Parameter estimation

Model parameters were estimated simultaneously to fit the t_mito values for all conditions using a least-squares optimization procedure. Different optimization algorithms (downhill simplex algorithm, Powell's method, Levenberg-Marquardt) gave similar quality of the fit. Initial and optimized parameter values can be found in Table 1. Figure 2 shows all t_mito values predicted by the model before and after parameter optimization for all conditions. After fitting, the model correctly predicts a quicker energy supply-demand signaling when CK is inhibited by 98%, causing weaker ADP/ATP buffering by CK. In the optimization procedure, the maximum velocities of the Mi-CK and the MM-CK enzyme were decreased by 12 and 36%, respectively, from their initial literature values. These literature enzyme activities for MM-CK and Mi-CK had been taken from the same experimental model, but without inhibition of glycolysis by IAA [8]. The experimental data used in the present analysis was measured in the presence of IAA which was found to impede CK activity by 20% [9]. The drop in estimated CK activity is therefore plausible. Other parameters which are altered significantly by the optimization are the apparent Michaelis constant for inorganic phosphate in the mitochondrion, K_pi, which drops from 800 to 347 µM, and the apparent K_M for ADP, K_adp, which increases from 25 to 36 µM. Both parameters occur in the model equation determining the rate of oxidative phosphorylation, which may explain the inverse variation. There exist in vitro measurements of K_pi that are lower than the initial value used in this analysis [18]: Stoner & Sirak for instance measured K_pi to be 360 µM [23] which is close to our optimized value. Likewise, reported values for K_adp vary between 20 and 30 µM [24], [25], corroborating the values obtained by the fit.

Monte Carlo sampling of parameter sets

Starting from the optimized parameter set (see Table 1), we sampled the parameter space to generate an ensemble of 658 independent parameter sets using the Metropolis-Hastings algorithm. The parameter set yielding the lowest cost in the complete ensemble was this optimized parameter set. The distributions of all parameters in the ensemble are shown in Figure 3. The nine kinetic parameters which had known error values (see Table 1) show a mean value in the ensemble close to the measured value and a standard deviation close to their reported measurement error from the literature, which was to be expected given the prior information in the cost function. However, the parameters for which there was no standard error value available from the literature in general gave a standard deviation in the ensemble which was smaller than the default assigned large standard error (see Table 1). We tested the effect of different assumptions on the default prior standard deviations on posterior parameter distributions and ensemble predictions, reported in Text S1 which shows that the conclusions reported here are not changed by larger or smaller values on the default prior.

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Figure 3. Distributions of individual parameters in the ensemble generated by the Metropolis-Hastings algorithm.

Plots show histograms of all values in the ensemble for the given parameter. The ensemble consists of 658 parameter sets. Plotted in red is the probability density function of the lognormal distribution with mean and standard deviation of each parameter scaled to the observed frequencies.

https://doi.org/10.1371/journal.pcbi.1002130.g003

The mean value of PS_mom,AdN in the ensemble is 31.7 s⁻¹, which is larger than the optimized value of 13.3 s⁻¹ found previously [18]. The distribution of PS_mom,AdN shows substantial skewing with a minimum value of 7.4 s⁻¹, and a rather sharp exclusion of small values which give slow response times of the system. Based on experiments in isolated permeabilized cardiomyocytes, Sepp et al. ([26]) estimated a value for MOM permeability to adenine nucleotides of 1833 nmol/min/mg protein per mM concentration difference. Converting this value expressed per mg tissue protein, assuming 150 mg protein per gram wet weight, this corresponds to PS_mom,AdN  =  7.45±1.89 s⁻¹. This is virtually the same as the minimum estimated in our ensemble analysis.

Predicting the contribution of PCr and ATP to energy transport

The contribution of PCr to intracellular HEP transfer, R_diff,PCr, is quantified by the ratio of PCr diffusion (J_diff,PCr) to the total phosphate group diffusion through the MOM:(2)

An ensemble of simulations based on the parameter ensemble described above allows evaluation of the confidence region for the model prediction. In the ensemble, R_diff,PCr is on average 0.17±0.09 (mean±SD) at heart rate 160 bpm and 0.15±0.08 at 220 bpm in the case of active CK. Figure 4 shows the 95% confidence interval between upper and lower bound of the ensemble prediction for R_diff,PCr for IAA and IA conditions in steady state at heart rate 220 bpm. The small oscillations during CK inhibition are due to the 2% residual activity of CK. The upper bound of the 95% confidence interval remains below 0.44 during the cardiac cycle for all simulated conditions.

