An approach to multiscale modelling with graph grammars

Background and Aims

Functional–structural plant models (FSPMs) simulate biological processes at different spatial scales. Methods exist for multiscale data representation and modification, but the advantages of using multiple scales in the dynamic aspects of FSPMs remain unclear. Results from multiscale models in various other areas of science that share fundamental modelling issues with FSPMs suggest that potential advantages do exist, and this study therefore aims to introduce an approach to multiscale modelling in FSPMs.

Methods

A three-part graph data structure and grammar is revisited, and presented with a conceptual framework for multiscale modelling. The framework is used for identifying roles, categorizing and describing scale-to-scale interactions, thus allowing alternative approaches to model development as opposed to correlation-based modelling at a single scale. Reverse information flow (from macro- to micro-scale) is catered for in the framework. The methods are implemented within the programming language XL.

Key Results

Three example models are implemented using the proposed multiscale graph model and framework. The first illustrates the fundamental usage of the graph data structure and grammar, the second uses probabilistic modelling for organs at the fine scale in order to derive crown growth, and the third combines multiscale plant topology with ozone trends and metabolic network simulations in order to model juvenile beech stands under exposure to a toxic trace gas.

Conclusions

The graph data structure supports data representation and grammar operations at multiple scales. The results demonstrate that multiscale modelling is a viable method in FSPM and an alternative to correlation-based modelling. Advantages and disadvantages of multiscale modelling are illustrated by comparisons with single-scale implementations, leading to motivations for further research in sensitivity analysis and run-time efficiency for these models.     

Kinetic modelling of plant metabolic pathways

This paper provides a review of kinetic modelling of plant metabolic pathways as a tool for analysing their control and regulation. An overview of different modelling strategies is presented, starting with those approaches that only require a knowledge of the network stoichiometry; these are referred to as structural. Flux-balance analysis, metabolic flux analysis using isotope labelling, and elementary mode analysis are briefly mentioned as three representative examples. The main focus of this paper, however, is a discussion of kinetic modelling, which requires, in addition to the stoichiometry, a knowledge of the kinetic properties of the constituent pathway enzymes. The different types of kinetic modelling analysis, namely time-course simulation, steady-state analysis, and metabolic control analysis, are explained in some detail. An overview is presented of strategies for obtaining model parameters, as well as software tools available for simulation of such models. The kinetic modelling approach is exemplified with discussion of three models from the general plant physiology literature. With the aid of kinetic modelling it is possible to perform a control analysis of a plant metabolic system, to identify potential targets for biotechnological manipulation, as well as to ascertain the regulatory importance of different enzymes (including isoforms of the same enzyme) in a pathway. Finally, a framework is presented for extending metabolic models to the whole-plant scale by linking biochemical reactions with diffusion and advective flow through the phloem. Future challenges include explicit modelling of subcellular compartments, as well as the integration of kinetic models on the different levels of the cellular and organizational hierarchy.     

Modeling and simulation: tools for metabolic engineering.

Mathematical modeling is one of the key methodologies of metabolic engineering. Based on a given metabolic model different computational tools for the simulation, data evaluation, systems analysis, prediction, design and optimization of metabolic systems have been developed. The currently used metabolic modeling approaches can be subdivided into structural models, stoichiometric models, carbon flux models, stationary and nonstationary mechanistic models and models with gene regulation. However, the power of a model strongly depends on its basic modeling assumptions, the simplifications made and the data sources used. Model validation turns out to be particularly difficult for metabolic systems. The different modeling approaches are critically reviewed with respect to their potential and benefits for the metabolic engineering cycle. Several tools that have emerged from the different modeling approaches including structural pathway synthesis, stoichiometric pathway analysis, metabolic flux analysis, metabolic control analysis, optimization of regulatory architectures and the evaluation of rapid sampling experiments are discussed.     

