Using L-systems for modeling source-sink interactions, architecture and physiology of growing trees: the L-PEACH model

Functional-structural plant models simulate the development of plant structure, taking into account plant physiology and environmental factors. The L-PEACH model is based on the development of peach trees. It demonstrates the usefulness of L-systems in constructing functional-structural models. L-PEACH uses L-systems both to simulate the development of tree structure and to solve differential equations for carbohydrate flow and allocation. New L-system-based algorithms are devised for simulating the behavior of dynamically changing structures made of hundreds of interacting, time-varying, nonlinear components. L-PEACH incorporates a carbon-allocation model driven by source-sink interactions between tree components. Storage and mobilization of carbohydrates during the annual life cycle of a tree are taken into account. Carbohydrate production in the leaves is simulated based on the availability of water and light. Apices, internodes, leaves and fruit grow according to the resulting local carbohydrate supply. L-PEACH outputs an animated three-dimensional visual representation of the growing tree and user-specified statistics that characterize selected stages of plant development. The model is applied to simulate a tree's response to fruit thinning and changes in water stress. L-PEACH may be used to assist in horticultural decision-making processes after being calibrated to specific trees.     

Neural networks for perceptual processing: from simulation tools to theories

Neural networks are modelling tools that are, in principle, able to capture the input-output behaviour of arbitrary systems that may include the dynamics of animal populations or brain circuits. While a neural network model is useful if it captures phenomenologically the behaviour of the target system in this way, its utility is amplified if key mechanisms of the model can be discovered, and identified with those of the underlying system. In this review, we first describe, at a fairly high level with minimal mathematics, some of the tools used in constructing neural network models. We then go on to discuss the implications of network models for our understanding of the system they are supposed to describe, paying special attention to those models that deal with neural circuits and brain systems. We propose that neural nets are useful for brain modelling if they are viewed in a wider computational framework originally devised by Marr. Here, neural networks are viewed as an intermediate mechanistic abstraction between 'algorithm' and 'implementation', which can provide insights into biological neural representations and their putative supporting architectures.     

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.     

The state of plant population modelling in light of environmental change

Plant population modelling has been around since the 1970s, providing a valuable approach to understanding plant ecology from a mechanistic standpoint. It is surprising then that this area of research has not grown in prominence with respect to other approaches employed in modelling plant systems. In this review, we provide an analysis of the development and role of modelling in the field of plant population biology through an exploration of where it has been, where it is now and, in our opinion, where it should be headed. We focus, in particular, on the role plant population modelling could play in ecological forecasting, an urgent need given current rates of regional and global environmental change. We suggest that a critical element limiting the current application of plant population modelling in environmental research is the trade-off between the necessary resolution and detail required to accurately characterize ecological dynamics pitted against the goal of generality, particularly at broad spatial scales. In addition to suggestions how to overcome the current shortcoming of data on the process-level we discuss two emerging strategies that may offer a way to overcome the described limitation: (1) application of a modern approach to spatial scaling from local processes to broader levels of interaction and (2) plant functional-type modelling. Finally we outline what we believe to be needed in developing these approaches towards a ‘science of forecasting’.     

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.     

Chromosome numbers in the algae: Phaeophyta II

L-systems provide a method for modelling seaweed branching patterns in terms of simple rules. These rules can be used to generate computer graphics images of an alga's basic architecture, which, of course, is often obscured by adventiltious growth in living plants. L-systems provide a mathematical abstraction that allows us to compare species by focusing on essential differences in their substitution rules rather than relying entirely on the visual appearance of branching patterns. Two species of red algae from closely related genera in the Dasyaceae—Dasya rigidula and Dasysiphonia concinna—are virtually identical in pseudolateral branching detail and appear to diverge solely in terms of pseudolateral position. L-systems models and their graphical representations are presented to demonstrate that the two-dimensional, alternate-distichous pattern of pseudolateral position for Dasysiphonia concinna can be derived in a straightforward way from that of the three-dimensional, spiralled pattern of Dasya rigidula. These models illustrate a case in which generic distinction is based upon an easily observable but relatively trivial feature, differences other than branching pattern being quite subtle.     

