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-cell
1Parallel Scientific Computing Institute/Computational Biology and Neurocomputing, Computer Science and Communication, Royal Institute of Technology, Stockholm, Sweden; 2Institute for Theoretical Biology, Humboldt University Berlin, Berlin, Germany; 3Department of Molecular Medicine, Endocrine and Diabetes Unit, Karolinska Institute and Hospital, Stockholm, Sweden; and 4Institute of Cancer Research and Molecular Medicine, Medical Faculty, Norwegian University of Science and Technology, Trondheim, Norway
Submitted 28 November 2005 ; accepted in final form 13 July 2006
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ABSTRACT |
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 TOP ABSTRACT GlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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-cells respond to an increased glycolytic flux by secreting insulin. The signal propagation goes via mitochondrial metabolism, which relays the signal to different routes. One route is an increased ATP production that, via ATP-sensitive K+ (KATP) channels, modulates the cell membrane potential to allow calcium influx, which triggers insulin secretion. There is also at least one other "amplifying" route whose nature is debated; possible candidates are cytosolic NADPH production or malonyl-CoA production. We have used mathematical modeling to analyze this relay system. The model comprises the mitochondrial NADH shuttles and the mitochondrial metabolism. We found robust signaling toward ATP, malonyl-CoA, and NADPH production. The signal toward NADPH production was particularly strong. Furthermore, the model reproduced the experimental findings that blocking the NADH shuttles attenuates the signaling to ATP production while retaining the rate of glucose oxidation (Eto K, Tsubamoto Y, Terauchi Y, Sugiyama T, Kishimoto T, Takahashi N, Yamauchi N, Kubota N, Murayama S, Aizawa T, Akanuma Y, Aizawa S, Kasai H, Yazaki Y, Kadowaki T. Science 283: 981–985, 1999) and provides an explanation for this apparent paradox. The model also predicts that the mitochondrial malate dehydrogenase reaction may proceed backward, toward malate production, if the activity of malic enzyme is sufficiently high. An increased fatty acid oxidation rate was found to attenuate the signaling strengths. This theoretical study has implications for our understanding of both the healthy and the diabetic -cell. systems biology; potassium-dependent adenosine triphosphate channel-independent pathway of insulin secretion; reduced nicotinamide adenine dinucleotide; diabetes; fatty acid oxidation
-cell is to secrete insulin into the blood in response to a raised blood glucose level. The -cell is thus crucial for maintaining blood glucose homeostasis. This is underscored by the fact that -cell disorders are an important factor behind non-insulin-dependent diabetes mellitus (type 2 diabetes). The signal transduction in the pancreatic -cell that mediates glucose-stimulated insulin secretion (GSIS) involves the relay of an increased glycolytic flux to a raised ATP-to-ADP ratio (ATP/ADP) and to at least one other yet-undetermined factor (23, 100). In this study, we have used mathematical modeling to explore this relay system theoretically and to evaluate the signal propagation from glucose to different putative candidates for the undetermined factor as well as to ATP production. A raised ATP/ADP ratio triggers insulin secretion via an increased intracellular calcium concentration. This is mediated by the closure of ATP/ADP-sensitive potassium channels, which results in the depolarization of the cell membrane and the opening of voltage-sensitive calcium channels (67). This mechanism is essentially necessary for insulin secretion and is referred to as the KATP-dependent pathway, or triggering pathway (40). The identity of the undetermined factor has been proposed to be either long-chain acyl-CoA (LC-CoA), NADPH, glutamate, or nucleotides (GTP, ATP, ADP) acting in some way to promote insulin secretion (100). Whatever the identity of this factor is, it is not alone sufficient for insulin secretion, but acts synergistically with the KATP-dependent pathway. This has been termed the KATP-independent pathway, or amplifying pathway (40).
Relaying involves the transfer of glycolytically produced carbon (pyruvate) and charge (NADH) to the mitochondria. NADH is transported to the mitochondrial matrix via the malate-aspartate (MA) shuttle (57) or directly to the respiratory chain via the glycerol-3-phosphate dehydrogenase (G3PDH) shuttle (56). An increased glycolytic flux is thus translated to a raised ATP/ADP ratio via an increased respiratory activity resulting from increased mitochondrial metabolism and shuttle activity. The relaying of the signal to the putative candidates for the KATP-independent pathway takes place via the mitochondrial metabolism (113). Cytosolic LC-CoA is proposed to increase via an increased citrate production. Citrate is cleaved by ATP-citrate synthase (ACS) to oxaloacetate and acetyl-CoA (ac-CoA), the latter of which is transformed to malonyl-CoA, which inhibits the mitochondrial carnitine palmitoyltransferase I transporter, causing LC-CoA to accumulate in the cytosol (23). NADPH is formed in the reactions catalyzed by cytosolic isocitrate dehydrogenase (IDHPc) and malic enzyme (ME) (59, 60). Glutamate has been proposed to be produced in the mitochondrial matrix by glutamate dehydrogenase (GDH) (64). Interestingly, ME and GDH catalyze, if operating in the directions suggested, cataplerotic reactions; i.e., they drain the tricarboxylic acid (TCA) cycle of carbons. Cataplerotic reactions have to be counterbalanced by anaplerotic reactions, and indeed the -cell has been shown to host a large quantity of the mitochondrial anaplerotic enzyme pyruvate carboxylase (PC) (59, 90).
One result from our modeling studies is that the glucose signal propagates strongly to ME and NADPH production, even at moderate ME activities. By doing so, the signal propagation to the respiratory chain weakens proportionally. The signal to ACS is more moderate but robust. Furthermore, there was a small net glutamate production in most scenarios evaluated here.
Another result concerns the findings of Eto et al. (26). Those investigators found that blocking of the NADH shuttles drastically impaired GSIS, whereas the glycolytic flux remained unchanged. Since the glycolysis demands continuous NADH reoxidation, this was taken as an indication of an unknown cytosolic factor reoxidizing the cytosolic NADH when the shuttles are blocked. We have simulated the experiments of Eto et al., and we show that it is not necessary to assume an unknown factor in order to explain their results.
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Glossary |
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TOPABSTRACTGlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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METHODS AND MODEL |
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TOPABSTRACTGlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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Model Description
The present model considers glycolysis as the influx and describes the NADH shuttles, TCA cycle, and outfluxes via respiratory NADH consumption, and NADPH, ac-CoA, and glutamate production. An earlier study (65) incorporated a rudimentary model for the energy coupling between the cytosolic and mitochondrial metabolism in the -cell; however, it is far too simplified to be used to address the questions posed in the present study. Other previous models concerning the mitochondrial shuttles and the TCA cycle (21, 44, 71) were for other cell types and do not include all enzymatic reactions relevant to -cell biochemistry.
