Towards aspect-oriented functional--structural plant modelling

A multiscale mathematical model describing the growth and development of bambara groundnut

A principal objective in agriculture is to maximise food production; this is particularly relevant with the added demands of an ever increasing population, coupled with the unpredictability that climate change brings. Further improvements in productivity can only be achieved with an increased understanding of plant and crop processes. In this respect, mathematical modelling of plants and crops plays an important role. In this paper we present a two-scale mathematical model of crop yield that accounts for plant growth and canopy interactions. A system of nonlinear ordinary differential equations (ODEs) is formulated to describe the growth of each individual plant, where equations are coupled via a term that describes plant competition via canopy-canopy interactions. A crop of greenhouse plants is then modelled via an agent based modelling approach in which the growth of each plant is described via our system of ODEs. The model is formulated for the African drought tolerant legume bambara groundnut (Vigna subterranea), which is currently being investigated as a food source in light of climate change and food insecurity challenges. Our model allows us to account for plant diversity and also investigate the effect of individual plant traits (e.g. plant canopy size and planting distance) on the yield of the overall crop. Informed with greenhouse data, model results show that plant positioning relative to other plants has a large impact on individual plant yield. Variation in physiological plant traits from genetic diversity and the environmental effects lead to experimentally observed variations in crop yield. These traits include plant height, plant carrying capacity, leaf accumulation rate and canopy spread. Of these traits plant height and ground cover growth rates are found to have the greatest impact on crop yield. We also consider a range of different planting arrangements (uniform grid, staggered grid, circular rings and random allocation) and find that the staggered grid leads to the greatest crop yield (6% more compared to uniform grid). Whilst formulated specifically for bambara groundnut, the generic formulation of our model means that with changes to certain parameter's, it may be extended to other crop species that form a canopy.

Fruit tree crop models: an update

Functional structural plant models of tree crops are useful tools that were introduced more than two decades ago. They can represent the growth and development of a plant through the in silico simulation of the 3D architecture in connection with physiological processes. In tree crops, physiological processes such as photosynthesis, carbon allocation and growth are usually integrated into these models, although other functions such as water and nutrient uptake are often disregarded. The implementation of the 3D architecture involves different techniques such as L-system frameworks, pipe model concepts and Markovian models to simulate branching processes, bud fates and elongation of stems based on the production of metamers. The simulation of root architecture is still a challenge for researchers due to a limited amount of information and experimental issues in dealing with roots, because root development is not based on the production of metamers. This review aims to focus on functional-structural models of fruit tree crops, highlighting their physiological components. The potential and limits of these tools are reviewed to point out the topics that still need more attention.

Computational Models of Auxin-Driven Patterning in Shoots

Auxin regulates many aspects of plant development and behavior, including the initiation of new outgrowth, patterning of vascular systems, control of branching, and responses to the environment. Computational models have complemented experimental studies of these processes. We review these models from two perspectives. First, we consider cellular and tissue-level models of interaction between auxin and its transporters in shoots. These models form a coherent body of results exploring different hypotheses pertinent to the patterning of new outgrowth and vascular strands. Second, we consider models operating at the level of plant organs and entire plants. We highlight techniques used to reduce the complexity of these models, which provide a path to capturing the essence of studied phenomena while running simulations efficiently.

Functional-Structural Plant Models Mission in Advancing Crop Science: Opportunities and Prospects

Functional-structural plant models (FSPMs) have been evolving for over 2 decades and their future development, to some extent, depends on the value of potential applications in crop science. To date, stabilizing crop production by identifying valuable traits for novel cultivars adapted to adverse environments is topical in crop science. Thus, this study will examine how FSPMs are able to address new challenges in crop science for sustainable crop production. FSPMs developed to simulate organogenesis, morphogenesis, and physiological activities under various environments and are amenable to downscale to the tissue, cellular, and molecular level or upscale to the whole plant and ecological level. In a modeling framework with independent and interactive modules, advanced algorithms provide morphophysiological details at various scales. FSPMs are shown to be able to: (i) provide crop ideotypes efficiently for optimizing the resource distribution and use for greater productivity and less disease risk, (ii) guide molecular design breeding via linking molecular basis to plant phenotypes as well as enrich crop models with an additional architectural dimension to assist breeding, and (iii) interact with plant phenotyping for molecular breeding in embracing three-dimensional (3D) architectural traits. This study illustrates that FSPMs have great prospects in speeding up precision breeding for specific environments due to the capacity for guiding and integrating ideotypes, phenotyping, molecular design, and linking molecular basis to target phenotypes. Consequently, the promising great applications of FSPMs in crop science will, in turn, accelerate their evolution and vice versa.

