MBI Videos

Workshop 1: Mathematical Modeling of Plant Development

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    Jan Vos
    Models are simplified representations of a system, i.e. a limited part of reality. Structure and properties of specific models are chosen depending on the purpose they serve. These purposes include: summarizing data, assisting in the analysis of experimental data, testing of hypotheses, extrapolation of system behaviour beyond the conditions that were covered experimentally, decision-support in practice.

    Over the last decade we have been involved in the development of functional-structural plant models (FSPM) in the domain of plant production. FSPM treat plants as a proliferation of elementary units explicitly describing 3D plant structure, and include physiological processes (e.g., photosynthesis and/or transport of substances through the plant structure). The main systems of study were tillering in wheat (an annual field crop) and flower cane production in glasshouse-grown cut roses (an intensively manipulated perennial crop system).

    Using wheat as an example, we shall outline the different steps to construct and parameterize an FSPM. Even though, in the first instance, such models remain quite descriptive, there is a range of applications of models that generate a realistic representation of the 3D plant structure (e.g., light distribution in relation to structural properties, remote-sensing research, pest and disease dynamics in relation to structure).

    In the wheat study it was tested whether the cessation of tillering, i.e. the arrest of 'bud break', could be explained from changes in light absorption (quantity) and signal perception (red/far-red ratio). This study was only a first step towards integrating knowledge on how light signals affect the structural development of a collection of plants. Subsequent steps taken include modelling the hormonal network regulating branching as modulated by the red/far-red ratio. Some issues that need to be addressed in order to make further progress will be discussed.

    Rose growers try to manage and manipulate their crops such as to maintain the production of a high number of flower canes of a particular quality (i.e. essentially a mixture of morphological, biometrical and flower yield-related traits) over a prolonged period of time. The number of canes produced depends on the number of buds that break. Understanding bud break is a central issue for FSPM because quiescence or breaking of buds directly affects the overall 3D structure of the crop, as well as the local light climate. Some data on bud break will be presented and the question will be raised on how to design and parameterize a 'decision tree', adequately describing bud fate over time in relation to the position of the bud in the structure and its environment (e.g. degree of illumination). Similar questions pertain to modelling signal transduction and to modelling of source-sink interaction and the associated flows of carbon in the 3D structure.

    Modelling structure and volume poses specific questions like: to which extent are growth in length and volume coupled with dry matter allocation? In other words: what are the temporal dynamics of growth in weight and volume and to what extent are these synchronized, and modified by internal and external factors? Results from detailed measurements on rose internodes show that the link between internode volume and fresh weight is very tight and largely independent from growth temperature, developmental age and phytomer rank.

    Properties of the shoot that grow from a broken bud may be predetermined from bud development. Stresses a plant experiences result in changed structural properties (e.g. leaf size) long after recovery from stress. Apparently, mechanisms exist of 'predetermination' of properties of organs. These are issues that need to be explored further to make FSPM able to cope with changing environments.
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    Ted DeJong
    Perennial deciduous fruit trees are very complex organisms that are governed and influenced by a multitude of factors. Empirical research approaches are generally limited to dealing with a couple factors at a time and integration of the effects of multiple factors affecting tree growth and productivity are generally limited to verbal descriptions and displaying data with two or three dimensional diagrams. The goal of functional-structural plant models (FSPM) is to simulate plant growth in silico on a computer with as much functional realism as is possible. The L-PEACH model is an example of an FSPM that simultaneously incorporates physiological (photosynthesis, transpiration, respiration, phloem transport, xylem water potential) architectural (bud fates and leaf, stem, branch and trunk structure) growth (stem extension and diameter, leaf area and thickness, fruit size) and productivity (fruit sizes and numbers) responses to environmental stimuli (light, temperature, water availability) and management (irrigation, pruning and fruit thinning) practices. The model can also be adapted relatively easily to simulate effects of specific genetic traits of scions and rootstocks on long term tree performance. In the L-PEACH model the plant is expressed in terms of modules that represent plant organs (including stems). An organ is represented as an elementary source/sink for carbohydrates and the whole plant is modeled as a branching network of organs that conduct carbohydrates and water. An analogy to an electrical network is used to calculate the flow and partitioning of carbohydrates between the individual organs in the current model but now that water uptake and transport are incorporated in the model we are attempting to simulate phloem transport directly using principles of hydrostatics.