Our model comprises the biochemical reactions drawn in Fig. 1. This system consists of 19 enzyme-catalyzed reactions and 10 metabolites represented by the numbers listed in Table 1. Glycolytic flux is controlled by the enzyme glyceraldehyde-3-phosphate dehydrogenase (GAPDH), which produces pyruvate and NADH. Pyruvate may be consumed by lactate dehydrogenase (LDH), reoxidizing NADH, or it is transported into the mitochondria and is either decarboxylated by pyruvate dehydrogenase (PDH), producing ac-CoA and NADH, or carboxylated by PC, producing oxaloacetate. Fatty acid oxidation (FO) also produces ac-CoA and NADH. The TCA cycle is represented by the reactions catalyzed by citrate synthase (CS; citrate is considered to be in quasi-equilibrium with isocitrate via the aconitase reaction), the isocitrate dehydrogenases (IDHs; NADPH-producing IDHs in the cytosol and mitochondria, IDHPc and IDHPm; and NADH-producing IDH in the mitochondria, IDHm), 2-oxoglutarate dehydrogenase or 
-ketoglutarate (OGDH; here we have for simplicity lumped the four reactions catalyzed by OGDH, succinate-CoA ligase, succinate dehydrogenase, and fumarate hydratase into a reaction block with a rate controlled by the physiologically irreversible enzyme OGDH), and mitochondrial malate dehydrogenase (MDHm). The NADH produced in the TCA cycle is reoxidized by respiration (Resp). Also, the reaction catalyzed by GDH is included in the model. We have assumed that the transports of pyruvate, citrate, isocitrate, 2-oxoglutarate, and malate across the mitochondrial inner membrane are fast compared with the enzymatic reactions (61) and thus are at quasi-equilibrium. Furthermore, alanine transaminase was not included in the model because it seems to have a low activity in -cells (97).
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-cell (59). Another cycle that has been suggested to be important in the -cell is the pyruvate-citrate shuttle, which involves the reactions catalyzed by ACS, MDHc, ME, PC, PDH, and CS (30). In this cycle, both the putative second messengers ac-CoA and NADPH are produced. However, in our analysis, we found the reaction cycle comprising PC, AATm, AATc, MDHc, and ME a cycle, which we denote the pyruvate-malate-aspartate (PMA) shuttle, to be the most prominent, as reported below.
Our model represents a module of the -cell biochemistry. By necessity, certain simplifications have to be made regarding the reactions on the boundary of the module. Strictly, the results will only hold for the module in isolation (i.e., in a constant surrounding). However, it is reasonable to believe that the behavior of the module is robust enough for an analysis of it to yield experimentally testable and meaningful predictions and insights. Important boundary parameters for the model assumed to be constant are glutamate and aspartate concentrations (affecting the AATs and GDH), ammonia concentration (affecting GDH), cytosolic ac-CoA/CoA ratio (affecting ACS), and factors affecting the respiratory rate. Also, we neglect the modulatory influence of adenine nucleotides on some enzymes (CS, IDH, GDH).
The dynamics of the model described graphically in Fig. 1 may be represented mathematically by a system of ordinary differential equations (ODEs), which is given in the APPENDIX. The different fluxes are described mathematically with rate equations, also given in the APPENDIX. Throughout this study, we use millimoles as the concentration unit, and millimoles per second as the unit for the rates of biochemical reactions, if nothing else is explicitly stated.
All rates are linearly dependent on the activities of the catalyzing enzymes, represented by the limiting rate V (often in the literature denoted Vmax). The limiting rates in our model were estimated from an extensive literature survey. The estimations are listed in Table 2, along with references to the literature. To transform the units given in the literature sources to our standard unit millimoles per second, either data directly from the same sources were used, or, if not given there, data from the review by Erecinska et al. (25) were used. Our approach was to examine different scenarios rather than a fixed parameter set, since living cells are variable chemical environments, enzyme concentrations are variable, and the experimental measurements are afflicted with significant measurement errors (1).
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We will begin our analysis of the model we have built up with an attempt to reproduce the results of Eto et al. (26). We will then proceed with a more general analysis of the signal propagation from the glycolytic flux to 1) the respiratory chain, 2) the cytosolic ac-CoA production by ACS, and 3) cytosolic NADPH production by ME and IDHPc. Glutamate production will also be discussed. To measure the coupling between the glycolytic signal and the four fluxes mentioned above, we introduce the gain, denoted , defined
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-cell. In this context, we used the gain as a measure of system response to a glucose stimulus. ATP production is measured as being proportional to the mitochondrial membrane potential . This is not a dynamic variable in our model, but we make a crude estimate and consider it to be proportional to jG3PDH and jResp, i.e.,
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from mitochondrial FADH2 production.
Cytosolic LC-CoA production is measured as the flux via ACS (jACS). Cytosolic NADPH is produced by ME and IDHPc; hence,
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Net mitochondrial glutamate production was measured as
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Finally, for the reproduction of the results of Eto et al. (26), we need the glucose oxidation rate jGO, which we measure as
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RESULTS |
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TOPABSTRACTGlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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As an integral part of the model-building process, we investigated the ability of our model to reproduce the findings of the important study by Eto et al. These authors investigated different aspects of mouse -cell metabolism in four genotypes. These were the four combinations that arise from the ability to treat wild-type (wt) and G3PDH knockout (mut) mice with AOA (aoa– or aoa+), which inhibits AATc and AATm.1 The main aspects of these experimental findings that we consider in the context of the present model are:
Mice exhibit the special trait of having no -cell ME activity (60); hence, we here set VME = 0. We defined the G3PDH mutant by letting VG3PDH = 0. We modeled the degree of AOA inhibition by reducing VAATc and VAATm by different percentages; in particular, the aoa+ genotype was defined by a 96% reduction of the limiting rates, as achieved in the experimental study of Eto et al. We then calculated corresponding steady-state solutions, i.e., the solutions to Eq. system A2.
