tissue in 3D using a mass-spring system approach and discrete element method

A hybrid model based on a mass-spring system methodology coupled with the discrete element method (DEM) was implemented to simulate the deformation of cellular structures in 3D. Models of individual cells were constructed using the particles which cover the surfaces of cell walls and are interconnected in a triangle mesh network by viscoelastic springs. The spatial arrangement of the cells required to construct a virtual tissue was obtained using Poisson-disc sampling and Voronoi tessellation in 3D space. Three structural features were included in the model: viscoelastic material of cell walls, linearly elastic interior of the cells (simulating compressible liquid) and a gas phase in the intercellular spaces. The response of the models to an external load was demonstrated during quasi-static compression simulations. The sensitivity of the model was investigated at fixed compression parameters with variable tissue porosity, cell size and cell wall properties, such as thickness and Young's modulus, and a stiffness of the cell interior that simulated turgor pressure. The extent of the agreement between the simulation results and other models published is discussed. The model demonstrated the significant influence of tissue structure on micromechanical properties and allowed for the interpretation of the compression test results with respect to changes occurring in the structure of the virtual tissue. During compression virtual structures composed of smaller cells produced higher reaction forces and therefore they were stiffer than structures with large cells. The increase in the number of intercellular spaces (porosity) resulted in a decrease in reaction forces. The numerical model was capable of simulating the quasi-static compression experiment and reproducing the strain stiffening observed in experiment. Stress accumulation at the edges of the cell walls where three cells meet suggests that cell-to-cell debonding and crack propagation through the contact edge of neighboring cells is one of the most prevalent ways for tissue to rupture.

A continuous growth model for plant tissue

Morphogenesis in plants and animals involves large irreversible deformations. In plants, the response of the cell wall material to internal and external forces is determined by its mechanical properties. An appropriate model for plant tissue growth must include key features such as anisotropic and heterogeneous elasticity and cell dependent evaluation of mechanical variables such as turgor pressure, stress and strain. In addition, a growth model needs to cope with cell divisions as a necessary part of the growth process. Here we develop such a growth model, which is capable of employing not only mechanical signals but also morphogen signals for regulating growth. The model is based on a continuous equation for updating the resting configuration of the tissue. Simultaneously, material properties can be updated at a different time scale. We test the stability of our model by measuring convergence of growth results for a tissue under the same mechanical and material conditions but with different spatial discretization. The model is able to maintain a strain field in the tissue during re-meshing, which is of particular importance for modeling cell division. We confirm the accuracy of our estimations in two and three-dimensional simulations, and show that residual stresses are less prominent if strain or stress is included as input signal to growth. The approach results in a model implementation that can be used to compare different growth hypotheses, while keeping residual stresses and other mechanical variables updated and available for feeding back to the growth and material properties.

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.

Hybrid vertex-midline modelling of elongated plant organs

We describe a method for the simulation of the growth of elongated plant organs, such as seedling roots. By combining a midline representation of the organ on a tissue scale and a vertex-based representation on the cell scale, we obtain a multiscale method, which is able to both simulate organ growth and incorporate cell-scale processes. Equations for the evolution of the midline are obtained, which depend on the cell-wall properties of individual cells through appropriate averages over the vertex-based representation. The evolution of the organ midline is used to deform the cellular-scale representation. This permits the investigation of the regulation of organ growth through the cell-scale transport of the plant hormone auxin. The utility of this method is demonstrated in simulating the early stages of the response of a root to gravity, using a vertex-based template acquired from confocal imaging. Asymmetries in the concentrations of auxin between the upper and lower sides of the root lead to bending of the root midline, reflecting a gravitropic response.

Shifting foundations: the mechanical cell wall and development

The cell wall has long been acknowledged as an important physical mediator of growth in plants. Recent experimental and modelling work has brought the importance of cell wall mechanics into the forefront again. These data have challenged existing dogmas that relate cell wall structure to cell/organ growth, that uncouple elasticity from extensibility, and those which treat the cell wall as a passive and non-stressed material. Within this review we describe experiments and models which have changed the ways in which we view the mechanical cell wall, leading to new hypotheses and research avenues. It has become increasingly apparent that while we often wish to simplify our systems, we now require more complex multi-scale experiments and models in order to gain further insight into growth mechanics. We are currently experiencing an exciting and challenging shift in the foundations of our understanding of cell wall mechanics in growth and development.

