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Mathematical modelling of metabolism
Christoph Giersch
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 Biology 2000, 3:249–253
1369-5266/00/$ — see front matter
© 2000 Elsevier Science Ltd. All rights reserved.
Abbreviations
BPGS/Pase
NMR
PFK
Rubisco
SBPase
2,3-bisphosphoglycerate synthase/phosphatase
nuclear magnetic resonance
phosphofructokinase
ribulose 1,5-bisphosphate carboxylase/oxygenase
sedoheptulose 1,7-bisphosphatase
the limited space available, this review is focussed on the
biochemistry of metabolism. Nevertheless, recent models of
antenna organisation [4], induction of chlorophyll a fluorescence [5], herbicide action [6], diffusive CO2 exchange [7,8],
and an estimation of the diffusive resistance of bundlesheath 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 literature. 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.
Introduction
Mathematical modelling is the art of describing the essentials 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 keyword 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 specific organism (e.g. NMR, modelling life based solely on
physical attributes like metabolism and entropy reduction), 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 special 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 topics 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 work by Pettersson [11] extends an earlier model of
Calvin cycle metabolism [12] by including the oxygenase
activity of ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco). Their model is helpful in understanding
the dependence of photosynthetic rate on CO2 and O2
concentrations, and on the gradients of inorganic phosphate and triose phosphates across the chloroplast
envelope. Fridlyand and Scheibe [13] consider the importance 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 minimising 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 sedoheptulose1,7-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 models 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.
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Physiology and metabolism
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 presented 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 reader may be surprised to find this reference here, 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 transport are reported. The experiments employ in vivo NMR
and reveal that the Km of BPGS/Pase for 2,3-BPG is significantly higher in vivo than in vitro [19•]. Using these
data, the kinetics of the BPGS/Pase reaction were modelled, 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 concentrations, a convincing agreement between modelling results
and published data was obtained, and topics for further
model refinement were identified (e.g. phosphofructokinase [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 vivo kinetics of PFK). It is clear that
modellers must live with this difference between in vitro
measurements and in vivo kinetics; 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 oscillations took a new turn when it was noted that oscillations can
be a first step on the route to chaotic phenomena. Glycolytic
oscillations were among the first to be recognised as biochemical 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 classical period-doubling bifurcations of bioelectric responses to
light–dark cycles in several plants. Moreover, a period-doubling scenario was identified for oscillations of K+ flux in the
mesophyll tissue of bean plants (S Shabala, personal communication). Lüttge and co-workers ([23] and references
therein) used an experimental analysis of transitions
between regular and irregular (chaotic?) gas-exchange patterns in crassulacean acid metablism plants to develop
models that describe the mechanism controlling the gas
exchange oscillations. According to their models the mechanism 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, doubling 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 edition 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 provides 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 applications. 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 consequence 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 carboxylase and diacylglycerol acyltransferase over lipid synthesis,
thereby taking the first step towards extending metabolic
control analysis to lipid metabolism.
Metabolic control analysis is a local linear approach that
produces meaningless results if used to describe the effect
of large changes in model parameters. Large changes, however, are the focus of interest of genetic manipulation and
metabolic engineering studies. Hatzimanikatis and Bailey
[32] present a (log)linear approach that uses ‘the best of
both worlds’ (i.e. linearity from metabolic control analysis,
parameters that are valid over a wide range from biochemical systems theory [33]). It remains to be seen whether
Mathematical modelling of metabolism Giersch
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 quasisteady state matches a true steady state always remains.
This problem and its consequences for interpreting measured 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 understanding of metabolism and its regulation in the long run.
On the other hand, metabolic control analysis is an excellent tool for describing and analysing the theoretical
aspects of regulation. Thus, there is now general agreement 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 regulation of numerous enzymes (in form of multisite
modulations) [36] forms the basis of another recent
advance that illuminates the integrative potential of metabolic 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 quantify 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 measurement 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 emerging and involve isotopic labelling (see below).
Somehow related to flux analysis is the theoretical
approach of decomposing metabolite networks into ‘elements’. 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 compartments that are connected by a limited number of
metabolite exchanges. When crassulacean acid metabolism
251
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 the tracer 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 molecule with n carbon atoms can occur in 2n different 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 isotopmeres. 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 problem. Recently, it was shown that the equations can be
solved analytically by defining ‘cumomers’ (from cumulative 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 optimisation 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://websites.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.
