## INTRODUCTION

Plant metabolic networks are substantially more complex than those of other organisms. This is because of several interlinked aspects of plant life: being sessile, ectothermic and autotrophic, and having vast chemical repertoires and a high degree of subcellular compartmentation. Therefore, it is not surprising that metabolic engineering (especially in primary metabolism) has had a low success rate with, for example, single gene alterations usually resulting in little of the desired change in composition, yield or growth. The relationship between phenotype and genotype is also inherently complex because the functioning of individual proteins or even pathways depends on the operational state of the larger metabolic network (Kruger & Ratcliffe 2008; Moreno-Sanchez *et al.* 2008). Genetic manipulation, followed by phenotypic – even ‘omic’– analyses, is therefore also limited as an approach to understanding how plant metabolism works, and there is a clear need for tools to measure and model metabolic function (as distinct from metabolic components) at the network level.

Metabolic flux analysis (MFA) provides such tools. MFA quantifies the flow of material through metabolism, yielding flux maps and can aid engineering efforts by explaining phenotypes in detail. Experience in the metabolic engineering of improved productivity by bacteria, shows that MFA can make substantial contributions to biotechnology. Used in a cycle of genetic alteration followed by subsequent analysis, MFA has allowed bacterial strain improvement for industrial purposes (Kim, Kim & Lee 2008) by, for example, highlighting potentially wasteful metabolic processes (Petersen *et al.* 2000, 2001; Nielsen 2001; Koffas, Jung & Stephanopoulos 2003; Koffas & Stephanopoulos 2005). MFA has already yielded substantial new insights into the structure and function of plant metabolic networks and holds promise for guiding successful engineering for practical purposes in the coming years.

MFA approaches of interest here are those that focus on obtaining estimates and models of multiple fluxes through a metabolic network or more commonly, a sub-network, and include systems in steady state (a constant set of metabolic fluxes) and ones whose fluxes may be changing. MFA approaches can be divided into several categories that differ in the information required, the kind of model used and the type of information obtained and are outlined in Table 1. Whether the biological system can be evaluated in steady state, the (sub)network size and the level of detail with which reactions are described will determine which MFA approaches are best suited to each system. Flux analyses all begin with a reaction network description (set of equations) that stoichiometrically relate the substrates of each reaction to its products. Given that there are multiple routes through the network, the stoichiometric description represents the full range of feasible metabolic behaviours. Elementary mode analysis (EMA; Schuster, Dandekar & Fell 1999; Schuster, Fell & Dandekar 2000; Poolman, Fell & Raines 2003) and extreme pathway analysis (EPA; Schilling, Letscher & Palsson 2000) are structural methods that are used to explore this range and define the boundaries of feasible steady-state flux distributions.

Method | Conditions and information required | Information yielded | Technique reference For overviews see: Stephanopoulos et al. (1998), Ratcliffe & Shachar-Hill (2006) | Recent plant applications For overviews see: Libourel & Shachar-Hill (2008); Rios-Estepa & Lange (2007), Poolman et al. (2004) | |||||||
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Steady state conditions | Typical network size analysed | Stoichiometric matrix | Pool size measurements | Labelling measurements | Objective function | Net flux | Reversibility | Predictive | |||