Investigating tree and fruit growth through functional-structural modelling: implications of carbon autonomy at different scales

BACKGROUND AND AIMS

Many experimental studies assume that some topological units are autonomous with regard to carbon because it is convenient. Some plant models simulate carbon allocation, employing complex approaches that require calibration and fitted parameters. For whole-tree canopy simulations, simpler carbon allocation models can provide useful insights.

METHODS

We propose a new method for simulating carbon allocation in the whole tree canopy considering various scales of carbon autonomy, i.e. branchlets, branches, limbs, and no autonomy. This method was implemented in a functional-structural plant model of growth of individual organs for studying macadamia tree growth during one growing season.

KEY RESULTS

This model allows the simulation of various scales of carbon autonomy in a simple tree canopy, showing organ within-tree variability according to the scale of autonomy. Using a real tree canopy, we observed differences in growth variability within the tree and in tree growth, with several scales of carbon autonomy. The simulations that assumed autonomy at branch scale, i.e. 2-year-old wood, showed the most realistic results.

CONCLUSIONS

Simulations using this model were employed to investigate and explain aspects of differences in carbon autonomy between trees, organ growth variability, competition between shoot and fruit growth, and time of autonomy.

An overview of agent-based models in plant biology and ecology

Agent-based modelling (ABM) has become an established methodology in many areas of biology, ranging from the cellular to the ecological population and community levels. In plant science, two different scales have predominated in their use of ABM. One is the scale of populations and communities, through the modelling of collections of agents representing individual plants, interacting with each other and with the environment. The other is the scale of the individual plant, through the modelling, by functional-structural plant models (FSPMs), of agents representing plant building blocks, or metamers, to describe the development of plant architecture and functions within individual plants. The purpose of this review is to show key results and parallels in ABM for growth, mortality, carbon allocation, competition and reproduction across the scales from the plant organ to populations and communities on a range of spatial scales to the whole landscape. Several areas of application of ABMs are reviewed, showing that some issues are addressed by both population-level ABMs and FSPMs. Continued increase in the relevance of ABM to environmental science and management will be helped by greater integration of ABMs across these two scales.

Parameter estimation for functional-structural plant models when data are scarce: using multiple patterns for rejecting unsuitable parameter sets

BACKGROUND AND AIMS

Functional-structural plant (FSP) models provide insights into the complex interactions between plant architecture and underlying developmental mechanisms. However, parameter estimation of FSP models remains challenging. We therefore used pattern-oriented modelling (POM) to test whether parameterization of FSP models can be made more efficient, systematic and powerful. With POM, a set of weak patterns is used to determine uncertain parameter values, instead of measuring them in experiments or observations, which often is infeasible.

METHODS

We used an existing FSP model of avocado (Persea americana 'Hass') and tested whether POM parameterization would converge to an existing manual parameterization. The model was run for 10 000 parameter sets and model outputs were compared with verification patterns. Each verification pattern served as a filter for rejecting unrealistic parameter sets. The model was then validated by running it with the surviving parameter sets that passed all filters and then comparing their pooled model outputs with additional validation patterns that were not used for parameterization.

KEY RESULTS

POM calibration led to 22 surviving parameter sets. Within these sets, most individual parameters varied over a large range. One of the resulting sets was similar to the manually parameterized set. Using the entire suite of surviving parameter sets, the model successfully predicted all validation patterns. However, two of the surviving parameter sets could not make the model predict all validation patterns.

CONCLUSIONS

Our findings suggest strong interactions among model parameters and their corresponding processes, respectively. Using all surviving parameter sets takes these interactions into account fully, thereby improving model performance regarding validation and model output uncertainty. We conclude that POM calibration allows FSP models to be developed in a timely manner without having to rely on field or laboratory experiments, or on cumbersome manual parameterization. POM also increases the predictive power of FSP models.

