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CALL FOR PAPERS
Special Section On Mitochondrial Modeling and Function
Data Integration, Analysis and Logistics (DIAL) Project, Centre for Medical Systems Biology, Leiden, Rotterdam, and Amsterdam; and VU University Medical Centre, Amsterdam, The Netherlands
Submitted 27 June 2006 ; accepted in final form 13 June 2007
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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14 heartbeats; 3) if the muscle isoform of CK is overexpressed, OxPhos reacts slower to changing workload; and 4) if mitochondrial CK is overexpressed, OxPhos reacts faster. systems biology; computational model; creatine kinase; phosphocreatine shuttle; regulatory module; mitochondrial membrane permeability; oxygen consumption
To analyze the dynamic adaptation of cardiac ATP synthesis to ATP hydrolysis, a "skeleton model" approach was chosen, incorporating key elements of biochemical reaction and transport of phosphoryl groups in the muscle cell (29, 41). These key processes were chosen from the thousands of molecules at work in the cell and were hypothesized to form the core module that regulates the fast dynamic adaptation of oxidative phosphorylation (OxPhos) to ATP hydrolysis, which varies with the load on the heart. The model is simplified as much as possible.
A minimal requirement for resynthesis of ATP after its hydrolysis to ADP and Pi is that ADP and Pi are transferred to the mitochondria to serve as substrates for OxPhos. ATP can also be resynthesized from phosphocreatine (PCr) via isoforms of the creatine kinase (CK) enzyme. The high-energy phosphate groups of ATP synthesized in the mitochondria may be transferred to creatine to yield PCr, in a reaction catalyzed by the mitochondrial isoform of CK, Mi-CK, located in the mitochondrial intermembrane space (IMS). Phosphate group transfer between adenine nucleotide and creatine (Cr) on the one hand and transport of these metabolites between cytosol and IMS on the other constitute the adenine nucleotide-creatine-phosphate (ACP) module, which is the focus of the present study. It is investigated here whether this module can explain experimental results on the fast dynamic adaptation of OxPhos to ATP hydrolysis.
It was proposed that ADP and ATP are not transferred directly between cytosol and mitochondria under normal conditions, but instead there exists a more or less obligatory "PCr shuttle," which transfers the "high-energy" phosphoryl groups after phosphate group transfer from ATP to Cr in the mitochondrial intermembrane space (5, 42). In the myofibrils the phosphoryl group of PCr would then be transferred back to ATP by the muscular isoform of CK (MM-CK) to energize myofibrillar contraction. Conversely, creatine may then also be important in the intracellular signaling pathway, which regulates dynamic adaptation of OxPhos to changing ATP hydrolysis in the myofibrils and at ion pumps. The permeability of the mitochondrial outer membrane (MOM) to ADP and ATP is a key parameter determining the importance of the PCr shuttle.
The ACP module is computationally modeled to represent the time course and dynamics of the high-energy metabolites, which contain phosphate groups. Adjacent modules, which produce and consume ATP in exchange with the ACP module, are not described extensively. These modules are represented by relatively simple equations that summarize the functioning of the ATP splitting module in the cytosol and ATP production by the mitochondria. ATP exported from the mitochondria across the adenine nucleotide translocator in exchange for ADP together with the uptake of Pi across the mitochondrial inner membrane form the interface between the ACP module and the module for mitochondrial ATP production. The time course of ATP hydrolysis in the cytosol is given by a forcing function determined from experimental measurements.
The ACP module is investigated here by isolating it as much as possible experimentally. Cytosolic ATP production by glycolysis is inhibited chemically at the level of glyceraldehyde-3-phosphate dehydrogenase in the experiments used to test the model (17). Pyruvate is given as a carbon source for the mitochondria to bypass glycolysis. The response of OxPhos was shown to be about twice as slow when glycolysis is active (17). In the present study, the situation is investigated where glycolysis is inactive, both in the model and in the experiments.
In addition to ADP and Pi, the direct substrates for oxidative phosphorylation, and PCr and Cr as intermediates in phosphate group transfer, other factors have been proposed to regulate oxidative phosphorylation during work transitions. Intracellular Ca2+ was proposed as an important regulator, but it turns out that it stimulates OxPhos with a time constant of 20–25 s, leading to increased NADH synthesis in a second, slow phase of the response (7). In contrast, the first phase of the response of OxPhos is characterized by a fast time constant of 4 s, which is accompanied by enhanced NADH oxidation rather than reduction (7, 10). Therefore, the effect of Ca2+ stimulation on the mitochondria is omitted from the model because it only plays a role at a slower time scale.
