shows the components of the respiratory chain connected in the extended model. Respiratory complexes I to III are components of the electron transport chain connected through ubiquinone. Complex III is linked to complex IV through reduction/oxidation of cytochrome c. NADH, which is a substrate for complex I, is produced in the TCA cycle. Since the total concentration of NAD+ and NADH is conserved, NADH consumption, which fuels electron transport in the respiratory chain, defines the levels of NAD+, which is a substrate for several reactions in the TCA cycle. In this way, the extended model links electron transport with central energy metabolism, in particular with the reactions of the TCA cycle.
Scheme for mitochondrial respiration and linked processes simulated in the model.
Determination of parameters by fitting experimental data
As described in the Methods
section, the model of the respiratory chain and linked substrate transport and TCA cycle reactions contains 51 parameters. Out of 22 parameters of complex III, six ratios for forward and reverse rate constants were expressed through midpoint potentials. The order of magnitude for the rate constants for forward electron transport reactions in complex III can be estimated based on previous studies 
. A qualitative reproduction of measured triphasic dynamics of cytochrome bH
reduction by succinate in isolated cytochrome bc
1 complex 
, as described in Text S1
and Figures S1
, provides some restrictions for rate constants for binding/dissociation of complex III with ubiquinone species.
The rates of respiration in the presence of ADP (state 3) or an uncoupler characterize the maximal capacity of the respiratory chain. In the absence of ADP (state 4), the respiration rate is characterized by proton leaks, which must be compensated by respiration. According to our measurements, the respiration rate is 480±40 and 170±30 ng atom O/min/mg protein in the uncoupled and in state 4 in succinate-fueled mitochondria, and 410±30 and 80±20 ng atom O/min/mg protein in mitochondria fueled by pyruvate and malate, respectively.
If mitochondria fueled by succinate do not expend the energy of the transmembrane electrochemical potential on ATP synthesis (state 4), succinate oxidation results in fast reduction of intramitochondrial NAD+
. In the presence of rotenone, an inhibitor of electron transport in complex I, NAD+
reduction is characterized by NAD+
-dependent reactions of the TCA cycle and in particular the forward respiratory flux resulting from succinate oxidation. In the absence of rotenone, reverse electron transport 
also participates in NAD+
reduction, which makes the process much faster (). These data define the rate constants for reverse electron transport.
Simulation of forward and reverse electron flows in the respiratory chain.
While succinate fuels complex III through succinate dehydrogenase, the oxidation of malate and pyruvate in the TCA cycle fuels complex I by reducing NAD+ to NADH. Respiration under such conditions defines the characteristics of complex I.
To evaluate the model parameters, we used a procedure that simulates all the different types of data listed above for the same set of parameters. The ratio of forward and reverse constants defined by a known midpoint potential or dissociation constant was kept fixed, and the conditions of substrate supply or membrane permeability for protons were changed in accordance with experimental conditions. The procedure fitted all the data by changing the free parameters within the order of magnitude indicated in 
, summarizing and minimizing the deviations in several calculations that simulated measurements. Minimization was performed using a standard stochastic procedure in the global space of parameters as described in Methods
The best fit reproduces well the dynamics of NAD+
reduction measured in brain mitochondria in the presence and absence of rotenone using the same set of parameters (). The insets in show respiration rates and ΔΨ in the presence and absence of rotenone. These characteristics remain practically the same in both conditions. Without rotenone inhibition reversible electron flow through complex I, which fits the experimental data shown in , is directed to NAD+
reduction (is negative) only during a short period of time (), although ROS are constantly produced for a much longer time under such conditions 
. Reverse electron flow is believed to induce excessive ROS production, but evidently these two processes are not correlated.
Rotenone essentially changes the dynamics of NADH measured before succinate addition. It is slightly oxidized by the RC in the absence of rotenone, but slowly reduced in its presence. This reduction is a result of oxidation of internal substrates while electron flow through the RC is blocked. We found that the metabolites of TCA cycle cannot be substrates that provide NADH reduction, because oxidation of TCA cycle metabolites results in much faster initial reduction of NADH. If the parameters of TCA reactions are changed to slow down and reproduce the initial dynamics of NADH, maximal respiration rate with pyruvate becomes inconsistent with experimental data (not shown). Rather, slow oxidation of other metabolites, probably aminoacids or lipids, contributes to NADH reduction. The simulation of such slow oxidation did not prevent NADH oxidation in the absence of rotenone, and reproduced NADH reduction in its presence.
