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.

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.

Averting robo-bees: why free-flying robotic bees are a bad idea.

Food security and the sustainability of native ecosystems depends on plant-insect interactions in countless ways. Recently reported rapid and immense declines in insect numbers due to climate change, the use of pesticides and herbicides, the introduction of agricultural monocultures, and the destruction of insect native habitat, are all potential contributors to this grave situation. Some researchers are working towards a future where natural insect pollinators might be replaced with free-flying robotic bees, an ecologically problematic proposal. We argue instead that creating environments that are friendly to bees and exploring the use of other species for pollination and bio-control, particularly in non-European countries, are more ecologically sound approaches. The computer simulation of insect-plant interactions is a far more measured application of technology that may assist in managing, or averting, 'Insect Armageddon' from both practical and ethical viewpoints.

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.

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.

A Generic Individual-Based Spatially Explicit Model as a Novel Tool for Investigating Insect-Plant Interactions: A Case Study of the Behavioural Ecology of Frugivorous Tephritidae.

Computational modelling of mechanisms underlying processes in the real world can be of great value in understanding complex biological behaviours. Uptake in general biology and ecology has been rapid. However, it often requires specific data sets that are overly costly in time and resources to collect. The aim of the current study was to test whether a generic behavioural ecology model constructed using published data could give realistic outputs for individual species. An individual-based model was developed using the Pattern-Oriented Modelling (POM) strategy and protocol, based on behavioural rules associated with insect movement choices. Frugivorous Tephritidae (fruit flies) were chosen because of economic significance in global agriculture and the multiple published data sets available for a range of species. The Queensland fruit fly (Qfly), Bactrocera tryoni, was identified as a suitable individual species for testing. Plant canopies with modified architecture were used to run predictive simulations. A field study was then conducted to validate our model predictions on how plant architecture affects fruit flies' behaviours. Characteristics of plant architecture such as different shapes, e.g., closed-canopy and vase-shaped, affected fly movement patterns and time spent on host fruit. The number of visits to host fruit also differed between the edge and centre in closed-canopy plants. Compared to plant architecture, host fruit has less contribution to effects on flies' movement patterns. The results from this model, combined with our field study and published empirical data suggest that placing fly traps in the upper canopy at the edge should work best. Such a modelling approach allows rapid testing of ideas about organismal interactions with environmental substrates in silico rather than in vivo, to generate new perspectives. Using published data provides a saving in time and resources. Adjustments for specific questions can be achieved by refinement of parameters based on targeted experiments.

Surface reconstruction of wheat leaf morphology from three-dimensional scanned data.

Realistic virtual models of leaf surfaces are important for several applications in the plant sciences, such as modelling agrichemical spray droplet movement and spreading on the surface. In this context, the virtual surfaces are required to be smooth enough to facilitate the use of the mathematical equations that govern the motion of the droplet. Although an effective approach is to apply discrete smoothing D2-spline algorithms to reconstruct the leaf surfaces from three-dimensional scanned data, difficulties arise when dealing with wheat (Triticum aestivum L.) leaves, which tend to twist and bend. To overcome this topological difficulty, we develop a parameterisation technique that rotates and translates the original data, allowing the surface to be fitted using the discrete smoothing D2-spline methods in the new parameter space. Our algorithm uses finite element methods to represent the surface as a linear combination of compactly supported shape functions. Numerical results confirm that the parameterisation, along with the use of discrete smoothing D2-spline techniques, produces realistic virtual representations of wheat leaves.

Generalist insects behave in a jasmonate-dependent manner on their host plants, leaving induced areas quickly and staying longer on distant parts.

Plants are sessile, so have evolved sensitive ways to detect attacking herbivores and sophisticated strategies to effectively defend themselves. Insect herbivory induces synthesis of the phytohormone jasmonic acid which activates downstream metabolic pathways for various chemical defences such as toxins and digestion inhibitors. Insects are also sophisticated animals, and many have coevolved physiological adaptations that negate this induced plant defence. Insect behaviour has rarely been studied in the context of induced plant defence, although behavioural adaptation to induced plant chemistry may allow insects to bypass the host's defence system. By visualizing jasmonate-responsive gene expression within whole plants, we uncovered spatial and temporal limits to the systemic spread of plant chemical defence following herbivory. By carefully tracking insect movement, we found induced changes in plant chemistry were detected by generalist Helicoverpa armigera insects which then modified their behaviour in response, moving away from induced parts and staying longer on uninduced parts of the same plant. This study reveals that there are plant-wide signals rapidly generated following herbivory that allow insects to detect the heterogeneity of plant chemical defences. Some insects use these signals to move around the plant, avoiding localized sites of induction and staying ahead of induced toxic metabolites.