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Figure 4. Prediction of energy transport from mitochondria to cytosol by PCr.

(A) Forcing function of pulsatile cytosolic ATP hydrolysis for the last two cardiac cycles of a simulation over 60 s. (B) Prediction of the relative PCr contribution to high-energy phosphate flux across the mitochondrial outer membrane (R_diff,PCr) at heart rate 220 bpm. The shaded region gives the central 95% confidence interval of the R_diff,PCr trajectories derived from ensemble simulations of 658 parameter sets. Solid lines depict a single simulation of the best scoring parameter set. Blue color indicates the condition with CK active. Simulations with CK inhibited by 98% by IA are plotted in orange. Note that two cardiac cycles are plotted after a steady state was reached.

https://doi.org/10.1371/journal.pcbi.1002130.g004

R_diff,PCr decreases during the peaks in ATP hydrolysis and even becomes negative for the lowest trajectories in the ensemble, which indicates that PCr diffuses back to the mitochondria at the end of systole (Figure 4). The simulations show for these cases that ADP diffuses into the IMS during the peaks of ATP hydrolysis, stimulating a reversal of the mitochondrial CK reaction to produce ATP from PCr, exactly as happens in the cytosol. For these lowest trajectories in the ensemble the CK activity per unit volume of the intermembrane space is higher than the CK activity per unit volume of the cytosol, causing the PCr to go down more steeply in the intermembrane space. This causes the cytosolic PCr concentration to exceed the PCr concentration in the IMS, and a negative gradient forces PCr to diffuse back into the IMS. However, when averaged over the cardiac cycle, R_diff,PCr is always positive, indicating net flux of PCr from the mitochondria to the cytosol, and for the vast majority of the ensemble PCr diffusion flux never becomes negative during the entire cardiac cycle. Simulations suggested that the relative importance of the PCr shuttle becomes less with higher ATP hydrolysis at heart rates of 160, 190 and 220 bpm. We tested this hypothesis by predicting R_diff,PCr for a range of heart rates from 60 to 300 bpm. The ensemble simulations reveal that R_diff,PCr continuously drops for increasing heart rates for all sampled parameter combinations (see Figure 5A). The predicted decline in R_diff,PCr and increase in P_i concentration agrees with results of a recent study on perfused rat hearts [27]. Increased energy demand induces an increased ATP gradient between both compartments. At 160 bpm, the average difference between the ATP concentration in IMS and cytosol is 18.6 µmol*l⁻¹, at 220 bpm it becomes 22.3 µmol*l⁻¹ for the optimal parameter set. The increased ATP gradient across the MOM induces direct ATP transport instead of facilitated transport via PCr.

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Figure 5. Dependence of PCr diffusion flux on heart rate and mitochondrial membrane permeability to adenine nucleotides.

Prediction of the PCr contribution to high-energy phosphate flux across the mitochondrial outer membrane (R_diff,PCr), averaged over the cardiac cycle, as a function of (A) heart rate and (B) mitochondrial outer membrane permeability for adenine nucleotides (PS_mom,AdN), respectively. Values for (A) Steady state values for R_diff,PCr as a function of heart rate (B) Steady state values for R_diff,PCr as a function of PS_mom,AdN at fixed heart rate of 220 bpm. We performed simulations for the ensemble of Figure 3, with the heart rate or PS_mom,AdN set according to the x-axis. Blue shaded regions depict the 95% confidence interval of the prediction, black solid lines show the prediction for the optimized parameters (see Table 1).

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In order to demonstrate the dependence of shuttle utilization on the membrane conductance for adenine nucleotides, we predicted R_diff,PCr as a function of PS_mom,AdN for the ensemble. The predicted range shown in Figure 5B indicates that only for very small ATP permeability, PCr contribution becomes high. Even for the minimum value for PS_mom,AdN in the ensemble (7.35 s⁻¹), the entire 95% confidence interval of R_diff,PCr remains below 0.5. Low MOM permeability to adenine nucleotides causes high-energy phosphate group transport via PCr, and that PS_mom,AdN is never lower than 7.35 s⁻¹ therefore argues against a predominant phosphocreatine transport. Also when the value PS_mom,AdN  =  7.45 s⁻¹ estimated from Sepp et al. ([26]), see above, is incorporated as prior knowledge, the analysis still yields similar predictions of R_diff,PCr, which stays with 95% confidence between 0.16 and 0.46 at heart rate 220 bpm.