Cardiac mitochondria are exposed to time-varying loads at two levels: pulsatile ATP hydrolysis occurs during systole and is low during diastole in the cardiac cycle (41, 45), and at a slower time scale heart rate varies. In the present study the model is tested by simulating the response to dynamic variations in ATP hydrolysis caused by heart rate steps and comparing the response time with experimental measurements. The role of CK is tested further by model simulation of the effect of lower CK activity on dynamic adaptation and comparing this with experimental results of inhibition of CK. The analysis demonstrates that the chemical reactions of ATP and ADP, with Cr and PCr, respectively, catalyzed by two CK isoforms which are linked by diffusion through the cytosol and permeation of the MOM are sufficient to explain the time course of dynamic adaptation of ATP synthesis to time-varying ATP hydrolysis.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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 | (1) |
ATP hydrolysis takes place in the myofibrils and at ion pumps during contraction, producing ADP plus Pi:
 | (2) |
Another major simplification is the omittance of special forms of functional coupling via a microcompartment between Mi-CK and the adenine nucleotide translocator (41). It is emphasized here that such special coupling may be important to explain measurements in isolated mitochondria. However, several considerations argue for omitting the microcompartment. First, in isolated rabbit heart mitochondria the coupling strength was shown to be inversely proportional to ATP concentration and is very low for >0.7 mM ATP (14), a condition which exists in heart muscle cells. Second, simulations with the full model (41) and with the reduced model yield similar results irrespective of functional coupling (23). Restrictions in membrane permeability between IMS and cytosol allowed in the reduced model already lead to a limited form of functional coupling between Mi-CK and adenine nucleotide translocator. To preserve the "skeleton model" approach and limit the number of parameters, special forms of functional coupling are omitted from the model, although we reiterate that functional coupling may be important to explain studies on isolated mitochondria and skinned fibers (14, 21).
An assumption for the ATP production module is that the concentrations of ADP and Pi in the IMS are the only variables determining the rate of OxPhos. ATP production is given by a simple Michaelis-Menten-type equation as a function of the ADP and Pi concentrations (see APPENDIX A). This equation lumps the complex processes involving the mitochondrial adenine nucleotide translocator, mitochondrial matrix, inner membrane proton pumps, and electrical potential in a simple overall equation which has been tested extensively on isolated mitochondria (16, 36) and is used often for modeling (19, 25). The ATP production enters the IMS compartment of the ACP module as a flux of ATP exported from the mitochondrial matrix in exchange for ADP. The response of ATP production to ADP and Pi in the IMS is assumed to be instantaneous in the model. Experimental evidence shows that the mitochondrial response to ADP is fast as indicated by a measured half time of 0.07 s at 26°C for the response of cytochrome b in the electron transport chain to external ADP addition (9). Model simulations, which I performed with the model of Cortassa et al. (10), suggest that the response of ATP export from the mitochondrion into the intermembrane space in response to ADP concentration changes in the IMS at the adenine nucleotide translocator are very fast, <1 ms, while mitochondrial O2 consumption follows with a response time of 3 ms in this model simulation. ATP production (i.e., export from the mitochondrial matrix), ATP synthesis, OxPhos, and mitochondrial O2 consumption are very closely coupled in time.
The ACP module is a simplification of the model of Vendelin et al. (41). In the original model, the transfer of energy was simulated by a spatially inhomogeneous reaction-diffusion process allowing for diffusion gradients in the cytosol, which were, however, shown to be very small in those simulations. The steps taken in simplification were discussed and the simplified model gave results which are very close to results of the extensive reaction-diffusion model in a preliminary analysis of this model presented at a conference (23).
Estimation of diffusion and membrane permeation parameters.
In the model simulations use is made of estimates of conductance parameters for diffusion in the myofibrillar and cytosolic space and for permeation of the MOM, taking place in series. For the cytosol, the conductance parameter is calculated from diffusion coefficients measured from diffusion-sensitized NMR spectra in muscle (11) based on a cylindrical geometry of the myofibril and surrounding cytosolic space (see 
Table 2 and APPENDIX B, parameter set I).
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In addition to diffusion, the MOM is interposed between cytosol and IMS, causing transport impediment which is not negligible (41). On the basis of the membrane permeability coefficients for ADP and ATP used in the model of Vendelin et al. (41), the conductance of MOM, PSmom, is calculated to be 0.10 s–1 for ADP and ATP; see APPENDIX B. Given the high diffusivity in the cytosol, the total conductance PStot between myofibril and intermembrane space with diffusion in the cytosol and permeation of the MOM in series is therefore virtually the same as PSmom. For PCr and Cr, the outer membrane conductance PSmom = 162.5 s–1 and total conductance PStot = 155 s–1; for Pi PSmom = 204 s–1 and PStot = 194 s–1 (APPENDIX B, parameter set II).
Beard (28) used the same membrane permeabilities, except for ADP and ATP where a higher permeability was estimated based on measurements on isolated mitochondria. This results in PSmom = 53.1 s–1 and PStot = 51.9 s–1 for ADP and ATP (APPENDIX B, parameter set III).
Saks et al. (32) reported in a later experimental study on permeabilized fibers that diffusibility in the cytosol was 6% of the previously assumed diffusibility (41), and membrane permeability P was 3% of previous reference values 260 µm/s for PCr and Cr, 327 µm/s for Pi, and 145 µm/s for ADP and ATP. These values were thought to explain the experimentally determined ADP affinity of the mitochondria in permeabilized fibers (32). This results in parameter set IV which is also tested in the present model: PStot,ATP = PStot,ADP = 2.6 s–1, PStot,PCr = PStot,Cr = 4.6 s–1, and PStot,Pi = 5.8 s–1.