Models of long-distance transport: how is carrier-dependent auxin transport regulated in the stem?

• This paper presents two models of carrier-dependent long-distance auxin transport in stems that represent the process at different scales. • A simple compartment model using a single constant auxin transfer rate produced similar data to those observed in biological experiments. The effects of different underlying biological assumptions were tested in a more detailed model representing cellular and intracellular processes that enabled discussion of different patterns of carrier-dependent auxin transport and signalling. • The output that best fits the biological data is produced by a model where polar auxin transport is not limited by the number of transporters/carriers and hence supports biological data showing that stems have considerable excess capacity to transport auxin. • All results support the conclusion that auxin depletion following apical decapitation in pea (Pisum sativum) occurs too slowly to be the initial cause of bud outgrowth. Consequently, changes in auxin content in the main stem and changes in polar auxin transport/carrier abundance in the main stem are not correlated with axillary bud outgrowth.

Towards aspect-oriented functional--structural plant modelling.

BACKGROUND AND AIMS: Functional-structural plant models (FSPMs) are used to integrate knowledge and test hypotheses of plant behaviour, and to aid in the development of decision support systems. A significant amount of effort is being put into providing a sound methodology for building them. Standard techniques, such as procedural or object-oriented programming, are not suited for clearly separating aspects of plant function that criss-cross between different components of plant structure, which makes it difficult to reuse and share their implementations. The aim of this paper is to present an aspect-oriented programming approach that helps to overcome this difficulty. METHODS: The L-system-based plant modelling language L+C was used to develop an aspect-oriented approach to plant modelling based on multi-modules. Each element of the plant structure was represented by a sequence of L-system modules (rather than a single module), with each module representing an aspect of the element's function. Separate sets of productions were used for modelling each aspect, with context-sensitive rules facilitated by local lists of modules to consider/ignore. Aspect weaving or communication between aspects was made possible through the use of pseudo-L-systems, where the strict-predecessor of a production rule was specified as a multi-module. KEY RESULTS: The new approach was used to integrate previously modelled aspects of carbon dynamics, apical dominance and biomechanics with a model of a developing kiwifruit shoot. These aspects were specified independently and their implementation was based on source code provided by the original authors without major changes. CONCLUSIONS: This new aspect-oriented approach to plant modelling is well suited for studying complex phenomena in plant science, because it can be used to integrate separate models of individual aspects of plant development and function, both previously constructed and new, into clearly organized, comprehensive FSPMs. In a future work, this approach could be further extended into an aspect-oriented programming language for FSPMs.

A functional-structural kiwifruit vine model integrating architecture, carbon dynamics and effects of the environment.

BACKGROUND AND AIMS: Functional-structural modelling can be used to increase our understanding of how different aspects of plant structure and function interact, identify knowledge gaps and guide priorities for future experimentation. By integrating existing knowledge of the different aspects of the kiwifruit (Actinidia deliciosa) vine's architecture and physiology, our aim is to develop conceptual and mathematical hypotheses on several of the vine's features: (a) plasticity of the vine's architecture; (b) effects of organ position within the canopy on its size; (c) effects of environment and horticultural management on shoot growth, light distribution and organ size; and (d) role of carbon reserves in early shoot growth. METHODS: Using the L-system modelling platform, a functional-structural plant model of a kiwifruit vine was created that integrates architectural development, mechanistic modelling of carbon transport and allocation, and environmental and management effects on vine and fruit growth. The branching pattern was captured at the individual shoot level by modelling axillary shoot development using a discrete-time Markov chain. An existing carbon transport resistance model was extended to account for several source/sink components of individual plant elements. A quasi-Monte Carlo path-tracing algorithm was used to estimate the absorbed irradiance of each leaf. KEY RESULTS: Several simulations were performed to illustrate the model's potential to reproduce the major features of the vine's behaviour. The model simulated vine growth responses that were qualitatively similar to those observed in experiments, including the plastic response of shoot growth to local carbon supply, the branching patterns of two Actinidia species, the effect of carbon limitation and topological distance on fruit size and the complex behaviour of sink competition for carbon. CONCLUSIONS: The model is able to reproduce differences in vine and fruit growth arising from various experimental treatments. This implies it will be a valuable tool for refining our understanding of kiwifruit growth and for identifying strategies to improve production.

A functional-structural modelling approach to autoregulation of nodulation.

