Stucki, Department of Pharmacology, University of Bern, Friedbu¨hlstrasse 49, CH-3010 Bern, Switzerland Fax: +41 31 632 4992 Tel: +41 31 632 3281 E-mail: joerg.stucki@pki.unibe.ch Websit
Trang 1What is optimized at steady state?
Jo¨rg W Stucki and Robert Urbanczik
Department of Pharmacology, University of Bern, Switzerland
Mitochondria are generally regarded as the
power-house of the cell having the main task of supplying
energy in the form of ATP produced during oxidative
phosphorylation from ADP and Pi Furthermore, it is
well known that mitochondria take up Ca2+ ions in
an energy-dependent manner Other important
reac-tions are the Krebs cycle, oxidation of fatty acids and
the urea cycle [1,2] One may ask, which of these
reac-tions are optimized and play the most important role?
There are probably no dominant tasks in liver
mito-chondria and we are therefore faced with multiple
optimizations, depending on the cellular demands
Mitochondria from liver and kidney contain
pyru-vate carboxylase and are actively involved in
gluconeo-genesis [3] The aim of this study was to discover the
importance of this first step in gluconeogenenesis, and
special conditions were therefore chosen to allow us to study the carboxylation of pyruvate, the Krebs cycle and ketone body production by ignoring the many other reactions also present in these organelles In this sense, the model and experiments of mitochondrial pyruvate metabolism are biased and do not consider the typical intracellular environment of hepatocytes Genomics has led to construction of the stoichio-metry matrices of several simple organisms such as Escherichia coli and Saccharomyces cerevisiae Apply-ing linear programmApply-ing methods, researchers have analysed the optimizations of different goals, the most prominent being the maximization of cellular growth [4–7] It goes without saying that this cannot be the major task of liver mitochondria Therefore, we inves-tigated the optimization of metabolic functions In
Keywords
dominant reactions; mitochondria;
optimization of metabolism; pyruvate
metabolism; stoichiometric network analysis
Correspondence
J W Stucki, Department of Pharmacology,
University of Bern, Friedbu¨hlstrasse 49,
CH-3010 Bern, Switzerland
Fax: +41 31 632 4992
Tel: +41 31 632 3281
E-mail: joerg.stucki@pki.unibe.ch
Website: http://www.cx.unibe.ch/pki/
index.html
(Received 17 August 2005, revised 28
September 2005, accepted 4 October 2005)
doi:10.1111/j.1742-4658.2005.05005.x
A representative model of mitochondrial pyruvate metabolism was broken down into its extremal independent currents and compared with experimen-tal data obtained from liver mitochondria incubated with pyruvate as a substrate but in the absence of added adenosine diphosphate Assuming no regulation of enzymatic activities, the free-flow prediction for the output of the model shows large discrepancies with the experimental data To study the objective of the incubated mitochondria, we calculate the conversion cone of the model, which describes the possible input⁄ output behaviour of the network We demonstrate the consistency of the experimental data with the model because all measured data are within this cone Because they are close to the boundary of the cone, we deduce that pyruvate is converted very efficiently (93%) to produce the measured extramitochondrial metabolites We find that the main function of the incubated mitochondria
is the production of malate and citrate, supporting the anaplerotic path-ways in the cytosol, notably gluconeogenesis and fatty acid synthesis Finally, we show that the major flow through the enzymatic steps of the mitochondrial pyruvate metabolism can be reliably predicted based on the stoichiometric model plus the measured extramitochondrial products A major advantage of this method is that neither kinetic simulations nor radioactive tracers are needed
Abbreviations
ACAC, acetoacetate; AcCoA, acetyl-CoA; AKG, 2-oxoglutarate; BOB, 3-OH-butyrate; CIT, citrate; FUM, fumarate; ICI, isocitrate; MAL, malate; OAA, oxaloacetate; SucCoA, succinyl-CoA; SUC, succinate.
