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.     

The regulatory design of an allosteric feedback loop: the effect of saturation by pathway substrate

Aspects of metabolic regulation can be fruitfully studied with a combination of generic modelling, control analysis and graphical analysis using rate characteristics. This paper analyses a prototypical supply-demand system consisting of a biosynthetic subsystem subject to allosteric inhibition by its product and a demand process that consumes this product. The effect of changes in affinity of the committing supply enzyme for the pathway substrate on the regulatory properties of the supply subsystem is compared for the Monod-Wyman-Changeux and the reversible Hill allosteric enzyme models. We found that the Hill model has a distinct advantage in that the steady-state concentration at which it maintains the product is set by the half-saturating product concentration and is independent of changes in the degree of saturation for substrate. In contrast, with the Monod-Wyman-Changeux model this set point varies with affinity for substrate. Explicitly incorporating reversibility in all rate equations made it possible to distinguish between kinetic and thermodynamic aspects of regulation. Combining the supply and demand rate characteristics allows us to explore both the control distribution at steady state and the regulatory performance of the system over a wide range of demand activities.     

Steady-state analysis of metabolic pathways: comparing the double modulation method and the lin-log approach

The increasing interest in studying enzyme kinetics under in vivo conditions requires practical methods to estimate control parameters from experimental data. In contrast to currently established approaches of dynamic modelling, this paper addresses the steady-state analysis of metabolic pathways. Within the framework of metabolic control analysis (MCA), elasticity coefficients are used to describe the control properties of a local enzyme reaction. The double modulation method is one of the first experimental approaches to estimate elasticity coefficients from measurements of steady-state flux rates and metabolite concentrations. We propose a generalized form of the double modulation method and compare it to the recently developed linear-logarithmic approach.     

Strategies for investigating the plant metabolic network with steady-state metabolic flux analysis: lessons from an Arabidopsis cell culture and other systems

Steady-state (13)C metabolic flux analysis (MFA) is currently the experimental method of choice for generating flux maps of the compartmented network of primary metabolism in heterotrophic and mixotrophic plant tissues. While statistically robust protocols for the application of steady-state MFA to plant tissues have been developed by several research groups, the implementation of the method is still far from routine. The effort required to produce a flux map is more than justified by the information that it contains about the metabolic phenotype of the system, but it remains the case that steady-state MFA is both analytically and computationally demanding. This article provides an overview of principles that underpin the implementation of steady-state MFA, focusing on the definition of the metabolic network responsible for redistribution of the label, experimental considerations relating to data collection, the modelling process that allows a set of metabolic fluxes to be deduced from the labelling data, and the interpretation of flux maps. The article draws on published studies of Arabidopsis cell cultures and other systems, including developing oilseeds, with the aim of providing practical guidance and strategies for handling the issues that arise when applying steady-state MFA to the complex metabolic networks encountered in plants.     

A kinetic model of the branch-point between the methionine and threonine biosynthesis pathways in Arabidopsis thaliana.

This work proposes a model of the metabolic branch-point between the methionine and threonine biosynthesis pathways in Arabidopsis thaliana which involves kinetic competition for phosphohomoserine between the allosteric enzyme threonine synthase and the two-substrate enzyme cystathionine gamma-synthase. Threonine synthase is activated by S-adenosylmethionine and inhibited by AMP. Cystathionine gamma-synthase condenses phosphohomoserine to cysteine via a ping-pong mechanism. Reactions are irreversible and inhibited by inorganic phosphate. The modelling procedure included an examination of the kinetic links, the determination of the operating conditions in chloroplasts and the establishment of a computer model using the enzyme rate equations. To test the model, the branch-point was reconstituted with purified enzymes. The computer model showed a partial agreement with the in vitro results. The model was subsequently improved and was then found consistent with flux partition in vitro and in vivo. Under near physiological conditions, S-adenosylmethionine, but not AMP, modulates the partition of a steady-state flux of phosphohomoserine. The computer model indicates a high sensitivity of cystathionine flux to enzyme and S-adenosylmethionine concentrations. Cystathionine flux is sensitive to modulation of threonine flux whereas the reverse is not true. The cystathionine gamma-synthase kinetic mechanism favours a low sensitivity of the fluxes to cysteine. Though sensitivity to inorganic phosphate is low, its concentration conditions the dynamics of the system. Threonine synthase and cystathionine gamma-synthase display similar kinetic efficiencies in the metabolic context considered and are first-order for the phosphohomoserine substrate. Under these conditions outflows are coordinated.     

