Pyruvate metabolism in rat liver mitochondria What 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 P i . Furthermore, it is well known that mitochondria take up Ca 2+ 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 programming 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. 6244 FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works previous 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- 14 C] 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. No Reaction Enzyme(s) EC No 1{}fi Pyruvate Influx 2 Pyruvate + CoASH + NAD + fi Acetyl-CoA + NADH + H + +CO 2 Pyruvate dehydrogenase 1.2.4.1 3 2 Acetyl-CoA fi Acetoacetate +2 CoASH Acetyltransferase, AcetoacetylCoA 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. 6 Acetyl-CoA + Oxaloacetate + H 2 O fi Citrate + CoASH Citrate synthase 2.3.3.1. 7 Pyruvate + ATP + CO 2 fi Oxaloacetate + ADP + P i Pyruvate carboxylase 6.4.1.1. 8 Citrate fi Isocitrate Aconitate hydratase 4.2.1.3 9 Isocitrate fi Citrate Aconitate hydratase 4.2.1.3. 10 Isocitrate + NAD + fi 2 -Oxoglutarate + NADH + H + +CO 2 Isocitrate dehydrogenase 1.1.1.41. 11 2 -Oxoglutarate + NAD + + CoASH fi Succinyl-CoA + NADH + H + +CO 2 Oxoglutarate dehydrogenase 1.2.4.2. 12 SuccinylCoA + GDP + P i fi Succinate + GTP + CoASH Succinyl-CoA synthetase 6.2.1.4. 13 Succinate + FAD fi Fumarate + FADH 2 Succinate dehydrogenase 1.3.99.1. 14 Fumarate + H 2 O fi Malate Fumarase 4.2.1.2. 15 Malate fi Fumarate + H 2 O Fumarase 4.2.1.2. 16 Malate + NAD + fi Oxaloacetate + NADH + H + Malate dehydrogenase 1.1.1.37. 17 Oxaloacetate + NADH + H + fi Malate + NAD + Malate dehydrogenase 1.1.1.37. 18 3 ADP + 3 P i + NADH + H + +1⁄ 2O 2 fi 3 ATP + NAD + +H 2 O Oxidative phosphorylation 19 2 ADP + 2 P i + FADH 2 +1⁄ 2O 2 fi 2 ATP + FAD + H 2 O Oxidative phosphorylation 20 ADP + GTP fi ATP + GDP Nucleoside-diphosphate kinase 2.7.4.6. 21 ATP fi ADP + P i ,ATPase’ 22 Acetoacetate fi { } Efflux 23 3 -OH-Butyrate fi { } Efflux 24 Citrate fi { } Efflux 25 Isocitrate fi { } Efflux 26 2 -Oxoglutarate fi { } Efflux 27 Succinate fi { } Efflux 28 Fumarate fi { } Efflux 29 Malate fi { } Efflux 30 Oxaloacetate fi { } Efflux J. W. Stucki and R. Urbanczik Pyruvate metabolism in rat liver mitochondria FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works 6245 program 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 polytope 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 P i . 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 P i ,O 2 ,H 2 O and CO 2 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 CO 2 and H 2 O but consumes pyruvate and dissipates ATP in reaction 21. Because, as mentioned above we are ignoring CO 2 , H 2 O and P i , 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 Pyruvate metabolism in rat liver mitochondria J. W. Stucki and R. Urbanczik 6246 FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works free-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 ()160000000015), showing that 16 PYR fi 15 OAA is one of the RYP CcAoA A CACBBO ICI GKA AoCcuSC U S M UF