Trang 2previous studies we analysed the efficiency of oxidative
phosphorylation and found different degrees of
coup-ling [8], which could be regulated by the metabolic
states of the liver, for example, feeding and starvation
[9]
In this study, we investigated whether the first step
of gluconeogenesis may also be a possible target for
optimization Finally, we wanted to see whether any
reasonable predictions about the behaviour of the
sys-tem could be made from knowledge of the
stoichio-metry matrix alone without any experimental data or
additional assumptions in what we called the free-flow
system
Experimental data and computational
procedures
Our experimental data were taken from a previous
publication in which computer-simulated fluxes were
compared with experimentally measured values [10] These procedures are not repeated here because an exhaustive description already exists Suffice it to men-tion that sodium [2-14C] pyruvate was used as a sub-strate, and allowed measurement of the pertinent metabolic flows in incubated mitochondria from rat liver The model investigated was somewhat simplified
by omitting activation of the pyruvate carboxylase by acetyl-CoA and its inhibition by ADP in order to get
an idea of the general properties of the unconstrained free-flow system and compare it with incubated mito-chondria The remaining reactions considered are listed
in Table 1 From these reactions the stoichiometry matrix was set up and the extremal currents were cal-culated using the mathematica program [11] Note that this program can deal directly only with irrevers-ible reactions and yields all extremal currents as defined by Clarke [12] It is based on the Nullspace approach and it is, in fact, an early version of a
Table 1 Major reactions involved in mitochondrial pyruvate metabolism rat liver The reactions are taken from a previously published model [10] and simplified as described in the text In addition to the numbering scheme the reactions and the enzymes catalysing them are listed together with their corresponding EC numbers The extramitochondrial pool is indicated by curly brackets.
2 Pyruvate + CoASH + NAD + fi Acetyl-CoA + NADH + H + + CO2 Pyruvate dehydrogenase 1.2.4.1
hydrolase
2.3.1.9 3.1.2.11
4 Acetoacetate + NADH + H + fi 3 -OH-Butyrate + NAD + 3-Hydroxybutyrate dehydrogenase 1.1.1.30.
5 3 -OH-Butyrate + NAD + fi Acetoacetate + NADH + H + 3-Hydroxybutyrate dehydrogenase 1.1.1.30.
10 Isocitrate + NAD + fi 2 -Oxoglutarate + NADH + H + + CO2 Isocitrate dehydrogenase 1.1.1.41.
11 2 -Oxoglutarate + NAD + + CoASH fi Succinyl-CoA + NADH + H + + CO2 Oxoglutarate dehydrogenase 1.2.4.2.
16 Malate + NAD+fi Oxaloacetate + NADH + H +
18 3 ADP + 3 P i + NADH + H + + 1 ⁄ 2 O 2 fi 3 ATP + NAD + + H 2 O Oxidative phosphorylation
19 2 ADP + 2 P i + FADH 2 + 1 ⁄ 2 O 2 fi 2 ATP + FAD + H 2 O Oxidative phosphorylation
Trang 3program by Urbanczik and Wagner [13] that handles
reversible reactions without splitting them into two
irreversible ones
Elementary flux modes [14], by contrast to extremal
currents, are able to deal with both reversible and
irre-versible reactions The extremal currents represent the
full solution of the system, whereas elementary flux
modes are generally a subset of them Only when all
reactions are irreversible, are the elementary flux
modes identical to the extremal currents, otherwise
they are a projection from the extremal current
poly-tope into a polypoly-tope with a lower dimension Because
a proper transformation operator exists [13], one can
switch from one representation to the other without
losing information
In order to arrive at a consistent presentation,
we use the same abbreviations for the metabolites
throughout
The free-flow system
As a first step we wanted to get a general idea about
mitochondrial pyruvate metabolism without any
experimental information, and to verify what
conclu-sions could be drawn at that stage Table 1 lists the
reactions we considered for pyruvate metabolism in
isolated mitochondria from rat liver Several reactions
were omitted for clarity First, the regulation of
pyru-vate carboxylase by acetyl-CoA and ADP is ignored
Second, different exchangers such as the citrate–malate
antiporter and the malate–oxoglutarate exchanger were
also omitted The main reason for this was not only to
simplify the model, but also because
extramitochond-rial counter ions were not added to the incubation
medium in the experiment, with the exception of Pi
Hence, all metabolites leaving the mitochondria do so
by simple efflux The same applies to pyruvate uptake,
because its exact mechanism remains unclear The
advantage of this procedure is that it provides