The ModelMaker software (Cherwell Scientific, Oxford,
http://www.modelmanager.com/) is a commercial product
that allows graphical input into the model, and features
optimisation and fitting algorithms. I found it quite useful
for small models (up to about six variables). If flexibility is
important, it may be the best to use algebraic software
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Physiology and metabolism
tools such as Maple (http://www.maplesoft.com),
Mathcad (http://www.mathcad.com), Mathematica (http://
www.mathematica.com) or Matlab (http://www.mathworks.com).
6.
Lazar D, Brokes M, Naus J, Dvorak L: Mathematical modelling of
3-(3′′,4′′-dichlorophenyl)-1,1-dimethylurea action in plant leaves.
J Theor Biol 1998, 191:79-86.
7.
Aalto T, Vesala T, Mattila T, Simbierowicz P, Hari P: A three-dimensional
stomatal CO2 exchange model including gaseous phase and leaf
mesophyll separated by irregular interface. J Theor Biol 1999,
196:115-128.
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.
8.
Lushnikov AA, Ahonen T, Vesala T, Juurola E, Nikinmaa E, Hari P:
Modelling of light-driven RuBP regeneration, carboxylation and
CO2 diffusion for leaf photosynthesis. J Theor Biol 1997,
188:143-151.
9.
He D, Edwards GE: Estimation of diffusive resistance of bundle
sheath cells to CO2 from modeling of C4 photosynthesis.
Photosynth Res 1996, 49:195-208.
Conclusions
Only few specific models of plant metabolism have been
considered in this review. This is only partially the consequence 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, modelling 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 especially 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.
References and recommended reading
Papers of particular interest, published within the annual period of review,
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• of special interest
•• of outstanding interest
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•
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•
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results more convincing by far than could have been the case for conventional kinetic modelling alone.
15. Pearcy RW, Gross LJ, He D: An improved dynamic model of
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Mathematical modelling of metabolism
Christoph Giersch
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 Biology 2000, 3:249–253
1369-5266/00/$ — see front matter
© 2000 Elsevier Science Ltd. All rights reserved.
Abbreviations
BPGS/Pase
NMR
PFK
Rubisco
SBPase
2,3-bisphosphoglycerate synthase/phosphatase
nuclear magnetic resonance
phosphofructokinase
ribulose 1,5-bisphosphate carboxylase/oxygenase
sedoheptulose 1,7-bisphosphatase
the limited space available, this review is focussed on the
biochemistry of metabolism. Nevertheless, recent models of
antenna organisation [4], induction of chlorophyll a fluorescence [5], herbicide action [6], diffusive CO2 exchange [7,8],
and an estimation of the diffusive resistance of bundlesheath 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 literature. 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.
Introduction
Mathematical modelling is the art of describing the essentials 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 keyword 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 specific organism (e.g. NMR, modelling life based solely on
physical attributes like metabolism and entropy reduction), 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 special 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 topics 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 work by Pettersson [11] extends an earlier model of
Calvin cycle metabolism [12] by including the oxygenase
activity of ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco). Their model is helpful in understanding
the dependence of photosynthetic rate on CO2 and O2
concentrations, and on the gradients of inorganic phosphate and triose phosphates across the chloroplast
envelope. Fridlyand and Scheibe [13] consider the importance 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 minimising 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 sedoheptulose1,7-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 models 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.
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Physiology and metabolism
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 presented 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 reader may be surprised to find this reference here, 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 transport are reported. The experiments employ in vivo NMR
and reveal that the Km of BPGS/Pase for 2,3-BPG is significantly higher in vivo than in vitro [19•]. Using these
data, the kinetics of the BPGS/Pase reaction were modelled, 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 concentrations, a convincing agreement between modelling results
and published data was obtained, and topics for further
model refinement were identified (e.g. phosphofructokinase [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 vivo kinetics of PFK). It is clear that
modellers must live with this difference between in vitro
measurements and in vivo kinetics; 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 oscillations took a new turn when it was noted that oscillations can
be a first step on the route to chaotic phenomena. Glycolytic
oscillations were among the first to be recognised as biochemical 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 classical period-doubling bifurcations of bioelectric responses to
light–dark cycles in several plants. Moreover, a period-doubling scenario was identified for oscillations of K+ flux in the
mesophyll tissue of bean plants (S Shabala, personal communication). Lüttge and co-workers ([23] and references
therein) used an experimental analysis of transitions
between regular and irregular (chaotic?) gas-exchange patterns in crassulacean acid metablism plants to develop
models that describe the mechanism controlling the gas
exchange oscillations. According to their models the mechanism 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, doubling 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 edition 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 provides 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 applications. 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 consequence 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 carboxylase and diacylglycerol acyltransferase over lipid synthesis,
thereby taking the first step towards extending metabolic
control analysis to lipid metabolism.