Mechanistic modelling of coupled phloem/xylem transport for L-systems: combining analytical and computational methods

Background and Aims

Transport of carbohydrates and water are essential aspects of plant function. The aim of this study was to develop and test the methods for mechanistic modelling of quasi-stationary coupled phloem/xylem transport in the context of functional-structural plant modelling.

Methods

The novelty of this approach is in combining analytical and computational methods. The plant structure is modelled at a metamer level with the internodes represented by conduit elements and the lateral organs represented by sources and sinks. Transport equations are solved analytically for each internode and then the solutions are adjusted and 'sewn' together using an iterative computational procedure taking into account concentration-dependent sinks and sources. The model is implemented in L-studio and uses the aspect-oriented modelling approach for phloem/xylem coupling.

Key Results

To our knowledge, this is the first transport model that provides continuous distributions of the system variables in a complex developing structure. The model takes into account non-linear dependence of phloem resistance and osmotic potential on the local carbohydrate concentration. The model solutions show excellent agreement with the existing results of other analytical and numerical models. These comparisons confirm the validity of the approximations made in the model. Combining analytical and computational methods made it possible to take into account continuous sink/source distribution within internodes without much increase in the complexity of the computational procedure, because the necessary changes in the model were mostly in the analytical part. The results emphasize sensitivity of phloem flux and lateral xylem flux to the presence of distributed sinks and sources along the transport system.

Conclusions

The presented approach provides a new insight into mechanistic modelling of phloem/xylem transport in growing plants. It will be useful for both fine-scale modelling of carbohydrate dynamics and for creating simpler models at a growth unit level.

Pattern-oriented modelling as a novel way to verify and validate functional-structural plant models: a demonstration with the annual growth module of avocado

Background and Aims

Functional-structural plant (FSP) models have been widely used to understand the complex interactions between plant architecture and underlying developmental mechanisms. However, to obtain evidence that a model captures these mechanisms correctly, a clear distinction must be made between model outputs used for calibration and thus verification, and outputs used for validation. In pattern-oriented modelling (POM), multiple verification patterns are used as filters for rejecting unrealistic model structures and parameter combinations, while a second, independent set of patterns is used for validation.

Methods

To test the potential of POM for FSP modelling, a model of avocado (Persea americana 'Hass') was developed. The model of shoot growth is based on a conceptual model, the annual growth module (AGM), and simulates photosynthesis and adaptive carbon allocation at the organ level. The model was first calibrated using a set of observed patterns from a published article. Then, for validation, model predictions were compared with a different set of empirical patterns from various field studies that were not used for calibration.

Key Results

After calibration, our model simultaneously reproduced multiple observed architectural patterns. The model then successfully predicted, without further calibration, the validation patterns. The model supports the hypothesis that carbon allocation can be modelled as being dependent on current organ biomass and sink strength of each organ type, and also predicted the observed developmental timing of the leaf sink-source transition stage.

Conclusions

These findings suggest that POM can help to improve the 'structural realism' of FSP models, i.e. the likelihood that a model reproduces observed patterns for the right reasons. Structural realism increases predictive power so that the response of an AGM to changing environmental conditions can be predicted. Accordingly, our FSP model provides a better but still parsimonious understanding of the mechanisms underlying known patterns of AGM growth.

Simulation of carbon allocation and organ growth variability in apple tree by connecting architectural and source-sink models

BACKGROUND AND AIMS

Plant growth depends on carbon availability and allocation among organs. QualiTree has been designed to simulate carbon allocation and partitioning in the peach tree (Prunus persica), whereas MappleT is dedicated to the simulation of apple tree (Malus × domestica) architecture. The objective of this study was to couple both models and adapt QualiTree to apple trees to simulate organ growth traits and their within-tree variability.

METHODS

MappleT was used to generate architectures corresponding to the 'Fuji' cultivar, accounting for the variability within and among individuals. These architectures were input into QualiTree to simulate shoot and fruit growth during a growth cycle. We modified QualiTree to account for the observed shoot polymorphism in apple trees, i.e. different classes (long, medium and short) that were characterized by different growth function parameters. Model outputs were compared with observed 3D tree geometries, considering shoot and final fruit size and growth dynamics.

