Primary root growth: a biophysical model of auxin-related control

The Dynamics of Root Growth: A Geometric Model

A new model for macroscopic root growth based on a dynamical Riemannian geometry is presented. Assuming that the thickness of the root is much less than its length, the model is restricted to growth in one dimension (1D). We treat 1D tissues as continuous, deformable, growing geometries for sizes larger than 1 mm. The dynamics of the growing root are described by a set of coupled tensor equations for the metric of the tissue and velocity field of material transport in non-Euclidean space. These coupled equations represent a novel feedback mechanism between growth and geometry. We compare 1D numerical simulations of these tissue growth equations to two measures of root growth. First, sectional growth along the simulated root shows an elongation zone common to many species of plant roots. Second, the relative elemental growth rate calculated in silico exhibits spatio-temporal dynamics recently characterized in high-resolution root growth studies but which thus far lack a biological hypothesis to explain them. In our model, these dynamics are a direct consequence of considering growth as both a geometric reaction-diffusion process and expansion due to a distributed source of new materials.

Mechanical Behavior of Cells within a Cell-Based Model of Wheat Leaf Growth

Understanding the principles and mechanisms of cell growth coordination in plant tissue remains an outstanding challenge for modern developmental biology. Cell-based modeling is a widely used technique for studying the geometric and topological features of plant tissue morphology during growth. We developed a quasi-one-dimensional model of unidirectional growth of a tissue layer in a linear leaf blade that takes cell autonomous growth mode into account. The model allows for fitting of the visible cell length using the experimental cell length distribution along the longitudinal axis of a wheat leaf epidermis. Additionally, it describes changes in turgor and osmotic pressures for each cell in the growing tissue. Our numerical experiments show that the pressures in the cell change over the cell cycle, and in symplastically growing tissue, they vary from cell to cell and strongly depend on the leaf growing zone to which the cells belong. Therefore, we believe that the mechanical signals generated by pressures are important to consider in simulations of tissue growth as possible targets for molecular genetic regulators of individual cell growth.

A quantitative report on the impact of chloride on the kinetic coefficients of auxin-induced growth: a numerical contribution to the "acid growth hypothesis"

This work presents the application of several our own novel methods of analysing the kinetics of plant growth, which create, among others, a common platform for the comparison of experimental results. A relatively simple formula is used to parameterize the wide range of data that has been obtained for Zea mays L. in the literature, though it can also be used for different species. A biophysical/biochemical interpretation of the parameters was obtained from a theoretical model that is based on a modified Lockhart equation. The derived formula, which was extended for practical use in Zajdel et al. (Acta Physiol Plant 38:5, 2016), and which was implemented in the attached computer program (ibid.), allowed the data that was obtained from the growth-related problems to be parameterized in a simple way. As a working example that shows the robustness of our approach, we comment in detail on the qualitative assessments of the impact of chloride ions on auxin-induced growth. We note that calculated continuous curves (fits), which are rooted in the growth functional that was introduced by Pietruszka (J Theor Biol 315:119-127, 2012), were in a perfect agreement (R2 ~ 0.99998) with the raw experimental data that was published recently by Burdach et al. (Ann Bot 114:1023-1034, 2014). This fact justified the use of this strict technique, which allows for the determination of kinetic coefficients, to critically evaluate the results and suppositions (claims) therein. Moreover, we calculated the time-delay derivative of elongation growth-pH cross-correlations, and validated the "acid growth hypothesis" in figures by considering, amongst others, the magnitude of the H+-activity of elongation growth (per μm). An empirical constant (field strength), EH+ = Em/(log10 1/aH+ ∙ μm) = 0.157 ± 0.009 [V/mm] was obtained, where Em [mV] is the membrane potential in the perenchymal coleoptile cells of Zea mays L. When this relation is known, the membrane potential can not only be determined for intact growth, but also for different intervening substances exclusively from growth (or growth rate) and pH measurements, i.e. without performing electrophysiological measurements. However, the question of whether this constant is universal remains open.

The significance of image compression in plant phenotyping applications

We are currently witnessing an increasingly higher throughput in image-based plant phenotyping experiments. The majority of imaging data are collected using complex automated procedures and are then post-processed to extract phenotyping-related information. In this article, we show that the image compression used in such procedures may compromise phenotyping results and this needs to be taken into account. We use three illuminating proof-of-concept experiments that demonstrate that compression (especially in the most common lossy JPEG form) affects measurements of plant traits and the errors introduced can be high. We also systematically explore how compression affects measurement fidelity, quantified as effects on image quality, as well as errors in extracted plant visual traits. To do so, we evaluate a variety of image-based phenotyping scenarios, including size and colour of shoots, leaf and root growth. To show that even visual impressions can be used to assess compression effects, we use root system images as examples. Overall, we find that compression has a considerable effect on several types of analyses (albeit visual or quantitative) and that proper care is necessary to ensure that this choice does not affect biological findings. In order to avoid or at least minimise introduced measurement errors, for each scenario, we derive recommendations and provide guidelines on how to identify suitable compression options in practice. We also find that certain compression choices can offer beneficial returns in terms of reducing the amount of data storage without compromising phenotyping results. This may enable even higher throughput experiments in the future.