Variability and Constancy in Cellular Growth of Arabidopsis Sepals

Growth of tissues is highly reproducible; yet, growth of individual cells in a tissue is highly variable, and neighboring cells can grow at different rates. We analyzed the growth of epidermal cell lineages in the Arabidopsis (Arabidopsis thaliana) sepal to determine how the growth curves of individual cell lineages relate to one another in a developing tissue. To identify underlying growth trends, we developed a continuous displacement field to predict spatially averaged growth rates. We showed that this displacement field accurately describes the growth of sepal cell lineages and reveals underlying trends within the variability of in vivo cellular growth. We found that the tissue, individual cell lineages, and cell walls all exhibit growth rates that are initially low, accelerate to a maximum, and decrease again. Accordingly, these growth curves can be represented by sigmoid functions. We examined the relationships among the cell lineage growth curves and surprisingly found that all lineages reach the same maximum growth rate relative to their size. However, the cell lineages are not synchronized; each cell lineage reaches this same maximum relative growth rate but at different times. The heterogeneity in observed growth results from shifting the same underlying sigmoid curve in time and scaling by size. Thus, despite the variability in growth observed in our study and others, individual cell lineages in the developing sepal follow similarly shaped growth curves.

Multiscale models in the biomechanics of plant growth

Plant growth occurs through the coordinated expansion of tightly adherent cells, driven by regulated softening of cell walls. It is an intrinsically multiscale process, with the integrated properties of multiple cell walls shaping the whole tissue. Multiscale models encode physical relationships to bring new understanding to plant physiology and development.

Oriented cell division shapes carnivorous pitcher leaves of Sarracenia purpurea

Complex morphology is an evolutionary outcome of phenotypic diversification. In some carnivorous plants, the ancestral planar leaf has been modified to form a pitcher shape. However, how leaf development was altered during evolution remains unknown. Here we show that the pitcher leaves of Sarracenia purpurea develop through cell division patterns of adaxial tissues that are distinct from those in bifacial and peltate leaves, subsequent to standard expression of adaxial and abaxial marker genes. Differences in the orientation of cell divisions in the adaxial domain cause bifacial growth in the distal region and adaxial ridge protrusion in the middle region. These different growth patterns establish pitcher morphology. A computer simulation suggests that the cell division plane is critical for the pitcher morphogenesis. Our results imply that tissue-specific changes in the orientation of cell division underlie the development of a morphologically complex leaf.

The pitcher-shaped leaf of the carnivorous plant Sarracenia purpurea acts as a pitfall trap to capture small animals. Here, Fukushima et al. analyse pitcher leaf development and propose that this unusual shape evolved from ancestral planar leaves through changes in the orientation of cell division.

A computational framework for 3D mechanical modeling of plant morphogenesis with cellular resolution

The link between genetic regulation and the definition of form and size during morphogenesis remains largely an open question in both plant and animal biology. This is partially due to the complexity of the process, involving extensive molecular networks, multiple feedbacks between different scales of organization and physical forces operating at multiple levels. Here we present a conceptual and modeling framework aimed at generating an integrated understanding of morphogenesis in plants. This framework is based on the biophysical properties of plant cells, which are under high internal turgor pressure, and are prevented from bursting because of the presence of a rigid cell wall. To control cell growth, the underlying molecular networks must interfere locally with the elastic and/or plastic extensibility of this cell wall. We present a model in the form of a three dimensional (3D) virtual tissue, where growth depends on the local modulation of wall mechanical properties and turgor pressure. The model shows how forces generated by turgor-pressure can act both cell autonomously and non-cell autonomously to drive growth in different directions. We use simulations to explore lateral organ formation at the shoot apical meristem. Although different scenarios lead to similar shape changes, they are not equivalent and lead to different, testable predictions regarding the mechanical and geometrical properties of the growing lateral organs. Using flower development as an example, we further show how a limited number of gene activities can explain the complex shape changes that accompany organ outgrowth.

Putting theory to the test: which regulatory mechanisms can drive realistic growth of a root?