KEY RESULTS

The modelling approach connecting MappleT and QualiTree was appropriate to the simulation of growth and architectural characteristics at the tree scale (plant leaf area, shoot number and types, fruit weight at harvest). At the shoot scale, mean fruit weight and its variability within trees was accurately simulated, whereas the model tended to overestimate individual shoot leaf area and underestimate its variability for each shoot type. Varying the parameter related to the intensity of carbon exchange between shoots revealed that behaviour intermediate between shoot autonomy and a common assimilate pool was required to properly simulate within-tree fruit growth variability. Moreover, the model correctly dealt with the crop load effect on organ growth.

CONCLUSIONS

This study provides understanding of the integration of shoot ontogenetic properties, carbon supply and transport between entities for simulating organ growth in trees. Further improvements regarding the integration of retroaction loops between carbon allocation and the resulting plant architecture are expected to allow multi-year simulations.

A modelling framework to simulate foliar fungal epidemics using functional-structural plant models

BACKGROUND AND AIMS

Sustainable agriculture requires the identification of new, environmentally responsible strategies of crop protection. Modelling of pathosystems can allow a better understanding of the major interactions inside these dynamic systems and may lead to innovative protection strategies. In particular, functional-structural plant models (FSPMs) have been identified as a means to optimize the use of architecture-related traits. A current limitation lies in the inherent complexity of this type of modelling, and thus the purpose of this paper is to provide a framework to both extend and simplify the modelling of pathosystems using FSPMs.

METHODS

Different entities and interactions occurring in pathosystems were formalized in a conceptual model. A framework based on these concepts was then implemented within the open-source OpenAlea modelling platform, using the platform's general strategy of modelling plant-environment interactions and extending it to handle plant interactions with pathogens. New developments include a generic data structure for representing lesions and dispersal units, and a series of generic protocols to communicate with objects representing the canopy and its microenvironment in the OpenAlea platform. Another development is the addition of a library of elementary models involved in pathosystem modelling. Several plant and physical models are already available in OpenAlea and can be combined in models of pathosystems using this framework approach.

KEY RESULTS

Two contrasting pathosystems are implemented using the framework and illustrate its generic utility. Simulations demonstrate the framework's ability to simulate multiscaled interactions within pathosystems, and also show that models are modular components within the framework and can be extended. This is illustrated by testing the impact of canopy architectural traits on fungal dispersal.

CONCLUSIONS

This study provides a framework for modelling a large number of pathosystems using FSPMs. This structure can accommodate both previously developed models for individual aspects of pathosystems and new ones. Complex models are deconstructed into separate 'knowledge sources' originating from different specialist areas of expertise and these can be shared and reassembled into multidisciplinary models. The framework thus provides a beneficial tool for a potential diverse and dynamic research community.

Testing the ontogenetic base for the transient model of inflorescence development

BACKGROUNDS AND AIMS

Current research in plant science has concentrated on revealing ontogenetic processes of key attributes in plant evolution. One recently discussed model is the 'transient model' successful in explaining some types of inflorescence architectures based on two main principles: the decline of the so called 'vegetativeness' (veg) factor and the transient nature of apical meristems in developing inflorescences. This study examines whether both principles find a concrete ontogenetic correlate in inflorescence development.

METHODS

To test the ontogenetic base of veg decline and the transient character of apical meristems the ontogeny of meristematic size in developing inflorescences was investigated under scanning electron microscopy. Early and late inflorescence meristems were measured and compared during inflorescence development in 13 eudicot species from 11 families.

KEY RESULTS

The initial size of the inflorescence meristem in closed inflorescences correlates with the number of nodes in the mature inflorescence. Conjunct compound inflorescences (panicles) show a constant decrease of meristematic size from early to late inflorescence meristems, while disjunct compound inflorescences present an enlargement by merging from early inflorescence meristems to late inflorescence meristems, implying a qualitative change of the apical meristems during ontogeny.