Parameter set V is the same as parameter sets II and III, with the exception of the PStot for ADP and ATP, which is optimized to yield the experimentally determined tmito, 3.7 s. APPENDIX B gives details on the parameter sets and their calculations.
Model calculations. The model was implemented in "R" (37), which is an object-oriented high-level computer language and analysis environment often used for scientific computing, in particular for statistical analysis of microarray data (http://www.r-project.org). R was also applied to simulation of ecological models (31). The R language supports flexible data structures, and is open source, allowing extensions and modifications by the user. R is free of charge with a large user community that supplies a vast library of analysis routines. Model simulations were run on 1.5–3 GHz desktop personal computers and a 1.7-GHz Pentium laptop computer with 512–1,024 kB RAM on a Windows XP platform.
The ordinary differential equations in the model are integrated using the Livermore solver developed by Petzold et al. (30), which switches algorithms automatically during execution to handle stiff and non-stiff conditions. As an additional check to assess whether the equations were correctly integrated the concentrations of conserved moieties (Eqs. 18–20 in APPENDIX A) were calculated and found to change by <10 µM during the simulation of a response, demonstrating good numerical accuracy.
Optimized parameter values were obtained using the R-routine "optim," based on the "L-BFGS-B" algorithm (8), which allows box constraints on the parameters which are optimized. For optimization of a single parameter a golden section search and successive parabolic interpolation, implemented in the R routine "optimize" was used. Computation times for optimization with the pulsatile forcing function were an order of magnitude larger than when ATP hydrolysis was constant over the cardiac cycle. Parameter optimization was therefore done with the latter approach. Response results are very similar, with tmito usually 0.1–0.2 s larger for the pulsatile than the continuous forcing function.
Experimental data on dynamic response of oxidative phosphorylation.
The dynamic response of oxidative phosphorylation has been determined under a great number of experimental conditions for heart muscle. The time course of the increase of O2 uptake in the mitochondria in response to step changes in cardiac pacing rate was determined. These changes in workload resulted in immediate changes in ATP hydrolysis in the myofibrils and at ion pumps, reflected by systolic peaks of initial heat generation (45). However, OxPhos is reflected by slow recovery heat generation and responds with clearly measurable delay to changes in ATP hydrolysis. An increase of O2 consumption in the muscle cells results in an increase in O2 uptake from the perfusate flowing through the capillaries, which after a delay caused by intravascular convective transport results in a decrease in cardiac venous O2 concentration, measurable with O2 electrodes in isolated hearts. The time course of O2 uptake measured at the level of the whole heart was corrected for the O2 transport delay between the mitochondrial site of O2 consumption and the cardiac venous O2 electrode. This deconvolution can be accurately done for the average response time (40), but the full time course of O2 consumption following a heart rate step cannot be accurately reconstructed. The definition of the response time tmito to a step function increase in cardiac pacing is
 | (3) |
In experiments the integral of Eq. 3 is taken for a limited period of time, 1 min, because drift of the O2 electrode and the biological preparation preclude longer measurements. For direct comparison with the experimentally measured tmito, tmito-exp is calculated from the simulations by replacing VO2end in numerator and denominator of Eq. 3 with VO2 calculated at 60 s in the simulation and integrating over 60 s. The true tmito value was calculated from the simulations by integrating until the steady state had been virtually fully reached. Simulations demonstrate that tmito-exp may underestimate the true tmito in some cases. If the response is close to an exponential response, Eq. 3 integrated over 60 s gives adequate results. However, the simulations showed that for low CK activities, the response of ATP synthesis is initially very quick but shows a tail, which deviates for a long time from the asymptotic end value.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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23 times at the level of ATP synthesis, demonstrating the damping characteristics of the system. A substantial pulsatile component remains, with amplitude about one-fourth of the average ATP synthesis. Despite the clearly pulsatile response of ATP synthesis to each systolic pulse of ATP hydrolysis, the response to a step in heart rate has a slow component with a response time ("time constant") equivalent to 14 cardiac cycles. ATP production averaged over the cardiac cycle equals average ATP hydrolysis in the steady state before the change in heart rate at t = 0. Cardiac cycle-averaged ATP synthesis approaches the increased average ATP hydrolysis again at the end of the response shown in Fig. 2.
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Test of restricted diffusion across MOM. The effect of different sets of diffusion and membrane permeation parameters on the response time to a heart rate step is conveniently demonstrated in a graph based on the rectangular, rather than pulsatile, forcing function for ATP hydrolysis (Fig. 3). The response time for the pulsatile forcing function, calculated according to Eq. 3, is 0.1–0.2 s larger than the result for a continuous forcing function which is constant during the cardiac cycle. The dynamic response to a sudden change in cytosolic ATP hydrolysis was simulated (Fig. 3) with the five sets of membrane permeability parameters given in APPENDIX B, keeping the other model parameters constant.
