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

Auxin-mediated stress relaxation in pericycle and endoderm remodeling drives lateral root initiation

Plant development relies on the precise coordination of cell growth, which is influenced by the mechanical constraints imposed by rigid cell walls. The hormone auxin plays a crucial role in regulating this growth by altering the mechanical properties of cell walls. During the postembryonic formation of lateral roots, pericycle cells deep within the main root are triggered by auxin to resume growth and divide to form a new root. This growth involves a complex interplay between auxin, growth, and the resolution of mechanical conflicts with the overlying endodermis. However, the exact mechanisms by which this coordination is achieved are still unknown. Here, we propose a model that integrates tissue mechanics and auxin transport, revealing a connection between the auxin-induced relaxation of mechanical stress in the pericycle and auxin signaling in the endodermis. We show that the endodermis initially limits the growth of pericycle cells, resulting in a modest initial expansion. However, the associated stress relaxation is sufficient to redirect auxin to the overlying endodermis, which then actively accommodates the growth, allowing for the subsequent development of the lateral root. Our model uncovers that increased pericycle turgor and decreased endodermal resistance license expansion of the pericycle and how the topology of the endodermis influences the formation of the new root. These findings highlight the interconnected relationship between mechanics and auxin flow during lateral root initiation, emphasizing the vital role of the endodermis in shaping root development through mechanotransduction and auxin signaling.

Contact area and tissue growth dynamics shape synthetic juxtacrine signaling patterns

Cell-cell communication through direct contact, or juxtacrine signaling, is important in development, disease, and many areas of physiology. Synthetic forms of juxtacrine signaling can be precisely controlled and operate orthogonally to native processes, making them a powerful reductionist tool with which to address fundamental questions in cell-cell communication in vivo. Here, we investigate how cell-cell contact length and tissue growth dynamics affect juxtacrine signal responses through implementing a custom synthetic gene circuit in Drosophila wing imaginal discs alongside mathematical modeling to determine synthetic Notch (synNotch) activation patterns. We find that the area of contact between cells largely determines the extent of synNotch activation, leading to the prediction that the shape of the interface between signal-sending and signal-receiving cells will impact the magnitude of the synNotch response. Notably, synNotch outputs form a graded spatial profile that extends several cell diameters from the signal source, providing evidence that the response to juxtacrine signals can persist in cells as they proliferate away from source cells, or that cells remain able to communicate directly over several cell diameters. Our model suggests that the former mechanism may be sufficient, since it predicts graded outputs without diffusion or long-range cell-cell communication. Overall, we identify that cell-cell contact area together with output synthesis and decay rates likely govern the pattern of synNotch outputs in both space and time during tissue growth, insights that may have broader implications for juxtacrine signaling in general.

The Effects of Auxin Transport Inhibition on the Formation of Various Leaf and Vein Patterns

Polar auxin transport (PAT) is a known component controlling leaf complexity and venation patterns in some model plant species. Evidence indicates that PAT generates auxin converge points (CPs) that in turn lead to local leaf formation and internally into major vein formation. However, the role of PAT in more diverse leaf arrangements and vein patterns is largely unknown. We used the pharmacological inhibition of PAT in developing pinnate tomato, trifoliate clover, palmate lupin, and bipinnate carrot leaves and observed dosage-dependent reduction to simple leaves in these eudicots. Leaf venation patterns changed from craspedodromous (clover, carrot), semi-craspedodromous (tomato), and brochidodromous (lupin) to more parallel patterning with PAT inhibition. The visualization of auxin responses in transgenic tomato plants showed that discrete and separate CPs in control plants were replaced by diffuse convergence areas near the margin. These effects indicate that PAT plays a universal role in the formation of different leaf and vein patterns in eudicot species via a mechanism that depends on the generation as well as the separation of auxin CPs. Computer simulations indicate that variations in PAT can alter the number of CPs, corresponding leaf lobe formation, and the position of major leaf veins along the leaf margin in support of experimental results.

Contact area and tissue growth dynamics shape synthetic juxtacrine signaling patterns

Cell-cell communication through direct contact, or juxtacrine signaling, is important in development, disease, and many areas of physiology. Synthetic forms of juxtacrine signaling can be precisely controlled and operate orthogonally to native processes, making them a powerful reductionist tool with which to address fundamental questions in cell-cell communication in vivo. Here we investigate how cell-cell contact length and tissue growth dynamics affect juxtacrine signal responses through implementing a custom synthetic gene circuit in Drosophila wing imaginal discs alongside mathematical modeling to determine synthetic Notch (synNotch) activation patterns. We find that the area of contact between cells largely determines the extent of synNotch activation, leading to the prediction that the shape of the interface between signal-sending and signal-receiving cells will impact the magnitude of the synNotch response. Notably, synNotch outputs form a graded spatial profile that extends several cell diameters from the signal source, providing evidence that the response to juxtacrine signals can persist in cells as they proliferate away from source cells, or that cells remain able to communicate directly over several cell diameters. Our model suggests the former mechanism may be sufficient, since it predicts graded outputs without diffusion or long-range cell-cell communication. Overall, we identify that cell-cell contact area together with output synthesis and decay rates likely govern the pattern of synNotch outputs in both space and time during tissue growth, insights that may have broader implications for juxtacrine signaling in general.