BACKGROUND AND AIMS: Autoregulation of nodulation is a long-distance shoot-root signalling regulatory system that regulates nodule meristem proliferation in legume plants. However, due to the intricacy and subtleness of the signalling nature in plants, molecular and biochemical details underlying mechanisms of autoregulation of nodulation remain largely unknown. The purpose of this study is to use functional-structural plant modelling to investigate the complexity of this signalling system. There are two major challenges to be met: modelling the 3D architecture of legume roots with nodulation and co-ordinating signalling-developmental processes with various rates. METHODS: Soybean (Glycine max) was chosen as the target legume. Its root system was observed to capture lateral root branching and nodule distribution patterns. L-studio, a software tool supporting context-sensitive L-system modelling, was used for the construction of the architectural model and integration with the internal signalling. KEY RESULTS: A branching pattern with regular radial angles was found between soybean lateral roots, from which a root mapping method was developed to characterize the laterals. Nodules were mapped based on 'nodulation section' to reveal nodule distribution. A root elongation algorithm was then developed for simulation of root development. Based on the use of standard sub-modules, a synchronization algorithm was developed to co-ordinate multi-rate signalling and developmental processes. CONCLUSIONS: The modelling methods developed here not only allow recreation of legume root architecture with lateral branching and nodulation details, but also enable parameterization of internal signalling to produce different regulation results. This provides the basis for using virtual experiments to help in investigating the signalling mechanisms at work.

Computational complementation: a modelling approach to study signalling mechanisms during legume autoregulation of nodulation.

Autoregulation of nodulation (AON) is a long-distance signalling regulatory system maintaining the balance of symbiotic nodulation in legume plants. However, the intricacy of internal signalling and absence of flux and biochemical data, are a bottleneck for investigation of AON. To address this, a new computational modelling approach called "Computational Complementation" has been developed. The main idea is to use functional-structural modelling to complement the deficiency of an empirical model of a loss-of-function (non-AON) mutant with hypothetical AON mechanisms. If computational complementation demonstrates a phenotype similar to the wild-type plant, the signalling hypothesis would be suggested as "reasonable". Our initial case for application of this approach was to test whether or not wild-type soybean cotyledons provide the shoot-derived inhibitor (SDI) to regulate nodule progression. We predicted by computational complementation that the cotyledon is part of the shoot in terms of AON and that it produces the SDI signal, a result that was confirmed by reciprocal epicotyl-and-hypocotyl grafting in a real-plant experiment. This application demonstrates the feasibility of computational complementation and shows its usefulness for applications where real-plant experimentation is either difficult or impossible.

Computational modeling and molecular physiology experiments reveal new insights into shoot branching in pea.

Bud outgrowth is regulated by the interplay of multiple hormones, including auxin, cytokinin, strigolactones, and an unidentified long-distance feedback signal that moves from shoot to root. The model of bud outgrowth regulation in pea (Pisum sativum) includes these signals and a network of five RAMOSUS (RMS) genes that operate in a shoot-root-shoot loop to regulate the synthesis of, and response to, strigolactones. The number of components in this network renders the integration of new and existing hypotheses both complex and cumbersome. A hypothesis-driven computational model was therefore developed to help understand regulation of shoot branching. The model evolved in parallel with stepwise laboratory research, helping to define and test key hypotheses. The computational model was used to verify new mechanisms involved in the regulation of shoot branching by confirming that the new hypotheses captured all relevant biological data sets. Based on cytokinin and RMS1 expression analyses, this model is extended to include subtle but important differences in the function of RMS3 and RMS4 genes in the shoot and rootstock. Additionally, this research indicates that a branch-derived signal upregulates RMS1 expression independent of the other feedback signal. Furthermore, we propose xylem-sap cytokinin promotes sustained bud outgrowth, rather than acting at the earlier stage of bud release.

Computational analysis of flowering in pea (Pisum sativum).

During plant development, the transition from a vegetative to reproductive state is a critical event. For decades, pea (Pisum sativum) has been used as a model species to study this transition. These studies have led to a conceptual, qualitative model for the control of flower initiation, referred to as the 'classical' model. This model involves many inputs, namely photoperiod, genetic states and two mobile signals which interact to determine the first node of flowering. Here, we developed a computational model based on the hypotheses of the classical model. Accordingly, we converted qualitative hypotheses into quantitative rules. We found that new hypotheses, in addition to those already described for the classical model, were required that explicitly described the signals. In particular, we hypothesized that the key flowering gene HR interacts with the photoperiod pathway to control flowering. The computational model was tested against a wide range of biological data, including pre-existing and new experimental results presented here, and was found to be accurate. This computational model, together with ongoing experimental advances, will assist future modelling efforts to increase our understanding of flowering in pea.