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.

Modelling hormonal response and development

As our knowledge of the complexity of hormone homeostasis, transport, perception, and response increases, and their outputs become less intuitive, modelling is set to become more important. Initial modelling efforts have focused on hormone transport and response pathways. However, we now need to move beyond the network scales and use multicellular and multiscale modelling approaches to predict emergent properties at different scales. Here we review some examples where such approaches have been successful, for example, auxin-cytokinin crosstalk regulating root vascular development or a study of lateral root emergence where an iterative cycle of modelling and experiments lead to the identification of an overlooked role for PIN3. Finally, we discuss some of the remaining biological and technical challenges.

Vertex-element models for anisotropic growth of elongated plant organs

New tools are required to address the challenge of relating plant hormone levels, hormone responses, wall biochemistry and wall mechanical properties to organ-scale growth. Current vertex-based models (applied in other contexts) can be unsuitable for simulating the growth of elongated organs such as roots because of the large aspect ratio of the cells, and these models fail to capture the mechanical properties of cell walls in sufficient detail. We describe a vertex-element model which resolves individual cells and includes anisotropic non-linear viscoelastic mechanical properties of cell walls and cell division whilst still being computationally efficient. We show that detailed consideration of the cell walls in the plane of a 2D simulation is necessary when cells have large aspect ratio, such as those in the root elongation zone of Arabidopsis thaliana, in order to avoid anomalous transverse swelling. We explore how differences in the mechanical properties of cells across an organ can result in bending and how cellulose microfibril orientation affects macroscale growth. We also demonstrate that the model can be used to simulate growth on realistic geometries, for example that of the primary root apex, using moderate computational resources. The model shows how macroscopic root shape can be sensitive to fine-scale cellular geometries.

A model of crosslink kinetics in the expanding plant cell wall: yield stress and enzyme action

The plant primary cell wall is a composite material containing stiff cellulose microfibrils that are embedded within a pectin matrix and crosslinked through a network of hemicellulose polymers. This microstructure endows the wall with nonlinear anisotropic mechanical properties and allows enzymatic regulation of expansive cell growth. We present a mathematical model of hemicellulose crosslink dynamics in an expanding cell wall incorporating strain-enhanced breakage and enzyme-mediated crosslink kinetics. The model predicts the characteristic yielding behaviour in the relationship between stress and strain-rate seen experimentally, and suggests how the effective yield and extensibility of the wall depend on microstructural parameters and on the action of enzymes of the XTH and expansin families. The model suggests that the yielding behaviour encapsulated in the classical Lockhart equation can be explained by the strongly nonlinear dependence of crosslink breakage rate on crosslink elongation. The model also demonstrates how enzymes that target crosslink binding can be effective in softening the wall in its pre-yield state, whereas its post-yield extensibility is determined primarily by the pectin matrix.

A model of crosslink kinetics in the expanding plant cell wall: yield stress and enzyme action

The plant primary cell wall is a composite material containing stiff cellulose microfibrils that are embedded within a pectin matrix and crosslinked through a network of hemicellulose polymers. This microstructure endows the wall with nonlinear anisotropic mechanical properties and allows enzymatic regulation of expansive cell growth. We present a mathematical model of hemicellulose crosslink dynamics in an expanding cell wall incorporating strain-enhanced breakage and enzyme-mediated crosslink kinetics. The model predicts the characteristic yielding behaviour in the relationship between stress and strain-rate seen experimentally, and suggests how the effective yield and extensibility of the wall depend on microstructural parameters and on the action of enzymes of the XTH and expansin families. The model suggests that the yielding behaviour encapsulated in the classical Lockhart equation can be explained by the strongly nonlinear dependence of crosslink breakage rate on crosslink elongation. The model also demonstrates how enzymes that target crosslink binding can be effective in softening the wall in its pre-yield state, whereas its post-yield extensibility is determined primarily by the pectin matrix.