SimuCell3D: three-dimensional simulation of tissue mechanics with cell polarization

The three-dimensional (3D) organization of cells determines tissue function and integrity, and changes markedly in development and disease. Cell-based simulations have long been used to define the underlying mechanical principles. However, high computational costs have so far limited simulations to either simplified cell geometries or small tissue patches. Here, we present SimuCell3D, an efficient open-source program to simulate large tissues in three dimensions with subcellular resolution, growth, proliferation, extracellular matrix, fluid cavities, nuclei and non-uniform mechanical properties, as found in polarized epithelia. Spheroids, vesicles, sheets, tubes and other tissue geometries can readily be imported from microscopy images and simulated to infer biomechanical parameters. Doing so, we show that 3D cell shapes in layered and pseudostratified epithelia are largely governed by a competition between surface tension and intercellular adhesion. SimuCell3D enables the large-scale in silico study of 3D tissue organization in development and disease at a great level of detail.

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

DyCeModel: a tool for 1D simulation for distribution of plant hormones controlling tissue patterning

To study the mechanisms of growth and development, it is necessary to analyze the dynamics of the tissue patterning regulators in time and space and to take into account their effect on the cellular dynamics within a tissue. Plant hormones are the main regulators of the cell dynamics in plant tissues; they form gradients and maxima and control molecular processes in a concentration-dependent manner. Here, we present DyCeModel, a software tool implemented in MATLAB for one-dimensional simulation of tissue with a dynamic cellular ensemble, where changes in hormone (or other active substance) concentration in the cells are described by ordinary differential equations (ODEs). We applied DyCeModel to simulate cell dynamics in plant meristems with different cellular structures and demonstrated that DyCeModel helps to identify the relationships between hormone concentration and cellular behaviors. The tool visualizes the simulation progress and presents a video obtained during the calculation. Importantly, the tool is capable of automatically adjusting the parameters by fitting the distribution of the substance concentrations predicted in the model to experimental data taken from the microscopic images. Noteworthy, DyCeModel makes it possible to build models for distinct types of plant meristems with the same ODEs, recruiting specific input characteristics for each meristem. We demonstrate the tool's efficiency by simulation of the effect of auxin and cytokinin distributions on tissue patterning in two types of Arabidopsis thaliana stem cell niches: the root and shoot apical meristems. The resulting models represent a promising framework for further study of the role of hormone-controlled gene regulatory networks in cell dynamics.

FitMultiCell: simulating and parameterizing computational models of multi-scale and multi-cellular processes

MOTIVATION

Biological tissues are dynamic and highly organized. Multi-scale models are helpful tools to analyse and understand the processes determining tissue dynamics. These models usually depend on parameters that need to be inferred from experimental data to achieve a quantitative understanding, to predict the response to perturbations, and to evaluate competing hypotheses. However, even advanced inference approaches such as approximate Bayesian computation (ABC) are difficult to apply due to the computational complexity of the simulation of multi-scale models. Thus, there is a need for a scalable pipeline for modeling, simulating, and parameterizing multi-scale models of multi-cellular processes.

RESULTS

Here, we present FitMultiCell, a computationally efficient and user-friendly open-source pipeline that can handle the full workflow of modeling, simulating, and parameterizing for multi-scale models of multi-cellular processes. The pipeline is modular and integrates the modeling and simulation tool Morpheus and the statistical inference tool pyABC. The easy integration of high-performance infrastructure allows to scale to computationally expensive problems. The introduction of a novel standard for the formulation of parameter inference problems for multi-scale models additionally ensures reproducibility and reusability. By applying the pipeline to multiple biological problems, we demonstrate its broad applicability, which will benefit in particular image-based systems biology.

AVAILABILITY AND IMPLEMENTATION

FitMultiCell is available open-source at https://gitlab.com/fitmulticell/fit.

Computer models of cell polarity establishment in plants

Plant development is a complex task, and many processes involve changes in the asymmetric subcellular distribution of cell components that strongly depend on cell polarity. Cell polarity regulates anisotropic growth and polar localization of membrane proteins and helps to identify the cell's position relative to its neighbors within an organ. Cell polarity is critical in a variety of plant developmental processes, including embryogenesis, cell division, and response to external stimuli. The most conspicuous downstream effect of cell polarity is the polar transport of the phytohormone auxin, which is the only known hormone transported in a polar fashion in and out of cells by specialized exporters and importers. The biological processes behind the establishment of cell polarity are still unknown, and researchers have proposed several models that have been tested using computer simulations. The evolution of computer models has progressed in tandem with scientific discoveries, which have highlighted the importance of genetic, chemical, and mechanical input in determining cell polarity and regulating polarity-dependent processes such as anisotropic growth, protein subcellular localization, and the development of organ shapes. The purpose of this review is to provide a comprehensive overview of the current understanding of computer models of cell polarity establishment in plants, focusing on the molecular and cellular mechanisms, the proteins involved, and the current state of the field.

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.

Novel Ground-Up 3D Multicellular Simulators for Synthetic Biology CAD Integrating Stochastic Gillespie Simulations Benchmarked with Topologically Variable SBML Models

The elevation of Synthetic Biology from single cells to multicellular simulations would be a significant scale-up. The spatiotemporal behavior of cellular populations has the potential to be prototyped in silico for computer assisted design through ergonomic interfaces. Such a platform would have great practical potential across medicine, industry, research, education and accessible archiving in bioinformatics. Existing Synthetic Biology CAD systems are considered limited regarding population level behavior, and this work explored the in silico challenges posed from biological and computational perspectives. Retaining the connection to Synthetic Biology CAD, an extension of the Infobiotics Workbench Suite was considered, with potential for the integration of genetic regulatory models and/or chemical reaction networks through Next Generation Stochastic Simulator (NGSS) Gillespie algorithms. These were executed using SBML models generated by in-house SBML-Constructor over numerous topologies and benchmarked in association with multicellular simulation layers. Regarding multicellularity, two ground-up multicellular solutions were developed, including the use of Unreal Engine 4 contrasted with CPU multithreading and Blender visualization, resulting in a comparison of real-time versus batch-processed simulations. In conclusion, high-performance computing and client-server architectures could be considered for future works, along with the inclusion of numerous biologically and physically informed features, whilst still pursuing ergonomic solutions.