CONCLUSIONS

Partial confirmation was found for the transient model for inflorescence architecture in the ontogeny: the initial size of the apical meristem in closed inflorescences is consistent with the postulated veg decline mechanism regulating the size of the inflorescence. However, the observed biphasic kinetics of the development of the apical meristem in compound racemes offers the primary explanation for their disjunct morphology, contrary to the putative exclusive transient mechanism in lateral axes as expected by the model.

Elements of a dynamic systems model of canopy photosynthesis

Improving photosynthesis throughout the full canopy rather than photosynthesis of only the top leaves of the canopy is central to improving crop yields. Many canopy photosynthesis models have been developed from physiological and ecological perspectives, however most do not consider heterogeneities of microclimatic factors inside a canopy, canopy dynamics and associated energetics, or competition among different plants, and most models lack a direct linkage to molecular processes. Here we described the rationale, elements, and approaches necessary to build a dynamic systems model of canopy photosynthesis. A systems model should integrate metabolic processes including photosynthesis, respiration, nitrogen metabolism, resource re-mobilization and photosynthate partitioning with canopy level light, CO(2), water vapor distributions and heat exchange processes. In so doing a systems-based canopy photosynthesis model will enable studies of molecular ecology and dramatically improve our insight into engineering crops for improved canopy photosynthetic CO(2) uptake, resource use efficiencies and yields.

L-py: an L-system simulation framework for modeling plant architecture development based on a dynamic language

The study of plant development requires increasingly powerful modeling tools to help understand and simulate the growth and functioning of plants. In the last decade, the formalism of L-systems has emerged as a major paradigm for modeling plant development. Previous implementations of this formalism were made based on static languages, i.e., languages that require explicit definition of variable types before using them. These languages are often efficient but involve quite a lot of syntactic overhead, thus restricting the flexibility of use for modelers. In this work, we present an adaptation of L-systems to the Python language, a popular and powerful open-license dynamic language. We show that the use of dynamic language properties makes it possible to enhance the development of plant growth models: (i) by keeping a simple syntax while allowing for high-level programming constructs, (ii) by making code execution easy and avoiding compilation overhead, (iii) by allowing a high-level of model reusability and the building of complex modular models, and (iv) by providing powerful solutions to integrate MTG data-structures (that are a common way to represent plants at several scales) into L-systems and thus enabling to use a wide spectrum of computer tools based on MTGs developed for plant architecture. We then illustrate the use of L-Py in real applications to build complex models or to teach plant modeling in the classroom.

Computational models of plant development and form

The use of computational techniques increasingly permeates developmental biology, from the acquisition, processing and analysis of experimental data to the construction of models of organisms. Specifically, models help to untangle the non-intuitive relations between local morphogenetic processes and global patterns and forms. We survey the modeling techniques and selected models that are designed to elucidate plant development in mechanistic terms, with an emphasis on: the history of mathematical and computational approaches to developmental plant biology; the key objectives and methodological aspects of model construction; the diverse mathematical and computational methods related to plant modeling; and the essence of two classes of models, which approach plant morphogenesis from the geometric and molecular perspectives. In the geometric domain, we review models of cell division patterns, phyllotaxis, the form and vascular patterns of leaves, and branching patterns. In the molecular-level domain, we focus on the currently most extensively developed theme: the role of auxin in plant morphogenesis. The review is addressed to both biologists and computational modelers.

Using functional–structural plant models to study, understand and integrate plant development and ecophysiology

Functional–structural plant models (FSPMs) explore and integrate relationships between a plant’s structure and processes that underlie its growth and development. In recent years, the range of topics being addressed by scientists interested in functional–structural plant modelling has expanded greatly. FSPM techniques are now being used to dynamically simulate growth and development occurring at the microscopic scale involving cell division in plant meristems to the macroscopic scales of whole plants and plant communities. The plant types studied also cover a broad spectrum from algae to trees. FSPM is highly interdisciplinary and involves scientists with backgrounds in plant physiology, plant anatomy, plant morphology, mathematics, computer science, cellular biology, ecology and agronomy. This special issue of Annals of Botany features selected papers that provide examples of comprehensive functional–structural models, models of key processes such as partitioning of resources, software for modelling plants and plant environments, data acquisition and processing techniques and applications of functional–structural plant models for agronomic purposes.