Our starting point was to explore whether conditions 1 and 2 above may be fulfilled simultaneously in our model. Intuitively, it may seem contradictory that glucose oxidation is decoupled from 14CO2 production from [6-14C]glucose. This has even been considered to be "biochemically impossible" (91). Furthermore, we asked what keeps the cytosolic reoxidation of NADH going if the shuttles are blocked. LDH has been suggested (91), although it is uncertain whether the low activity of LDH in the -cell seen experimentally (7, 90, 93, 97) would suffice to handle the load normally taken care of by the two shuttles. This was also investigated theoretically here.
When checking for compliance with condition 1, we demanded that, for the mut and/or aoa+ genotypes, jGO be >90% of jGO for the wt/aoa– genotype. When checking for compliance to condition 2, we considered the 14CO2 production from the flux via the IDHs and OGDH, jDH, and demanded that jDH for the mut/aoa+ be <75% of jDH for the wt/aoa– genotype, while at the same time, jDH for the mut/aoa– and wt/aoa+ genotypes be >90% of jDH for the wt/aoa– genotype. We explored different scenarios characterized by different enzyme activities. Given a scenario where conditions 1 and 2 were fulfilled, we validated the model by checking whether conditions 3 and 4 also were fulfilled.
An exploration of the parameter space is, due to its dimensionality, by necessity not exhaustive. Our strategy here was first to restrict the investigation to a selected set of parameters. We chose to restrict it to a few thousand combinations of the limiting rates V of some of the enzymes. Either the enzymes chosen have an experimentally more poorly characterized V, or the system is sensitive to it in the sense that the enzymes have high flux control coefficients over jDH. We selected the parameters VFO and VACS because we were unable to find good recent measurements of these limiting rates [although FO may be estimated to exhibit a rate in the same order of magnitude as GO (14)]. VIDHPc was selected because of highly conflicting experimental results concerning the presence and activity of this enzyme in the -cell (60, 95). We further introduced another degree of freedom for the IDHs with the factor idhinh, which modulates all three IDH limiting rates, since the activity of IDHm is modulated by Ca2+ and since IDHPm exhibits a high flux control coefficient over jDH (cf. Table 3). VLDH was selected since available data on the limiting rate of this enzyme are of low resolution, pointing only to a generally low rate, and since one of the objectives of the study was to test whether LDH is needed to compensate for lost shuttle activity. Finally, we selected the parameter VOGDH due to a high flux control coefficient over jDH (see below and Table 3). We let each limiting rate take three to five values spanning many orders of magnitude, e.g., take the values 0, 0.001, and 0.01. For each enzyme, each value was assigned a category, used later in the statistical analysis of the results. Table 4 lists the values that we let the different limiting rates take, along with the categories.
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One of the combinations is marked in Table 4 with one asterisk (*). The other two combinations differed only in the value for VFO, which was 0 and 0.001 mM/s, respectively. The flux control coefficients over jDH at this parameter set are given in Table 3. With this parameter set, we took a bird's eye view of the successive inhibition of the AATs and calculated steady-state solutions, i.e., solutions to equation system A2, letting the inhibition of AATc and AATm rise from 0 to 100%. Figure 2 shows the results of these calculations. Figure 2A shows the steady-state glucose oxidation as a function of AOA inhibition. The dots represent a 96% inhibition of the AATs as achieved experimentally by Eto et al. To directly compare our results with that of the experimental study of Eto et al., the corresponding histogram, Fig. 2C, is presented. Figure 2, B and D, show the corresponding steady-state flux jDH as a function of AOA inhibition. The mut/aoa+ genotype exhibits a marked deviation in the steady-state jDH, which is the most significant deviation evident from the histograms in Fig. 2, C and D. The bird's eye view presented in Fig. 2B reveals that AOA affects jDH also in the wt genotype, although more moderately. It should be noted that, although our results are in excellent agreement with the experimental data considered (compare Fig. 2, B and D, in the study of Eto et al.), the parameter combination used to simulate the experiments should, due to the course-grained distribution of the varied parameters, be regarded only as an approximate pointer to a region in parameter space representing a scenario that agrees with experimental observations.
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for an increase of VGPDH from 0.005 to 0.025 mM/s. Again, the solid line represents the wt genotype, whereas the dashed line represents the mut genotype. We see that decreases in the mut genotype as the AAT inhibition approaches 100%. This may be considered as a validation of condition 3, although one has to keep in mind that the estimation of is crude. Furthermore, Fig. 5B shows a more marked decrease in [NAD(P)H]m at higher levels of AAT inhibition. This behavior is taken as a validation of condition 4.
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. We investigated the gains of two other output fluxes corresponding to signal candidates: jACS and jIDHPc. Figure 5C shows a consistently higher ACS for the mut genotype, whereas the opposite is true for IDHPc, as seen in Fig. 5D, especially at higher degrees of AAT inhibition. The prediction from this model is thus that abolished NADPH production via IDHPc may be involved in the impaired insulin response seen by Eto et al. We stress that we can assert only that our model is in qualitative agreement with the results of Eto et al.; the experiments were made only either with no AOA or with AOA applied in excess.
In summary, we have shown that our minimal model is capable of reproducing the behavior described by conditions 1 and 2. The set of enzyme limiting rates that were selected to satisfy these two conditions were also found to satisfy conditions 3 and 4. This provides a validation of the model. So far, the model suggests that ACS and MDH, but not LDH, are responsible for the maintenance of cytosolic NADH reoxidation when the shuttles are inhibited. Furthermore, the impaired insulin response of the mut/aoa+ genotype may be due to impaired responses in and in the NADPH-producing reaction catalyzed by IDHPc but not due to impaired citrate cleavage.
Investigations into Coupling Between Glycolytic Stimulus and Putative Response Fluxes and the Control Thereof
We here turn to the coupling between a glycolytic stimulus and the putative response fluxes. We investigated the more general case, applicable to, e.g., rat and human pancreatic -cells where ME is present, with the activity listed in Table 2 if nothing else is stated.
Parameter search and validation.
We applied the same strategy as above and considered all possible combinations of the limiting rates listed in Table 4. Here, we applied broader goodness criteria, following several studies of mostly rat -cells from several laboratories (30, 52, 54, 61, 62). We addressed the steady-state concentrations of the metabolites and required the citrate concentration and the [citrate]/[malate] ratio to be in reasonable accord with recent experimental observations made in these studies. We required citrate to be present in more than 0.05 mM and the [citrate]/[malate] ratio to exceed 0.5, at a high glycolytic influx rate VGPDH = 0.025. Furthermore, , ACS, and NADPHc had to be positive. Of the 2,880 parameter combinations, 645, or 22%, satisfied these requirements.