Introduction to Computational Modeling of Multicellular Tissues

The study of biological tissues is extremely complicated, as they comprise mechanisms and properties at many different temporal and spatial scales. For this reason, modeling is becoming one of the most active and important research fields for the analysis and understanding of tissues. However, this is not a simple task, as it requires mathematical and computational skills, as well as the development of software tools for its implementation. Here, we provide an introduction covering some of the most important and basic issues for modeling tissues. In particular, we focus on both the chemical and cellular properties of a tissue. We explain how to represent and couple these properties within a virtual tissue. All our examples were done using Multicell, a Python library that simplifies their reproducibility, even by readers with little experience in biological modeling.

Modeling Plant Tissue Development Using VirtualLeaf

Cell-based computational modeling and simulation are becoming invaluable tools in analyzing plant development. In a cell-based simulation model, the inputs are behaviors and dynamics of individual cells and the rules describing responses to signals from adjacent cells. The outputs are the growing tissues, shapes, and cell-differentiation patterns that emerge from the local, chemical, and biomechanical cell-cell interactions. In this updated and extended version of our previous chapter on VirtualLeaf (Merks and Guravage, Methods in Molecular Biology 959, 333-352), we present a step-by-step, practical tutorial for building cell-based simulations of plant development and for analyzing the influence of parameters on simulation outcomes by systematically changing the values of the parameters and analyzing each outcome. We show how to build a model of a growing tissue, a reaction-diffusion system on a growing domain, and an auxin transport model. Moreover, in addition to the previous publication, we demonstrate how to run a Turing system on a regular, rectangular lattice, and how to run parameter sweeps. The aim of VirtualLeaf is to make computational modeling more accessible to experimental plant biologists with relatively little computational background.

A rigid body framework for multicellular modeling

Off-lattice models are a well-established approach in multicellular modeling, where cells are represented as points that are free to move in space. The representation of cells as point objects is useful in a wide range of settings, particularly when large populations are involved; however, a purely point-based representation is not naturally equipped to deal with objects that have length, such as cell boundaries or external membranes. Here we introduce an off-lattice modeling framework that exploits rigid body mechanics to represent objects using a collection of conjoined one-dimensional edges in a viscosity-dominated system. This framework can be used to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, to model rod-shaped cells such as bacteria and to robustly represent membranes. We demonstrate that this approach offers solutions to the problems that limit the scope of current off-lattice multicellular models.

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.

Mechanical heterogeneity along single cell-cell junctions is driven by lateral clustering of cadherins during vertebrate axis elongation

Morphogenesis is governed by the interplay of molecular signals and mechanical forces across multiple length scales. The last decade has seen tremendous advances in our understanding of the dynamics of protein localization and turnover at subcellular length scales, and at the other end of the spectrum, of mechanics at tissue-level length scales. Integrating the two remains a challenge, however, because we lack a detailed understanding of the subcellular patterns of mechanical properties of cells within tissues. Here, in the context of the elongating body axis of Xenopus embryos, we combine tools from cell biology and physics to demonstrate that individual cell-cell junctions display finely-patterned local mechanical heterogeneity along their length. We show that such local mechanical patterning is essential for the cell movements of convergent extension and is imparted by locally patterned clustering of a classical cadherin. Finally, the patterning of cadherins and thus local mechanics along cell-cell junctions are controlled by Planar Cell Polarity signaling, a key genetic module for CE that is mutated in diverse human birth defects.

A Sight on Single-Cell Transcriptomics in Plants Through the Prism of Cell-Based Computational Modeling Approaches: Benefits and Challenges for Data Analysis

Single-cell technology is a relatively new and promising way to obtain high-resolution transcriptomic data mostly used for animals during the last decade. However, several scientific groups developed and applied the protocols for some plant tissues. Together with deeply-developed cell-resolution imaging techniques, this achievement opens up new horizons for studying the complex mechanisms of plant tissue architecture formation. While the opportunities for integrating data from transcriptomic to morphogenetic levels in a unified system still present several difficulties, plant tissues have some additional peculiarities. One of the plants' features is that cell-to-cell communication topology through plasmodesmata forms during tissue growth and morphogenesis and results in mutual regulation of expression between neighboring cells affecting internal processes and cell domain development. Undoubtedly, we must take this fact into account when analyzing single-cell transcriptomic data. Cell-based computational modeling approaches successfully used in plant morphogenesis studies promise to be an efficient way to summarize such novel multiscale data. The inverse problem's solutions for these models computed on the real tissue templates can shed light on the restoration of individual cells' spatial localization in the initial plant organ-one of the most ambiguous and challenging stages in single-cell transcriptomic data analysis. This review summarizes new opportunities for advanced plant morphogenesis models, which become possible thanks to single-cell transcriptome data. Besides, we show the prospects of microscopy and cell-resolution imaging techniques to solve several spatial problems in single-cell transcriptomic data analysis and enhance the hybrid modeling framework opportunities.

