Modelling of metabolism attempts to improve our
understanding of metabolic regulation by quantifying essential parts or aspects of the metabolic system. Three areas in which modelling has recently made considerable contributions toward this aim can be identified. First, the more detailed description of individual reactions and pathways; second, the analysis of relative flux limitations within a pathway by means of metabolic control analysis; and third, in vivo flux analysis using nuclear magnetic resonance or mass spectroscopic analysis in combination with positionally labelled carbon compounds.
Addresses
Institut für Botanik der Technischen Universität Darmstadt, Schnittspahnstrasse 3-5, D-64287 Darmstadt, Germany; e-mail: giersch@bio.tu-darmstadt.de
Current Opinion in Plant Biology2000, 3:249–253 1369-5266/00/$ — see front matter
© 2000 Elsevier Science Ltd. All rights reserved.
Abbreviations
BPGS/Pase 2,3-bisphosphoglycerate synthase/phosphatase
NMR nuclear magnetic resonance
PFK phosphofructokinase
Rubisco ribulose 1,5-bisphosphate carboxylase/oxygenase
SBPase sedoheptulose 1,7-bisphosphatase
Introduction
Mathematical modelling is the art of describing the essen-tials of a system in mathematical terms. Although modelling is not a new technique and has been applied to plant biology for a long time, models of plant metabolism are still scarce. Among the 96 papers containing the key-word combination ‘model(l)ing’ and ‘metabolism’ found by a recent Current Contents search, 71 papers were on humans/animals, ten on microorganisms, nine on no spe-cific organism (e.g. NMR, modelling life based solely on physical attributes like metabolism and entropy reduc-tion), and just six on plants.
Modelling of plant metabolism has not been reviewed before in this series. Therefore, work published during the past three years (1997–1999) is covered in this review, with spe-cial emphasis on work published in 1999. Moreover, because the field of modelling is largely concept-driven, approaches, concepts and algorithms rather than individual models are at the centre of my review. Two recent reviews on related top-ics though with focus on the metabolic engineering of microorganisms, have been written by Bailey [1•] and
Nielsen [2]. A recent book by Gershenfeld [3] is a valuable, broad compendium of modelling methods and algorithms.
Plant metabolism includes not just biochemical processes but also light absorption and the subsequent processes of exciton and electron transfer, and the regulation of gas exchange between the leaf and the atmosphere. Because of
the limited space available, this review is focussed on the biochemistry of metabolism. Nevertheless, recent models of antenna organisation [4], induction of chlorophyll a fluores-cence [5], herbicide action [6], diffusive CO2exchange [7,8],
and an estimation of the diffusive resistance of bundle-sheath cells to CO2[9] are listed in the references.
The straightforward approach: translating
biochemistry into mathematics
To construct a metabolic model of a pathway of known structure, the kinetics of the individual enzymes must be measured and/or these data must be recovered from the lit-erature. The kinetic data together with data on the effects of co-factors, pH, ions and so on are used to parameterise the model. This straightforward type of modelling means translating biochemistry into mathematics. There are examples in which this approach has produced meaningful non-trivial results [10]. The quality of a model, however, depends essentially on the quality (i.e. completeness, and uniformity of the experimental material and conditions) of the data, and complete data sets of good quality are more the exception than the rule. Nevertheless, this classical reductionistic method (i.e. understanding metabolism by quantifying its components) has been used in the past and will continue to be used in the future.