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Although dynamic/kinetic metabolic flux analysis (MFA) and MCA do not explicitly use the stoichiometric matrix, the same information, in the form of network structure and reaction definitions is needed. (MCA is mathematically parallel to Biochemical Systems Theory, which uses a power-law description of kinetics for the reaction steps in a biological network; both representing methods of sensitivity analysis). Definitions: metabolic steady state, when pool sizes and fluxes are constant; stoichiometric matrix, matrix containing the number of molecules of each metabolite made or consumed in each metabolic or transport reaction in the netwosrk; pool size, the concentration of an intracellular metabolite; objective function, a goal that the system is believed to maximize or minimize (usually growth rate) used in flux balance analysis (FBA) to predict the net fluxes through a system; net flux, overall flow of material from one metabolite to another (forward minus reverse fluxes); reversibility, the degree to which material flows in both directions through a reaction; predictive model: one that allows the calculation of fluxes under conditions different from those of the experiment(s) already performed; elementary mode analysis (EMA) and extreme pathway analysis (EPA), two ways to structurally analyse a network to obtain a subset of the infinite number of possible flux patterns through the network that is sufficient to describe the full range of possible patterns. Each elementary mode is a minimal set of enzymes that could operate together at metabolic steady state; elementary modes represent a superset of extreme pathways because non-decomposable modes in the interior (i.e. not on the boundary of the admissible flux region) are included in EMA; FBA, constraint-based approach that uses an objective function to restrict the range of possible flux patterns; steady-state isotopic labelling MFA, often called ‘MFA’ by its practitioners, an analysis that uses measurements of isotopic labelling made in a system at metabolic steady state after labelling patterns are no longer changing, together with direct measurements of uptake and output fluxes to generate a map of net and exchange fluxes; instationary state MFA, same as steady-state isotopic labelling MFA, except that labelling patterns are changing, requires differential equation description rather than linear equations for isotopomer balances, metabolic steady state is maintained; kinetic/dynamic MFA, analysis of metabolic fluxes in either steady state or not, using time-dependent measurements of pool sizes and (usually) isotopic labelling to define a predictive model and can be used for metabolic control analysis (MCA); MCA, quantifies the control/sensitivity of individual fluxes and concentrations throughout a pathway or network. MCA can be applied with or without a full kinetic analysis and is usually applied to individual pathways or small sub-networks.
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Elementary mode analysis and extreme pathway analysis | Y | Full network | Y | N | N | N | Y | N | N | Schuster et al. (2000); Schilling et al. (2000) | Schwender et al. (2004a)Poolman et al. (2003)Rohwer & Botha (2001) |

Flux balance analysis | Y | Full network | Y | N | N | Y | Y | N | Y | Varma & Palsson (1994) | Boyle & Morgan (2009) Grafahrend-Belau et al. (2009)Schwender (2008) Shastri & Morgan (2005) |

Steady state isotopic labelling MFA | Y | 50–100 reactions | Y | N | Y | N | Y | Y | N | Zupke & Stephanopoulos (1994) Schmidt, Carlsen, Nielsen, Villadsen (1997) Wiechert et al. (2001) | Allen et al. (2009)Williams et al. (2008)Alonso et al. (2007a)Junker et al. (2007) |

Instationary state-MFA | Y/N | 50–100 reactions | Y | Y | Y | N | Y | Y | Y | Noh et al. (2007)Antoniewicz et al. (2007)Zhao et al. (2008) | Shastri & Morgan (2007) |

Kinetic/dynamic MFA | N | 10–50 reactions | N* | Y | Usually | N | Y | Y | Y | Reviewed in Steuer (2007) | Matsuda et al. (2007)Heinzle et al. (2007)Uys et al. (2007) |

Metabolic control analysis | N | 5–20 reactions | N* | N | N | N | N | N | Y | Kacser et al. (1995) Heinrich & Rapoport (1974) | Moreno-Sanchez et al. (2008) Rohwer & Botha (2001) |

At steady state, the range of flux patterns is finite but still very large. With estimates of input and output flux values and an objective function (i.e. an optimization that maximizes or minimizes some particular goal such as maximal biomass production), flux balance analysis (FBA; Varma & Palsson 1994) yields a set of net flux values from the feasible ‘flux space’. The branching of networks creates more unknown fluxes than stoichiometric relationships, resulting in an under-determined system; the degree of under-determination is especially high in primary metabolism. The use of linear programming with an objective function is necessary to limit this range of solutions. Still, the result of FBA may be more than one equally optimal flux solution set, and energy balance analysis (EBA) and thermodynamics-based MFA (TMFA) are used to constrain FBA by imposing further thermodynamic considerations on the network. Together, they enforce free energy rules (e.g. the decrease in free energy through the network) and reduce the number of solutions obtained. In general, EMA, EPA and FBA approaches are often applied to the full metabolic network (∼1000 reactions for a microbe). This is possible because only net fluxes are considered, no pool size or labelling measurements are needed for the metabolites, and because an acceptable outcome may entail multiple solutions. Experimentally based methods deal with sub-networks of different sizes (Table 1).