Since there is no consensus regarding the question whether glutamate is a coupling factor (11, 17, 63, 64, 114), we did not constrain our model with respect to total glutamate production or consumption. However, we noted that, of the 645 accepted parameter combinations, 407 corresponded to a net glutamate production. The mean glutamate production rate in these 407 scenarios was 
2 µM/s. Furthermore, in all the 645 accepted parameter combinations, jGDH was positive with a low net flux with a mean of 0.6 µM/s.
Some statistics from this parameter inventory are given in Table 5. The distribution of the accepted parameter combinations is given under the heading "Counts". Each count for each enzyme falls under one of five categories, ordered according to increased parameter values (see Table 4). The lowest limiting rates of OGDH, FO, and ACS are distinctly absent from the group of accepted parameter combinations, and the limiting rate for IDHc seems to be either high or low, whereas the LDH limiting rate and idhinh seem to be evenly distributed. Further tendencies in the data are revealed by the correlation coefficients, also presented in Table 5. The correlation coefficients were taken between the numbers defining the categories, as given in Table 4. The strongest correlation by this measurement is that between the general inhibition of all the IDHs and IDHPc. There is also a negative correlation between OGDH and ACS, enzymes which are present in alternate pathways from citrate to malate. Furthermore, there is a positive correlation between FO and ACS. Correlation coefficients only measure the strength of a linear relationship between two variables and thus are blind to nonlinear tendencies. Therefore, we also list the correlation ratios between the categories in Table 5. The correlation ratio is equal to the absolute value of the correlation coefficient for a purely linear relationship, greater otherwise (47). In two cases, the correlation coefficients miss significant correlations. The first of these concerns the correlation between VIDHPc and idhinh. The counts of the categories for these parameters are therefore presented in Table 5. It is here revealed that idhinh more often attains a low value at either low or high values of VIDHPc, but more seldom at intermediate values of the latter. Thus a high value of VIDHPc is associated with high limiting rates for the two mitochondrial IDHs, and the same goes for low values of VIDHPc. The second markedly nonlinear correlation is the one between VIDHPc and VACS. The counts of the categories of these limiting rates are also presented in Table 5. Here, low and high values of VACS correspond to low values of VIDHPc, whereas intermediate values of VACS correspond to high values of VIDHPc.
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, ACS (LC-CoA production), and NADPHc.
The mean values of the gains of , ACS, and NADPHc for the 645 accepted parameter combinations were 0.52, 1.3, and 3.7, respectively, with respective standard deviations 0.25, 0.94, and 1.6. Hence, NADPHc exhibits a particularly strong gain and is therefore a strong output signal.
We examined the correlation coefficients between the categories of the parameters and the gains, with results presented in Table 5. First, there is a positive correlation between idhinh and ACS, meaning that a general inhibition of the IDHs correlates with a stronger output signal from ACS. Second, somewhat unexpectedly, there is a negative correlation between VIDHPc and NADPHc, whereas the correlation between VIDHPc and ACS is positive. Third, there are positive correlations between VOGDH and the gains of and NADPHc. Fourth, again unexpectedly, the correlation between VACS and ACS was negative. And finally, there are strong negative correlations between VFO and all gains. This was further investigated as reported below.
Distribution of fluxes. We now turn to the closer investigations of the model using the nonmouse parameter set, which will be used in all of the following, unless otherwise explicitly indicated. We again made a pair of snapshot surveys of the flux distribution in our model (Fig. 7). Two observations deserve mentioning. First, there is extensive cycling of carbons in the reaction cycle comprised by PC, AATm, AATc, MDHc, and ME, the PMA shuttle. Second, with a lower GDH activity (Fig. 7B), the MDHm reaction goes backward. This is investigated further below.
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, ACS, and NADPHc in response to increased glycolytic flux, decrease with increased VFO (Fig. 8), whereas the absolute fluxes increase with increased VFO (Fig. 8, insets). These results are suggestive, as several studies (92, 101, 117) show the same behavior of insulin secretion with regard to glucose concentration and fatty acid supply: the basal secretion, present at low glucose levels, is stimulated by a high fatty acid supply, but the glucose response is weaker at this high fatty acid supply; i.e., the relative change in insulin secretion in response to an increased glucose oxidation is attenuated. Also (Fig. 8D), the cytosolic citrate concentration increases when VFO increases, whereas the rate of the PDH reaction decreases (Fig. 8E). These two effects are classic components of the Randle cycle (84), which defines different mechanisms of downregulation of glucose oxidation in response to an increased fat metabolism. Citrate here is thought to modulate the glycolytic flux via inhibition of phosphofructokinase (PFK). Moreover, an increase in VFO stimulates the rate of the PC reaction (Fig. 8E). This is not surprising, since ac-CoA, the product of fatty acid oxidation, is an allosteric activator of PC but an inhibitor of PDH. PC is activated more than PDH is inhibited, and in the steady state this has to be balanced by an increased flux via ME. This is an important factor of the rise of jNADPHc.
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Effect of GDH on the gains. Leucine and its nonmetabolizable analog BCH have long been known to stimulate insulin secretion, presumably via the allosteric activation of GDH by these compounds (38, 94). Here, we investigated the influence of a modulated GDH activity both on the gains (i.e., on GSIS) and on the absolute fluxes (pertinent to leucine-stimulated insulin secretion). The fluxes and gains are calculated as functions of VGDH [the results are applicable to, e.g., leucine activation of GDH, since leucine increases the apparent limiting rate of GDH (29)], and the results are shown in Fig. 9, A–C.
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and NADPHc both rise as GDH is activated, while the gain of ACS reaches a maximum and then decreases slightly. However, as seen in the insets, it is only the rise in the gain of that is accompanied with a rise in the corresponding flux. Thus, when interpreting the present model, we may safely say only that leucine stimulation of GDH results in an augmentation of the KATP-dependent pathway; the effect on the NADPHc production is inconclusive (smaller fluxes but higher gain). Stimulation of GDH also decreases the citrate concentrations (Fig. 9D). This may be an important link to the glycolysis in the case that citrate modulation of PFK affects the glycolytic rate. Shown in Fig. 9E is the flux via GDH as a function of VGDH, and the flux is always toward glutamate production. In Fig. 9, the total mitochondrial and cellular glutamate production is shown. There is a net consumption of glutamate in the mitochondria, but when AATc is also taken into consideration, it is seen that there is a cellular net production, which is stimulated by an increased VGPDH. To get a summarizing picture of how increased GDH activity increases glutamate consumption and NADH production, we calculated some concentration and flux control coefficients, which are presented in Table 6. The increase in VGDH results in a reduced 2-oxoglutarate concentration and an increased backward flux in AATm, causing the mitochondrial glutamate consumption to rise. This is coupled to an increased flux via MDHm, which is the dominating factor behind the increased NADH production rate. The dependence of jMDHm on VGDH is shown in Fig. 9G, which interestingly predicts negative flux via MDHm at lower values of VGDH. We will investigate this further below.