The work by Pettersson [11] extends an earlier model of Calvin cycle metabolism [12] by including the oxygenase activity of ribulose 1,5-bisphosphate carboxylase/oxyge-nase (Rubisco). Their model is helpful in understanding the dependence of photosynthetic rate on CO2 and O2
concentrations, and on the gradients of inorganic phos-phate and triose phosphos-phates across the chloroplast envelope. Fridlyand and Scheibe [13] consider the impor-tance of turnover times (or pool sizes) of Calvin cycle intermediates in regulating metabolism. Whereas these two analyses are essentially restricted to the steady state, Poolman et al.[14•] have studied the control of the Calvin
cycle using a recently developed kinetic model. In this work [14•], an evolution-strategy algorithm was used to
maximise assimilation flux, whilst simultaneously min-imising total protein, and maintaining approximately equal triose phosphate export and starch synthesis fluxes. This strategy produced models in which control of assimilation was shared equally between Rubisco and sedoheptulose-1,7-bisphosphatase (SBPase). Interestingly, this optimised model was dynamically unstable, and the next best optima were found in areas in which either Rubisco or SBPase dominated assimilation. Two groups [15,16] studied mod-els of photosynthesis that describe the highly dynamic conditions found in sunflecks, that is brief time intervals for which the photon flux density exceeds the background shadelight. These models reliably reproduce the dynamics of photosynthesis in rapidly changing light.
Attempts to engineer metabolic pathways have led to the development of new models of the central metabolism in unicellular organisms. On the basis of their own data [17] and data retrieved from the literature, a detailed dynamic model of glycolysis in Saccharomyces cerevisiae was present-ed by Rizzi et al. [18]. In addition to the glycolytic pathway proper, Rizzi et al.[18] modelled mass transfer from the gas phase to the cell and mitochondrial metabolism. An impressive agreement between the measured time courses of intermediates following a short glucose pulse and the modelling results was obtained for most intermediates. Glycolysis in human erythrocytes was analysed in a series of three papers from Kuchel’s group [19•–21•]. (The
read-er may be surprised to find this refread-erence hread-ere, but this work is of general interest and may trigger comparable studies in plant biology.) In these papers [19•–21•], studies
of the 2,3-bisphosphoglycerate synthase/phosphatase (BPGS/Pase) reaction and its role in blood oxygen trans-port are retrans-ported. The experiments employ in vivo NMR and reveal that the Km of BPGS/Pase for 2,3-BPG is
sig-nificantly higher in vivo than in vitro [19•]. Using these
data, the kinetics of the BPGS/Pase reaction were mod-elled, and the model was subsequently employed as the core of a glycolysis model [20•]. With respect to kinetic
constants and the time course of metabolite concentra-tions, a convincing agreement between modelling results and published data was obtained, and topics for further model refinement were identified (e.g. phosphofructoki-nase [PFK] kinetics).
This work [20•] also highlights a reoccurring problem in
modelling: the work was initially motivated by inferred and observed discrepancies in BPGS/Pase kinetics between in vitro and in vivo studies, but the subsequently developed glycolysis model was formulated using mostly in vitro data. The in vitro data, however, may have little relevance to the in vivo situation. Thus, even such detailed and expert work, in the end, merely opens a door to another room (in this case probably the in vivokinetics of PFK). It is clear that modellers must live with this difference between in vitro measurements and in vivokinetics; hence, it is likely that model refinement will never come to a conclusive end.
Oscillations and chaos
Oscillations of biological systems (i.e. circadian rhythms, heartbeat and glycolytic oscillations) have attracted much experimental and theoretical interest. The study of oscilla-tions took a new turn when it was noted that oscillaoscilla-tions can be a first step on the route to chaotic phenomena. Glycolytic oscillations were among the first to be recognised as bio-chemical oscillations that can, via period doubling, lead to chaotic oscillations. Recently, period-doubling has been found also in higher plants: Shabala et al.[22] described clas-sical period-doubling bifurcations of bioelectric responses to light–dark cycles in several plants. Moreover, a period-dou-bling scenario was identified for oscillations of K+flux in the
mesophyll tissue of bean plants (S Shabala, personal com-munication). Lüttge and co-workers ([23] and references
therein) used an experimental analysis of transitions between regular and irregular (chaotic?) gas-exchange pat-terns in crassulacean acid metablism plants to develop models that describe the mechanism controlling the gas exchange oscillations. According to their models the mech-anism of oscillations resides in the way the malate flux across the tonoplast is triggered.