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A restriction in MOM permeation for ADP and ATP was added to the model, using membrane permeation parameter 0.16 µm/s for ADP and ATP as assumed in Vendelin et al. (41). This means that ADP and ATP transport is strongly restricted, simulating a condition in which a "phosphocreatine shuttle" operates (parameter set II, APPENDIX B). MOM permeabilities for Cr, PCr, and Pi corresponded with those in published mathematical models (4, 41) and are much higher than for ADP and ATP. The simulated response for this parameter set was characterized by tmito = 14.8 s (see Fig. 3), much slower than the experimental value of 3.7 s. Using the pulsatile forcing function (see Fig. 2), the cytosolic ADP concentration increased from a value oscillating from 134–217 µM to 500–650 µM (Fig. 4) for this low MOM permeability, which is much higher than derived from end-diastolic NMR measurements in this experimental model (13). The end-diastolic PCr concentration decreased from 3,230 to 990 µM, which is much lower than measured experimentally (13). End-diastolic Pi values changed from 3,470 to 6,060 µM, much higher than the experimental findings (13). A 20% reduction in CK activity led to a doubling of ADP concentration for these MOM permeabilities. It is clear that the results of the dynamic simulation with this parameter set with very low ATP and ADP permeability, reflecting a "phosphocreatine shuttle" condition, deviate substantially from experimental values.
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6% of previous values (41), yielded tmito = 7.25 s, which is still >10 times means ± SE removed from the experimental value (parameter set IV, APPENDIX B).
The membrane permeation parameter 85 µm/s for ADP and ATP from the model of Beard (4), based on measurements of Lee et al. (28), resulted in tmito = 2.7 s, which is 3 times the SE removed from the experimental value (parameter set III, APPENDIX B).
The MOM conductance for ADP and ATP is of major importance for theories about the phosphocreatine shuttle and regulation of mitochondrial ATP synthesis, but is apparently not well known, as demonstrated by the considerable differences in its value between various models and experimental results (4, 24, 25, 32, 41). The other parameters in the present model were either measured independently in the same experimental model in the same laboratory (CK activities, mitochondrial aerobic capacity, metabolite concentrations), or represent well established literature values for kinetic constants of enzyme processes (MM-CK, Mi-CK, oxidative phosphorylation). The permeability parameters for PCr, Cr, and Pi (PSPCr, PSCr, PSPi) were used in various models with some degree of consensus. The diffusivity and permeability for ADP and ATP are not well known and were therefore estimated from the dynamic response experiment (Fig. 3). PStot,ADP and PStot,ATP were set equal to each other and optimized with a golden section search to reproduce the experimental value of tmito = 3.7 s, which resulted in PStot,ADP = PStot,ATP = 13.3 s–1 (parameter set V, APPENDIX B). This permeability value, "reverse engineered" from the experimental response, is 130-fold higher than the PS value derived from the permeability in the model of Vendelin et al. (41) with severe ADP and ATP permeability restriction, about 4-fold higher than the ATP and ADP conductances resulting from the analysis of permeabilized fiber experiments by Saks et al. (32) and about 4-fold lower than the value based on the model of Beard (4).
Effects of changes in CK activity on dynamic response of oxidative phosphorylation. The two CK isoforms are key molecules in the "skeleton model." Therefore the dynamic response of OxPhos to the heart rate step was calculated as function of the CK activity, to simulate partial inhibition or overexpression of CK in the model. Figure 5 shows how tmito varies with the relative CK activity (VCK) when the mitochondrial and the cytosolic activity of CK were changed in parallel by the same factor. For relative CK activity VCK = 1, the CK activities are equal to those measured in rabbit hearts in the experimental group, resulting in tmito = 3.7 s. For VCK > 0.03 the response becomes faster if VCK increases. However, if VCK falls below 0.03, tmito drops steeply (Fig. 5A, continuous line). This applies to the true tmito, calculated according to Eq. 3 for a period of time sufficient to reach the new steady state. However, tmito-exp, calculated for a 1-min recording period as done experimentally (see Experimental data), is already affected at higher VCK values (Fig. 5A, dotted line). Figure 5B gives tmito also for high VCK where tmito decreases further.
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1 min in the experiments. The model analysis makes this weakness in the experimental design clear. The dotted lines in Fig. 5, A and B, give an approximation of tmito as actually calculated from the experimental measurement (see METHODS).
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To determine the contributions of each of the two isoforms, in Fig. 7, tmito is given as function of the activities of Mi-CK and MM-CK, varied independently of each other. If MM-CK activity is kept constant at its reference level VCK = 1, an increase in Mi-CK activity always leads to a decrease in tmito. If Mi-CK activity is kept constant at VCK = 1, increases in MM-CK activity always lead to increases in tmito. The decrease of tmito above VCK = 0.03 in Fig. 5 apparently is caused by the increase in mitochondrial CK activity. Increasing relative Mi-CK activity from 0 to 3 at constant VMM-CK = 1 causes tmito to decrease from 4.5 to 3.0 s, demonstrating the accelerating effect of phosphocreatine transport into the IMS on the dynamic adaptation of OxPhos to ATP hydrolysis. If the MM-CK activity reaches zero, this results in tmito = 2.4 s. The remaining Mi-CK activity is then still 7.7% of the normal MM-CK activity. If the Mi-CK activity is subsequently also completely abolished, tmito becomes 0.5 s.