infor-mation about the unconstrained flows possible in this
scheme In other words, this simplified model furnishes
an idea about the general behaviour of the model
Fur-thermore, the metabolites Pi, O2, H2O and CO2 were
treated as external molecules in large excess and thus
as being essentially constant Because we are not
inter-ested in tracking these four metabolites they were
ignored in the stoichiometry matrix
For the intramitochondrial reactions in Table 1, the
corresponding stoichiometry matrix was set up (not
shown) This matrix was then further processed to find
all extremal currents by using the program
mathemat-ica The resulting matrix of the extremal currents is
shown in Fig 1 The 30 reactions listed in Table 1
produced 37 extremal currents all fulfilling the steady-state condition Note that the algorithm used is based
on the Nullspace approach, which eliminates all inde-pendent species automatically Therefore conservation conditions like NADH + NAD+¼ constant could be ignored The same goes for the free and bound CoA The matrix in Fig 1 shows all 37 extremal currents possible for the reactions listed in Table 1 Column 1 shows the entry of pyruvate and the last nine columns represent the different outputs of the produced meta-bolites (reactions 22–30 in Table 1) Four of these extremal currents produce no output because they rep-resent reversible reactions For example, row 3 stands for the reversible reaction of the fumarase and row 5 represents the reversible conversion of citrate to iso-citrate Furthermore, row 2 represents the Krebs cycle, which generates no output except CO2 and H2O but consumes pyruvate and dissipates ATP in reaction 21 Because, as mentioned above we are ignoring CO2,
H2O and Pi, in our notation the Krebs cycle appears
as simply consuming pyruvate but generating no out-put
The numbers in Fig 1 are not easy to visualize They could, for example, be drawn as reaction dia-grams, as done previously [12,15] Here, by simply adding the numbers in each column of the extremal current matrix, we obtain the free-flow diagram shown
in Fig 2 This gives a general graphical picture of the mutual interconnections of the flows
Of course, assigning an equal weight of unity to every extremal current is of questionable physiological meaning However, in the absence of experimental data determining the true weights, constructing the free-flow system may still be the best thing one can do
The incubated mitochondria
How does the free-flow system compare with incuba-ted rat liver mitochondria? To this end, we used the experimental results from a previous publication in which mitochondria were incubated with 2-[14 C]-pyru-vate [10] This allowed measurement of all intramito-chondrial fluxes including the citric acid cycle and ketone body production The measurements of the extramitochondrial metabolites are given in Table 2 and are compared in Fig 3 together with the free flow data
Obviously, there is a discrepancy between the two data sets Notably, in the free-flow system there is a large production of oxaloacetate, whereas none was found in the incubation The same goes for isocitrate Furthermore, there is a large difference in the cit-rate produced during incubation compared with the
Trang 4free-flow system This emphasizes the need for the
experimental data to get closer to the realistic
beha-viour of mitochondrial pyruvate metabolism
The conversion cone
When describing the metabolic model by the cone of extremal currents, flows through the internal reactions play an important role However, the experimental results in Table 2 provide information about the flows through the extramitochondrial exchange reaction of the network only Hence, in analysing the model, we consider only conversions between the external metab-olites, in effect treating the mitochondria as a black box input⁄ output system
Obtaining a description of the possible input⁄ output behaviour is straightforward We delete the columns (2–21) in the current matrix that correspond to the internal reactions Then, as a matter of convention, we invert the sign of the first column, as this represents an input exchange The projected current matrix thus obtained, which we call P is shown in Table 3 For instance, from the first row of the current matrix we obtain the first row of P as ()16 0 0 0 0 0 0 0 0 15), showing that 16 PYRfi 15 OAA is one of the
R
I C I
G A
A o C c S C S M U
F
L
A
M
A
6
0
0 9
8
4
8 5
0
9
3
1
1
6
0 3
0
8
Fig 2 Free, unconstrained flows at steady state This diagram was
constructed by using the information contained in the extremal
cur-rents matrix in Fig 1 (see text).