    Other major contributors to this work: David Da Silva, Romeo Favreau, Inigo Auzmendi, Gerardo Lopez, Evelyne Costes and P. Prusinkiewicz
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    Véronique Letort
    No description available.
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    John Sperry
    Models of plant hydraulic resistance are useful because they provide mechanistically anchored predictions of plant gas exchange and survival in response to environmental stress and ontogeny. Flow resistance in soil and plant xylem can be modeled relatively easily because the processes are largely physical and linked to environment and plant size in ways that can be readily characterized. Three hydraulic models are discussed. A soil-plant-atmosphere-continuum model predicts the sensitivity of plant gas exchange and survival in response to water stress. An allometric model improves on previous "metabolic scaling theory" and predicts water use and productivity as a function of plant size and functional type. In development is a model linking vascular structure to its vulnerability to cavitation. The potential exists to combine all three in a comprehensive bottom-up representation of species-specific responses to environment and ontogeny.
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    Xavier Draye
    Due in part to recent progress in root genetics and genomics, increasing attention is being devoted to root system architecture (RSA) for the improvement of drought tolerance. The focus is generally set on deep roots, expected to improve access to soil water resources during water deficit episodes. Surprisingly, our quantitative understanding of the role of RSA in the uptake of soil water remains extremely limited, which is mainly due to the inherent complexity of the soil-plant continuum. Evidently, there is a need for plant biologists and hydrologists to develop together their understanding of water movement in the soil-plant system. Using recent quantitative models coupling the hydraulic behaviour of soil and roots in an explicit 3D framework, this paper illustrates that the contribution of RSA to root water uptake is hardly separable from the hydraulic properties of the roots and of the soil. We also argue that the traditionnal view that either the plant or the soil should be dominating the patterns of water extraction is not generally appropriate for crops growing with sub-optimal water supply. Hopefully, in silico experiments using this type of model will help explore how water fluxes driven by soil and plant processes affect soil water availability and uptake throughout a growth cycle and will embed the study of RSA within the domains of root hydraulic architecture and sub-surface hydrology.
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    Vince Gutschick
    As part of a selective review of progress in modelling of plant development, I wish to offer a larger context, extending to the evolutionary scale, and mostly to pose a set of questions or challenges to modellers.

    First, I would like to affirm some premises that I believe are widely shared by modellers. Why do we make models?: 1) to predict plant (or stand or ecosystem) performance, though this is relatively rare, because vast knowledge is required unless the universe of possible environments is much restricted; 2) to simplify the design of multi-factorial experiments, reducing their dimension by eliminating those factors or combinations of factors that are least likely to be informative; 3) to develop testable hypotheses of plant responses based on combining our firm knowledge with our shakier estimates; witness the successful erect-leaf hypothesis of the '60's or my own model prediction that reduced leaf chlorophyll allows gains in crop yield, and 4) to synthesize our knowledge, so that we may understand emergent properties, or be inspired to develop new hypotheses.

    We do aim, I hope, for ultimate utility of our models, both in applications and for use by other modellers who may develop a synergy with us. The essence of utility is often simplification. A maxim to keep in mind is that of Einstein, reported by chemist John Ross: "Simplify as much as possible, but no further." We may recall the chaos in modelling the biochemistry of photosyntheses through the '60's and '70's until Farquhar, von Caemmer, and Berry offered their elegant, simple, yet accurate model of C3 photosynthesis in 1980, followed by one for C4 photosynthesis. Necessarily elaborate and computationally-intensive models of radiative transport in plant stands became simplified (initially, over-simplified) as turbid-medium models or two-flux models, with a (final?), accurate simplification of the truly high-order processes via the concept of nested radiosity - alas, more applied by moviemakers than by us modellers. On larger spatial scales, modelling of biogeochemical fluxes must retain some empirical simplifications that need our attention. For estimating evapotranspiration (ET), is fractional cover or fractional PAR interception better than leaf area index when ET is mostly energy-limited... but is it much inferior to using LAI for estimating photosynthetic CO2 flux? Related to simplification but not identical to it is user-friendliness of our algorithms. A major practical use of models is in irrigation management, arising from the simple fact that 36% of all crop biomass is produced on the 16% of the land that is irrigated and thus 3-fold more productive than the rest. Farmers and water managers demand simple interfaces partly because of their lesser sophistication in process understanding but even more so because of the insupportability of data-hungry models and the diversity of other demands on their time. Even considering the community of other modellers, we must examine our models carefully for the tradeoffs between complexity (process-inclusiveness) and ability to parametrize models; the brief comparison by Vogel et al. (1995) is worth reading. We may ask, How do evaluate our models, for both accuracy and usability. We can do sensitivity analyses to see that parameters are most important for accuracy. To assess sensitivity to forcing variables, is it worth the effort to develop adjoint equations numerically? Are ensembles of models useful, as in climate research? Do we have the right statistics to express our sensitivity studies? It is harder yet to analyze what processes are most important - that is, What is a good model structure? Replacement of submodels with empirical (often algebraic) relations is a test, though not an easy one nor one with clear endpoints.