LAM A AOCTI 6 01 172286111 05 9 81 87 43 81 82 83 54 05 9 3 1 1 6 0134 46 6 0 1 82 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 columns 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). J. W. Stucki and R. Urbanczik Pyruvate metabolism in rat liver mitochondria FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works 6247 conversions 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 h ðiÞ c  0 for i ¼ 1; ; m ð1Þ For example the experimentally observed conversions in Table 2 has c ¼ð20:88; 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 010 0 00000 0 001 0 00000 0 000 1 00000 0 000 0 10000 0 000 0 01000 0 000 0 00100 0 000 0 00010 0 000 0 00001 0 000 0 00000 1 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 B B B B B B B B B B @ 1 C C C C C C C C C C C C C C C C C C C C C A 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 mitochondrial protein) were incubated with pyruvate-2-[ 14 C] and concentrations were measured after incubation at 37 °C for 10 min [10]. The pyru- vate used was calculated from the pyruvate added to the medium at the beginning and that found after 10 min of incubation. Extramitochondrial (used or found) lmol (10 min) Pyruvate used 20.88 Citrate found 4.56 Isocitrate found < 0.03 2-Oxoglutarate found 0.20 Succinate found 0.15 Fumarate found 0.91 Malate found 5.08 Oxaloacetate found < 0.03 Acetoacetate found 0.97 3-OH-Butyrate found 0.57 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. Pyruvate metabolism in rat liver mitochondria J. W. Stucki and R. Urbanczik 6248 FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works c ¼ðx; 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 x ¼ 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) c ¼ 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. P P reaction form Edge -160 000000015 16PYRfi 15 OAA + -1 0 0 0 0 0 0 0 0 0 PYR fi 0+ 00 000000 00 0fi 0 00 000000 00 0fi 0 00 000000 00 0fi 0 00 000000 00 0fi 0 -5 0 1 0 0 0 0 0 0 3 5 PYR fi BOB + 3 OAA + -2 0 1 0 0 0 0 0 0 0 2 PYR fi BOB + -8 1 0 0 0 0 0 0 0 6 8 PYR fi ACAC + 6 OAA + -2 1 0 0 0 0 0 0 0 0 2 PYR fi ACAC + -4 0 0 1 0 0 0 0 0 2 4 PYR fi CIT + 2 OAA + -2 0 0 1 0 0 0 0 0 0 2 PYR fi CIT + -4 0 0 0 1 0 0 0 0 2 4 PYR fi ICI + 2 OAA + -2 0 0 0 1 0 0 0 0 0 2 PYR fi ICI + -7 0 0 0 0 1 0 0 0 5 7 PYR fi AKG + 5 OAA + -2 0 0 0 0 1 0 0 0 0 2 PYR fi AKG + -110 00001009 11PYRfi 9 OAA + SUC + -2 0 0 0 0 0 1 0 0 0 2 PYR fi SUC + -130 000001011 13PYRfi FUM + 11 OAA -2 0 0 0 0 0 0 1 0 0 2 PYR fi FUM -130 000000111 13PYRfi MAL + 11 OAA -2 0 0 0 0 0 0 0 1 0 2 PYR fi MAL -19 0 0 0 0 0 0 0 15 0 19 PYR fi 15 MAL + -110 40000030 11PYRfi 4 BOB + 3 MAL + -7 2 0 0 0 0 0 0 3 0 7 PYR fi 2 ACAC + 3 MAL + -5 0 0 2 0 0 0 0 1 0 5 PYR fi 2 CIT + MAL + -5 0 0 0 2 0 0 0 1 0 5 PYR fi 2 ICI + MAL + -130 00040050 13PYRfi 4 AKG + 5 MAL + -170 00004090 17PYRfi 9 MAL + 4 SUC + -19 0 0 0 0 0 0 4 11 0 19 PYR fi 4 FUM + 11 MAL -19 0 0 0 0 0 0 15 0 0 19 PYR fi 15 FUM + -110 40000300 11PYRfi 4 BOB + 3 FUM + -7 2 0 0 0 0 0 3 0 0 7 PYR fi 2 ACAC + 3 FUM + -5 0 0 2 0 0 0 1 0 0 5 PYR fi 