Metabolic control analysis
Metabolic control analysis is basically the analysis of the sensitivity of metabolic systems. Meanwhile, an impressive collection of definitions (e.g. elasticities, control coefficients, response coefficients, co-control coefficients) and relations among these quantities has been established. This may obscure the fact that metabolic control analysis is basically a straightforward sensitivity analysis whose appeal resides largely in the fact that, for an ideal metabolic pathway, dou-bling all enzyme concentrations will double the flux through the pathway and leave the concentrations of metabolites unchanged. In this context, ‘ideal’ means that each reaction rate (vi) in the pathway is proportional to the concentration
of the enzyme (ei) that catalyses that reaction.
Two books, devoted either partially [24] or completely [25] to metabolic control analysis, and a chapter in the recent edi-tion of a textbook on enzyme kinetics [26] document that metabolic control analysis has left the workbench. The review by Fell [27] should also be mentioned here as it pro-vides an excellent introduction to the field of metabolic control analysis. The proceedings of a recent workshop on metabolic control analysis [28] document that metabolic control analysis has found numerous interesting applica-tions. Only a few of these applications, however, are devoted explicitly to plant metabolism. In addition to the sensitivity studies of the Calvin cycle model [14•] (referred to above)
there has been a re-analysis of data on glycolysis in the tuber tissue of transgenic potato using control analysis [29,30]. This work provides evidence that the low control coefficient of PFK over glycolytic flux, for many decades considered to be the ‘bottleneck’ of glycolysis, is likely to be a conse-quence of PFK inhibition by phosphoenolpyruvate. However, this work also illuminates the problems of missing and scattered data encountered if control analysis is applied to real experimental data. Harwood et al.[31] were able to estimate the flux control coefficients of acetyl-CoA carboxy-lase and diacylglycerol acyltransferase over lipid synthesis, thereby taking the first step towards extending metabolic control analysis to lipid metabolism.
their approach is the ideal tool for describing metabolic systems for the purpose of metabolic engineering. Metabolic control analysis is restricted to analysis of the steady state. True steady states, however, never occur in real experiments and the issue of how well the quasi-steady state matches a true quasi-steady state always remains. This problem and its consequences for interpreting mea-sured control coefficients or elasticities are largely ignored by modellers and experimenters [34]. For this and other [35] reasons, it is not yet clear what the experimental side of metabolic control analysis will contribute to our under-standing of metabolism and its regulation in the long run.
On the other hand, metabolic control analysis is an excel-lent tool for describing and analysing the theoretical aspects of regulation. Thus, there is now general agree-ment that the concept of a single ‘bottleneck’ in a pathway is obsolete and should be replaced by quantification of the flux control coefficients. The expectation that regulation of physiological processes requires the co-ordinated regu-lation of numerous enzymes (in form of multisite modulations) [36] forms the basis of another recent advance that illuminates the integrative potential of meta-bolic control analysis; so does theoretical evidence that removal of the end-product can be a highly effective means by which to increase the pathway flux, certainly more effective than over-expressing single enzymes [37].
Flux analysis and elementary modes
If the map of a metabolic pathway is known then it may be possible to carry out a flux analysis that attempts to quan-tify intracellular fluxes by measuring all fluxes between the metabolic system and the suspension medium. There are situations in which this analysis has a unique solution (e.g. when the pathway is a chain with one branch point, and at least two fluxes can be measured). For slightly more complicated pathways, however, and especially if the metabolic pathway has cycles, not all intracellular fluxes can be determined from the external fluxes [38]. The mea-surement of fluxes of stoichiometrically coupled cofactors may sometimes be helpful in these circumstances, but the classical cofactors (e.g. ATP, ADP, NAD[P]H) are usually involved in a number of pathways (which would all have to be included in the analysis) and the stoichiometry of adenylate–pyridine nucleotide coupling can be variable. Experimental methods that allow the identification of fluxes in such an under-determined system are just emerg-ing and involve isotopic labellemerg-ing (see below).
Somehow related to flux analysis is the theoretical approach of decomposing metabolite networks into ‘ele-ments’. These elements are called ‘elementary modes’ [39,40•], that is, parts of the pathway that cannot be
decomposed further. Elementary modes can be applied, for example, to analyse how metabolic pathways operate when they are divided between different cellular compart-ments that are connected by a limited number of metabolite exchanges. When crassulacean acid metabolism
is subjected to this analysis, it is found that there are six distinct modes of operation, using different combinations of enzymes and transporters, and producing a different main product (DA Fell, personal communication). The software packages Metatool and Empath (see below) return a list of the elementary modes upon entering a metabolic pathway.