Metabolic control analysis is a local linear approach that
produces meaningless results if used to describe the effect
of large changes in model parameters. Large changes, however, are the focus of interest of genetic manipulation and
metabolic engineering studies. Hatzimanikatis and Bailey
[32] present a (log)linear approach that uses ‘the best of
both worlds’ (i.e. linearity from metabolic control analysis,
parameters that are valid over a wide range from biochemical systems theory [33]). It remains to be seen whether
Mathematical modelling of metabolism Giersch
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 quasisteady state matches a true steady state always remains.
This problem and its consequences for interpreting measured 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 understanding of metabolism and its regulation in the long run.
On the other hand, metabolic control analysis is an excellent tool for describing and analysing the theoretical
aspects of regulation. Thus, there is now general agreement 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 regulation of numerous enzymes (in form of multisite
modulations) [36] forms the basis of another recent
advance that illuminates the integrative potential of metabolic 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 quantify 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 measurement 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 emerging and involve isotopic labelling (see below).
Somehow related to flux analysis is the theoretical
approach of decomposing metabolite networks into ‘elements’. 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 compartments that are connected by a limited number of
metabolite exchanges. When crassulacean acid metabolism
251
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 the tracer 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 molecule with n carbon atoms can occur in 2n different 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 isotopmeres. 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 problem. Recently, it was shown that the equations can be
solved analytically by defining ‘cumomers’ (from cumulative 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 optimisation 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://websites.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.
The ModelMaker software (Cherwell Scientific, Oxford,
http://www.modelmanager.com/) is a commercial product
that allows graphical input into the model, and features
optimisation and fitting algorithms. I found it quite useful
for small models (up to about six variables). If flexibility is
important, it may be the best to use algebraic software
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Physiology and metabolism
tools such as Maple (http://www.maplesoft.com),
Mathcad (http://www.mathcad.com), Mathematica (http://
www.mathematica.com) or Matlab (http://www.mathworks.com).
6.
Lazar D, Brokes M, Naus J, Dvorak L: Mathematical modelling of
3-(3′′,4′′-dichlorophenyl)-1,1-dimethylurea action in plant leaves.
J Theor Biol 1998, 191:79-86.
7.
Aalto T, Vesala T, Mattila T, Simbierowicz P, Hari P: A three-dimensional
stomatal CO2 exchange model including gaseous phase and leaf
mesophyll separated by irregular interface. J Theor Biol 1999,
196:115-128.
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.
8.
Lushnikov AA, Ahonen T, Vesala T, Juurola E, Nikinmaa E, Hari P:
Modelling of light-driven RuBP regeneration, carboxylation and
CO2 diffusion for leaf photosynthesis. J Theor Biol 1997,
188:143-151.
9.
He D, Edwards GE: Estimation of diffusive resistance of bundle
sheath cells to CO2 from modeling of C4 photosynthesis.
Photosynth Res 1996, 49:195-208.
Conclusions
Only few specific models of plant metabolism have been
considered in this review. This is only partially the consequence 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, modelling 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 especially 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.
References and recommended reading
Papers of particular interest, published within the annual period of review,
have been highlighted as:
• of special interest
•• of outstanding interest
1.
•
Bailey JE: Mathematical modeling and analysis of biochemical
engineering: past accomplishments and future opportunities.
Biotechnol Prog 1998, 14:8-20.
This is a vivid, personal review of mathematical modelling that addresses the
question of why models are useful in the modelling of metabolism. A hierarchy of modelling approaches in metabolic studies is also presented and a
pronounced gap between theory and practice is identified.
10. Cornish-Bowden A, Eisenthal R: Computer simulation as a tool for
studying metabolism and drug design. In Technological and
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12. Pettersson G, Ryde-Pettersson U: A mathematical model of the
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14. Poolman MG, Fell DA, Thomas S: Modelling photosynthesis and its
•
control. J Exp Botany 2000, 51:319-328.
This paper presents a fresh attempt to model the Calvin cycle and associated reactions by combining kinetic modelling with optimisation and metabolic control analysis. The combination of these methods make the modelling
results more convincing by far than could have been the case for conventional kinetic modelling alone.
15. Pearcy RW, Gross LJ, He D: An improved dynamic model of
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16. Kirschbaum MUF, Küppers M, Schneider H, Giersch C, Noe S:
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•
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