In recent years there has been a strong development of computational approaches to mechanistically understand organ growth regulation in plants. In this study, simulation methods were used to explore which regulatory mechanisms can lead to realistic output at the cell and whole organ scale and which other possibilities must be discarded as they result in cellular patterns and kinematic characteristics that are not consistent with experimental observations for the Arabidopsis thaliana primary root. To aid in this analysis, a ‘Uniform Longitudinal Strain Rule’ (ULSR) was formulated as a necessary condition for stable, unidirectional, symplastic growth. Our simulations indicate that symplastic structures are robust to differences in longitudinal strain rates along the growth axis only if these differences are small and short-lived. Whereas simple cell-autonomous regulatory rules based on counters and timers can produce stable growth, it was found that steady developmental zones and smooth transitions in cell lengths are not feasible. By introducing spatial cues into growth regulation, those inadequacies could be avoided and experimental data could be faithfully reproduced. Nevertheless, a root growth model based on previous polar auxin-transport mechanisms violates the proposed ULSR due to the presence of lateral gradients. Models with layer-specific regulation or layer-driven growth offer potential solutions. Alternatively, a model representing the known cross-talk between auxin, as the cell proliferation promoting factor, and cytokinin, as the cell differentiation promoting factor, predicts the effect of hormone-perturbations on meristem size. By down-regulating PIN-mediated transport through the transcription factor SHY2, cytokinin effectively flattens the lateral auxin gradient, at the basal boundary of the division zone, (thereby imposing the ULSR) to signal the exit of proliferation and start of elongation. This model exploration underlines the value of generating virtual root growth kinematics to dissect and understand the mechanisms controlling this biological system.

The growth of a plant root is driven by cell division and cell expansion occurring in spatially distinct developmental zones. Although these zones are in principle stable, depending on the conditions, their size and properties can be modulated. This has been meticulously described by kinematic studies, which have led to the proposal of mechanisms underpinning those observations. At the same time, much knowledge of the identities and interactions of molecules involved in these mechanisms has accumulated, in particular from the model species Arabidopsis thaliana. Here we attempt to resolve the longstanding question whether observed growth patterns can be explained by autonomous decision-making at the level of individual cells or if the aid of some external signal is required. We then ask, building on the accumulated molecular information, which minimal models can provide for stable growth while keeping sufficient flexibility to regulate growth. Therefore, we constructed computational models for different growth mechanisms operating in a virtual two-dimensional Arabidopsis root and compared their behaviour with biological experiments. The simulations provide strong indications that spatial signals are required for realistic and flexible root growth, likely orchestrated by the plant hormones auxin and cytokinin.

Physical models of plant development

The definition of shape in multicellular organisms is a major issue of developmental biology. It is well established that morphogenesis relies on genetic regulation. However, cells, tissues, and organism behaviors are also bound by the laws of physics, which limit the range of possible deformations organisms can undergo but also define what organisms must do to achieve specific shapes. Besides experiments, theoretical models and numerical simulations of growing tissues are powerful tools to investigate the link between genetic regulation and mechanics. Here, we provide an overview of the main mechanical models of plant morphogenesis developed so far, from subcellular scales to whole tissues. The common concepts and discrepancies between the various models are discussed.

Mechanical modelling quantifies the functional importance of outer tissue layers during root elongation and bending

Root elongation and bending require the coordinated expansion of multiple cells of different types. These processes are regulated by the action of hormones that can target distinct cell layers. We use a mathematical model to characterise the influence of the biomechanical properties of individual cell walls on the properties of the whole tissue. Taking a simple constitutive model at the cell scale which characterises cell walls via yield and extensibility parameters, we derive the analogous tissue-level model to describe elongation and bending. To accurately parameterise the model, we take detailed measurements of cell turgor, cell geometries and wall thicknesses. The model demonstrates how cell properties and shapes contribute to tissue-level extensibility and yield. Exploiting the highly organised structure of the elongation zone (EZ) of the Arabidopsis root, we quantify the contributions of different cell layers, using the measured parameters. We show how distributions of material and geometric properties across the root cross-section contribute to the generation of curvature, and relate the angle of a gravitropic bend to the magnitude and duration of asymmetric wall softening. We quantify the geometric factors which lead to the predominant contribution of the outer cell files in driving root elongation and bending.