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Energy Coupling of the System and the Direction of the MDHm Reaction
We addressed the general question of the coupling of glycolytically produced carbon (i.e., pyruvate) and charge (i.e., NADH) to mitochondrial NADH production and to cytosolic ac-CoA and NADPH production. As a measure of the degree of coupling between glycolytic flux and i, where i may be mitochondrial NADH production, ACS flux, or cytosolic NADPH production, we define
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5, the maximal NADH yield of one triose molecule. Since we have observed that in our model the MDH reaction can be made to go backward when the ME catalyzed reaction is effective, we were motivated to study whether this may be correlated with a weaker coupling between the glycolysis and the respiratory chain and with a stronger coupling between the glycolysis and NADPH production. ME could then essentially be directing the electron flux from respiration to cytosolic NADPH production. In Fig. 10A, the forward coupling strengths have been plotted as functions of VME. These curves show a remarkable reciprocal relationship between q and q
. For low values of VME, q lies around 1.5, whereas q is very low, around 0.1. As the ME activity increases, a dramatic shift occurs, where the roles are switched and glycolytic flux becomes most tightly coupled to cytosolic NADPH production: ME "overtakes" the forward coupling from respiration. This sliding from coupling via respiration to coupling via NADPH represents an important difference between mouse on one hand, which according to our model couples the glycolytic flux mainly to respiration, and rat and human on the other hand, which couple glycolytic flux mainly to NADPH production, at least when VME is sufficiently high. The snapshot in Fig. 7B further shows what happens: a new cycle consisting of the reactions catalyzed by PC, MDHm, and ME emerges due to the ME activity.
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The tug-of-war between ME and respiration is illustrated from a different angle in Fig. 11. In Fig. 11A, we see how an increased VME "pulls" the MDHm flux below zero, which happens more easily if VGDH is low. On the other hand, an increase in VResp "pulls" jMDHm in the other direction, toward higher values, which is shown in Fig. 11B. Again, lower VGDH values are associated with more negative MDHm fluxes. The correlation between jME and jMDHm is remarkably linear, as seen in Fig. 11C. The curves were virtually independent of whether VME or VResp was varied in the continuation algorithm. For a given MDHm flux, higher ME flux correlates with a lower GDH activity, and vice versa. This can also be seen in the flux distribution snapshots of Fig. 7. Also, jME and jResp were linearly negatively correlated (Fig. 11D), again robustly in the sense that the curves were independent of whether VME or VResp was varied in the continuation algorithm. This correlation represents a tug-of-war between mitochondrial NADH production and cytosolic NADPH production.
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DISCUSSION |
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TOPABSTRACTGlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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-cell TCA cycle and NADH shuttles operate as an isolated module. The model represents a fairly compact description of this module in the sense that it consists of only ten ODEs while still integrating much experimental data on the molecular level and having a behavior that agrees with many experimental observations on the systemic level. Furthermore, the model generated diverse experimentally testable predictions. It also fills a previously empty gap among the theoretical (biophysical) models of the -cell. We must emphasize that the parameters of the model in some cases are subject to considerable uncertainties. Hence, we evaluated a multitude of scenarios where some limiting rates were allowed to vary by several orders of magnitude.
It is our hope that the results of this parameter inventory are amenable for comparison to future experimental studies of the expression profile of the -cell. For instance, future experimental studies may compare expression profiles with the correlations between the ACS and IDH activities obtained in this study. Our parameter inventory was course-grained and should be regarded as a first step toward a more complete quantitative characterization of the -cell. In our opinion, the road toward this goal goes via successive steps of model refinements against experimental data.
The model successfully reproduced the results of Eto et al. (26) and suggested that the flux via ACS may compensate for lost AATm activity, allowing NADHc to be reoxidized by MDHc. An experimental test for this would be to inhibit ACS in addition to the AATs and G3PDH. If the glucose oxidation is then still not attenuated, our model would be falsified.
Our model predicts a particularly strong signal transmission from an increased glycolytic flux to NADPHc production. This is linked to the high flux through the PMA shuttle, the latter which in fact allows cytosolic NADH to be converted to NADPH via the reaction series catalyzed by MDHc and ME. On this basis, we conclude that investigations into possible effects of NADPH in the -cell should continue (50).
An increased fatty acid oxidation attenuated the gains of the output signals, and at the same time it increased the absolute value of the fluxes. This is a behavior that has experimental support (14, 53, 92, 117). Regarding the effects of an increased fatty acid oxidation in the -cell, one usually considers acute effects, occurring on a metabolic time scale, and of long-term effects, resulting, e.g., from altered gene expression (12). Our results support a view that acute effects play an important role. It is also relevant to mention here the Randle cycle (84), which is a general term for the feedback mechanisms originally thought to attenuate glucose oxidation in response to a raised fatty acid oxidation. Our results highlight the metabolic parts of the Randle cycle, i.e., the fatty acid oxidation-induced attenuation of jPDH and increased cytosolic citrate concentration. The latter effect may inhibit the catalytic rate of PFK and will be examined in the context of glycolytic oscillations elsewhere. Our model may also help in partially reinterpreting certain experimental results. For instance, Roduit et al. (88) found that lowering the malonyl-CoA concentration by overexpressing malonyl-CoA decarboxylase attenuated GSIS. They attributed this effect to the resultant lower LC-CoA concentration in the cytosol. Our results predict that the increased fatty acid oxidation caused by the lower malonyl-CoA levels may have contributed to the attenuated GSIS as well.