Fig 1 Extremal currents for the reactions listed in Table 1 These were calculated using the MATHEMATICA program [11] Because the 30 col-umns of this matrix correspond exactly to the numbering scheme used in Table 1 (from left to right) the last nine colcol-umns correspond to the metabolites leaving the mitochondria Similarly column 1 represents pyruvate uptake Note that of the 37 extremal currents, 5 produce
no output measured in the experiment (see text).
Trang 5conversions between the external metabolites that the
network can perform
Each row of P, the projected current matrix, thus
describes an allowed input⁄ output behaviour, and
indeed any possible input⁄ output behaviour is
obtained as a linear combination (with non-negative
scalar coefficients) of these 37 rows of P Such linear
combinations generate a convex cone, which we call the conversion cone C Unfortunately, although the conversion cone is obtained from P, this method of describing C is too complicated It would be simpler if
we just knew the edges of the cone C or, alternatively,
if we had a description of C by a set of linear inequal-ities, i.e a list of m vectors h(i)such that a point c is in
C if and only if
hðiÞc 0 for i ¼ 1; ; m ð1Þ For example the experimentally observed conversions
in Table 2 has
c¼ ð20:88;0:97;0:57;4:56;0:01;0:20;0:15;0:91;5:08;0:01Þ: The mathematical techniques used to obtain the edges
of the conversion cone C as well as the h(i) from the projected extremal current matrix P have been des-cribed elsewhere [16] Here, we state the results It turned out, as shown in Table 3, that of the 37 rows in
P only 28 are actually edges of C
The h(i) yielding the inequalities representation of C, Eqn (1) are given by the rows of the following matrix H:
H ¼
PYR ACAC BOB CIT ICI AKG SUC FUM MAL OAA
1 2 2 2 2 2 2 1 1 1
15 24 27 28 28 25 21 19 19 16
0 B B B B B B B B B B B
1 C C C C C C C C C C C
It is worth noting that the first nine rows of H state the obvious fact that all metabolites except pyruvate can only be produced but never consumed by the network Only the last two rows of H give nontrivial constraints on the allowed input⁄ output behaviour Using the matrix H makes it very easy to determine whether a given 10-dimensional point c lies in the con-version cone We just test if Hc ‡ 0 One easily verifies that the experimental result in Table 2 passes this test Hence, the experimental data lie within the conversion cone, thus demonstrating the consistency of our meta-bolic model However, although the experimentally observed conversion is in the interior of C, it actually lies quite close to the boundary of the conversion cone This can be illustrated by replacing the measured value
of 20.88 lmoles for the consumption of pyruvate in Table 2 by a variable uptake x while keeping the out-put metabolites at the measured values, i.e choosing
Table 2 Experimentally measured metabolites in the
extramitoch-ondrial medium Mitochondria from rat liver (18 mg mitochextramitoch-ondrial
protein) were incubated with pyruvate-2-[14C] and concentrations
were measured after incubation at 37 C for 10 min [10] The
pyru-vate used was calculated from the pyrupyru-vate added to the medium
at the beginning and that found after 10 min of incubation.
Extramitochondrial (used or found) lmol (10 min)
Fig 3 Comparison of extramitochondrial metabolites in incubated
mitochondria and in the free-flow system The data in Fig 2 and
Table 2 were converted into percentage on the basis of the
pyru-vate entering the mitochondria and plotted together in a bar graph.