It might be argued that the K_ia value of the mitochondrial CK should be set to 290 µM with oxidative phosphorylation active ([28]) to reflect functional coupling of CK to the adenine nucleotide translocator (ANT). Optimization based on this K_ia value gives as result that on average 18% of the high-energy phosphate flux at a heart rate of 220 beats/min is transported in the form of PCr, the rest as ATP. The parameter values for V_max,Mi,f calculated from rat heart mitochondria is 1609±113 µM/s in [28] and V_max,ATPsyn is 2960 µM/s which is about twice the value measured in the rabbit heart study analyzed here. When using the rat heart parameters combined with K_ia = 290 µM, the contribution of PCr to high-energy phosphate transport is estimated to be 25%. Further analysis of a model which incorporates a microcompartment which functionally couples the mitochondrial creatine kinase to the adenine nucleotide translocator ([6]) shows that it is difficult to explain the response time and molecular kinetic parameters simultaneously with this model. The results of this analysis can be found in Text S2. The conclusion that the contribution of PCr to high-energy phosphate transport is relatively modest appears to be robust, because the contribution was estimated to be 15–17% in the ensemble study with rabbit heart parameters, see above, and does not become substantially higher in analyses with other parameter sets.

Prediction of temporal energy buffering

The results described above indicate that direct ATP transport is predominant in working heart muscle. Given that PCr energy shuttling is of limited importance, we investigated another potential function of CK, i.e. temporal energy buffering. When ATP consumption by the myofibrils exceeds mitochondrial ATP production during muscle contraction, ATP homeostasis can be maintained by PCr [4]. Ensemble predictions for R_diff,PCr, concentrations of cytosolic ADP and P_i and ATP synthesis rate at relative CK activity of 2, 100, and 300% of wild type levels are shown in Figure 6. Note that Mi-CK and MM-CK activities are both changed by the same factor in this set of simulations. Even at 3-fold increased CK activity, R_diff,PCr does not increase dramatically (Figure 6F). However, oscillations of cytosolic ADP concentrations are significantly affected by the CK activity. The amplitude of the ADP oscillation is 77±11 µM at normal CK levels and becomes 146±1 µM if CK is inhibited by 98%, as is the case for IA treated perfused hearts (Figure 6K,J). At threefold increased CK activity this becomes 36±22 µM (Figure 6L). In simulations of a hypothetical case with 10000-fold increase of enzyme activity, oscillations of adenine nucleotide concentrations are almost fully damped to an amplitude of 2.6±0.2 µM (data not shown).

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Figure 6. Fluctuations of metabolite concentrations and fluxes during the cardiac cycle at three levels of CK activity.

Plots show (A–C) Trajectory of the forcing function of ATP hydrolysis and ensemble predictions of (D–F) R_diff,PCr, (G–I) mitochondrial ATP synthesis rate, (J–L) cytosolic ADP and (M–O) cytosolic P_i concentrations at heart rate 220 bpm. Mi-CK and MM-CK activities were set to 2, 100, and 300% of wildtype levels. Three cardiac cycles are shown at steady state. Solid lines show the simulated trajectory of the optimized parameter set (see Table 1). Shaded regions show the 95% confidence interval for all trajectories of the ensemble of 658 parameter sets. To alter CK activity, the parameters describing maximum enzyme velocity, V_max,Mif and V_max,MMf, are changed in parallel to the indicated percentage.

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The time course of mitochondrial ATP production oscillates with amplitudes of 566±31, 215±23 and 91±14 µM*s⁻¹ for 2, 100 and 300% relative CK activity, respectively (Figure 6G–I). The pulsatility of ATP and ADP concentrations and of ATP synthesis is synchronized to ATP hydrolysis in the myofibrils. The confidence regions for these trajectories are relatively narrow. By blocking CK by 98%, the average concentrations of ADP in the IMS increases to 64±9 µM from 56±9 µM at normal CK levels. In contrast to ADP, the amplitude of oscillations of cytosolic inorganic phosphate stays relatively constant at different CK activities at about 145 µM. This reflects that P_i is not directly buffered by CK. Surprisingly, average levels of cytosolic inorganic phosphate drop with CK activity. The average P_i concentration at 2% CK activity is 1618±97 µM and becomes 1416±80 µM for wild-type CK activity (Figure 6M,N). For all parameter sets in the ensemble the P_i concentration declines when CK activity is increased.