Issues in high-throughput comparative modelling: a case study using the ubiquitin E2 conjugating enzymes

Sequences of the ubiquitin-conjugating enzyme (UBC or E2) family were used as a test set to investigate issues associated with the high-throughput comparative modelling of protein structures. A semi-automatic method was initially developed with particular emphasis on producing models of a quality suitable for structural comparison. Structural and sequence features of the E2 family were used to improve the sequence alignment and the quality of the structural templates. Initially, failure to correct for subtle structural inconsistencies between templates lead to problems in the comparative analysis of the UBC electrostatic potentials. Modelling of known UBC structures using Modeller 4.0 showed that multiple templates produced, on average, no better models than the use of just one template, as judged by the root-mean-squared deviation between the comparative model and crystal structure backbones. Using four different quality-checking methods, for a given target sequence, it was not possible to distinguish the model most similar to the experimental structure. The UBC models were thus finally modelled using only the crystal structure template with the highest sequence identity to the target to be modelled, and producing only one model solution. Quality checking was used to reject models with obvious structural anomalies (e.g., bad side-chain packing). The resulting models have been used for a comparison of UBC structural features and of their electrostatic potentials. The work was extended through the development of a fully automated pipeline that identifies E2 sequences in the sequence databases, aligns and models them, and calculates the associated electrostatic potential.     

Mathematical Modelling of Anaerobic Reactors Treating Domestic Wastewater: Rational Criteria for Model Use

Anaerobic digestion modelling is an established method for assessing anaerobic wastewater treatment for design, systems analysis, operational analysis, and control. Anaerobic treatment of domestic wastewater is a relatively new, but rapidly maturing technology, especially in developing countries, where the combination of low cost, and moderate-good performance are particularly attractive. The key emerging technology is high-rate anaerobic treatment, particularly UASB reactors. Systems modelling can potentially offer a number of advantages to this field, and the key motivations for modelling have been identified as operational analysis, technology development, and model-based design. Design is particularly important, as it determines capital cost, a key motivation for implementers. Published modelling studies for anaerobic domestic sewage treatment are limited in number, but well directed at specific issues. Most have a low structural complexity, with first order kinetics, as compared to the more commonly used Monod kinetics. This review addresses the use of anaerobic models in general, application of models to domestic sewage systems, and evaluates future requirements for models that need to address the key motivations of operational analysis, technology development, and model-based design. For operational analysis and technology development, a complex model such as the ADM1 is recommended, with further extensions as required to address factors such as sulphate reduction. For design, the critical issSues are hydraulics and particles (i.e., biomass and solid substrate) modelling. Therefore, the kinetic structure should be relatively simple (at least two-step), but the hydraulic and particulate model should be relatively complex.     

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.     

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.     

Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?

What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.     

Modelling Primate Behavioral Ecology

Models play an important role in any mature science because they force us to make explicit our assumptions about how a phenomenon works and allow us to explore the way in which different variables influence a complex biological system. I review the principal kinds of models that could be used to study primate behavior and ecology: linear programming models, systems models, optimality models, stochastic dynamic programming models and agent-based simulation models. Although less use has been made of modelling in primatology than in some other areas of behavioral ecology, there is considerable scope for exploiting the predictive and explanatory power of models in the field.     

Structural modelling of the cardiovascular system

Computational modelling of the cardiovascular system offers much promise, but represents a truly interdisciplinary challenge, requiring knowledge of physiology, mechanics of materials, fluid dynamics and biochemistry. This paper aims to provide a summary of the recent advances in cardiovascular structural modelling, including the numerical methods, main constitutive models and modelling procedures developed to represent cardiovascular structures and pathologies across a broad range of length and timescales; serving as an accessible point of reference to newcomers to the field. The class of so-called hyperelastic materials provides the theoretical foundation for the modelling of how these materials deform under load, and so an overview of these models is provided; comparing classical to application-specific phenomenological models. The physiology is split into components and pathologies of the cardiovascular system and linked back to constitutive modelling developments, identifying current state of the art in modelling procedures from both clinical and engineering sources. Models which have originally been derived for one application and scale are shown to be used for an increasing range and for similar applications. The trend for such approaches is discussed in the context of increasing availability of high performance computing resources, where in some cases computer hardware can impact the choice of modelling approach used.     