PlantLayout pipeline to model tissue patterning

To study the mechanisms underlying developmental pattern formation in a tissue, one needs to analyze the dynamics of the regulators in time and space across the tissue of a specific architecture. This problem is essential for the developmental regulators (morphogens) that distribute over the tissues anisotropically, forming there maxima and gradients and guiding cellular processes in a dose-dependent manner. Here we present the PlantLayout pipeline for MATLAB software, which facilitates the computational studies of tissue patterning. With its help, one can build a structural model of a two-dimensional tissue, embed it into a mathematical model in ODEs, perform numerical simulations, and visualize the obtained results - everything on the same platform. As a result, one can study the concentration dynamics of developmental regulators over the cell layout reconstructed from the real tissue. PlantLayout allows studying the dynamics and the output of gene networks guided by the developmental regulator in specific cells. The gene networks could be different for different cell types. One of the obstacles that PlantLayout removes semi-automatically is the determination of the cell wall orientation which is relevant when cells in the tissue have a polarity. Additionally, PlantLayout allows automatically extracting other quantitative and qualitative features of the cells and the cell walls, which might help in the modeling of a developmental pattern, such as the length and the width of the cell walls, the set of the neighboring cells, cell volume and cell perimeter. We demonstrate PlantLayout performance on the model of phytohormone auxin distribution over the plant root tip.

Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning

Our current understanding of vein development in leaves is based on canalization of the plant hormone auxin into self-reinforcing streams which determine the sites of vascular cell differentiation. By comparison, how auxin biosynthesis affects leaf vein patterning is less well understood. Here, after observing that inhibiting polar auxin transport rescues the sparse leaf vein phenotype in auxin biosynthesis mutants, we propose that the processes of auxin biosynthesis and cellular auxin efflux work in concert during vein development. By using computational modeling, we show that localized auxin maxima are able to interact with mechanical forces generated by the morphological constraints which are imposed during early primordium development. This interaction is able to explain four fundamental characteristics of midvein morphology in a growing leaf: (i) distal cell division; (ii) coordinated cell elongation; (iii) a midvein positioned in the center of the primordium; and (iv) a midvein which is distally branched. Domains of auxin biosynthetic enzyme expression are not positioned by auxin canalization, as they are observed before auxin efflux proteins polarize. This suggests that the site-specific accumulation of auxin, as regulated by the balanced action of cellular auxin efflux and local auxin biosynthesis, is crucial for leaf vein formation.

Chaste: Cancer, Heart and Soft Tissue Environment

Chaste (Cancer, Heart And Soft Tissue Environment) is an open source simulation package for the numerical solution of mathematical models arising in physiology and biology. To date, Chaste development has been driven primarily by applications that include continuum modelling of cardiac electrophysiology ('Cardiac Chaste'), discrete cell-based modelling of soft tissues ('Cell-based Chaste'), and modelling of ventilation in lungs ('Lung Chaste'). Cardiac Chaste addresses the need for a high-performance, generic, and verified simulation framework for cardiac electrophysiology that is freely available to the scientific community. Cardiac chaste provides a software package capable of realistic heart simulations that is efficient, rigorously tested, and runs on HPC platforms. Cell-based Chaste addresses the need for efficient and verified implementations of cell-based modelling frameworks, providing a set of extensible tools for simulating biological tissues. Computational modelling, along with live imaging techniques, plays an important role in understanding the processes of tissue growth and repair. A wide range of cell-based modelling frameworks have been developed that have each been successfully applied in a range of biological applications. Cell-based Chaste includes implementations of the cellular automaton model, the cellular Potts model, cell-centre models with cell representations as overlapping spheres or Voronoi tessellations, and the vertex model. Lung Chaste addresses the need for a novel, generic and efficient lung modelling software package that is both tested and verified. It aims to couple biophysically-detailed models of airway mechanics with organ-scale ventilation models in a package that is freely available to the scientific community. Chaste is designed to be modular and extensible, providing libraries for common scientific computing infrastructure such as linear algebra operations, finite element meshes, and ordinary and partial differential equation solvers. This infrastructure is used by libraries for specific applications, such as continuum mechanics, cardiac models, and cell-based models. The software engineering techniques used to develop Chaste are intended to ensure code quality, re-usability and reliability. Primary applications of the software include cardiac and respiratory physiology, cancer and developmental biology.

PhysiBoSS: a multi-scale agent-based modelling framework integrating physical dimension and cell signalling

MOTIVATION

Due to the complexity and heterogeneity of multicellular biological systems, mathematical models that take into account cell signalling, cell population behaviour and the extracellular environment are particularly helpful. We present PhysiBoSS, an open source software which combines intracellular signalling using Boolean modelling (MaBoSS) and multicellular behaviour using agent-based modelling (PhysiCell).

RESULTS

PhysiBoSS provides a flexible and computationally efficient framework to explore the effect of environmental and genetic alterations of individual cells at the population level, bridging the critical gap from single-cell genotype to single-cell phenotype and emergent multicellular behaviour. PhysiBoSS thus becomes very useful when studying heterogeneous population response to treatment, mutation effects, different modes of invasion or isomorphic morphogenesis events. To concretely illustrate a potential use of PhysiBoSS, we studied heterogeneous cell fate decisions in response to TNF treatment. We explored the effect of different treatments and the behaviour of several resistant mutants. We highlighted the importance of spatial information on the population dynamics by considering the effect of competition for resources like oxygen.