Trang 6c¼ ðx; 0:97; 0:57; 4:56; 0:01; 0:20; 0:15; 0:91; 5:08; 0:01Þ:
We may then ask, for which minimal value of x this
choice of c is still within the conversion cone This
minimal pyruvate uptake is found to be x¼
19.25 lmoles because for this choice the last inequality
in H holds as an equality whereas the other
inequalit-ies are still strictly satisfied The minimal value of
19.25 lmoles is near the observed value of
20.88 lmoles and therefore shows that the
mitochon-dria are using pyruvate close to optimally (93%) in producing the metabolites measured in the experiment The remaining 1.63 lmoles of pyruvate are used to dissipate energy, probably to regulate the degree of coupling of oxidative phosphorylation (see below)
As already mentioned, the inequality given by the last row h(11)of H is the first one to be violated when decreasing x from 20.88 lmoles Now, the equation
h(11)c¼ 0 defines a subset of the 10-dimensional con-version cone C, viz a nine-dimensional facet of the
Table 3 The P matrix with its edges defining the conversion cone C The P matrix is obtained from columns 1 and 22–30 from the current matrix as explained in the text This matrix was then further processed as described previously [16] to find the edges of the conversion cone
C generated by its rows The 28 rows, which are edges, are marked with a + in the right column For example, 2 PYR fi FUM is not an edge because a multiple of this conversion is obtained as a linear combination with positive scalar coefficients from 19 PYR fi 15 FUM and PYR fi 0.
PYR ACAC BOB CIT ICI AKG SUC FUM MAL OAA
Trang 7conversion cone This situation is shown schematically
in Fig 4
Because this facet is the one closest to the
experimen-tal results we shall further pinpoint the location of the
observed conversion by computing its angles to the 21
edges of the facet The angles a were calculated
accord-ing to the standard formula from vector algebra
a b¼ jj a jj jj b jj cos a ð2Þ where a are vectors from the P matrix belonging to the
facet and b is the vector of the experimental data In
Table 4 we list the angles thus obtained and we also
show the angles between the free flow conversions and
the 21 edges A perusal of the angles of the free flow
system show that they are more or less uniformly
spread between 22 and 33 degrees and that none is
close to an edge This means, that there is no reaction
really dominating in this system, and all occur with
more or less the same probability By contrast, in
incu-bated mitochondria one conversion is 10 degrees closer
to one edge and the farthest edge is 46 degrees away
This means that the transformation number 1, two
citrates and one malate formed from five pyruvates,
dominates all other reactions, because it is closest to
the experiment This is also in accordance with the
data shown in Table 2 in which malate and citrate are
the main products This does not mean, however, that
no other conversions contribute to these metabolites,
as is evident from Table 4 in which these products
occur in different conversions
Predicting internal flows from extramitochondrial measurements
Given that the experimental findings for the extra-mitochondrial flows are consistent with our model, it
is interesting to ask in how many ways the model can reproduce these findings To this end, we modified the reaction system in Table 1, removing the 10 exchange
Table 4 Angles between edges and conversions in the facet for the free flow system as well as for the incubated mitochondria The calculation of the angles (in degrees) is described in the text.
Experiment
Free Flow
Fig 4 Schematic sketch of the conversion cone C in three
dimen-sions The vector lying squarely in the interior of the cone is
analog-ous to the conversion given by the free-flow system The second
vector that is close to the boundary of the cone lying nearly on the
front left facet corresponds to the experimental result In contrast
to this three-dimensional sketch in reality the nine-dimensional
facet is delimited by 21 of the 28 edges of the conversion cone.