Validation of the Greater Hunter Native Vegetation Mapping as it pertains to the Upper Hunter region of New South Wales

Reliable vegetation maps are an important component of any long‐term landscape planning initiatives. A number of approaches are available but one, in particular, pattern recognition (segmentation) combined with modelling from floristic site data, is currently being used to map vegetation across NSW. An independent assessment of this approach based on a review of the Greater Hunter Native Vegetation Mapping (GHM_v4) was undertaken in order to assess its ability to cater for regional, local, strategic and landscape planning. The validation process tested 2151 locations across the Upper Hunter Valley region of New South Wales (NSW), Australia. The results suggest that mapping at the coarsest level of NSW vegetation classification, the Formation, is generally poor, with only Dry Sclerophyll Forest and Woodland modelled with some level of reliability. The modelled mapping of individual plant community types (PCTs) was found to be highly inaccurate with only 17% of validation points attributed as ‘correct’ and a further 13% ‘essentially correct’. Therefore, a majority of PCTs were mapped with an accuracy of less than 30%. The results of this validation suggest that the GHM_v4 is of such a low level of accuracy within the upper Hunter as to be inherently unusable for broad‐scale regional and local landscape planning or environmental assessment, including locating compensatory offsets for the loss of native vegetation due to developments. The GHM_v4 methods of pattern recognition of mainly SPOT5 satellite imagery combined with modelling from plot data have not produced reliable vegetation maps of plant community types. Yet this mapping programme is extending across NSW and could be misused for environmental decisions or as a regulation.     

Modelling as a tool to redesign livestock farming systems: a literature review

Livestock farming has recently come under close scrutiny, in response especially to environmental issues. Farmers are encouraged to redesign their livestock farming systems in depth to improve their sustainability. Assuming that modelling can be a relevant tool to address such systemic changes, we sought to answer the following question: ‘How can livestock farming systems be modelled to help farmers redesign their whole farming systems?’ To this end, we made a literature review of the models of livestock farming systems published from 2000 to mid-2009 (n = 79). We used an analysis grid based on three considerations: (i) system definition, (ii) the intended use of the model and (iii) the way in which farmers’ decision-making processes were represented and how agricultural experts and farmers were involved in the modelling processes. Consistent rationales in approaches to supporting changes in livestock farming were identified in three different groups of models, covering 83% of the whole set. These could be defined according to (i) the way in which farmers’ decisions were represented and (ii) the model's type of contribution to supporting changes. The first type gathered models that dynamically simulated the system according to different management options; the farmers’ decision-making processes are assumed to consist in choosing certain values for management factors. Such models allow long-term simulations and endorse different disciplinary viewpoints, but the farmers are weakly involved in their design. Models of the second type can indicate the best combination of farm activities under given constraints, provided the farmers’ objectives are profit maximisation. However, when used to support redesigning processes, they address neither how to implement the optimal solution nor its long-term consequences. Models of the third type enable users to dynamically simulate different options for the farming system, the management of which is assumed to be planned according to the farmers’ general objectives. Although more comprehensive, these models do not easily integrate different disciplinary viewpoints and different subsystems, which limits their usefulness as support tools for redesigning processes. Finally, we concluded about what specific requirements should be for modelling approaches if farmers were to be supported in redesigning their whole livestock farming systems using models.     

Towards mechanistic models of plant organ growth

Modelling and simulation are increasingly used as tools in the study of plant growth and developmental processes. By formulating experimentally obtained knowledge as a system of interacting mathematical equations, it becomes feasible for biologists to gain a mechanistic understanding of the complex behaviour of biological systems. In this review, the modelling tools that are currently available and the progress that has been made to model plant development, based on experimental knowledge, are described. In terms of implementation, it is argued that, for the modelling of plant organ growth, the cellular level should form the cornerstone. It integrates the output of molecular regulatory networks to two processes, cell division and cell expansion, that drive growth and development of the organ. In turn, these cellular processes are controlled at the molecular level by hormone signalling. Therefore, combining a cellular modelling framework with regulatory modules for the regulation of cell division, expansion, and hormone signalling could form the basis of a functional organ growth simulation model. The current state of progress towards this aim is that the regulation of the cell cycle and hormone transport have been modelled extensively and these modules could be integrated. However, much less progress has been made on the modelling of cell expansion, which urgently needs to be addressed. A limitation of the current generation models is that they are largely qualitative. The possibilities to characterize existing and future models more quantitatively will be discussed. Together with experimental methods to measure crucial model parameters, these modelling techniques provide a basis to develop a Systems Biology approach to gain a fundamental insight into the relationship between gene function and whole organ behaviour.     