A controversial subject is the direction of the GDH flux in the -cell. This is related to the hypothesis that glutamate is a coupling factor in GSIS, where GDH has been proposed to mediate an increased glutamate concentration in response to a raised glucose consumption (64), a hypothesis later refuted (11, 63, 114). Our model predicts that the GDH flux is directed toward glutamate. However, the GDH flux is quite moderate, although the total glutamate production usually is larger due to cytosolic glutamate production via AATc (AATm is always in the direction of glutamate consumption due to the electrogenic aspartate transporter). A small increase in the net production of glutamate in response to a raised glycolytic flux is also predicted by our model, although probably not enough to change the glutamate pool significantly, in line with experimental observations (63). It is, however, possible that a more detailed analysis of glutamate metabolism may yield additional insights. We conclude that our results should be possible to reconcile with recent experimental data (16, 17, 51), which reveal a small increase in the glutamate pool and a small decrease in the aspartate pool when the glucose level is increased. One should note that Li et al. (51) attributed this to mitochondrial glutamate production via AATm, whereas our results point to AATc being the probable cause. Furthermore, that study proposed glutamine, which is produced from glutamate, ATP, and NH4+ by glutamine synthetase, as a putative coupling factor in insulin secretion. The present model could be extended in order to evaluate this hypothesis. A new result is that activation of GDH may result in a decreased cytosolic citrate concentration. Since citrate is an allosteric inhibitor of PFK, this has the potential of being a feedback loop of importance for leucine-stimulated insulin secretion. A fuller analysis of glutamate metabolism is a suitable subject for further modeling studies.
An interesting prediction from our model is that the presence of ME may pull the MDH reaction in the backward (NADH-consuming) direction. This would have the effect of creating a short PM cycle consisting of the reactions catalyzed by PC, MDHm, and ME, which would divert the mass flux from the classical TCA cycle. Lu et al. (55) found in a thorough study, where they used 13C isotopomer analysis to track carbon flow from pyruvate to the TCA cycle, that pyruvate to a large extent is involved in what they term "pyruvate cycling," essentially the same as the pyruvate-citrate cycle. Their data were found to fit only a model in which there exists two separate compartmentalized pyruvate pools. However, the mathematical model they used did not allow the possibility that the MDHm-catalyzed reaction might proceed backward. It is possible that the assumption of two separate pyruvate compartments is unnecessary if the short PM cycle and the PMA cycle are taken into consideration. The possibility that the MDH reaction may run backward in the presence of ME activity, implying that the TCA cycle is not operative in its classical sense, may be important for other cell types too. This relationship between MDH and ME has to our knowledge not been stated explicitly before.
The -cell has attracted much attention from theoretical investigators (110). The complex electrophysiology of the -cell has already been considerably elucidated by mathematical modeling efforts (10). The electrophysiology is coupled to the metabolism at least via the KATP channel. An interesting hypothesis with much experimental support is that oscillations in the glycolysis may modulate the electrophysiological behavior (111). A realistic model of the coupling between glycolysis and ATP production has, however, been lacking, and the present model should fill part of this gap. It should thus be suitable as a template that, maybe in a simplified form, could be integrated in an emerging full-scale model of GSIS, encompassing all steps from glucose to insulin secretion.
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APPENDIX: ORDINARY DIFFERENTIAL EQUATIONS, EQUILIBRIA, AND RATE EQUATIONS |
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TOPABSTRACTGlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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Each of the 10 metabolite concentrations modeled as dynamic variables is normalized according to Table 1, and we use the notation xi to designate the concentration of metabolite i divided by the corresponding normalization constant. The metabolites are produced and consumed by reactions with rates denoted vi, where i represents the catalyzing enzyme. The ODE system then becomes, with stoichiometry according to Fig. 1:
 | (A1) |
The different factors fi follow from the quasi-equilibria that are assumed in the model, which are described in detail in the next section along with the fi factors. All of the different rates are functions of metabolite concentrations, substrates, and products, as well as activators and inhibitors.
In the present work, we analyzed only the steady state, i.e., solutions to the equation system
 | (A2) |
This corresponds to most experimental settings, where measurements are made after incubation under different conditions. An analysis of the temporal aspects of the model has also been made and will be presented elsewhere. The solutions to equation system A2 were usually calculated as functions of some model parameters. For this we used a Moore-Penrose continuation algorithm implemented in the software package MATCONT (24).
Equilibria
To our best knowledge, the activities of the different mitochondrial carriers have not been determined quantitatively with high resolution but appear to be quite high (61). To keep our model minimal, we thus treat the transport reactions as being in quasi-equilibrium. This allows us to determine the partitioning between the cytosolic and mitochondrial compartments as discussed below. We have kept this analysis in line with the current knowledge of mitochondrial carriers, reviewed recently by Palmieri (75).
Pyruvate is a monovalent molecule and is transported across the mitochondrial inner membrane electroneutrally, in exchange with a hydroxide ion through an antiport. Equilibrium is thus displaced from unity only due to the pH difference of 0.3–0.4 (more acidic in the cytosolic and intermembrane compartments) that is "felt" by the anions subject to transport (103). This is described by the equation
 | (A3) |
Malate, which is divalent, is exchanged electroneutrally with divalent phosphate by means of the dicarboxylate carrier. Monovalent phosphate, necessary for ATP synthesis, is electroneutrally transported with a proton or in exchange with a hydroxide ion, thus assumed to obey the equation
 | (A4) |
If we equate the equilibria between monovalent and divalent phosphate in both compartments, i.e.,
 | (A5) |
 | (A6) |
Furthermore, malate and 2-oxoglutarate are carried across the mitochondrial membrane by means of an electroneutral antiport. Assuming that malate attains equilibrium according to Eq. A6, we may write
 | (A7) |
Malate is also electroneutrally exchanged with a trivalent (iso)citrate molecule and a proton through the tricarboxylate carrier, which allows us to write
 | (A8) |
The equilibria so far calculated roughly agree with experimental observations (103). Glutamate and aspartate, which are both monovalent, are not included as variables in the model. However, their concentrations have to be estimated to estimate, in turn, the effective equilibrium constants of AAT and GDH. Glutamate is transported with a proton or in exchange with a hydroxide ion through a port that is highly expressed in the pancreas (32). Here, we assume that glutamate attains equilibrium according to the pH difference,
 | (A9) |
Aspartate is transported through an electrogenic antiport in exchange with glutamate and a proton. We use the Nernst equation and write
 | (A10) |
Here, 
µ is the protonmotive force, which amounts to 200 mV (13), F is Faraday's constant, R is the gas constant, and T is the absolute temperature.