2 CIT + FUM + -5 0 0 0 2 0 0 1 0 0 5 PYR fi FUM + 2 ICI + -130 00040500 13PYRfi 4 AKG + 5 FUM + -170 00004900 17PYRfi 9 FUM + 4 SUC + PYR ACAC BOB CIT ICI AKG SUC FUM MAL OAA J. W. Stucki and R. Urbanczik Pyruvate metabolism in rat liver mitochondria FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works 6249 conversion 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 ab¼jja 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. No. Angle Conversion Experiment 1 10.4 5 PYR fi 2 CIT + MAL 2 18.4 5 PYR fi 2CIT+FUM 3 19.6 7 PYR fi 2 ACAC + 3 MAL 4 21.1 13 PYR fi 4 AKG + 5 MAL 5 21.7 11 PYR fi 4 BOB + 3 MAL 6 21.9 17 PYR fi 9 MAL + 4 SUC 7 24.8 5 PYR fi 2 ICI + MAL 8 27.5 19 PYR fi 15 MAL 9 28.1 11 PYR fi 4 BOB + 3 FUM 10 29.1 5 PYR fi FUM + 2 ICI 11 29.2 4 PYR fi CIT + 2 OAA 12 29.5 7 PYR fi 2 ACAC + 3 FUM 13 29.8 13 PYR fi 4 AKG + 5 FUM 14 32.8 17 PYR fi 9 FUM + 4 SUC 15 34.1 4 PYR fi ICI + 2 OAA 16 36.3 5 PYR fi BOB + 3 OAA 17 39.6 19 PYR fi 15 FUM 18 39.8 7 PYR fi AKG + 5 OAA 19 40.6 8 PYR fi ACAC + 6 OAA 20 42.9 11 PYR fi 9 OAA + SUC 21 46.2 16 PYR fi 15 OAA Free Flow 1 22.1 4 PYR fi CIT + 2 OAA 2 22.1 4 PYR fi ICI + 2 OAA 3 23.4 5 PYR fi BOB + 3 OAA 4 23.5 13 PYR fi 4 AKG + 5 MAL 5 23.8 11 PYR fi 4 BOB + 3 MAL 6 24.3 7 PYR fi 2 ACAC + 3 MAL 7 24.7 13 PYR fi 4 AKG + 5 FUM 8 24.7 11 PYR fi 4 BOB + 3 FUM 9 25.3 17 PYR fi 9 MAL + 4 SUC 10 25.5 7 PYR fi 2 ACAC + 3 FUM 11 25.6 5 PYR fi 2 CIT + MAL 12 25.6 5 PYR fi 2 ICI + MAL 13 26.2 5 PYR fi 2CIT+FUM 14 26.2 5 PYR fi FUM + 2 ICI 15 26.3 7 PYR fi AKG + 5 OAA 16 26.7 17 PYR fi 9 FUM + 4 SUC 17 27.5 8 PYR fi ACAC + 6 OAA 18 29.2 11 PYR fi 9 OAA + SUC 19 31.8 19 PYR fi 15 MAL 20 32.7 16 PYR fi 15 OAA 21 33.5 19 PYR fi 15 FUM 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. Pyruvate metabolism in rat liver mitochondria J. W. Stucki and R. Urbanczik 6250 FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works reactions for the external metabolites and replacing them with the single pseudoreaction 0:97 ACAC þ 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 )1 Æmg 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 (q ¼ 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 q ¼ 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 ACAC BOB AcCoA OAA CIT AKG SUCFUM 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 -1 Æmg 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. J. W. Stucki and R. Urbanczik Pyruvate metabolism in rat liver mitochondria FEBS Journal 272 (2005) 6244–6253 ª 2005 FEBS No claim to original US government works 6251 a degree of coupling of q ¼ 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 q ¼ 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. 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P i Pyruvate carboxylase 6.4.1.1. 8 Citrate fi Isocitrate Aconitate hydratase 4.2.1.3 9 Isocitrate fi Citrate Aconitate hydratase 4.2.1.3. 10 Isocitrate. isocitrate. Furthermore, there is a large difference in the cit- rate produced during incubation compared with the Pyruvate metabolism in rat liver mitochondria