Simulating the effects of localized red:far-red ratio on tillering in spring wheat (Triticum aestivum) using a three-dimensional virtual plant model

The outgrowth of tiller buds in Poaceae is influenced by the ratio of red to far-red light irradiance (R:FR). At each point in the plant canopy, R:FR is affected by light scattered by surrounding plant tissues. This paper presents a three-dimensional virtual plant modelling approach to simulate local effects of R:FR on tillering in spring wheat (Triticum aestivum). R:FR dependence of bud outgrowth was implemented in a wheat model, using three hypothetical responses of bud extension to R:FR (unit step, curvilinear and linear response). Bud break occurred when a threshold bud length was reached. Simulations were performed for three plant population densities. In accordance with experimental observations, fewer tillers per plant were simulated for higher plant population densities. The linear and curvilinear responses caused a delay in the increase in tiller number compared with experimental data. The unit step response approached experimental results best. It is suggested that a model based on relatively simple relations can be used to simulate degree of tillering. This study has shown that the virtual plant approach is a promising tool with which to address crop morphological and ecological research questions where the determining factors act at the level of the individual plant organ.     

Computer simulation: The imaginary friend of auxin transport biology

Regulated transport of the plant hormone auxin is central to many aspects of plant development. Directional transport, mediated by membrane transporters, produces patterns of auxin distribution in tissues that trigger developmental processes, such as vascular patterning or leaf formation. Experimentation has produced many, largely qualitative, data providing strong evidence for multiple feedback systems between auxin and its transport. However, the exact mechanisms concerned remain elusive and the experiments required to evaluate alternative hypotheses are challenging. Because of this, computational modelling now plays an important role in auxin transport research. Here we review some current approaches and underlying assumptions of computational auxin transport models. We focus on self‐organising models for polar auxin transport and on recent attempts to unify conflicting mechanistic explanations. In addition, we discuss in general how these computer simulations are proving to be increasingly effective in hypothesis generation and testing, and how simulation can be used to direct future experiments. Editor's suggested further reading in BioEssays Local auxin production: a small contribution to a big field Abstract     

Functional–structural plant models: a growing paradigm for plant studies

A number of research groups in various areas of plant biology as well as computer science and applied mathematics have addressed modelling the spatiotemporal dynamics of growth and development of plants. This has resulted in development of functional–structural plant models (FSPMs). In FSPMs, the plant structure is always explicitly represented in terms of a network of elementary units. In this respect, FSPMs are different from more abstract models in which a simplified representation of the plant structure is frequently used (e.g. spatial density of leaves, total biomass, etc.). This key feature makes it possible to build modular models and creates avenues for efficient exchange of model components and experimental data. They are being used to deal with the complex 3-D structure of plants and to simulate growth and development occurring at spatial scales from cells to forest areas, and temporal scales from seconds to decades and many plant generations. The plant types studied also cover a broad spectrum, from algae to trees. This special issue of Annals of Botany features selected papers on FSPM topics such as models of morphological development, models of physical and biological processes, integrated models predicting dynamics of plants and plant communities, modelling platforms, methods for acquiring the 3-D structures of plants using automated measurements, and practical applications for agronomic purposes.     

QTL methodology for response curves on the basis of non-linear mixed models, with an illustration to senescence in potato

The improvement of quantitative traits in plant breeding will in general benefit from a better understanding of the genetic basis underlying their development. In this paper, a QTL mapping strategy is presented for modelling the development of phenotypic traits over time. Traditionally, crop growth models are used to study development. We propose an integration of crop growth models and QTL models within the framework of non-linear mixed models. We illustrate our approach with a QTL model for leaf senescence in a diploid potato cross. Assuming a logistic progression of senescence in time, two curve parameters are modelled, slope and inflection point, as a function of QTLs. The final QTL model for our example data contained four QTLs, of which two affected the position of the inflection point, one the senescence progression-rate, and a final one both inflection point and rate.     

Modelling evolutionary cell behaviour using neural networks: Application to tumour growth

In this paper, we present a modelling framework for cellular evolution that is based on the notion that a cell’s behaviour is driven by interactions with other cells and its immediate environment. We equip each cell with a phenotype that determines its behaviour and implement a decision mechanism to allow evolution of this phenotype. This decision mechanism is modelled using feed-forward neural networks, which have been suggested as suitable models of cell signalling pathways. The environmental variables are presented as inputs to the network and result in a response that corresponds to the phenotype of the cell. The response of the network is determined by the network parameters, which are subject to mutations when the cells divide. This approach is versatile as there are no restrictions on what the input or output nodes represent, they can be chosen to represent any environmental variables and behaviours that are of importance to the cell population under consideration. This framework was implemented in an individual-based model of solid tumour growth in order to investigate the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. Our results show that the oxygen concentration affects the tumour at the morphological level, but more importantly has a direct impact on the evolutionary dynamics. When the supply of oxygen is limited we observe a faster divergence away from the initial genotype, a higher population diversity and faster evolution towards aggressive phenotypes. The implementation of this framework suggests that this approach is well suited for modelling systems where evolution plays an important role and where a changing environment exerts selection pressure on the evolving population.     