The specific role of the mitochondrial CK isoform

Transport of HEP by PCr from mitochondria to cytosol partially takes place via the circuit formed by both CK isoforms, but was predicted to be quantitatively not very important. On the other hand, temporal buffering of the systolic ATP hydrolysis burst needs only the MM-CK activity in the cytosol, which is much higher than the Mi-CK activity (see Table 1). It was therefore still unclear what the function of the mitochondrial CK isoform is.

In order to further elucidate the effect of the compartmentalized CK system on metabolism, we performed ensemble predictions with individual inhibition of both CK isoforms one by one. In Figure 7, we show the 95% confidence intervals of predicted metabolite concentrations and reaction fluxes. The amplitude of oscillations in mitochondrial ATP synthesis is predicted to rise from 215±23 µM*s⁻¹ at baseline CK activity to 278±33 with 98% Mi-CK inhibition, compared to 375±21 µM when MM-CK is inhibited by 98% (Figure 7I–K). Thus, despite its low activity, Mi-CK still has a small but clear effect on the ATP synthesis oscillation amplitude. Inhibition of Mi-CK has a larger effect when MM-CK is already inhibited (amplitude 565±31 µM*s⁻¹, Figure 7L). The damping of ADP oscillation is highly affected by MM-CK but not by Mi-CK: 98% inhibition of Mi-CK leads to an increase in the amplitude of systolic ADP oscilation from 77±11 to 83±11 µM (Figure 7M,N), whereas MM-CK inhibition doubles the amplitude to 146±1 µM (Figure 7O).

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Figure 7. Ensemble predictions of metabolite concentration and flux oscillations during the cardiac cycle for selective CK isoform inhibition.

In the first row (panels A–D), the pulsatile forcing function for ATP hydrolysis is plotted. Predictions of the time courses of (E–H) relative contribution of PCr to high-energy phosphate transport, R_diff,PCr, (I–L) ATP synthesis rate, (M–P) cytosolic ADP and (Q–T) P_i concentrations. Heart rate is 220 bpm. In the four columns we compare: no CK inhibition, 98% Mi-CK inhibition, 98% MM-CK, or both CK enzyme reactions inhibited by 98%. Black solid lines show the simulated trajectory of the optimized parameter set (Table 1). Blue shaded regions show the 95% central confidence interval for all trajectories of the ensemble of 658 parameter sets. To alter CK activity, the parameters describing maximum enzyme velocity, V_max,Mif and V_max,MMf, are changed to the indicated percentage. Three cardiac cycles are shown after a steady state was reached. Note that the first and the last column also appear in Figure 6 and are shown here for ease of comparison.

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Predictions of R_diff,PCr illustrate that both Mi-CK and MM-CK are required for a functioning phosphocreatine shuttle. PCr diffusion averaged over the cardiac cycle makes a very small contribution to total HEP delivered from the mitochondria when either Mi-CK or MM-CK is inhibited by 98%. With 98% inhibited Mi-CK activity, R_diff,PCr is even slightly below zero during diastole with low ATP hydrolysis, meaning that PCr is transported from cytosol to IMS (Figure 7F). Note that this situation is reversed with respect to normal Mi-CK and MM-CK activity where PCr diffusion is always positive during diastole and occasionally becomes negative during ATP hydrolysis peaks. For normal CK activity the explanation for reversed PCr diffusion during ATP hydrolysis (Figure 7E) was that the CK activity per unit volume is higher in the IMS than in the cytosol. During Mi-CK inhibition this is of course no longer the case and systolic PCr consumption in the cytosol leads to PCr diffusion from the IMS, explaining the reversal of PCr transport during systole. In contrast, with MM-CK inhibited, ATP is buffered by Mi-CK in the IMS and PCr diffuses to the IMS at the end of the ATP hydrolysis peaks. This explains why R_diff,PCr goes more negative during ATP hydrolysis peaks with MM-CK inhibition and its oscillation is stronger than for normal Mi-CK and MM-CK activity (Figure 7E, G). When inhibiting Mi-CK activity, our model predicts an increase in the amplitude of [ADP] oscillation in the IMS from 57±8 to 71±8 µM. Mi-CK therefore has a damping effect on oscillations of ADP concentrations in the IMS, which contributes to the damping of mitochondrial ATP synthesis.