WGCNA: an R package for weighted correlation network analysis

Background

Modelling the time-related behaviour of biological systems is essential for understanding their dynamic responses to perturbations. In metabolic profiling studies, the sampling rate and number of sampling points are often restricted due to experimental and biological constraints.

Results

A supervised multivariate modelling approach with the objective to model the time-related variation in the data for short and sparsely sampled time-series is described. A set of piecewise Orthogonal Projections to Latent Structures (OPLS) models are estimated, describing changes between successive time points. The individual OPLS models are linear, but the piecewise combination of several models accommodates modelling and prediction of changes which are non-linear with respect to the time course. We demonstrate the method on both simulated and metabolic profiling data, illustrating how time related changes are successfully modelled and predicted.

Conclusion

The proposed method is effective for modelling and prediction of short and multivariate time series data. A key advantage of the method is model transparency, allowing easy interpretation of time-related variation in the data. The method provides a competitive complement to commonly applied multivariate methods such as OPLS and Principal Component Analysis (PCA) for modelling and analysis of short time-series data.     

SURF’s UP! — Protein classification by surface comparisons

Large-scale genome sequencing and structural genomics projects generate numerous sequences and structures for 'hypothetical' proteins without functional characterizations. Detection of homology to experimentally characterized proteins can provide functional clues, but the accuracy of homology-based predictions is limited by the paucity of tools for quantitative comparison of diverging residues responsible for the functional divergence. SURF'S UP! is a web server for analysis of functional relationships in protein families, as inferred from protein surface maps comparison according to the algorithm. It assigns a numerical score to the similarity between patterns of physicochemical features(charge, hydrophobicity) on compared protein surfaces. It allows recognizing clusters of proteins that have similar surfaces, hence presumably similar functions. The server takes as an input a set of protein coordinates and returns files with "spherical coordinates" of proteins in a PDB format and their graphical presentation, a matrix with values of mutual similarities between the surfaces, and the unrooted tree that represents the clustering of similar surfaces, calculated by the neighbor-joining method. SURF'S UP! facilitates the comparative analysis of physicochemical features of the surface, which are the key determinants of the protein function. By concentrating on coarse surface features, SURF'S UP! can work with models obtained from comparative modelling. Although it is designed to analyse the conservation among homologs, it can also be used to compare surfaces of non-homologous proteins with different three-dimensional folds, as long as a functionally meaningful structural superposition is supplied by the user. Another valuable characteristic of our method is the lack of initial assumptions about the functional features to be compared. SURF'S UP! is freely available for academic researchers at http://asia.genesilico.pl/surfs_up/.     

A simplified method for power-law modelling of metabolic pathways from time-course data and steady-state flux profiles

Background  

In order to improve understanding of metabolic systems there have been attempts to construct S-system models from time courses. Conventionally, non-linear curve-fitting algorithms have been used for modelling, because of the non-linear properties of parameter estimation from time series. However, the huge iterative calculations required have hindered the development of large-scale metabolic pathway models. To solve this problem we propose a novel method involving power-law modelling of metabolic pathways from the Jacobian of the targeted system and the steady-state flux profiles by linearization of S-systems.     

Power-law models of signal transduction pathways

The mathematical modelling of signal transduction pathways has become a valuable aid to understanding the complex interactions involved in intracellular signalling mechanisms. An important aspect of the mathematical modelling process is the selection of the model type and structure. Until recently, the convention has been to use a standard kinetic model, often with the Michaelis-Menten steady state assumption. However this model form, although valuable, is only one of a number of choices, and the aim of this article is to consider the mathematical structure and essential features of an alternative model form--the power-law model. Specifically, we analyse how power-law models can be applied to increase our understanding of signal transduction pathways when there may be limited prior information. We distinguish between two kinds of power law models: a) Detailed power-law models, as a tool for investigating pathways when the structure of protein-protein interactions is completely known, and; b) Simplified power-law models, for the analysis of systems with incomplete structural information or insufficient quantitative data for generating detailed models. If sufficient data of high quality are available, the advantage of detailed power-law models is that they are more realistic representations of non-homogenous or crowded cellular environments. The advantages of the simplified power-law model formulation are illustrated using some case studies in cell signalling. In particular, the investigation on the effects of signal inhibition and feedback loops and the validation of structural hypotheses are discussed.     