AVAILABILITY AND IMPLEMENTATION

PhysiBoSS is freely available on GitHub (https://github.com/sysbio-curie/PhysiBoSS), with a Docker image (https://hub.docker.com/r/gletort/physiboss/). It is distributed as open source under the BSD 3-clause license.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

In silico study of the role of cell growth factors in photosynthesis using a virtual leaf tissue generator coupled to a microscale photosynthesis gas exchange model

Computational tools that allow in silico analysis of the role of cell growth and division on photosynthesis are scarce. We present a freely available tool that combines a virtual leaf tissue generator and a two-dimensional microscale model of gas transport during C3 photosynthesis. A total of 270 mesophyll geometries were generated with varying degrees of growth anisotropy, growth extent, and extent of schizogenous airspace formation in the palisade mesophyll. The anatomical properties of the virtual leaf tissue and microscopic cross-sections of actual leaf tissue of tomato (Solanum lycopersicum L.) were statistically compared. Model equations for transport of CO2 in the liquid phase of the leaf tissue were discretized over the geometries. The virtual leaf tissue generator produced a leaf anatomy of tomato that was statistically similar to real tomato leaf tissue. The response of photosynthesis to intercellular CO2 predicted by a model that used the virtual leaf tissue geometry compared well with measured values. The results indicate that the light-saturated rate of photosynthesis was influenced by interactive effects of extent and directionality of cell growth and degree of airspace formation through the exposed surface of mesophyll per leaf area. The tool could be used further in investigations of improving photosynthesis and gas exchange in relation to cell growth and leaf anatomy.

In silico study of the role of cell growth factors in photosynthesis using a virtual leaf tissue generator coupled to a microscale photosynthesis gas exchange model

Computational tools that allow in silico analysis of the role of cell growth and division on photosynthesis are scarce. We present a freely available tool that combines a virtual leaf tissue generator and a two-dimensional microscale model of gas transport during C3 photosynthesis. A total of 270 mesophyll geometries were generated with varying degrees of growth anisotropy, growth extent, and extent of schizogenous airspace formation in the palisade mesophyll. The anatomical properties of the virtual leaf tissue and microscopic cross-sections of actual leaf tissue of tomato (Solanum lycopersicum L.) were statistically compared. Model equations for transport of CO2 in the liquid phase of the leaf tissue were discretized over the geometries. The virtual leaf tissue generator produced a leaf anatomy of tomato that was statistically similar to real tomato leaf tissue. The response of photosynthesis to intercellular CO2 predicted by a model that used the virtual leaf tissue geometry compared well with measured values. The results indicate that the light-saturated rate of photosynthesis was influenced by interactive effects of extent and directionality of cell growth and degree of airspace formation through the exposed surface of mesophyll per leaf area. The tool could be used further in investigations of improving photosynthesis and gas exchange in relation to cell growth and leaf anatomy.

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.

Adapting a Plant Tissue Model to Animal Development: Introducing Cell Sliding into VirtualLeaf

Cell-based, mathematical modeling of collective cell behavior has become a prominent tool in developmental biology. Cell-based models represent individual cells as single particles or as sets of interconnected particles and predict the collective cell behavior that follows from a set of interaction rules. In particular, vertex-based models are a popular tool for studying the mechanics of confluent, epithelial cell layers. They represent the junctions between three (or sometimes more) cells in confluent tissues as point particles, connected using structural elements that represent the cell boundaries. A disadvantage of these models is that cell-cell interfaces are represented as straight lines. This is a suitable simplification for epithelial tissues, where the interfaces are typically under tension, but this simplification may not be appropriate for mesenchymal tissues or tissues that are under compression, such that the cell-cell boundaries can buckle. In this paper, we introduce a variant of VMs in which this and two other limitations of VMs have been resolved. The new model can also be seen as on off-the-lattice generalization of the Cellular Potts Model. It is an extension of the open-source package VirtualLeaf, which was initially developed to simulate plant tissue morphogenesis where cells do not move relative to one another. The present extension of VirtualLeaf introduces a new rule for cell-cell shear or sliding, from which cell rearrangement (T1) and cell extrusion (T2) transitions emerge naturally, allowing the application of VirtualLeaf to problems of animal development. We show that the updated VirtualLeaf yields different results than the traditional vertex-based models for differential adhesion-driven cell sorting and for the neighborhood topology of soft cellular networks.

Constructing living buildings: a review of relevant technologies for a novel application of biohybrid robotics

Biohybrid robotics takes an engineering approach to the expansion and exploitation of biological behaviours for application to automated tasks. Here, we identify the construction of living buildings and infrastructure as a high-potential application domain for biohybrid robotics, and review technological advances relevant to its future development. Construction, civil infrastructure maintenance and building occupancy in the last decades have comprised a major portion of economic production, energy consumption and carbon emissions. Integrating biological organisms into automated construction tasks and permanent building components therefore has high potential for impact. Live materials can provide several advantages over standard synthetic construction materials, including self-repair of damage, increase rather than degradation of structural performance over time, resilience to corrosive environments, support of biodiversity, and mitigation of urban heat islands. Here, we review relevant technologies, which are currently disparate. They span robotics, self-organizing systems, artificial life, construction automation, structural engineering, architecture, bioengineering, biomaterials, and molecular and cellular biology. In these disciplines, developments relevant to biohybrid construction and living buildings are in the early stages, and typically are not exchanged between disciplines. We, therefore, consider this review useful to the future development of biohybrid engineering for this highly interdisciplinary application.