The concentration of cytosolic P_i is predicted to be lowered by mitochondrial creatine kinase activity. Blocking Mi-CK leads to a P_i increase by about 18% from 1416±80 to 1670±167 µM (Figure 7Q, R). If Mi-CK is inhibited by 100%, the steady state P_i concentration becomes 1678±173 µM (data not shown). MM-CK inhibition decreases the P_i concentration; a combination of Mi-CK and MM-CK inhibition leads to a slightly higher P_i level compared to the wildtype (Figure 7S, T).

Computational model

For our analysis, we employed a previously published computational model [18]. It is available in various formats and can be found in the BioModels database [47] as well as in the CellML model repository [48]. The model incorporates the key elements of the CK system with ATP synthesis in the mitochondria and pulsatile ATP hydrolysis in the cytosol (see Figure 1). The input of the model is a forcing function of cytosolic ATP usage catalyzed by myosin-ATPase and ion pumps. The model contains ten ordinary differential equations (ODEs) describing the rate of change of each metabolite concentration (ADP, ATP, PCr, Cr, P_i) in two compartments over time. These equations were extensively described previously [18]. Model dynamics depend on 22 kinetic parameters retrieved from the literature which are listed in Table 1. In general the kinetic constants retrieved from the literature have relatively modest standard errors. However, for the permeability of the MOM to ATP and ADP (assumed to be equal in the model analysis; cf. [6]), reported values differed from 0.16 [6] to 85 µm*s⁻¹ in the model of Beard [14] based on measurements of Lee et al. [49]. This large variation is possibly due to mitochondrial isolation or cell membrane permeabilization procedures.

The mitochondrial outer membrane permeability-surface product parameter PS_mom,AdN influences the response time for dynamic adaptation of oxidative phosphorylation strongly. Therefore the dynamic measurements of venous oxygen outflow in the heart as a whole in response to an increase of heart rate allow estimating the mitochondrial membrane permeability at the organellar level. The whole heart measurements were corrected for oxygen transport delay to reflect events at the level of the mitochondria (see below). The mitochondrial response time t_mito is defined as the generalized time constant of the time-course of oxygen consumption (defined to be equivalent to the first central statistical moment of the impulse response function in case the system is linear), previously described in [18], [50]-[52]. From a model simulation, t_mito is calculated as follows:(3)

Where J_ATPhyd,basis and J_ATPhyd,test are the values for the ATP hydrolysis rates for the two electrically paced heart rates at baseline and test level, averaged over the cardiac cycle; J_ATPsyn denotes the time course of ATP synthesis in the mitochondrion. t_step is the time point when the heart rate is increased and t_end is the time point of the final oxygen measurement. Note that the average J_ATPsyn in the steady state before and at the end of a test challenge equals J_ATPhyd,basis and J_ATPhyd,test, respectively.

In order to correspond with the experimental conditions in [9], t_end was set to 60 seconds with t_step = 0 seconds; an initial run for 40 seconds before the heart rate step ensures that ATP synthesis has adapted to ATP hydrolysis and is found to be in steady state at this stage. In order to investigate the damping capabilities of the modeled system, ATP hydrolysis is simulated as a pulsatile function representing the alternating nature of energy demand in systole and diastole as described in [18]. Figure 8 shows the dynamic response of mitochondrial ATP production in response to a step in heart rate and ATP hydrolysis.

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Figure 8. Pulsatile nature of energy production and consumption in the beating heart and the response to a step in heart rate.

Shown are the time courses of (A) ATP hydrolysis and (B) synthesis simulated with the model of Figure 1. At time 0 s, average ATP hydrolysis rate was increased from 486.5 to 627.6 µM*s⁻¹ corresponding to an increase in heart rate from 135 to 220 bpm, as was imposed in the experiments which were simulated in this study. Please note the difference in scale of the y-axis between panels A and B.