Growth-induced hormone dilution can explain the dynamics of plant root cell elongation

In the elongation zone of the Arabidopsis thaliana plant root, cells undergo rapid elongation, increasing their length by ∼10-fold over 5 h while maintaining a constant radius. Although progress is being made in understanding how this growth is regulated, little consideration has been given as to how cell elongation affects the distribution of the key regulating hormones. Using a multiscale mathematical model and measurements of growth dynamics, we investigate the distribution of the hormone gibberellin in the root elongation zone. The model quantifies how rapid cell expansion causes gibberellin to dilute, creating a significant gradient in gibberellin levels. By incorporating the gibberellin signaling network, we simulate how gibberellin dilution affects the downstream components, including the growth-repressing DELLA proteins. We predict a gradient in DELLA that provides an explanation of the reduction in growth exhibited as cells move toward the end of the elongation zone. These results are validated at the molecular level by comparing predicted mRNA levels with transcriptomic data. To explore the dynamics further, we simulate perturbed systems in which gibberellin levels are reduced, considering both genetically modified and chemically treated roots. By modeling these cases, we predict how these perturbations affect gibberellin and DELLA levels and thereby provide insight into their altered growth dynamics.

Multiscale modelling of auxin transport in the plant-root elongation zone

In the root elongation zone of a plant, the hormone auxin moves in a polar manner due to active transport facilitated by spatially distributed influx and efflux carriers present on the cell membranes. To understand how the cell-scale active transport and passive diffusion combine to produce the effective tissue-scale flux, we apply asymptotic methods to a cell-based model of auxin transport to derive systematically a continuum description from the spatially discrete one. Using biologically relevant parameter values, we show how the carriers drive the dominant tissue-scale auxin flux and we predict how the overall auxin dynamics are affected by perturbations to these carriers, for example, in knockout mutants. The analysis shows how the dominant behaviour depends on the cells' lengths, and enables us to assess the relative importance of the diffusive auxin flux through the cell wall. Other distinguished limits are also identified and their potential roles discussed. As well as providing insight into auxin transport, the study illustrates the use of multiscale (cell to tissue) methods in deriving simplified models that retain the essential biology and provide understanding of the underlying dynamics.

VirtualLeaf: an open-source framework for cell-based modeling of plant tissue growth and development

Plant organs, including leaves and roots, develop by means of a multilevel cross talk between gene regulation, patterned cell division and cell expansion, and tissue mechanics. The multilevel regulatory mechanisms complicate classic molecular genetics or functional genomics approaches to biological development, because these methodologies implicitly assume a direct relation between genes and traits at the level of the whole plant or organ. Instead, understanding gene function requires insight into the roles of gene products in regulatory networks, the conditions of gene expression, etc. This interplay is impossible to understand intuitively. Mathematical and computer modeling allows researchers to design new hypotheses and produce experimentally testable insights. However, the required mathematics and programming experience makes modeling poorly accessible to experimental biologists. Problem-solving environments provide biologically intuitive in silico objects ("cells", "regulation networks") required for setting up a simulation and present those to the user in terms of familiar, biological terminology. Here, we introduce the cell-based computer modeling framework VirtualLeaf for plant tissue morphogenesis. The current version defines a set of biologically intuitive C++ objects, including cells, cell walls, and diffusing and reacting chemicals, that provide useful abstractions for building biological simulations of developmental processes. We present a step-by-step introduction to building models with VirtualLeaf, providing basic example models of leaf venation and meristem development. VirtualLeaf-based models provide a means for plant researchers to analyze the function of developmental genes in the context of the biophysics of growth and patterning. VirtualLeaf is an ongoing open-source software project (http://virtualleaf.googlecode.com) that runs on Windows, Mac, and Linux.

Modelling in vitro growth of dense root networks

Hairy roots are plants genetically transformed by Agrobacterium rhizogenes, which do not produce shoots and are composed mainly by roots. Hairy roots of Ophiorrhiza mungos Linn. are currently gaining interest of pharmacologists, since a secondary product of their metabolism, camptothecin, is used in chemotherapy. To optimize the production of valuable secondary metabolites it is necessary to understand the metabolism and growth of these roots systems. In this work, a mathematical model for description of apical growth of a dense root network (e.g. hairy roots) is derived. A continuous approach is used to define densities of root tips and root volume. Equations are posed to describe the evolution of these and are coupled to the distribution of nutrient concentration in the medium and inside the network. Following the principles of irreversible thermodynamics, growth velocity is defined as the sum over three different driving forces: nutrient concentration gradients, space gradients and root tip diffusion. A finite volume scheme was used for the simulation and parameters were chosen to fit experimental data from O. mungos Linn. hairy roots. Internal nutrient concentration determines short-term growth. Long-term behavior is limited by the total nutrient amount in the medium. Therefore, mass yield could be increased by guaranteeing a constant supply of nutrients. Increasing the initial mass of inoculation did not result in higher mass yields, since nutrient consumption due to metabolism also rose. Four different growth strategies are compared and their properties discussed. This allowed to understand which strategy might be the best to increase mass production optimally. The model is able to describe very well the temporal evolution of mass increase and nutrient uptake. Our results provide further understanding of growth and density distribution of hairy root network and therefore it is a sound base for future applications to describe, e.g., secondary metabolite production.