While, in the absence of rotenone, succinate induced much faster NADH reduction due to reverse electron transport, the steady state levels are lower than in the presence of rotenone. The steady state levels are defined by NADH production and consumption in respiration. Rotenone blocks the consumption, therefore NADH levels further increase when rotenone is added after succinate. The model parameters were adjusted without considering subsequent NADH increase and the reproduction of this phenomenon validates the model.
The model reproduces measured maximal and state 4 respiratory electron flows for succinate-fueled mitochondria, as well as for mitochondria fueled by pyruvate/malate (). The change in ΔΨ in the same simulations qualitatively corresponds to known changes measured under such conditions (). The parameters for simulations shown in are listed in (column indicated as best fit).
The 99% confidence intervals of parameters and levels of free radicals.
These simulations of measured data provide an insight into important hidden characteristics, such as the capacity of ROS production. ROS are produced by the respiratory chain as a consequence of one-electron transfer directly to oxygen from free radicals of electron transporters such as the semiquinone radical (SQ) at the Qo site in complex III 
or FMNH 
, SQ bound to complex I 
, or N2 centers 
in complex I. Simulating the experimental data as presented in the model at the same time simulates the dynamics of these free radicals.
Qualitative analysis of associations between the overall ROS production and individual radicals
The model describes various states of respiratory complexes formed in the process of electron transport, including those containing free radicals. Such radicals could be responsible for passing unpaired electrons to oxygen thus forming superoxide radicals and other forms of ROS. The contributions of various radicals to ROS production remain unknown; to clarify it we compared measured ROS production and the levels of various free radicals predicted by the model for the same conditions. A similar change in radical content and measured ROS production indicates qualitative accordance between the model and the described process and thus validates the model.
It is generally known that inhibition of reverse electron transport by rotenone decreases ROS production in succinate-fueled brain mitochondria 
, . In our measurements, ROS accumulation was inhibited immediately after rotenone addition (). The model predicts that the SQ content at site Qo in complex III in succinate-fueled mitochondria is practically unchanged by the presence of rotenone () and this remains valid for simulations with any set of parameters describing the data well. This is the reason for the coincidence of intervals for SQ at Qo for the first two types of simulation shown in . Thus, the ROS-stimulating role of reverse electron transport and the ROS-inhibitory effect of rotenone cannot be explained at the level of complex III. Apparently, reverse electron flow mainly affects complex I by increasing the concentrations of free radicals able to pass electrons to oxygen.
Effect of rotenone on ROS production in mitochondria fueled by 0.5 mM succinate.
The model predicts that rotenone essentially decreases initial levels of SQ bound on site Qn (), FMNH (), and the content of reduced N2 centers (inset). After an initial decrease, levels of SQ and FMNH increase, in agreement with the acceleration of ROS production measured after initial inhibition induced by rotenone (). The reason for accumulation of free radicals and acceleration of ROS production is the production of malate from succinate, which then reduces NAD+ in malate dehydrogenase reactions. This supply of substrate for complex I increases ROS production in rotenone inhibited mitochondria. The increase in the rate constant for malate–succinate exchange eliminates a slow increase in free radical content when rotenone is present (dashed curve in ). It should be noted that the acceleration of ROS accumulation is not always observed experimentally and this agrees with the predicted disappearance of this slow component after acceleration of malate-succinate exchange. Such similarity of experimental and simulated behavior supports the mechanism accepted for its simulation and in this way validates the model.
The fact that the species of complex I can contain more than one radical makes it more difficult to understand the contribution of each site. In particular, the species 1101001 (the positions of digits correspond to Qp-Qp-Qn-Qn-N2-FMN-FMN), which contain SQ and FMNH radicals, slowly accumulate after inhibition by rotenone. This accumulation defines the dynamics of SQ and FMNH, whereas 1101100 defines the fast component in levels of Qn-bound SQ and reduced N2 (inset in ). It is possible that only one of coupled radicals makes the major contribution to ROS production, but in this case the levels of other radicals would also correlate with ROS production. On the other hand, radicals situated inside the same species could interact, so that the specie as a whole produce superoxide. In the considered example the behavior of the whole ensemble of radicals in complex I agrees with the observed effect of rotenone, and this validates the model.