    One key use of models is optimization. I risk an over-emphasis on the topic here, even without expanding the discussion to mathematical methods. Optimization of plant performance is a tool for crop management, for ideotype development, and for exploring the evolutionary physiology and ecology of plants. The first two uses are perhaps obvious and familiar. The third use, in evolutionary studies, presents interesting opportunities. Of course, to assume that plants evolved to optimal function is to risk the Panglossian fallacy of adaptationism, if only because all species necessarily lag in adapting to neighbors' evolution (the Red Queen phenomenon). Moreover, we are not privy to the full range of selection pressures and (phylogenetic) constraints faced by organisms. Nevertheless, evidence of near-optimization is abundant, as in the Ci/Ca setpoint for leaf gas exchange or the distribution of leaf N within a canopy. Assuming optimization is a starting point, first, to fill out our process understanding, and, second, to discover the constraints and competing selection pressures. - e.g., the compromise of a higher specific leaf area than that which optimizes canopy photosynthesis, in order to shade competitors (Gutschick and Wiegel, 1988; Schieving and Poorter, 1999). I now offer several questions about optimization or adaptiveness of key plant traits: 1) Why do plants not have higher N content, particularly the ruderals? It is generally not that costs of metabolizing N exceed photosynthetic benefit. Is it risk of herbivory, lack of metabolic "room," or other constraints?; 2) What is the tradeoff between high photosynthetic rate per mass and strong depletion of seasonal water supplies?; 3) How far have we progressed in understanding trait selection, especially developmental times (anthesis... ) in environments with stochasticity (risks), that are abiotic, such as frosts, or biotic, such as herbivory and disease?; 4) What would a comprehensive look like for explaining the set-points of major plant traits - physiological, such as Ci/Ca, developmental, such as allocation (leaf/shoot, hydraulic conductivity attributes, phenology etc.), and ecological, such as disease susceptibility?; 5) What combination of trait complexes and environmental variation (space and time) explain coexistence of species, given that most models of extinction (under competition or other pressures) generate a winner-take-all result. Multiple resource competition and spatial patchiness only expand the number of species modestly, and alternating selection only delays the winnowing of species.

    The mechanistic description of development demands that we understand the proximate signals that trigger developmental events - direct environmental cues (here, much is known) and integrated cues such as C and N reserves (here, less is known). The signals for root development - elongation rates, branching, gravitropic patterns - that arise from water and nutrient status (plant and soil) are probably the most poorly known, if only because of the difficulties of studying roots in soil. We need process models for these signals from experiment and, conversely, we need to help interpret experimental studies. Among the pressing questions economically and intellectually are, What causes alternate bearing in tree crops?, and, How does elevated CO2 cause shifts in N status of diverse tissues in a species-specific manner?

    For plant breeders, evolutionary biologists, and evolutionary ecologists, the Holy Grail is understanding the links from genes and genomes to phenotype and performance. Advances have occurred in quantitative genetics and the study of regulatory networks. Challenges remain in understanding epigenetic effects and, at the population level, the constraints to genetic adaptation posed by population genetic structure, sensu Lande. Expanding upon the genetic constraints, we may profitably pay attention to the degree to which populations have lost adaptive genetic variation for growth at elevated CO2 over the past 20 My.

    Selected references:

    * V. P.Gutschick. 1984. Photosynthesis model for C3 leaves incorporating CO2 transport, radiation propagation, and biochemistry.. 2. Ecological and agricultural utility. Photosynthetica 18: 569-595.
    * W. T. Pettigrew, J. D. Hesketh, D. B. Peters, et al. 1989. Characterization Of canopy photosynthesis of chlorophyll-deficient soybean isolines. Crop Sci. 29: 1025-1029.
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    Hamlyn (Lyn) Jones
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    Jan Kozlowski
    There are annual and perennial plants; perennials may reproduce once in life (semelparity, monocarpic perennials) or repeatedly (iteroparity). Perennial herbs lose virtually all vegetative parts but storage organs before winter, whereas shrubs and trees retain large part or even most of vegetative mass. Theory of optimal resource allocation helps to predict important features of the two strategies. Annuals must chose the best time during a year for the switch from vegetative to generative growth and "decide" about the proportion of seeds germinating in the next season and amounting to a seed bank. Semelparous perrenials must choose the year for flowering. Iteroporous perennials must choose when to allocate assimilated resources to growth and when to reproduction, and they must optimize investments to storage. If perennials reproduce vegetatively, they must also optimize division between generative and vegetative reproduction. Grazers are everywhere, but defense is costly: plants must optimize the intensity of protection. Understanding these complex problems an
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    Ilya Ioslovich
    A dynamical biological system containing a vegetable crop and control tools for protected and intensive cultivation is considered. To optimize an economical criterion along the growing season, model based control must be designed. A special simplified biological model was developed for the purpose of determining the control inputs. This model uses the main biological properties of plant growth: the three stage process of growth (vegetative, mixed, and reproductive) resulted from the adaptation to the natural selection ontogeny, and the maintenance of the balanced sink/source ratio in relation to the growth and development processes. The appropriate optimal control problem was investigated by means of the sufficient conditions of optimality and it was found that, independently of weather inputs, the time-invariant parameter, 'optimal control intensity', can be determined analytically. A previously calibrated generic comprehensive multidimensional model of the tomato plant was used as a generator of data for simulation.
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    Evelyne Costes
    Fruit trees present developmental characteristics which are similar to other perennials, such as the existence of ontogenetic gradients and dependencies between consecutive growth. However, because of their agronomical use, fruit tree species also raise specific issues. For instance, the practice of grafting on dwarfing rootstock considerably reduces the juvenile period during which trees do not flower. During the mature phase, flower formation is often instable from one year to the next, leading to an irregular fruit bearing. This represents a major disability for fruit production that is usually prevented through a large number of manipulations such as grafting, pruning, chemical applications for fruit thinning, water and nitrogen supply. Presently, fruit production is facing a number of new challenges such as climatic changes or the necessary reduction of incomes in orchards, which may endanger its performance. A deep renewal of practices has to be engaged, involving a large set of competences. Indeed, numerous factors (genetic, climatic, abiotic and biotic stresses) interacting in a complex network must be optimised to find innovative answers to the current economical and environmental challenges. The role of modelling will thus be crucial in the formalisation of this complexity and the exploration of innovative scenarios. In this talk, we will attempt to highlight key ideas and concepts that underline our research and that aim to face the different levels of complexity. We will also submit our current open questions for discussion.

    Our approach consists in the exploration of plant structure from cellular to whole plant scale, taking into account one or several scales depending on the target question. For the exploration of tree perennial development over several years and dependencies between consecutive growth, macroscopic scales of description, such as annual shoots or growth units, are usually considered. Mixed probabilistic/structural models have been built in which a decomposition approach was applied to separate the growth components due to ontogeny or environmental factors in the measured plant development. Their application in apple tree has led us to identity unexpected patterns, in which two successive phases corresponding to different patterns of alternations between flowering and vegetative growth units were identified. The current challenge concerns the analysis of the influence of genetic and environmental factors on the development of perennial plant structures over several years.

    Such mixed probabilistic/structural models have also been combined with different physical and eco-physiological models (carbon acquisition, transpiration, carbon and water transport, effect of gravity, etc.) in fruit tree simulation systems. However, the integration of different sub-models, either mechanistic or stochastic, into a global simulation model has generated new issues due to the high complexity of the whole system which needs to be carefully addressed. Moreover, due to the large number of potential sources of variation (parameters, environmental variables, sub-model choices, ...), exploring the behaviour of such models is not possible analytically, and in silico experiments combined with sensitivity analyses are required. However, the strategy and intermediate objectives that must be defined to progress towards the build of efficient and integrated perennial plant models at macroscopic scales remain open questions.

    At the microscopic scale, the understanding of the relative contribution of mechanisms at different scales, from molecules, cells and organs to the construction of individual plant structures, is a major challenge for biologists. Following pioneer studies on model plants, projects have recently emerged for agronomic species. After having demonstrated the genetic control of architectural traits in segregating populations of apple, we have engaged in the description of the cellular patterning of apple tree organs, internodes and leaves, in different allelic combinations. The relative contribution of the number of cells and cell size to the final internode shape was shown to be genotype-depend and affected by a period of soil water deficit. Similarly, the relative contribution of the number of cells and cell volumes in apple leaf histogenesis is currently under study. The large databases generated by these studies will be presented and are likely to lead to new 3D modelling developments with the objective to link cellular to organ scale.
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    Thierry Simonneau
    Modeling plant growth and development underwent considerable development with strong incentives from various consortia. It emerged as an efficient tool in ecology and genetics to face new challenges raised by competition for resources and to benefit from breakthroughs in biotechnology. In this presentation, we propose a classification of approaches used in modeling plants based on our experience at the Laboratoire d'Ecophysiologie des Plantes sous Stress Environnementaux (Montpellier, France). We present 5 types of models and discuss how they participate in a general workflow aimed at designing plant ideotypes for future climates.