Application of Petri net theory for modelling and validation of the sucrose breakdown pathway in the potato tuber

MOTIVATION: Because of the complexity of metabolic networks and their regulation, formal modelling is a useful method to improve the understanding of these systems. An essential step in network modelling is to validate the network model. Petri net theory provides algorithms and methods, which can be applied directly to metabolic network modelling and analysis in order to validate the model. The metabolism between sucrose and starch in the potato tuber is of great research interest. Even if the metabolism is one of the best studied in sink organs, it is not yet fully understood. RESULTS: We provide an approach for model validation of metabolic networks using Petri net theory, which we demonstrate for the sucrose breakdown pathway in the potato tuber. We start with hierarchical modelling of the metabolic network as a Petri net and continue with the analysis of qualitative properties of the network. The results characterize the net structure and give insights into the complex net behaviour.     

A simple and highly accurate numerical differentiation method for sensitivity analysis of large-scale metabolic reaction systems

Numerical differentiation is known to be one of the most difficult numerical calculation methods to obtain reliable calculated values at all times. A simple numerical differentiation method using a combination of finite-difference formulas, derived by approximation of Taylor-series equations, is investigated in order to efficiently perform the sensitivity analysis of large-scale metabolic reaction systems. A result of the application to four basic mathematical functions reveals that the use of the eight-point differentiation formula with a non-dimensionalized stepsize close to 0.01 mostly provides more than 14 digits of accuracy in double precision for the numerical derivatives. Moreover, a result of the application to the modified TCA cycle model indicates that the numerical differentiation method gives the calculated values of steady-state metabolite concentrations within a range of round-off error and also makes it possible to transform the Michaelis-Menten equations into the S-system equations having the kinetic orders whose accuracies are mostly more than 14 significant digits. Because of the simple structure of the numerical differentiation formula and its promising high accuracy, it is evident that the present numerical differentiation method is useful for the analysis of large-scale metabolic reaction systems according to the systematic procedure of BST.     

Dynamic flux balance analysis of the metabolism of Saccharomyces cerevisiae during the shift from fully respirative or respirofermentative metabolic states to anaerobiosis

Dynamic flux balance analysis was utilized to simulate the metabolic behaviour of initially fully respirative and respirofermentative steady-state cultures of Saccharomyces?cerevisiae during sudden oxygen depletion. The hybrid model for the dynamic flux balance analysis included a stoichiometric genome-scale metabolic model as a static part and dynamic equations for the uptake of glucose and the cessation of respirative metabolism. The yeast consensus genome-scale metabolic model [Herrg?rd MJ et?al. (2008) Nat Biotechnol26, 1155-1160; Dobson PD et?al. (2010) BMC Syst Biol4, 145] was refined with respect to oxygen-dependent energy metabolism and further modified to reflect S.?cerevisiae anabolism in the absence of oxygen. Dynamic flux balance analysis captured well the essential features of the dynamic metabolic behaviour of S.?cerevisiae during adaptation to anaerobiosis. Modelling and simulation enabled the identification of short time-scale flux distribution dynamics under the transition to anaerobic metabolism, during which the specific growth rate was reduced, as well as longer time-scale process dynamics when the specific growth rate recovered. Expression of the metabolic genes was set into the context of the identified dynamics. Metabolic gene expression responses associated with the specific growth rate and with the cessation of respirative metabolism were distinguished.     

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.     