Unravelling the regulatory structure of biochemical networks using stimulus response experiments and large-scale model selection

To unravel the complex in vivo regulatory interdependences of biochemical networks, experiments with the living organism are absolutely necessary. Stimulus response experiments (SREs) have become increasingly popular in recent years. The response of metabolite concentrations from all major parts of the central metabolism is monitored over time by modem analytical methods, producing several thousand data points. SREs are applied to determine enzyme kinetic parameters and to find unknown enzyme regulatory mechanisms. Owing to the complex regulatory structure of metabolic networks and the amount of measured data, the evaluation of an SRE has to be extensively supported by modelling. If the enzyme regulatory mechanisms are part of the investigation, a large number of models with different enzyme kinetics have to be tested for their ability to reproduce the observed behaviour. In this contribution, a systematic model-building process for data-driven exploratory modelling is introduced with the aim of discovering essential features of the biological system. The process is based on data pre-processing, correlation-based hypothesis generation, automatic model family generation, large-scale model selection and statistical analysis of the best-fitting models followed by an extraction of common features. It is illustrated by the example of the aromatic amino acid synthesis pathway in Escherichia coli.     

On the use of qualitative reasoning to simulate and identify metabolic pathways

MOTIVATION: Perhaps the greatest challenge of modern biology is to develop accurate in silico models of cells. To do this we require computational formalisms for both simulation (how according to the model the state of the cell evolves over time) and identification (learning a model cell from observation of states). We propose the use of qualitative reasoning (QR) as a unified formalism for both tasks. The two most commonly used alternative methods of modelling biochemical pathways are ordinary differential equations (ODEs), and logical/graph-based (LG) models. RESULTS: The QR formalism we use is an abstraction of ODEs. It enables the behaviour of many ODEs, with different functional forms and parameters, to be captured in a single QR model. QR has the advantage over LG models of explicitly including dynamics. To simulate biochemical pathways we have developed 'enzyme' and 'metabolite' QR building blocks that fit together to form models. These models are finite, directly executable, easy to interpret and robust. To identify QR models we have developed heuristic chemoinformatics graph analysis and machine learning procedures. The graph analysis procedure is a series of constraints and heuristics that limit the number of ways metabolites can combine to form pathways. The machine learning procedure is generate-and-test inductive logic programming. We illustrate the use of QR for modelling and simulation using the example of glycolysis. AVAILABILITY: All data and programs used are available on request.     

Reconstruction of Danio rerio Metabolic Model Accounting for Subcellular Compartmentalisation

Plant and microbial metabolic engineering is commonly used in the production of functional foods and quality trait improvement. Computational model-based approaches have been used in this important endeavour. However, to date, fish metabolic models have only been scarcely and partially developed, in marked contrast to their prominent success in metabolic engineering. In this study we present the reconstruction of fully compartmentalised models of the Danio rerio (zebrafish) on a global scale. This reconstruction involves extraction of known biochemical reactions in D. rerio for both primary and secondary metabolism and the implementation of methods for determining subcellular localisation and assignment of enzymes. The reconstructed model (ZebraGEM) is amenable for constraint-based modelling analysis, and accounts for 4,988 genes coding for 2,406 gene-associated reactions and only 418 non-gene-associated reactions. A set of computational validations (i.e., simulations of known metabolic functionalities and experimental data) strongly testifies to the predictive ability of the model. Overall, the reconstructed model is expected to lay down the foundations for computational-based rational design of fish metabolic engineering in aquaculture.