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10% with increasing ATP hydrolysis, in agreement with NMR measurements in rabbits at two heart rates (13). This decrease depends very little on CK activity, except at very low VCK values. ADP in the intermembrane space increases slightly with ATP hydrolysis, 39 to 43 µM for the heart rate step at VCK = 1. In the cytosol ADP increases from 64 to 77 µM. The major part of the increase in ATP synthesis is caused by the increase of Pi concentration in the IMS from 910 to 1,552 µM. These values are only 3 µM lower than in the cytosol. The Pi increase with heart rate mirrors the decrease in PCr concentration.
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For the rat heart parameter set, with higher Mi-CK and MM-CK activities than for rabbit, the flux carried by PCr is 349 and by ATP 138 µM/s. This becomes 430 µM/s for PCr and 57 µM/s for ATP if CK activities are three times higher than the normal rat values.
Predictions for metabolite oscillations as function of MM-CK and Mi-CK activities. The model predicts that ADP levels and ATP production oscillate appreciably (see Fig. 10) because of the pulsatile hydrolysis of ATP. For Pi, the average level is increased at low VCK, with little change in the amplitude of pulsation, which is 141–146 µM for VCK 0–3. This lack of change in Pi oscillation reflects that CK does not buffer Pi directly. When CK activity is increased above its normal level the oscillations become less strong, and the average Pi concentration drops further. For low CK activity, at a similar level as in the experiment with CK inhibition (17), the predicted oscillation of ATP synthesis and of cytosolic ADP concentration becomes very strong.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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Given that the complete system contains thousands of molecular species and that these molecules tend to be very densely connected in extensive networks, it is not a priori clear to what extent a modular approach is possible (38). However, although the structure of the system in terms of possible molecular interactions shows very high connectivity that makes it difficult to discern modules at the structural level, it may be feasible to discern modules at the kinetic level (D. A. Fell, unpublished communication). The module investigated here was tested under conditions of blocked glycolysis and exclusively for the first phase of the dynamic response of OxPhos to isolate it experimentally as much as possible from the complete biomolecular system.
The tests performed in this paper suggest that the ACP module may be sufficient to explain the first phase of fast regulation of OxPhos. Predictions were made for further testing of the model. In the module, use was made of kinetic parameters, which had been determined independently by measurements on isolated components of the system. However, the permeability of the MOM to ADP and ATP was optimized from measurements on the behavior of the system as a whole because there was no consensus in the literature on this parameter. Thus a constraint based on a higher-level function was used in conjunction with knowledge on the molecular constituents of the model. It is possible to improve the description of the modules which communicate with the ACP module, and retest the integrated model in an iterative cycle. Although further experimental testing with new experimental data and with improved ATP production and cytosolic ATP hydrolysis models is desirable, these first results suggest that the present model of the ACP module is sufficient to explain the first phase of the dynamic adaptation of OxPhos to ATP hydrolysis in cardiomyocytes.
The alternative to working with a modular "skeleton model" is to start directly with large models containing a large number of molecular processes. Many of the details in such a model will probably be wrong to a smaller or larger extent. Furthermore, models with all molecular details are hard to understand and debug. A small model containing only the key processes determining the overall process is preferable to understand the essential characteristics of the process. A beautiful example of a "skeleton model" approach is a model for an ecological system (6) with just three variables (vegetation, consumers, and predators), explaining the chaotic variation in the amplitude of peaks in the populations of hares and lynxes while the periodicity of the peaks is regular and peaks are synchronized between neighboring populations. Weaker interactions with many other species were ignored and nevertheless the essential behavior of the ecological system was retrieved.
Topdown experimental systems biology usually means fishing for relations amongst a large number of high density molecular measurements obtained for a limited number of experimental conditions. This usually does not lead to immediate improvement of mechanistic models of the system (20, 38), among others because testing of many relationships in such data sets leads to high false positive rates. Such an inductive approach of purely data-driven discovery of facts cannot replace the scientific cycle of testing and improving mechanistic models of the system (38). Large scale models (e-cells, silicon cells) that contain all mechanisms of interactions are not yet feasible because kinetic parameters and descriptive equations for many of the interactions are lacking. Designing and testing models for modules that may be assembled in larger models in subsequent steps may be a feasible strategy to build models of the whole system.
Regulation of oxidative phosphorylation via ADP and Pi.
It has been debated whether mitochondrial ATP production in cardiac muscle is regulated via concentration changes in ADP and Pi, or that additional mechanisms play a role (19). Ca2+ entry into the mitochondria, where it stimulates matrix dehydrogenases and may regulate other sites in oxidative phosphorylation may play an important regulatory role (7, 10), although at longer time scales (time constant 20–25 s) than considered here (4 s). An additional calcium signaling module would be needed to simulate this slow regulation. Even slower regulatory changes are conceivably involved, such as changes in synthesis and breakdown of the mitochondria. However, the present study is limited to the fast phase only.