METAMOD: software for steady-state modelling and control analysis of metabolic pathways on the BBC microcomputer

METAMOD, a BBC microcomputer-based software package for steady-statemodelling and control analysis of model metabolic pathways,is described, The package consists of two programs. METADEFallows the user to define the pathway in terms of reactions,rate equations and initial concentrations of metabolites. METACALuses one of two algorithms to calculate the steady-state concentrationsand fluxes. One algorithm uses the current ratio of productionand consumption rates of variable metabolites to adjust iterativelytheir concentrations in such a way that they converge towardsthe steady state. The other algorithm solves the roots of thesystem equations by means of a quasi-Newtonian procedure. Controlanalysis allows the calculation of elasticity, control and responsecoefficients, by means of finite difference approximation. METAMODis interactive and easy to use, and suitable for teaching andresearch purposes. Received on January 17, 1986; accepted on June 2, 1986     

Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-kappaB signalling pathway

Mathematical modelling offers a variety of useful techniques to help in understanding the intrinsic behaviour of complex signal transduction networks. From the system engineering point of view, the dynamics of metabolic and signal transduction models can always be described by nonlinear ordinary differential equations (ODEs) following mass balance principles. Based on the state-space formulation, many methods from the area of automatic control can conveniently be applied to the modelling, analysis and design of cell networks. In the present study, dynamic sensitivity analysis is performed on a model of the IkappaB-NF-kappaB signal pathway system. Univariate analysis of the Euclidean-form overall sensitivities shows that only 8 out of the 64 parameters in the model have major influence on the nuclear NF-kappaB oscillations. The sensitivity matrix is then used to address correlation analysis, identifiability assessment and measurement set selection within the framework of least squares estimation and multivariate analysis. It is shown that certain pairs of parameters are exactly or highly correlated to each other in terms of their effects on the measured variables. The experimental design strategy provides guidance on which proteins should best be considered for measurement such that the unknown parameters can be estimated with the best statistical precision. The whole analysis scheme we describe provides efficient parameter estimation techniques for complex cell networks.     

Control of threonine pathway in E. coli. application to biotechnologies

Threonine is an essential amino acid for mammals and birds and an adequate supply is necessary for growth and maintenance. Its production has become the aim of metabolic bioengineering and genetic manipulations. We propose in this paper a rational approach for increasing threonine production in anE. coli strain based on metabolic control theory. We have derived a way to measure the control coefficients of threonine pathwayin vivo. The method consists in modelling the results of presteady-state experiments. Thein vivo concentrations and activities of the enzymes can then be measured and introduced into the model, so that thein vivo steady-state of the pathway can be evaluated. With such a model it is possible to calculate the theoretical values of the control coefficients of the threonine synthesis fluxin vivo.     

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.     

A gradual update method for simulating the steady-state solution of stiff differential equations in metabolic circuits

Numerical simulation of differential equation systems plays a major role in the understanding of how metabolic network models generate particular cellular functions. On the other hand, the classical and technical problems for stiff differential equations still remain to be solved, while many elegant algorithms have been presented. To relax the stiffness problem, we propose new practical methods: the gradual update of differential-algebraic equations based on gradual application of the steady-state approximation to stiff differential equations, and the gradual update of the initial values in differential-algebraic equations. These empirical methods show a high efficiency for simulating the steady-state solutions for the stiff differential equations that existing solvers alone cannot solve. They are effective in extending the applicability of dynamic simulation to biochemical network models. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A structured approach for the engineering of biochemical network models, illustrated for signalling pathways

Quantitative models of biochemical networks (signal transduction cascades, metabolic pathways, gene regulatory circuits) are a central component of modern systems biology. Building and managing these complex models is a major challenge that can benefit from the application of formal methods adopted from theoretical computing science. Here we provide a general introduction to the field of formal modelling, which emphasizes the intuitive biochemical basis of the modelling process, but is also accessible for an audience with a background in computing science and/or model engineering. We show how signal transduction cascades can be modelled in a modular fashion, using both a qualitative approach--qualitative Petri nets, and quantitative approaches--continuous Petri nets and ordinary differential equations (ODEs). We review the major elementary building blocks of a cellular signalling model, discuss which critical design decisions have to be made during model building, and present a number of novel computational tools that can help to explore alternative modular models in an easy and intuitive manner. These tools, which are based on Petri net theory, offer convenient ways of composing hierarchical ODE models, and permit a qualitative analysis of their behaviour. We illustrate the central concepts using signal transduction as our main example. The ultimate aim is to introduce a general approach that provides the foundations for a structured formal engineering of large-scale models of biochemical networks.