Trang 8reactions for the external metabolites and replacing
them with the single pseudoreaction
0:97 ACACþ 0:2 AKG þ 0:57 BOB þ 4:56 CIT
þ 0:91 FUM þ 0:01 ICI þ 5:08 MAL þ 0:01 OAA
þ 0:15 SUC ! 20:88 PYR
Note, that this pseudoreaction is the experimentally
observed conversion (Table 2) with the roles of input
and output interchanged A study of this modified
network reveals that it has only five extremal currents
The first four are the futile cycles observed in the
ori-ginal model and only the fifth extremal current has a
nonzero flow through the above pseudoreaction But
because all other extremal currents of the modified
model are futile cycles that, based on thermodynamic
considerations, cannot run in a steady state, the fifth
extremal current is the only way by which the model
can explain the behaviour observed in the experiment
It is surprising that only one extremal current dictates
the behaviour of the system The flows for some
reac-tions were measured previously [10], and the measured
values are compared with the prediction from our
model in Fig 5 This shows that the extremal current
reliably describes the major flows
Furthermore, in the extremal current reaction 21 the
‘ATPase’ dissipates 158.7 nmoles ATPÆmin)1Æmg
mito-chondrial protein)1 (not shown in Fig 5) As
men-tioned above, the utilization of 19.25 lmoles of
pyruvate exactly fulfils inequality h(11), i.e the point c
lies precisely on the facet Taking this limiting value of
pyruvate utilization instead of the measured one
sur-prisingly shows that the ‘ATPase’ vanishes completely,
and no ATP is then dissipated So at this limiting
point there can no longer be any flow through the
complete Krebs cycle Hence, in the experiment the
excess 1.63 lmoles are destroyed by the Krebs cycle
One might, therefore, speculate about the physiological
role of these 1.63 lmoles of pyruvate leading to the
observed dissipation of ATP
In a previous study [8], we investigated the optimal
degrees of coupling q of oxidative phosphorylation
and found that in all cases q must be smaller than 1
In other words, full coupling (q¼ 1) was incompatible
with optimal efficiency at finite speed of oxidative
phosphorylation In these experiments an external load
utilizing ATP was present; this is not the case here
Furthermore, the efficiency of oxidative
phosphoryla-tion was defined as output power divided by input
power, calculated from the input and output reactions
of the mitochondria treated as a black box These
ingredients are absent in our experiment, but can we
still say something about the efficiency and the degree
of coupling of oxidative phosphorylation in the present system?
Such estimation requires some assumptions First,
we have to assume that oxidative phosphorylation is working at optimal efficiency, i.e that conductance matching is fulfilled [8] Second, we need to estimate the efficiency of oxidative phosphorylation as the ratio
of ATP utilized (reaction 7) divided by ATP produced The latter quantity can be obtained by adding the flows through reactions 7 plus 21, because these are the only ones consuming ATP which first must have been produced Taking the limiting value of 19.25 lmoles pyruvate used yields an efficiency of 1 and a degree of coupling q¼ 1 In other words, oxida-tive phosphorylation is fully coupled under these circumstances, which is incompatible with optimal effi-ciency
By contrast, doing the same calculation for the measured pyruvate utilization in the experiment yields
an efficiency of oxidative phosphorylation of 0.30 and
PYR AcCoA ACAC BOB
AKG
SUC FUM
MAL
(63.2) 62.2
(67.2) 68.1
(47.5) 42.9
(19.5) 14.4
(18.5) 13,2
(15.7) 19.2
(3.7) 3.5
(17.6) 12.2 (11.4)
6.5
(19.7) 25.1
Fig 5 Predicted and measured flows through some of the reac-tions The numbers in brackets are the measured flows (in nmolÆmin-1Æmg protein-1) taken from Stucki and Walter [10] In addition to these, the predictions of the extremal current analysis described in the text are given The extremal current was normal-ized to obtain the experimentally determined pyruvate uptake One
of the key junctions in the pathway is given by the flow of oxalo-acetate either directly to malate or, alternatively, into the Krebs cycle via citrate The predicted flow from oxaloacetate to citrate deviates from the measured value by 10% In absolute terms this relative error corresponds to some 5 units, and this absolute differ-ence must stay essentially the same throughout the Krebs cycle because the outflows are predetermined But this constant abso-lute error leads to increasingly large relative errors in the down-stream part of the Krebs cycle since the magnitude of the flows in the cycle decreases due to the outflows of the intermediates.