The ACP module may be sufficient to explain the first phase of stimulation of oxidative phosphorylation, as shown in the present study. However, the membrane permeation parameter for ADP and ATP transport across the MOM was unknown and had to be estimated from the overall response of the system. To corroborate this finding, independent studies on MOM permeability are desirable. The model may be further tested based on its predictions, for instance on the oscillation of cytosolic metabolites and metabolite levels for different CK activities (Fig. 10) due to overexpression or inhibition of the mitochondrial and cytosolic isoforms of CK. For the moment, the hypothesis that the ACP module can explain the fast response of oxidative phosphorylation appears plausible. Note in particular that the model indicated that processes in the mitochondrial matrix play a negligible role in the delay of ATP production in response to changing ATP hydrolysis, as detailed in METHODS.
The glycolytic chain seems to play an important role in the dynamic response, increasing the response time from 4 with blocked glycolysis to 8 s if glycolysis is active (17). A remarkable finding by NMR spectroscopy during heart rate steps is that Pi, PCr, and by inference ADP, change with a time constant of roughly 2.5 ± 2 s, while in the same study tmito was 11 s. In these experiments, showing the delayed response of OxPhos with respect to the changes of the phosphate metabolites located mainly in the cytosol (13), glycolysis was fully functional. Note that the isolation procedure of the hearts in these experiments entailed a period of ischemia, which may increase tmito (46). Based on the simultaneous NMR and tmito data, we hypothesized that glycolysis is able to delay changes of ADP and Pi in the mitochondrial intermembrane space relative to the myofibrillar core. Increased glycolytic capacity after a brief period of ischemia may then further delay the dynamic response of OxPhos (46). Thus the ACP module probably shows a strong interaction with glycolysis, a complexity which was experimentally removed from the present study by blocking glycolysis and giving a substrate for the TCA cycle which bypasses glycolysis.
With the estimated permeability parameter PSATP = PSADP = 13.3 s–1 the concentration difference across the outer mitochondrial membrane is 56 µM at half of the maximum ATP synthesis rate of the mitochondria. Adding this to the KM for mitochondrial ATP production for ADP in the intermembrane space the apparent KADP for intact mitochondria is predicted to be 80 µM. This is higher than the maximum KADP value estimated for mitochondria in skinned cardiac muscle fiber experiments (25) and compatible with the suggestion that isolation of mitochondria (32) or perhaps even skinning of muscle fibers leads to a decrease of diffusional restriction for ADP across the MOM, which is present in the mitochondria in situ.
Although other regulatory factors, such as Ca2+, very likely play a role at longer time scales (7, 10), the fast adaptation of OxPhos to cytosolic demand is explained by regulation via changes in ADP and Pi in the intermembrane space. The lack of appreciable changes in PCr value at increased cardiac workload was often regarded as a sign that ADP and Pi are not the prime regulators of OxPhos. Figure 8 demonstrates that the decline of PCr depends on the parameter values and can be small for some parameter sets, even under the assumption that only ADP and Pi regulate OxPhos. ADP changes in the intermembrane space can be almost absent and Pi is then the main regulator of OxPhos for some parameter combinations. Stimulation of mitochondrial ATP synthesis via calcium ions is not included in the simulations for Fig. 8 and leads to even smaller changes in PCr. Figure 8 demonstrates that altered conditions of the system, for instance caused by altered levels of gene expression, can lead to a rather broad range of system behavior.
Role of CK in cardiac muscle cell. CK has been proposed to play several roles in muscle energy metabolism. CK was thought to be an emergency ATP synthesis system if oxidative phosphorylation and/or glycolytic ATP synthesis fail. PCr was proposed to be the major carrier for transport of "high-energy" phosphoryl groups in the cell between mitochondria and sites of ATP hydrolysis (5, 29, 41).
Figure 10 shows that CK activity reduces the amplitude of the oscillation in cytosolic ADP concentration, by 60% for a VCK change from 0.02 to 1. The oscillation in ATP synthesis is reduced even more. The Pi oscillation is virtually unaltered, because it is not buffered by the CK reaction. However, the Pi concentration is decreased from 2.1 mM for VCK = 0.02 to 1.55 mM for VCK = 1. Damping the ADP oscillation and decreasing the Pi concentration in the cytosol by CK activity is of importance to muscle cell function, because these breakdown products of the myosin ATPase might inhibit contractile function.
The contribution of PCr to transport of "high-energy" phosphate groups is only 32% at an ATP hydrolysis rate of 486.5 µM s–1 for the rabbit heart parameters, which is modest. However, this contribution depends on the parameters chosen: it becomes 72% for the rat heart parameters and becomes 88% if the CK activity is further increased by threefold relative to the "rat heart" reference condition. The contribution of PCr to high-energy phosphate transport determined in the present study is somewhat lower than predicted in other models (29).
Conclusion. A model for the adenine nucleotide-creatine phosphate module in the metabolic system was derived by selecting key processes involved in the fast regulation of oxidative phosphorylation and in transfer of high-energy phosphoryl groups. This model explains measurements on the fast regulation of oxidative phosphorylation in isolated hearts with glycolysis inactivated. The addition of a glycolytic module will be necessary to explain the slower regulation of oxidative phosphorylation in vivo with glycolysis active. The advantage of the "skeleton model" approach is that compact modules are tested while detail and size of the model are kept to a minimum. The model predicts results of experimental interventions useful for further tests, and can be easily incorporated in larger models because it is compact. The model demonstrates effective buffering of metabolite oscillations, reduction of the peak value of ADP and reduction of the average value of Pi by CK activity.