https://doi.org/10.1371/journal.pcbi.1002130.g008

Sloppy ensemble modeling

Almost all models in systems biology contain parameters that cannot be determined precisely. It is common practice to estimate missing parameter values by a parameter fit to experimental data. After the fit, one can make model predictions and analyze the underlying biological processes. This, however, is dangerous because a range of parameter combinations may agree with the available data equally well, potentially leading to deviating model predictions of new experimental situations. Directions in parameter space where parameter changes do change the simulation outcome very little were termed ‘sloppy’ by Brown et al., whereas directions where small changes in parameter values affect the dynamic behavior of the system strongly were termed ‘stiff’ [21]. Sloppy parameter sensitivity spectra have been identified for numerous biological models by the analysis of the eigenvectors and eigenvalues of a sensitivity matrix calculated from the chi-square cost function [22]. Sloppy models exhibit a characteristic pattern with the logarithms of eigenvalues approximately uniformly distributed over a large range. A sensitivity analysis of the CK model revealed the presence of both stiff and sloppy parameter combinations and a ‘sloppy’ sensitivity spectrum [53]. Since our model shows sloppy parameter sensitivities and is based on data subject to experimental variation, drawing predictions from an ensemble of parameter sets is preferable to merely relying on one parameter set fit to experimental data. According to the sloppy modeling paradigm ([21], [22]), the probability of a set of model parameters to be included in the ensemble is proportional to its likelihood to describe given experiment data multiplied by the likelihood of prior experimental information about the parameter values themselves. The sampling process is thus based on Bayesian inference of a posterior distribution of parameter sets , where is the likelihood of the data given a parameter set, is the prior probability of the parameter set based on experimental prior knowledge on single parameter values and the posterior is the probability of a parameter set to describe the given experimental data. The construction of the ensemble with a Markov-Chain Monte Carlo (MCMC) method was done with the Metropolis-Hastings algorithm [54]. The Sloppy cell software environment, used for the analysis, was adapted to process all operators which were in the SBML file describing the model. The modified version is provided in Dataset S1. To speed up convergence, Sloppy Cell takes larger steps along sloppy directions and smaller steps along stiff directions in parameter space; this ‘importance sampling’ is described in [20], [21].

Experimental data

Measured values of molecular model parameters and their provenance, extracted from the scientific literature, are listed in Table 1. For nine of the 22 parameters reliable standard measurement errors could be found. In addition to the direct measurements on molecular parameters, we employ t_mito values from a study by Harrison et al. where the effects of inhibiting creatine kinase and different sizes of electrically paced heart rate jumps in rabbit hearts were investigated [9]. Isolated hearts were perfused with Tyrode's solution containing among others glucose and pyruvate to provide substrates for energy metabolism. In our dataset we include two experimental conditions where hearts were exposed to either (i) iodoacetic acid (IAA) to block glyceraldehyde-3-phosphate dehydrogenase (GAPDH) or (ii) iodacetamide (IA) to inhibit both CK and GAPDH. In order to provide a sufficient amount of reducing equivalents to fuel aerobic respiration despite the inhibition of glycolysis, the buffer also contained pyruvate.

Adenosine was added to the Tyrode buffer to ensure that oxygen supply is non-limiting when oxygen consumption is recorded. The whole heart measurements were corrected for the O₂ transport time in the coronary vessels based on a model of oxygen transport by convection in blood vessels and diffusion through tissue. The t_mito therefore reflects the response time at the level of the mitochondria (cf. [9] and references cited there). The mean response time was also corrected for a small deviation from an ideal step in beat-to-beat ATP hydrolysis measured as an initial overshoot in rate-pressure product [50]. For each condition, steps in heart rate were imposed from 135 to 160, 190 and 220 beats per minute, respectively, using electrical pacing. Note that glycolysis is always inactive when the dynamic response is measured, which corresponds to the absence of glycolysis in the computational model. This approach made it possible to isolate the contribution of the CK system from the contribution of glycolysis, which removes substantial complexity from the model analysis.

A step in ATP hydrolysis from 486.5 to 627.6 µmol*l⁻¹ cell water*s⁻¹ corresponds to a step in the electrically paced heart rate from 135 to 220 bpm, as was estimated from measurements of myocardial oxygen consumption [18]. From these values, we linearly interpolated hydrolysis rates of 531.4 and 579.5 µmol*l⁻¹ cell water*s⁻¹ for heart rates 160 and 190 bpm, respectively. To simulate CK inhibition by IA the model parameters for the maximum velocities of both enzyme reactions were set to 2.3% of their original values, corresponding to the CK activity measured for the inhibited hearts. Note that the enzyme activities, the mitochondrial capacities and the whole organ dynamic response times were all measured in the same experimental model by the same laboratory.