Incorporation of isotopic labelling data allows one to model the transition not only of metabolites, but also individual atoms through metabolism. Mapping the transition of atoms (almost always carbon) greatly enhances the information content for MFA. For steady-state isotopic labelling MFA, all fluxes are unvarying and labelling patterns in intermediates and products are allowed to reach stable values. Label distribution in the end products can then be used along with mass balances to determine fluxes throughout the network. Isotopic labelling-based steady-state MFA results in a mathematically over-determined system (for sub-networks of metabolism), with more information than flux parameters and allows both net and some exchange fluxes to be obtained without assuming an objective function. Fluxes are obtained by fitting their values in a stoichiometric model to the labelling data and uptake/efflux measurements through quadratic programming. For those tissues that can be maintained at or close to a metabolic steady state, this approach is attractive and indeed the term ‘MFA’ is currently often applied exclusively to steady-state analyses. Though steady-state MFA does not produce a predictive model, it is appealing because it yields flux maps without requiring measurements of metabolite pool sizes or the estimation of kinetic parameters, which are often difficult to obtain but which are required for dynamic (kinetic) MFA. Additionally, any set of reactions between branch points is combined into one step in steady-state MFA, which dramatically reduces the number of variables to be determined. In a kinetic/dynamic or unsteady-state approach, fluxes need not be constant (so metabolite pool sizes can change) and fluxes are established from time course measurements of pool sizes and labelling. The use of dynamic labelling experiments for single flux evaluations and for pathway elucidation is well established in plant biochemistry and has made enormous contributions, but is beyond the scope of this review. With dynamic MFA, a much larger number of independent parameters is used because individual enzymatic and transport steps are modelled, each involving multiple concentrations and rate constant values.

For steady-state MFA using labelling experiments, the system must be in a metabolic steady state long enough to reach isotopic steady state (a stable labelling pattern in metabolites). However, many plant tissues do not show steady-state metabolism and/or cannot be labelled to isotopic steady state under physiologically relevant conditions. In these cases, dynamic MFA is needed to quantify multiple fluxes though networks, and this approach has the further advantage of yielding models that can be used to predict the effects of genetic or other changes on metabolic fluxes and pool sizes (Morgan & Rhodes 2002; Poolman, Assmus & Fell 2004). Dynamic MFA is also important to identifying and analysing regulatory points within these pathways which is performed using metabolic control analysis (MCA; Rees & Hill 1994; Moreno-Sanchez *et al.* 2008). In MCA, the control of flux along pathways is quantitatively assigned to different enzymes (Heinrich & Rapoport 1974; Kacser & Burns 1981; Fell 1998). By revealing which enzymes maintain greatest control over flux, this method is a powerful predictive guide for metabolic engineering efforts. In Top Down Control Analysis (Hafner, Brown & Brand 1990), enzymatic reactions are grouped into blocks and the MCA analysis yields information on control of flux between, but not within those blocks.

Recent successes in MFA have been possible because of modern tools: nuclear magnetic resonance (NMR) and mass spectroscopies, the availability of a range of substrates positionally labelled with stable isotopes, and crucially, the development of modelling theory and computational methods. Detailed accounts of how different types of MFA analyses are performed using these tools are given in the literature: general overview (Stephanopoulos, Aristidou & Nielsen 1998), FBA (Schilling & Palsson 1998; Edwards, Covert & Palsson 2002), EMA (Schuster *et al.* 1999, 2000), EPA (Schilling *et al.* 2000), dynamic and steady-state MFA (Zupke & Stephanopoulos 1994; Schmidt *et al.* 1997; Wiechert *et al.* 2001; Ratcliffe & Shachar-Hill 2006; Rios-Estepa & Lange 2007; Steuer 2007) and MCA (Fell 1992; Kacser, Burns & Fell 1995). As shown in Fig. 1, the outcome of MFA may vary from a single set of flux values to a range of possible values in ‘flux space’. The degree to which a range or absolute set of values is established is defined by the number and scope of constraints and experimental measurements available, as well as the network complexity and choice of MFA strategy. Different ‘omics’ data can establish a feasible range of values for fluxes, but for networks of any complexity, a substantial number of functional measurements are necessary to determine flux values accurately. From an MFA perspective, ‘omics’ data serve as input and constraints for model building and solving. From an ‘omics’ perspective, the fluxome is simply another level of system-wide description. From a Systems Biology perspective, MFA is qualitatively different from ‘omics’ in its attempt to describe function rather than structure and its aim of building predictive models. In this sense of quantitative modelling to make predictions, MFA is more like informatics, though in the latter the models are statistical and in MFA they are mechanistic. Others have recently provided surveys and guides to plant-based MFA (Ratcliffe & Shachar-Hill 2006; Rios-Estepa & Lange 2007; Schwender 2008; Sweetlove, Fell & Fernie 2008). Here, we discuss a small number of studies to illustrate the contributions of MFA to our understanding of the complexities of plant metabolism and then focus on the challenges facing MFA of plant systems, strategies to overcome them and new methods for future work.