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APPENDIX A |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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A glossary of the variables in the model, which are dynamically changing during a simulation run, is given in Table 1. The subscript ims denotes the concentrations in the IMS, the subscript cyt denotes the concentrations in the cytosol (see Fig. 1). The parameters in the model, which are kept constant during a simulation run, are summarized in Table 2. The time derivatives of the concentrations (C) of five metabolites (ATP, ADP, PCr, Cr, and Pi) are calculated, each for two compartments: the cytosol and the intermembrane space (IMS):
 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
 | (A8) |
 | (A9) |
 | (A10) |
The CK reaction rate JCK is described by the following equation from Ref. 33, which describes a sequential, rapid equilibrium, random bi-bi enzymatic reaction mechanism that corresponds well with experimental data on the CK enzymes (33, 34)
 | (A11) |
 | (A12) |
Equations A11 and A12 govern the rates of both JCK,Mi and JCK,MM with separate kinetic constants for the mitochondrial or MM isoenzyme, respectively; see Table 2. Please note that the kinetic equations for the CK reactions reproduce extensive measurements on the isolated enzymes very well, both for the MM-CK (34) and Mi-CK (21, 33) isoforms.
The cytosolic and IMS compartments are separated by a partly permeable diffusion barrier formed by MOM. Metabolites permeating the outer mitochondrial membrane, thereby leaving the IMS and entering the cytosolic space, are taken into account in the mass balances of the IMS and cytosol. Diffusional fluxes, Jdiff, between the intermembrane space and the myofibrillar space across the MOM depend on the permeability surface product (PS), expressed per unit volume of total intracellular water to be consistent with the unit for fluxes (see above):
 | (A13) |
 | (A14) |
 | (A15) |
 | (A16) |
 | (A17) |
The metabolites obey the following moiety conservation relations:
 | (A18) |
 | (A19) |
 | (A20) |
The ATP exiting the mitochondrial matrix enters the IMS and thus enters the ACP module in Eq. A6 with flux Jsyn. ADP and Pi is exchanged with the ATP production module, as shown in Eqs. A6, A7, and A10, respectively. Hydrolysis flux Jhyd gives the exchange with the ATP hydrolysis module in the cytosol in Eqs. A1, A2, and A5.
Simple representation of ATP hydrolysis module communicating with the ACP module. The equations above describe the ACP module proper. Although we concentrate on the analysis of this ACP module, it communicates with two adjacent modules, representing ATP consumption in the cytosol and mitochondrial ATP production, respectively. These contacts embed the ACP module in the complete system.
First, we will discuss the ATP hydrolysis module. ATP hydrolysis in the cytosolic compartment, Jhyd, is found in Eqs. A1, A2, and A5 of the ACP module. ATP consumption can be very efficiently represented by the time course of ATP cleavage during steps in heart rate, derived from experimental measurements. ATP turnover in the steady state is stoichiometrically related to O2 consumption (12). Measurements of the rate of heat release show a sharp peak because of ATP splitting during systole, while ATP splitting is minimal during diastole (45).
Part of the simulations is therefore run with a pulsatile ATP hydrolysis rate, peaking during systole, as forcing function. The mathematical expressions for time-dependent ATP hydrolysis were described previously (41). The ATP hydrolysis rate, Jhyd, increases linearly during the first one-sixth of the cardiac cycle to peak value HATP(max)
 | (A21) |
During the next sixth of the cardiac cycle, Jhyd decreases from HATP(max) to zero
 | (A22) |
 | (A23) |
From the steady-state basal level of ATP hydrolysis (Jhyd,basis) derived from O2 uptake measurements at 135 beats/min (17), at t = 0 a step is made in heart rate to 220 beats/min and ATP hydrolysis rate Jhyd,test; these values are averaged over the cardiac cycle. ATP hydrolysis is calculated from the steady-state O2 consumption measured in the experiments in the author's laboratory which are simulated here (17), assuming 4.8 molecules of ATP synthesized per molecule of O2 consumed, a value measured experimentally for heart muscle (22). HATP(max) = 6 Jhyd,basis for a heart rate of 135 beats/min and HATP(max) = 6 Jhyd,test for 220 beats/min, based on Eqs. A21–A23.
Simple representation of ATP production module communicating with the ACP module. ATP production consists of ATP synthesis (syn) in the mitochondrial matrix by oxidative phosphorylation followed by ATP export across the mitochondrial inner membrane into the IMS. ADP leaves the IMS in immediate exchange for ATP across the adenine nucleotide translocator in the mitochondrial inner membrane. Pi leaves the IMS via the phosphate carrier across the mitochondrial inner membrane. The stoichiometry of oxidative phosphorylation dictates that for each molecule of ATP transferred from matrix to IMS, one molecule of ADP and one of Pi are transferred in the reverse direction, reflected by Jsyn in Eqs. A6, A7, and A10. The ACP module therefore exchanges ADP and Pi for ATP with a module representing mitochondrial ATP production.