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.

An Integrative Systems Perspective on Plant Phosphate Research

The case for improving crop phosphorus-use-efficiency is widely recognized. Although much is known about the molecular and regulatory mechanisms, improvements have been hampered by the extreme complexity of phosphorus (P) dynamics, which involves soil chemistry; plant-soil interactions; uptake, transport, utilization and remobilization within plants; and agricultural practices. The urgency and direction of phosphate research is also dependent upon the finite sources of P, availability of stocks to farmers and reducing environmental hazards. This work introduces integrative systems approaches as a way to represent and understand this complexity, so that meaningful links can be established between genotype, environment, crop traits and yield. It aims to provide a large set of pointers to potential genes and research practice, with a view to encouraging members of the plant-phosphate research community to adopt such approaches so that, together, we can aid efforts in global food security.

A Dynamic Non-Manifold Mesh Data Structure to Represent Biological Materials

Computational models of biological materials enable researchers to gain insight and make testable predictions of quantitative dynamic responses to stimuli. These models are particularly challenging to develop because biological materials are (1) highly heterogeneous containing both biological cells and complex substances such as extra-cellular medium, (2) undergo structural rearrangement (3) couple biological cells with their environment via chemical and mechanical processes. Existing numerical approaches excel at either describing biological cells or solids and fluids, but have difficulty integrating them into a single simulation approach. We present a novel dynamic non-manifold mesh data structure that naturally represents biological materials with coupled chemical and mechanical processes and structural rearrangement in a unified way.

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.

Algorithm for a particle-based growth model for plant tissues

We have developed an algorithm for a particle-based model for the growth of plant tissues in three dimensions in which each cell is represented by a single particle, and connecting cell walls are represented as permanent bonds between particles. A sample of plant tissue is represented by a fixed network of bonded particles. If, and only if a cell divides, this network is updated locally. The update algorithm is implemented in a model where cell growth and division gives rise to forces between the cells, which are relaxed in steepest descent minimization. The same forces generate a pressure inside the cells, which moderates growth. The local nature of the algorithm makes it efficient computationally, so the model can deal with a large number of cells. We used the model to study the growth of plant tissues for a variety of model parameters, to show the viability of the algorithm.

Multiscale Modeling for Food Drying: State of the Art

Plant-based food materials are mostly porous in nature and heterogeneous in structure with huge diversity in cellular orientation. Different cellular environments of plant-based food materials, such as intercellular, intracellular, and cell wall environments, hold different proportions of water with different characteristics. Due to this structural heterogeneity, it is very difficult to understand the drying process and associated morphological changes during drying. Transport processes and morphological changes that take place during drying are mainly governed by the characteristics of and the changes in the cells. Therefore, to predict the actual heat and mass transfer process that occurs in the drying process and associated morphological changes, development of multiscale modeling is crucial. Multiscale modeling is a powerful approach with the ability to incorporate this cellular structural heterogeneity with microscale heat and mass transfer during drying. However, due to the huge complexity involved in developing such a model for plant-based food materials, the studies regarding this issue are very limited. Therefore, we aim in this article to develop a critical conceptual understanding of multiscale modeling frameworks for heterogeneous food materials through an extensive literature review. We present a critical review on the multiscale model formulation and solution techniques with their spatial and temporal coupling options. Food structure, scale definition, and the current status of multiscale modeling are also presented, along with other key factors that are critical to understanding and developing an accurate multiscale framework. We conclude by presenting the main challenges for developing an accurate multiscale modeling framework for food drying.

Genetic and Developmental Basis for Increased Leaf Thickness in the Arabidopsis Cvi Ecotype

Leaf thickness is a quantitative trait that is associated with the ability of plants to occupy dry, high irradiance environments. Despite its importance, leaf thickness has been difficult to measure reproducibly, which has impeded progress in understanding its genetic basis, and the associated anatomical mechanisms that pattern it. Here, we used a custom-built dual confocal profilometer device to measure leaf thickness in the Arabidopsis Ler × Cvi recombinant inbred line population and found statistical support for four quantitative trait loci (QTL) associated with this trait. We used publically available data for a suite of traits relating to flowering time and growth responses to light quality and show that three of the four leaf thickness QTL coincide with QTL for at least one of these traits. Using time course photography, we quantified the relative growth rate and the pace of rosette leaf initiation in the Ler and Cvi ecotypes. We found that Cvi rosettes grow slower than Ler, both in terms of the rate of leaf initiation and the overall rate of biomass accumulation. Collectively, these data suggest that leaf thickness is tightly linked with physiological status and may present a tradeoff between the ability to withstand stress and rapid vegetative growth. To understand the anatomical basis of leaf thickness, we compared cross-sections of Cvi and Ler leaves and show that Cvi palisade mesophyll cells elongate anisotropically contributing to leaf thickness. Flow cytometry of whole leaves show that endopolyploidy accompanies thicker leaves in Cvi. Overall, our data suggest that mechanistically, an altered schedule of cellular events affecting endopolyploidy and increasing palisade mesophyll cell length contribute to increase of leaf thickness in Cvi. Ultimately, knowledge of the genetic basis and developmental trajectory leaf thickness will inform the mechanisms by which natural selection acts to produce variation in this adaptive trait.