Modeling the fitness of plant morphologies across three levels of complexity.

Linking together disparate biological models has the potential to provide insights that cannot be gained from studying the models in isolation. Computational modeling provides the means to integrate and test theories that span different temporal and spatial scales, and thus provides an ideal platform for forging this link. This paper describes a model that combines three key levels of biological theory. A computational model of gene regulation is integrated with a model of early land plant morphology, and placed in an evolutionary environment of competing fitness pressures. Simulation results indicate that each of these sub-models constrain possible phenotypic fitness. This work provides a methodological scaffold upon which increasingly complex models at each level of genotype, phenotype and environment can be built.

A canopy architectural model to study the competitive ability of chickpea with sowthistle.

BACKGROUND AND AIMS: Improving the competitive ability of crops is a sustainable method of weed management. This paper shows how a virtual plant model of competition between chickpea (Cicer arietinum) and sowthistle (Sonchus oleraceus) can be used as a framework for discovering and/or developing more competitive chickpea cultivars. METHODS: The virtual plant models were developed using the L-systems formalism, parameterized according to measurements taken on plants at intervals during their development. A quasi-Monte Carlo light-environment model was used to model the effect of chickpea canopy on the development of sowthistle. The chickpea-light environment-sowthistle model (CLES model) captured the hypothesis that the architecture of chickpea plants modifies the light environment inside the canopy and determines sowthistle growth and development pattern. The resulting CLES model was parameterized for different chickpea cultivars (viz. 'Macarena', 'Bumper', 'Jimbour' and '99071-1001') to compare their competitive ability with sowthistle. To validate the CLES model, an experiment was conducted using the same four chickpea cultivars as different treatments with a sowthistle growing under their canopy. RESULTS AND CONCLUSIONS: The growth of sowthistle, both in silico and in glasshouse experiments, was reduced most by '99071-1001', a cultivar with a short phyllochron. The second rank of competitive ability belonged to 'Macarena' and 'Bumper', while 'Jimbour' was the least competitive cultivar. The architecture of virtual chickpea plants modified the light inside the canopy, which influenced the growth and development of the sowthistle plants in response to different cultivars. This is the first time that a virtual plant model of a crop-weed interaction has been developed. This virtual plant model can serve as a platform for a broad range of applications in the study of chickpea-weed interactions and their environment.

Quasi-Monte Carlo simulation of the light environment of plants.

The distribution of light in the canopy is a major factor regulating the growth and development of a plant. The main variables of interest are the amount of photosynthetically active radiation (PAR) reaching different elements of the plant canopy, and the quality (spectral composition) of light reaching these elements. A light environment model based on Monte Carlo (MC) path tracing of photons, capable of computing both PAR and the spectral composition of light, was developed by Mech (1997), and can be conveniently interfaced with virtual plants expressed using the open L-system formalism. To improve the efficiency of the light distribution calculations provided by Mech's MonteCarlo program, we have implemented a similar program QuasiMC, which supports a more efficient randomised quasi-Monte Carlo sampling method (RQMC). We have validated QuasiMC by comparing it with MonteCarlo and with the radiosity-based CARIBU software (Chelle et al. 2004), and we show that these two programs produce consistent results. We also assessed the performance of the RQMC path tracing algorithm by comparing it with Monte Carlo path tracing and confirmed that RQMC offers a speed and/or accuracy improvement over MC.

Analysis of maize canopy development under water stress and incorporation into the ADEL-Maize model.

Substantial progress in modelling crop architecture has been made under optimal watering conditions; however, crop production is often exposed to water stress. In this research, we develop methods for implementing the simulation of maize (Zea mays L.) canopy architectural development under water stress using data from a maize field trial in 2006-07. Data of leaf number, leaf and internode extension were collected using non-destructive and destructive sampling at 2-3 day intervals. Water stress reduced the extension rate of organs and, therefore, their final length, the reduction being greater as severity of water stress increased. The duration of extension of organs in most phytomers was not significantly affected by water stress. Also, the rate of extension during the linear phase responded linearly to fraction of extractable soil water. An existing 3-D architectural model ADEL-Maize was revised using relationships developed in this study to better incorporate effects of water stress on organ extension and production. Simulated canopy production under three water regimes was validated by comparing predicted final leaf and internode length, plant height and leaf area to independent observations. The analysis and simulation showed that maize organ extension and final length under water stress can be adequately represented by simple linear patterns that are easily integrated into models.