Spatio-temporal quantification of differential growth processes in root growth zones based on a novel combination of image sequence processing and refined concepts describing curvature production

Differential growth processes in root and shoot growth zones are governed by the transport kinetics of auxin and other plant hormones. While gene expression and protein localization of hormone transport facilitators are currently being unraveled using state-of-the-art techniques of live cell imaging, the quantitative analysis of growth reactions is lagging behind because of a lack of suitable methods. A noninvasive technique, based on digital image sequence processing, for visualizing and quantifying highly resolved spatio-temporal root growth processes was applied in the model plant Arabidopsis thaliana and was adapted to provide precise information on differential curvature production activity within the root growth zone. Comparison of root gravitropic curvature kinetics in wild-type and mutant plants altered in a facilitator for auxin translocation allowed the determination of differences in the location and in the temporal response of curvature along the growth zone between the investigated plant lines. The findings of the quantitative growth analysis performed here confirm the proposed action of the investigated transport facilitator. The procedure developed here for the investigation of differential growth processes is a valuable tool for characterizing the phenomenology of a wide range of shoot and root growth movements and hence facilitates elucidation of their molecular characterization.

Hormone signalling and root development: an update on the latest Arabidopsis thaliana research

Plants are sessile organisms whose developmental programs depend mainly on environmental cues that are sensed and interpreted through hormonal signalling pathways. Roots are specialised plant organs that are instrumental during water and nutrient uptake, biotic interactions, stress responses and for mechanical support. Our knowledge about the basic molecular events shaping root patterning and growth has advanced significantly in the past few years thanks to the use of Arabidopsis thaliana (L.) Heynh. as a model system. In this review, I will discuss recent findings that indicate crosstalk between growth regulators and hormone signalling pathways during primary root development. Further comparative research using non-model species will shed light on the conserved developmental modules among distant lineages involved in root architecture.

Anisotropic plant growth due to phototropism

Phototropism--the directional curvature of organs in response to lateral differences in light intensity and/or quality--represents one of the most rapid and visually obvious reaction of plants to changes in their light environment. It is a topic of fundamental interest to understand the mechanics of plants during growth. We propose a generalization of the scalar Lockhart model (1965) to three dimensional deformation, solve the new equation in two particular cases and compare results with empirical data. We believe that carefully designed experiments linked to our model will provide (by determining the active transport coefficient) a new method for qualitative description of auxin redistribution during phototropism. The proposed method supplements very recent investigations concerning specific auxin-influx and -efflux carriers (LAX and PIN proteins).

Functional dynamics of plant growth and photosynthesis--from steady-state to dynamics--from homogeneity to heterogeneity

Plants are much more dynamic than we usually expect them to be. This dynamic behaviour is of paramount importance for their performance under natural conditions, when resources are distributed heterogeneously in space and time. However, plants are not only the cue ball of their physical and chemical environment. Endogenous rhythms and networks controlling photosynthesis and growth buffer plant processes from external fluctuations. This review highlights recent evidence of the importance of dynamic temporal and spatial organization of photosynthesis and of growth in leaves and roots. These central processes for plant performance differ strongly in their dependence on environmental impact and endogenous properties, respectively. Growth involves a wealth of processes ranging from the supply of resources from external and internal sources to the growth processes themselves. In contrast, photosynthesis can only take place when light and CO2 are present and thus clearly requires 'input from the environment'. Nevertheless, growth and photosynthesis are connected to each other via mechanisms that are still not fully understood. Recent advances in imaging technology have provided new insights into the dynamics of plant-environment interactions. Such processes do not only play a crucial role in understanding stress response of plants under extreme environmental conditions. Dynamics of plants under modest growth conditions rise from endogenous mechanisms as well as exogenous impact too. It is thus an important task for future research to identify how dynamic external conditions interact with plant-internal signalling networks to optimize plant behaviour in real time and to understand how plants have adapted to characteristic spatial and temporal properties of the resources from their environment, on which they depend on.

Modeling plant morphogenesis

Applications of computational techniques to developmental plant biology include the processing of experimental data and the construction of simulation models. Substantial progress has been made in these areas over the past few years. Complex image-processing techniques are used to integrate sequences of two-dimensional images into three-dimensional descriptions of development over time and to extract useful quantitative traits. Large amounts of data are integrated into empirical models of developing plant organs and entire plants. Mechanistic models link molecular-level phenomena with the resulting phenotypes. Several models shed light on the possible properties of active auxin transport and its role in plant morphogenesis.