Mechanical regulation of organ asymmetry in leaves

How appendages, such as plant leaves or animal limbs, develop asymmetric shapes remains a fundamental question in biology. Although ongoing research has revealed the genetic regulation of organ pattern formation, how gene activity ultimately directs organ shape remains unclear. Here, we show that leaf dorsoventral (adaxial-abaxial) polarity signals lead to mechanical heterogeneity of the cell wall, related to the methyl-esterification of cell-wall pectins in tomato and Arabidopsis. Numerical simulations predicate that mechanical heterogeneity is sufficient to produce the asymmetry seen in planar leaves. Experimental tests that alter pectin methyl-esterification, and therefore cell wall mechanical properties, support this model and lead to polar changes in gene expression, suggesting the existence of a feedback mechanism for mechanical signals in morphogenesis. Thus, mechanical heterogeneity within tissue may underlie organ shape asymmetry.

Virtual Plant Tissue: Building Blocks for Next-Generation Plant Growth Simulation

Motivation: Computational modeling of plant developmental processes is becoming increasingly important. Cellular resolution plant tissue simulators have been developed, yet they are typically describing physiological processes in an isolated way, strongly delimited in space and time. Results: With plant systems biology moving toward an integrative perspective on development we have built the Virtual Plant Tissue (VPTissue) package to couple functional modules or models in the same framework and across different frameworks. Multiple levels of model integration and coordination enable combining existing and new models from different sources, with diverse options in terms of input/output. Besides the core simulator the toolset also comprises a tissue editor for manipulating tissue geometry and cell, wall, and node attributes in an interactive manner. A parameter exploration tool is available to study parameter dependence of simulation results by distributing calculations over multiple systems. Availability: Virtual Plant Tissue is available as open source (EUPL license) on Bitbucket (https://bitbucket.org/vptissue/vptissue). The project has a website https://vptissue.bitbucket.io.

Comparing individual-based approaches to modelling the self-organization of multicellular tissues

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

A cell-based computational model of early embryogenesis coupling mechanical behaviour and gene regulation

The study of multicellular development is grounded in two complementary domains: cell biomechanics, which examines how physical forces shape the embryo, and genetic regulation and molecular signalling, which concern how cells determine their states and behaviours. Integrating both sides into a unified framework is crucial to fully understand the self-organized dynamics of morphogenesis. Here we introduce MecaGen, an integrative modelling platform enabling the hypothesis-driven simulation of these dual processes via the coupling between mechanical and chemical variables. Our approach relies upon a minimal 'cell behaviour ontology' comprising mesenchymal and epithelial cells and their associated behaviours. MecaGen enables the specification and control of complex collective movements in 3D space through a biologically relevant gene regulatory network and parameter space exploration. Three case studies investigating pattern formation, epithelial differentiation and tissue tectonics in zebrafish early embryogenesis, the latter with quantitative comparison to live imaging data, demonstrate the validity and usefulness of our framework.

Regulation of vascular cell division

Vascular tissue, comprising xylem and phloem, is responsible for the transport of water and nutrients throughout the plant body. Such tissue is continually produced from stable populations of stem cells, specifically the procambium during primary growth and the cambium during secondary growth. As the majority of plant biomass is produced by the cambium, there is an obvious demand for an understanding of the genetic mechanisms that control the rate of vascular cell division. Moreover, wood is an industrially important product of the cambium, and research is beginning to uncover similar mechanisms in trees such as poplar. This review focuses upon recent work that has identified the major molecular pathways that regulate procambial and cambial activity.

Theoretical approaches to understanding root vascular patterning: a consensus between recent models

The root vascular tissues provide an excellent system for studying organ patterning, as the specification of these tissues signals a transition from radial symmetry to bisymmetric patterns. The patterning process is controlled by the combined action of hormonal signaling/transport pathways, transcription factors, and miRNA that operate through a series of non-linear pathways to drive pattern formation collectively. With the discovery of multiple components and feedback loops controlling patterning, it has become increasingly difficult to understand how these interactions act in unison to determine pattern formation in multicellular tissues. Three independent mathematical models of root vascular patterning have been formulated in the last few years, providing an excellent example of how theoretical approaches can complement experimental studies to provide new insights into complex systems. In many aspects these models support each other; however, each study also provides its own novel findings and unique viewpoints. Here we reconcile these models by identifying the commonalities and exploring the differences between them by testing how transferable findings are between models. New simulations herein support the hypothesis that an asymmetry in auxin input can direct the formation of vascular pattern. We show that the xylem axis can act as a sole source of cytokinin and specify the correct pattern, but also that broader patterns of cytokinin production are also able to pattern the root. By comparing the three modeling approaches, we gain further insight into vascular patterning and identify several key areas for experimental investigation.

Computational Modeling of Auxin: A Foundation for Plant Engineering

Since the development of agriculture, humans have relied on the cultivation of plants to satisfy our increasing demand for food, natural products, and other raw materials. As we understand more about plant development, we can better manipulate plants to fulfill our particular needs. Auxins are a class of simple metabolites that coordinate many developmental activities like growth and the appearance of functional structures in plants. Computational modeling of auxin has proven to be an excellent tool in elucidating many mechanisms that underlie these developmental events. Due to the complexity of these mechanisms, current modeling efforts are concerned only with single phenomena focused on narrow spatial and developmental contexts; but a general model of plant development could be assembled by integrating the insights from all of them. In this perspective, we summarize the current collection of auxin-driven computational models, focusing on how they could come together into a single model for plant development. A model of this nature would allow researchers to test hypotheses in silico and yield accurate predictions about the behavior of a plant under a given set of physical and biochemical constraints. It would also provide a solid foundation toward the establishment of plant engineering, a proposed discipline intended to enable the design and production of plants that exhibit an arbitrarily defined set of features.