We assume that the aconitase reaction is in quasi-equilibrium, i.e.,
 | (A11) |
We also assume that OGDH, succinate-CoA ligase, succinate dehydrogenase, and fumarate hydratase reactions may be described as a reaction block. This always holds in the steady state since the OGDH reaction is irreversible. In a dynamic description, one may as an ansatz approximate the four enzymatic reactions in the block as being in quasi-equilibrium. One may derive the factors fi in the ODE system 10 for completeness:
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Equilibrium Constants
All equilibrium constants of enzyme-catalyzed reactions were, in order to obtain consistent values, calculated from the standard transformed Gibbs free energies of formation of the products and reactants of the reactions, as tabulated by Alberty (2). Some of these contain concentrations of biochemical species not included in the model as dynamic variables. In these cases, an effective equilibrium constant Keff, with the constant concentrations factorized out, was calculated. Thus,
 | (A12) |
 | (A13) |
20 (30, 39). In the case of ME, the equilibrium constant had to be divided by the tissue partial pressure of carbon dioxide (2), which amounts to 0.06 bar (6), and then multiplied by the NADPc/NADPHc ratio. The rate of the ME-catalyzed reaction was found to be very insensitive to this ratio in our simulations. In the case of ACS, we have to multiply the equilibrium constant with
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4,000 M–1 (25, 81). The quote will vary with the ATP/ADP ratio, but we found the system to be quite insensitive to qACS. In the case of GDH, we multiply the equilibrium constant with the concentration of ammonia, and we found our model to be insensitive to this concentration. All equilibrium constants are collected in Table 7, and the estimations of the concentrations of metabolites here considered to be constant are found in Table 8.
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To obtain a compact notation for all rate equations, we use a shorthand notation for all chemical species included in the model as dynamic variables, which are defined in Table 1. In all rate equations, 
i stands for the concentration of species i, considered to be a substrate of the enzyme in question, normalized by the half-saturation point S
for enzyme E of this substrate. The same definition applies for 
i, with the difference that species i in this case is considered to be a product of the reaction, and the concentration is in this case normalized by the half-saturation point P of this product. In the same manner, if the enzyme is subject to activation or inhibition, we use the notation 
i as a representation of the concentration of modifier i normalized by the corresponding dissociation constant X. The situation is slightly more complicated in the case of ac-CoA and the reduced pyridines. Here, we define the conversion factors Si = [i]tot/S, where [i]tot represents the total concentration of the conserved moiety of the species in question, e.g., (NAD) + (NADH). The corresponding holds for the conversion factors Pi = [i]tot/P and Xi = [i]tot/X. We adopt the practice described, e.g., by Hofmeyr and Cornish-Bowden (42) when writing reversible rate equations and use the ratio 
/Keq, where = 
iPi/iSi, in which Pi and Si stand for the concentrations of product and substrate i, respectively, is the mass action ratio, and where Keq is the equilibrium constant of the reaction. The form of the rate equations conforms to the generalized reversible Hill equation (109), in which the parameter represents the alteration of the apparent limiting rate caused by an effector and where a represents the alteration of the apparent half-saturation point. The other constants, defined in the rate equations below, are listed in Table 9.
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 | (A14) |
This, of course, means that certain feedback mechanisms, such as the inhibition of PFK by citrate, have been neglected. However, it has been suggested that glucokinase has a very high degree of control over the glycolytic flux in the -cell (67, 111), which may justify our ansatz, which still takes into account that if the cytosolic NAD concentration reaches sufficiently low levels the GPDH rate will fall, causing intermediate metabolites upstream in the glycolysis to either accumulate or, by means of product inhibition, slow the rate of glucose consumption.
G3PDH.
The dependence of cytosolic G3PDH on NADH is modeled as an irreversible MM equation:
 | (A15) |
LDH.
LDH was assumed to follow irreversible Michaelis-Menten kinetics:
 | (A16) |
AAT.
AAT exists as two different isozymes: one cytosolic (AATc) and one mitochondrial (AATm). These differ in terms of kinetic constants, but neither of them is considered allosterically regulated (15), and studies of the kinetic behavior (41) motivate the use of the reversible MM equation. We do not include aspartate and glutamate as dynamic variables in the model; these compounds are thus incorporated in the effective equilibrium constant:
 | (A17) |
 | (A18) |
ME.
There are two isozymes of ME, a cytosolic and a mitochondrial. Only the former is considered in our model, since it has been shown that this isozyme is responsible for 90% of the ME activity in the -cell (39). ME is described using the reversible Michaelis-Menten equation. The enzyme is inhibited by oxaloacetate, but only at high concentrations compared with the ones in the present model (115), leading us to ignore this property. Also, NADPc and NADPHc are not included as dynamic variables in the model and are here included in the effective equilibrium constant for the reaction. This was justified by our finding that, in all our simulations, the model including the ME rate was found to be quite insensitive to variations of the NADPH/NADP ratio. The rate equation becomes
 | (A19) |
ACS.
Cytosolic CoA is not included as a dynamic variable in the model; hence, the ratio between ac-CoA and CoA is included in the equilibrium constant. The same applies to ATP and ADP. These constants were lumped together in the parameter qACS described above, and the model was found to be very insensitive to thisparameter. Studies of the kinetics of the reaction (77, 79) motivate the use of a reversible Michaelis-Menten equation for ACS:
 | (A20) |
PDH.
The PDH complex exhibits complex regulatory patterns involving activation by Ca2+ as well as regulation through phosphorylation and dephosphorylation (85). These aspects are not considered here, since this study does not aim to model the regulation by Ca2+ and since the phosphorylation-dephosphorylation regulation would be overly complex to be included in a model of this scope and also probably proceeds at a slower time scale than the events considered here. The enzyme is subject to competitive product inhibition by ac-CoA with respect to CoA and by NADH with respect to NAD (48, 85), and we write the rate equation as
 | (A21) |
PC.
The reaction catalyzed by PC uses, in addition to pyruvate, both MgATP and HCO3– as substrates, the two latter species not being included in the present model and thus assumed to be constant. A prominent property of PC is the strong activation of the enzyme by ac-CoA (regarded as a cofactor of the reaction), which both lowers the effective Km value of pyruvate and increases the limiting rate VPC (3, 4). Moreover, the product oxaloacetate is a competitive inhibitor with respect to pyruvate (4). This leads us to formulate the rate equation as
 | (A22) |
CS.