Cost function

Model parameters are fitted to experimental data using a modified Levenberg-Marquardt least-squares procedure in logarithmic parameter space, which is part of the SloppyCell modeling environment. For our model and data we calculate the cost for a given parameter set as follows:(4)with y_c being the model prediction of t_mito (Eq. 3) as a function of the parameter value θ and d_c the measured value for condition c with standard error . The first term of the cost function takes into account the experimental data on the whole heart level, whereas the second term represents prior experimental information about parameter values found in the literature or measured in conjunction with the modeled experiments. The prior cost, which gives a penalty for a parameter θ_i for drifting to far from its measured value θ_i^*, is calculated as in [54]:(5)

Note how the prior is used to enter experimentally measured information on parameters measured at the molecular level in the second term of Eq. 4, while the first term contributes measured information on the whole system response. The deviations of the predicted response times from their measured values are penalized relative to their measured standard errors and the deviation of the molecular parameters from measured values are penalized relative to their reported standard errors. Values for molecular parameters reported in the literature are usually given as mean and standard error. However, in the sloppy modeling framework, it is preferable to choose a normal distribution in log space [20], [22], [54]. A Gaussian distribution of logarithmic parameters has been proposed to be biologically plausible [55]. This forms a convenient way to deal with dimensionless positive quantities as parameter values [56].

In order to calculate the σ value for a parameter θ in log space from its reported standard error (considering the span of a 95% confidence region), we set the value as follows:(6)where SE is the absolute standard error of parameter θ_i. If the standard error is small relative to the mean of the parameter, the shapes of the prior distributions become approximately normal (see Figure 3). Since standard errors for only nine of all 22 system parameters could be found, we chose the default value for the remaining parameters to be at the maximum of all values for parameters with known error. This maximum was the error of the parameter for the binary dissociation constant for creatine from Mi-CK (K_ib,Mi, and see Table 1). In order to investigate the effect of altered default prior standard deviation on posterior parameter distributions and ensemble predictions, we performed several additional ensemble simulations with lower and higher default values. Results of these simulations can be found in Text S1. The parameter describing MOM conductance for adenine nucleotides, PS_mom,AdN, could not be reliably determined by experiments on the organellar level and was therefore not constrained by a prior.

Determining prediction uncertainty: Ensemble simulations

A first estimate of parameter values was determined by a least-squares fit to the data, using the cost function of equation 4. This initial best parameter estimate resulting from the optimization is used as the starting point for a walk through the parameter space using the Metropolis-Hastings algorithm. Starting the random walk from the optimized set of parameters made the algorithm converge more quickly to the posterior distribution. We use the algorithm's implementation in SloppyCell to sample parameter sets with probability density proportional to . All scripts to reproduce the presented calculations can be found in Dataset S2. To ensure that the members of the ensemble are statistically independent, we ‘prune’ the ensemble by taking only every n^th sample, where n is the maximum correlation time of all parameters. The correlation time of a parameter is defined as the time constant of its autocorrelation function. For our model, taking 50000 steps in the random walk is sufficient to obtain more than 600 independent parameter sets. The independent parameter sets in the ensemble provide the final estimate of the parameters, not only characterized by a mean but also by a standard deviation which reflects the spread of the estimation. Calculations were executed in parallel on a ClusterVision parallel machine with 16 nodes of four 3GHz processors with 4GB RAM. For computational performance reasons, we calculated model simulations for parameter estimation and ensemble sampling with an ATP hydrolysis rate averaged over the cardiac cycle rather than the pulsatile pattern shown in Figure 8. This reduced the time needed for calculations tremendously, making it feasible to do the ensemble calculations in several hours.

However, to investigate the damping characteristics of the system, we use a pulsatile forcing function of ATP hydrolysis (see Figure 8A) [18]. To assess the differences in metabolite levels and fluxes caused by replacing the pulsatile function with a time-averaged continuous function, 1000 parameter sets were randomly drawn from all parameter sets tried in the Monte-Carlo random walk, to compare the values of model results between pulsatile and nonpulsatile simulations. The variables most affected by the pulsatile approximation are R_diff,PCr and t_mito. The difference between pulsatile vs. nonpulsatile simulations of all 1000 parameter sets is 7.6±4.3 and 6.8±1.5% (mean±SD), respectively. t_mito values from nonpulsatile simulations are always slightly smaller than values from a pulsatile simulation, but their deviation is smaller than the standard error of the experimental t_mito data. The difference between pulsatile and non-pulsatile model results for other variables is below 4.5% of their average values in a nonpulsatile setting.