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.

Integrating Physiology and Architecture in Models of Fruit Expansion

Architectural properties of a fruit, such as its shape, vascular patterns, and skin morphology, play a significant role in determining the distributions of water, carbohydrates, and nutrients inside the fruit. Understanding the impact of these properties on fruit quality is difficult because they develop over time and are highly dependent on both genetic and environmental controls. We present a 3D functional-structural fruit model that can be used to investigate effects of the principle architectural properties on fruit quality. We use a three step modeling pipeline in the OpenAlea platform: (1) creating a 3D volumetric mesh representation of the internal and external fruit structure, (2) generating a complex network of vasculature that is embedded within this mesh, and (3) integrating aspects of the fruit's function, such as water and dry matter transport, with the fruit's structure. We restrict our approach to the phase where fruit growth is mostly due to cell expansion and the fruit has already differentiated into different tissue types. We show how fruit shape affects vascular patterns and, as a consequence, the distribution of sugar/water in tomato fruit. Furthermore, we show that strong interaction between tomato fruit shape and vessel density induces, independently of size, an important and contrasted gradient of water supply from the pedicel to the blossom end of the fruit. We also demonstrate how skin morphology related to microcracking distribution affects the distribution of water and sugars inside nectarine fruit. Our results show that such a generic model permits detailed studies of various, unexplored architectural features affecting fruit quality development.

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.

Mathematical study on robust tissue pattern formation in growing epididymal tubule

Tissue pattern formation during development is a reproducible morphogenetic process organized by a series of kinetic cellular activities, leading to the building of functional and stable organs. Recent studies focusing on mechanical aspects have revealed physical mechanisms on how the cellular activities contribute to the formation of reproducible tissue patterns; however, the understanding for what factors achieve the reproducibility of such patterning and how it occurs is far from complete. Here, I focus on a tube pattern formation during murine epididymal development, and show that two factors influencing physical design for the patterning, the proliferative zone within the tubule and the viscosity of tissues surrounding to the tubule, control the reproducibility of epididymal tubule pattern, using a mathematical model based on experimental data. Extensive numerical simulation of the simple mathematical model revealed that a spatially localized proliferative zone within the tubule, observed in experiments, results in more reproducible tubule pattern. Moreover, I found that the viscosity of tissues surrounding to the tubule imposes a trade-off regarding pattern reproducibility and spatial accuracy relating to the region where the tubule pattern is formed. This indicates an existence of optimality in material properties of tissues for the robust patterning of epididymal tubule. The results obtained by numerical analysis based on experimental observations provide a general insight on how physical design realizes robust tissue pattern formation.

Localized auxin peaks in concentration-based transport models of the shoot apical meristem

We study the formation of auxin peaks in a generic class of concentration-based auxin transport models, posed on static plant tissues. Using standard asymptotic analysis, we prove that, on bounded domains, auxin peaks are not formed via a Turing instability in the active transport parameter, but via simple corrections to the homogeneous steady state. When the active transport is small, the geometry of the tissue encodes the peaks' amplitude and location: peaks arise where cells have fewer neighbours, that is, at the boundary of the domain. We test our theory and perform numerical bifurcation analysis on two models that are known to generate auxin patterns for biologically plausible parameter values. In the same parameter regimes, we find that realistic tissues are capable of generating a multitude of stationary patterns, with a variable number of auxin peaks, that can be selected by different initial conditions or by quasi-static changes in the active transport parameter. The competition between active transport and production rate determines whether peaks remain localized or cover the entire domain. In particular, changes in the auxin production that are fast with respect to the cellular life cycle affect the auxin peak distribution, switching from localized spots to fully patterned states. We relate the occurrence of localized patterns to a snaking bifurcation structure, which is known to arise in a wide variety of nonlinear media, but has not yet been reported in plant models.

Automatic analysis of the 3-D microstructure of fruit parenchyma tissue using X-ray micro-CT explains differences in aeration

BACKGROUND

3D high-resolution X-ray imaging methods have emerged over the last years for visualising the anatomy of tissue samples without substantial sample preparation. Quantitative analysis of cells and intercellular spaces in these images has, however, been difficult and was largely based on manual image processing. We present here an automated procedure for processing high-resolution X-ray images of parenchyma tissues of apple (Malus × domestica Borkh.) and pear (Pyrus communis L.) as a rapid objective method for characterizing 3D plant tissue anatomy at the level of single cells and intercellular spaces.

RESULTS

We isolated neighboring cells in 3D images of apple and pear cortex tissues, and constructed a virtual sieve to discard incorrectly segmented cell particles or unseparated clumps of cells. Void networks were stripped down until their essential connectivity features remained. Statistical analysis of structural parameters showed significant differences between genotypes in the void and cell networks that relate to differences in aeration properties of the tissues.

CONCLUSIONS

A new model for effective oxygen diffusivity of parenchyma tissue is proposed that not only accounts for the tortuosity of interconnected voids, but also for significant diffusion across cells where the void network is not connected. This will significantly aid interpretation and analysis of future tissue aeration studies. The automated image analysis methodology will also support pheno- and genotyping studies where the 3D tissue anatomy plays a role.

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.