    Type 1 are frameworks for spatiotemporal analysis of growth and development. They consist in quantitative description of when and where growth occurs in a specific organ, or how growth is synchronized between organs at whole plant level or, more generally, how development occurs. They are developed almost systematically when a novel species or organ are to be studied in the lab, as they are critical to set up experimental designs (what to measure, when, ...) or to reason the sampling of tissues for omics studies (which part of the organ should be taken, when).

    Type 2 are data-driven (statistical) representations of the responses to the environment and of determinant of growth. Often based on specialized statistical methods (multivariate analysis, structural equation models, mixed models,...), they allow us to formalize responses, reveal linkages and hierarchies between processes or multiple factors affecting growth, and demonstrate causalities between them. They are used as such to produce synthetic representations of growth processes or decipher between genetic and environmental effects. They are also used as bases for building other types of models.

    Type 3 are empirical models of response to environmental factors specially designed for parameterizations in high throughput phenotyping. This entails parsimony in the parameterization (hence in input variables), high heritability of parameters (traits) and compatibility with high throughput measurements. Such models have proved to be very efficient in genetic studies, for characterizing allelic diversity or finding QTLs. They also are ready to use and genetically well parameterized components for more integrative models.

    Type 4 are computer-assisted simulation models, used to make predictions a priori. They are based on hypothetical frames of integration of biological processes within the plant, and on physical models for the interactions with soil and atmospheric compartments. They allow simulation of complex system (crop models). They are used in the lab to simulate the behavior of plants in different climatic scenarios and thus to assess the relevance of a given set of traits for a given location. They are also produced to help reasoning agricultural practices, and especially irrigation.

    Type 5 are computer simulation models, used to reconstruct or interpret experiments a posteriori. Such models are designed to be fitted to experimental data, either to compute hidden variables (like sink strengths, radiation interception and efficiency), interpolate variables of interest from fragmental inputs in time or space (e.g. LAI time course) or interpret a macroscopic response by extracting meaningful parameters (e.g. hydraulic characteristics from difference in growth kinetics).

    During this presentation, new challenges are proposed to modelers in each type of models. They consist in bottlenecks which currently hamper plant scientists from achieving multiple goals in biology, ecology, agronomy or breeding.

    Work done in collaboration with Christian Fournier, Francois Tardieu, Denis Vile, Angelique Christophe, Eric Lebon, Christine Granier and Bertrand Muller.
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    Michael Chelle
    Fungal diseases are a major concern for crop production. Up to now crop protection has mainly relied on fungicide sprays and host genetic resistance. However, their intensive use boosts the adaptation of fungal populations, causing the decrease of pathogen sensivity to fungicides and the breakdown of host resistance. Therefore, novel crop management techniques have to be developed to be used complementary to fungicides and resistant cultivars in sustainable farming. To achieve that, new levers have to be identified.

    An epidemic causes crop damages if several steps act in sequence:

    * Initial inoculum survives on soil and stubble during winter, or is spread by wind from remote survival locations, and produces spores that are spread upwards.
    * A leaf is reached by pathogen spores
    * This leaf provides a physical and physiological environment that favors epidemiological processes (infection, penetration, lesion growth, sporulation).
    * The lesions alter the plant functioning, whose consequences are a quantitative and qualitative decrease of yield. Knowing how a plant population influences these steps enables the definition of three types of plant self-defense against airborne fungal diseases: resistance, tolerance, and escape, resistance being subdivided in genetic (the usual one), phylloclimatic and physiological resistance. Regarding the ability to characterize and model the feedback action of disease on canopy architecture as well as on biomass and grain production, the function-structure plant modeling (FSPM) approach seems well suited. FSPM is the combination of a 3D description of the canopy morphogenesis with a functional plant model, where the basic unit is the individual organ.

    Introducing fungi-leaf interactions in such model also requres coupling it with phylloclimate models. Phylloclimate corresponds to the physical environment actually perceived by each individual aerial organ of a plant population, and is described by physical variables such as spectral irradiance, temperature, on-leaf water and features of around-organ air (wind speed, temperature, humidity, etc.).Characterizing phylloclimate variables, using experimental work or modeling, raises many questions such as the choice of suitable space- and time-scale as well as the ability to individualize plant organs within a canopy. Recent trends and challenging questions in phylloclimate research are discussed, as well as their possible applications in plant epidemiology. Finally, a brief review of emerging needs in terms of collaborations with mathematicians and computer scientists has been adressed.

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