CS is regulated by many species, including inhibition by ATP and succinyl-CoA, which are not included in the present model. In addition, it is subject to effective product inhibition by CoA, which is a competitive inhibitor for ac-CoA (68). These conditions are conveyed by the following rate equation:
 | (A23) |
IDH.
IDH is in human tissue present in three variants; an NAD-linked mitochondrial enzyme and cytosolic and mitochondrial NADP-linked enzymes. NAD-dependent IDH is an allosterically regulated enzyme, activated by Ca2+ (89) and ADP, and inhibited by ATP and NADH (78). The first three of these regulating species are not accounted for in the model, but NADH is. The inhibition exerted by NADH is competitive with respect to NAD (78), which is reflected in the rate equation below. Moreover, according to the study by Rutter and Denton (89), IDHm exhibits cooperativity with a Hill coefficient of almost 3. This behavior is, however, absent in the results of Plaut and Aogaichi (78). We use the irreversible Hill equation with allosteric modifiers to be able to account for the possible cooperativity and write
 | (A24) |
NADP-dependent IDH (here abbreviated IDHPm) is also present in the mitochondria of -cells (95). This enzyme is not generally considered to be allosterically regulated, but it exhibits competitive product inhibition of NADPH with respect to NADP (106). Thus we write
 | (A25) |
Cytosolic NADP-dependent IDH (IDHPc) was assumed to follow Michaelis-Menten kinetics:
 | (A26) |
OGDH.
The OGDH complex is allosterically regulated by Ca2+, a property that is not, however, considered here explicitly. It is also subject to product inhibition by succinyl-CoA and NADH (98), the former of which is not included in our model and thus is not considered here. Inhibition by NADH is close to noncompetitive (98), i.e., 8 
1, and on the basis of that study we write
 | (A27) |
MDH.
There are two isozymes of MDH, one cytosolic (MDHc) and one mitochondrial (MDHm). MDHc is not allosterically regulated and is here modeled using the reversible Michaelis-Menten equation, ignoring inhibition by oxaloacetate, which occurs only at very high and unphysiological concentrations (9):
 | (A28) |
9 = P9(1 – S9
9).
In contrast to the cytosolic isozyme, MDHm is allosterically regulated in a peculiar manner. Experiments (34) indicate that the allosteric effector citrate may act as an inhibitor or an activator for the malate oxidation reaction, depending on the NAD concentration. A detailed analysis (34) showed that citrate increases the effective limiting rate VMDHm while at the same time also increasing the half-activation points for malate and NAD. In the reverse direction, citrate increases the half-activation point for NADH while leaving the half-activation point for oxaloacetate unchanged, as well as the backward limiting rate. Here, we model this behavior by allowing the modifier citrate to affect the dissociation constants of an enzyme-substrate complex and an enzyme-product complex differently. If we, as usual, assume the independence of binding constants of substrates and products, we, using the method of Cha (20), arrive at the following rate equation:
 | (A29) |
9 = P9(1 – S99). The constants denoted represent the factors by which the binding of effector molecule alters the enzyme-product dissociation constants, and, due to detailed balance, vice versa, analogous to the definitions of Hofmeyr and Cornish-Bowden (42). The constant 1+ represents the factor by which citrate increases the forward limiting rate. There is also a factor 1– representing the factor by which citrate increases the backward limiting rate, which, since citrate does not affect the backward limiting rate, is equal to 1. It is very important to note here that the constants , , and are not independent but are constrained by detailed balance. The relationship between the constants is simply
 | (A30) |
1– and the -factors. These constraints, of course, demand that experimental data on the kinetics of the enzyme agree with the assumptions. This is indeed the case, as may be verified from the values in Table 10, which were inferred from the study of Gelpí et al. (34). Some properties of MDHm were not considered here; first, the citrate-dependent substrate activation by malate, which occurs only at NAD concentrations higher than those assumed in our study; and second, the substrate inhibition by oxaloacetate, which occurs only at much higher oxaloacetate concentrations than those assumed here (9, 34).
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 | (A31) |
8 = s8(1 – p
8), and where the values 8 and 10 (both smaller than 1) control the amount of negative cooperativity. K
is an effective rate constant including the concentration of ammonia as discussed in the equilibria section. The model was very insensitive to variations in the ammonia concentration. We could not use the reversible Hill equation in this case, since negative cooperativity applies only to the oxidized pyridine species (the reversible Hill equation assigns the same Hill coefficient to both substrate and product). Correspondingly, the NADP-dependent reaction flux is written
 | (A32) |
In these equations, g stands for glutamate, whose concentration is given as
 |
Resp.
The regulation of the respiratory rate is distributed over cytochrome c oxidase, the ATP/ADP translocator, proton leak, and NADH supply (36). Detailed descriptions of all these processes are beyond the scope here, and we model the dependence of the respiratory rate of NADH phenomenologically in the simplest possible way:
 | (A33) |
denotes reduced cytochrome a3, (1 – x8)/x8 = [NADH]mit/[NAD]mit, and VResp is implicitly a function of the concentration of the cytochromes, oxygen concentration, the mitochondrial ATP/(ADP x Pi) ratio, and other factors. This ansatz was adapted from Wilson et al. (112) and builds on the assumption that the electron transport from NADH to cytochrome a3 is in near-equilibrium. This rate equation qualitatively reproduces experimental data on the dependence of the respiratory rate on the mitochondrial NADH/NAD ratio (49, 112) but should ultimately be regarded as phenomenological.
FO.
FO was modeled phenomenologically as
 | (A34) |
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GRANTS |
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TOPABSTRACTGlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
1 Strictly, the AOA-treated -cell is not a genotype. Nevertheless, for practical purposes it may be considered as such, and we will use this nomenclature for the purpose of clarity of our treatment. 
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TOPABSTRACTGlossaryMETHODS AND MODELRESULTSDISCUSSIONAPPENDIX: ORDINARY DIFFERENTIAL...GRANTSREFERENCES |
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-cell mitochondrial calcium handling and electrical activity. I. Cytoplasmic variables. Am J Physiol Cell Physiol 274: C1158–C1173, 1998.-cells. Diabetes Rev 4: 130–144, 1996.This article has been cited by other articles:
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  Y. Li, R. K. Dash, J. Kim, G. M. Saidel, and M. E. Cabrera Role of NADH/NAD+ transport activity and glycogen store on skeletal muscle energy metabolism during exercise: in silico studies Am J Physiol Cell Physiol, January 1, 2009; 296(1): C25 - C46. [Abstract] [Full Text] [PDF]
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