A mathematical framework for modelling cambial surface evolution using a level set method

Background and Aims

During their lifetime, tree stems take a series of successive nested shapes. Individual tree growth models traditionally focus on apical growth and architecture. However, cambial growth, which is distributed over a surface layer wrapping the whole organism, equally contributes to plant form and function. This study aims at providing a framework to simulate how organism shape evolves as a result of a secondary growth process that occurs at the cellular scale.

Methods

The development of the vascular cambium is modelled as an expanding surface using the level set method. The surface consists of multiple compartments following distinct expansion rules. Growth behaviour can be formulated as a mathematical function of surface state variables and independent variables to describe biological processes.

Key Results

The model was coupled to an architectural model and to a forest stand model to simulate cambium dynamics and wood formation at the scale of the organism. The model is able to simulate competition between cambia, surface irregularities and local features. Predicting the shapes associated with arbitrarily complex growth functions does not add complexity to the numerical method itself.

Conclusions

Despite their slenderness, it is sometimes useful to conceive of trees as expanding surfaces. The proposed mathematical framework provides a way to integrate through time and space the biological and physical mechanisms underlying cambium activity. It can be used either to test growth hypotheses or to generate detailed maps of wood internal structure.     

A mathematical modelling framework for understanding chemorepulsive signal transduction in Dictyostelium

Chemorepulsion is the process by which an organism or a cell moves in the direction of decreasing chemical concentration. While a few experimental studies have been performed, no mathematical models exist for this process. In this paper we have modelled gradient sensing, the first subprocess of chemorepulsion, in Dictyostelium discoideum-a well characterized model eukaryotic system. We take the first steps towards achieving a comprehensive mechanistic understanding of chemorepulsion in this system. We have used, as a basis, the biochemical network of the Keizer-Gunnink et al. (2007) to develop the mathematical modelling framework. This network describes the underlying pathways of chemorepellent gradient sensing in D. discoideum. Working within this modelling framework we address whether the postulated interactions of the pathways and species in this network can lead to a chemorepulsive response. We also analyse the possible role of additional regulatory effects (such as additional receptor regulation of enzymes in this network) and if this is necessary to achieve this behaviour. Thus we have investigated the receptor regulation of important enzymes and feedback effects in the network. This modelling framework generates important insights into and testable predictions regarding the role of key components and feedback loops in regulating chemorepulsive gradient sensing, and what factors might be important for generating a chemorepulsive response; it serves as a first step towards a comprehensive mechanistic understanding of this process.     

Linking water stress effects on carbon partitioning by introducing a xylem circuit into L-PEACH

Background and Aims

Many physiological processes such as photosynthesis, respiration and transpiration can be strongly influenced by the diurnal patterns of within-tree water potential. Despite numerous experiments showing the effect of water potential on fruit-tree development and growth, there are very few models combining carbohydrate allocation with water transport. The aim of this work was to include a xylem circuit into the functional–structural L-PEACH model.

Methods

The xylem modelling was based on an electrical circuit analogy and the Hagen–Poisseuille law for hydraulic conductance. Sub-models for leaf transpiration, soil water potential and the soil–plant interface were also incorporated to provide the driving force and pathway for water flow. The model was assessed by comparing model outputs to field measurements and published knowledge.

Key Results

The model was able to simulate both the water uptake over a season and the effect of different irrigation treatments on tree development, growth and fruit yield.

Conclusions

This work opens the way to a new field of modelling where complex interactions between water transport, carbohydrate allocation and physiological functions can be simulated at the organ level and describe functioning and behaviour at the tree scale.     

Plants in silico: why,why now and what?—an integrative platform for plant systems biology research

A paradigm shift is needed and timely in moving plant modelling from largely isolated efforts to a connected community endeavour that can take full advantage of advances in computer science and in mechanistic understanding of plant processes. Plants in silico (Psi) envisions a digital representation of layered dynamic modules, linking from gene networks and metabolic pathways through to cellular organization, tissue, organ and whole plant development, together with resource capture and use efficiency in dynamic competitive environments, ultimately allowing a mechanistically rich simulation of the plant or of a community of plants in silico. The concept is to integrate models or modules from different layers of organization spanning from genome to phenome to ecosystem in a modular framework allowing the use of modules of varying mechanistic detail representing the same biological process. Developments in high‐performance computing, functional knowledge of plants, the internet and open‐source version controlled software make achieving the concept realistic. Open source will enhance collaboration and move towards testing and consensus on quantitative theoretical frameworks. Importantly, Psi provides a quantitative knowledge framework where the implications of a discovery at one level, for example, single gene function or developmental response, can be examined at the whole plant or even crop and natural ecosystem levels.