Oxidative phosphorylation and metabolism in the mitochondrial matrix are complex processes requiring a separate model which merits extensive analysis. Several such models have been devised and analyzed (4, 10, 26, 27). To focus the present study on the "skeleton model" of the ACP module, we do not analyze the applicability to rabbit cardiomyocytes of various competing complex models of the mitochondria and oxidative phosphorylation. Instead the communication between ACP module and ATP production module is represented by a simple Michaelis-Menten-type equation (Eq. A24). The ATP transfer rate, Jsyn, from the mitochondrial matrix into the IMS is determined by a Michaelis-Menten-type equation depending only on ADPims and Piims:
 | (A24) |
 | (A25) |
This reflects ATP synthesis as measured in experiments on isolated mitochondria. The crucial point is that this equation has been experimentally validated (16, 36). It was also used in other models, see for instance (19, 25). The ATP amount transferred per unit time from the ATP production module to the ACP module is therefore completely determined by the ADP and Pi concentrations in the IMS in this model. Under stress conditions (hypoxia, low carbon substrate supply, presence of uncoupling substances) the parameters of Eq. A24 may be affected because ATP synthesis also depends on the energization state of the mitochondria reflected in the mitochondrial membrane potential (27, 43). However, for the present study it is assumed that mitochondrial energization is not compromised, e.g., the heart is not ischemic or carbon substrate limited, and data on well-energized isolated mitochondria represented by Eq. A24 are used. Because Eq. A24 has been well tested experimentally, it forms an efficient and simple representation of the ATP production module, adequate to test and investigate the ACP module. However, in future studies Eq. A24 may be replaced by more complex computational modules for ATP production by the mitochondria.
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APPENDIX B |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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PSdiff is calculated from diffusion coefficients measured from diffusion-sensitized NMR spectra in muscle (11). The myofibril and its small surrounding cytosolic space is approximated with a cylinder with radius a = 1.2 µm (29, 41). The solution for the concentration profile for a metabolite diffusing into this cylinder is (29)
 | (B1) |
 | (B2) |
Jdiff of the metabolite, which balances the consumption rate Q, is expressed as
 | (B3) |
 | (B4) |
The average diffusional transit time tdiff for a metabolite is defined by analogy to the mean transit time for a metabolized indicator, which was shown to equal the change in amount of metabolite in the system divided by the change in metabolic rate (40):
 | (B5) |
This tdiff for the cytosolic space is for ATP 0.340 ms, for ADP 0.346 ms, for PCr 0.243 ms, for Cr 0.197 ms and for Pi 0.189 ms. Diffusional transit times in the intermembrane space are smaller than in the myofibrillar space. The diffusion times in cytosol and intermembrane space (tenths of milliseconds) are so small relative to the total response time of oxidative phosphorylation (several seconds) that diffusional delays and gradients can be neglected. The diffusion gradients in the cytosol during the cardiac cycle were extensively modelled in (41) and shown to be very small.
Parameter PSdiff only takes diffusion through the myofibril and cytosol into account. However, the outer mitochondrial membrane is interposed between cytosol and IMS and its permeability is sometimes considered to be a major limiting factor (41). The permeation flux across the MOM is given by
 | (B6) |
The permeability coefficient P for ADP and ATP is assumed to be 0.16 µm/s in the model of Vendelin et al. (41). The conductance of the MOM, PSmom, is then calculated to be 0.10 s–1 for ADP and ATP, taking the dimensions of the model into account (41). The total conductance, PStot, between myofibril and intermembrane space via cytosol and outer mitochondrial membrane is given by 1/(PS)tot = 1/(PS)diff + 1/(PS)mom, which is 0.099995 s–1 for ADP and ATP. For PCr and Cr the outer membrane permeability coefficient is 260 µm/s, yielding PSmom = 162.5 and PStot = 155 s–1; for Pi with permeability coefficient 327 µm/s, PSmom = 204.4 s–1 and PStot = 194 s–1. This set of PStot values constitutes parameter set II reflecting low MOM permeability for ADP and ATP.
Beard uses the same permeability coefficients (4), except for ADP and ATP where 85 µm/s is taken from measurements on isolated mitochondria (28). This results in PSmom = 53.1 s–1 and PStot = 51.9 s–1. This set of PS values constitutes parameter set III.
In a later experimental study on permeabilized fibers (32), the permeabilities in Ref. 41 were modified: diffusibility in the cytosol was estimated to be 6% of the previously assumed diffusibility, and membrane permeability P was 3% of previous values, the latter being 260 µm/s for PCr and Cr, 327 µm/s for Pi and 145 µm/s for ADP and ATP. These values were thought to explain the experimentally determined ADP affinity of the mitochondria in permeabilized fibers. This results in the following parameter set: PStot,ATP = PStot,ADP = 2.6 s–1, PStot,PCr = PStot,Cr = 4.6 s–1 and PStot,Pi = 5.8 s–1. This is parameter set IV.
Finally, optimization to obtain correspondence with the measured response time of 3.7 s yielded 13.3 s–1 for PStot for ADP and ATP (see RESULTS), keeping the other PS values the same as in parameter sets II and III. This is parameter set V.
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GRANTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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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.
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REFERENCES |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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