Trang 9a degree of coupling of q¼ 0.849 This degree of
coupling is between the values of optimal power
out-put and optimal flow of ATP production in oxidative
phosphorylation [8] An independent check with the
data in Stucki and Walter [10] (Table 3), yields an
effi-ciency of 0.293 with a value of q¼ 0.838 These results
show that the theoretical predictions and the data
taken from the experiment yield very similar values
This indicates that 1.63 lmoles of pyruvate are used
for the regulation of the degree of coupling, provided
that the assumptions mentioned above are indeed
valid Note, that the ‘ATPase’ is not a clearly defined
chemical reactions because it contains slips and leaks
of oxidative phosphorylation as well as the breakdown
of intra- and extramitochondrial ATP by unknown
ATPases, Thus we are not able to identify a single
pro-cess that would be responsible for the regulation of the
degree of coupling
Concluding remarks
The main results of this study are: first, the
transfor-mation of five pyruvates into two citrates plus one
malate is the dominating reaction of the system; and
second, the conversion of pyruvate into its products is
nearly optimal with 93% efficiency Hence, only 7%
of pyruvate is used for the dissipation of ATP,
prob-ably to regulate the degree of coupling of oxidative
phosphorylation
Predicting the dominating reactions in a network
is difficult This study has shown that a free-flow
diagram, although yielding the correct factors for a
steady state, can say nothing about which reactions
are important and which are not, thus there is (yet)
no simple recipe of reducing an extremal currents
matrix to its essential parts It is the impression of
the authors that there exist only two reasonable
pro-cedures possible at present to solve this question:
(a) imposing or assuming external constraints or
(b) the measurement of metabolite turnover in vitro,
or better still in vivo, under different metabolic
conditions
To illustrate this difficulty, we consider the study by
Stelling et al [4] as an example Successful splitting of
a large metabolic network into its independent currents
or elementary modes usually yields too much
informa-tion Thus Klamt and Stelling found 507 632
element-ary flux modes in a stylised partial model of E coli
[17] and one might indeed ask how to proceed further
From a practical point of view, it is convenient to
con-centrate on the functional aspects only, as done here,
and restrict analysis to the input⁄ output relationship
Note that by constructing the conversion cone one
loses no information because all other, less interesting, details are contained in the current matrix
Assuming external constraints makes sense for autonomous organisms such as E coli or S cerevisiae One might then ask under what conditions there is maximal growth [4–7] or maximal production of eth-anol, for example This approach, however, fails com-pletely for organelles such as mitochondria, which are
an integral part of a cell in an organ such as the liver
As already mentioned, such biochemical entities are tightly integrated in a constantly changing cellular environment In other words, the major role of these organelles is to act as servants rather than as inde-pendent, autonomic entities Mitochondria not only have to produce ATP but they play also an essential part in anaplerotic functions Under certain circum-stances, in starved rats for example, it is reasonable that they can take part in glucose production by pro-ducing malate Malate contains not only the carbon moieties for glucose synthesis, but it also shuttles the reducing equivalents into the cytosol where it is needed for glucose synthesis [3]
In concluding, it is instructive to compare our approach for estimating intramitochondrial flows from experimental observations with the full dynamic mod-elling of the reaction system employed previously [10] Whereas the latter approach yields somewhat more accurate results, it is not only much more involved as
it requires more computer time for the simulations, but it also needs a more detailed knowledge of the reaction kinetics By contrast, our approach is straight-forward and quick, needing only a minimum of bench work, notably without the use of radioactive tracers
Acknowledgements
The Swiss National Science Foundation has supported this study It is a pleasure to thank Dr Clemens Wag-ner for helpful comments
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