Positional carbon labelling for flux analysis
To overcome the shortfalls of flux analysis, isotopic tracers (mainly 13C) have been used and the positions of thetrac-er within individual carbon compounds have been measured by nuclear magnetic resonance (NMR) or mass spectroscopy [41,42]. In such NMR or mass spectroscopy analyses, individual isotopomeres are measured — a mole-cule with ncarbon atoms can occur in 2ndifferent 12C/13C
labelling patterns which are called isotopomers. Following the positional labelling, it is frequently (but not always) feasible to calculate the fluxes of the individual isotop-meres. Contrasting with the case for molecules, however, flux-balance equations for isotopomeres are nonlinear and cannot be solved analytically. So far, various numerical methods [43,44] have been employed to solve this prob-lem. Recently, it was shown that the equations can be solved analytically by defining ‘cumomers’ (from cum ula-tive isotopomer fraction). An isomer of a molecule is defined by indicating for any of its atoms whether that atom is labelled or not, whereas a cumomer is defined as being labelled in certain positions irrespective of whether or not it is labelled in other positions [45••,46•]. There is a
one-to-one correspondence between isotopomer and cumomer fractions. The flux-balance equations for the cumomers can be decomposed into linear subsets, which can be solved analytically in a sequence of about ten steps.
Software tools for metabolic modelling
A number of excellent tools are now available for developing and analysing metabolic models. GEPASI (http://www.ncgr. org/software/gepasi/index.html; developed by P Mendes) is specifically designed for the analysis of biochemical systems. This software package calculates the control coefficients and elasticities of biochemical systems, and includes various opti-misation algorithms [47]. Two other software tools with comparable scope are SCAMP (http://www.brookes.ac.uk/ bms/research/molcell/fell/mca_rg/sware.html#SCAMP; developed by HM Sauro [48]), and DBSolve (http://web-sites.ntl.com/~igor.goryanin; developed recently by I Goryanin and others [49]). The software tools Empath (ftp://bmshuxley.brookes.ac.uk/pub/mca/software/ibmpc/em path/) and Metatool (ftp://bmshuxley.brookes.ac.uk/pub/ mca/software/ibmpc/metatool/) calculate elementary modes.
tools such as Maple (http://www.maplesoft.com), Mathcad (http://www.mathcad.com), Mathematica (http:// www.mathematica.com) or Matlab (http://www.math-works.com).
Although these tools are helpful in setting up and analysing metabolic models, the Kyoto Encyclopaedia of Genes and Genomes (KEGG) [50•] takes a step further by
providing a database of all known metabolic pathways (in the form of maps). Combined with genomic sequence data, this project aims to facilitate pathway computation using both genomic and pathway data.
Conclusions
Only few specific models of plant metabolism have been considered in this review. This is only partially the conse-quence of my preference for concepts; rather my impression is that the scarcity of new significant metabolic models reflects a change in the role of modelling. In the past, mod-elling of plant metabolism mostly attempted to create a mathematical plant or at least some parts thereof. Because the existing models of plant metabolism give only partial and limited answers, and because new concepts (e.g. control analysis, flux analysis and optimisation [24,47,51]) and espe-cially new experimental methods (e.g. tracer NMR) are being developed, we can expect the development of a new generation of models. It will be some time before these appear in large numbers, as the new methods must first be proven with simpler systems (e.g. microorganisms) before being applied to complex organisms such as plants.
Acknowledgements
I am grateful to David Fell, Klaus Mauch, Mark Poolman, Sergey Shabala and Wolfgang Wiechert for providing unpublished material and/or for comments on drafts of this manuscript. Part of this review was written while I was a visiting fellow with Graham Farquhar and Murray Badger at the Research School of Biological Sciences, ANU, Canberra. The financial support of the School is gratefully acknowledged.
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