Overall, according to the model predictions, rotenone hardly affects SQ levels in complex III, but initially it significantly decreases the levels of free radicals produced in complex I; this is the reason for the decrease in ROS production induced by rotenone in succinate-fueled mitochondria. The model also explains the subsequent increase in ROS production as a result of the formation of malate in rotenone-inhibited mitochondria.
Rotenone induces a large increase in ROS production in pyruvate/malate-fueled mitochondria (). The corresponding simulations show that rotenone greatly increases the levels of FMNH and SQ at Qn site, but decreases the levels of reduced N2 (). Since the changes in N2 disagree with measured ROS production, probably N2 center does not make essential contribution in ROS production under the considered conditions. The same species (1101001) that defined the slow component in the increase in free radicals now change faster and defines the main part of the response to rotenone. Species 1101100 also constitute an essential part of the total radical content, but their levels decrease in response to rotenone in accordance with the decrease of total N2 radical levels.
Effect of rotenone on ROS production in mitochondria fueled by 5 mM pyruvate and 5 mM malate.
Stimulation of electron transport by addition of ADP or an uncoupler such as FCCP to succinate-fueled mitochondria results in a decrease in ROS production (). This generally known phenomenon 
validates the model prediction that the levels of all free radicals decrease when electron transport is stimulated by addition of ADP or an uncoupler.
Effect of acceleration of electron transport on ROS production inmitochondria fueled by 3 mM succinate.
Mitochondria fueled by pyruvate/malate also produce less ROS when electron transport is stimulated by an uncoupler (). Such measurements also validate the model, which predicts a decrease in the levels of free radicals ().
Effect of stimulation of electron transport on ROS production in mitochondria fueled by 5 mM pyruvate and 5 mM malate.
At high succinate concentrations, brain mitochondria produce much more ROS than those fueled by pyruvate (). The model also predicts higher levels of free radicals in complex III, as well as in complex I, for mitochondria fueled by succinate ().
Substrate dependence of ROS production and levels of free radicals.
Thus, the study of associations between measured ROS production and predicted radical levels in RC revealed qualitative consistency of measurements with all types of radicals and therefore validated the model, or showed a way of discrimination between possible sites of ROS production, and even between possible ROS producing species. However, in the latter case, a special, quantitative study is needed, which currently is beyond of the scope of presented study.
Prediction of bistability for the whole respiratory chain
It has been predicted that the Q-cycle mechanism of complex III can in principle induce bistable behavior 
. The whole respiratory chain considered here, with the parameters that fit the experimental data, also has two different steady states for the same parameters. shows that the SQ content at site Qo in complex III could persist at different values, depending on whether the respiratory chain is initially reduced or oxidized. shows how the steady states for free radicals of complexes I and III change with the external succinate concentration for a set of parameters that reproduces the experimental data described above.
Bistable behavior of the respiratory chain.
With increase in succinate concentration at some point the system switches to the state with the highest levels of semiquinone radicals at Qo site of complex III. The difference here from the similar curve in is that pyruvate is present in addition to succinate. Once the system is switched to the state of highest SQ content at Qo, it remains in this state even if the succinate concentration decreases back to low values. Thus, if the system is initially in an oxidized state, the steady state SQ levels at Qo depend on the succinate concentration, in accordance with the blue curve in . If the system is initially in a reduced state, it remains in this state until succinate concentrations decrease to the micromolar range. Since complex III is directly connected to complex I through a common substrate (ubiquinone), the bistable behavior of complex III induces bistability in complex I. However, when complex III enters the state with high SQ levels at Qo, SQ levels at Qn decrease (), as well as the levels of other free radicals in complex I (not shown). In some range of succinate concentrations total amount of radicals in the two presented steady states can be similar, but this does not necessary means similar ROS production in the two states since the probability of ROS production can be different for various radicals.
Thus, bistable behavior remains valid for the extended model of the RC with proton translocation and transmembrane potential (ΔΨ) generation, and with parameters defined by fitting the experimental data and validated by qualitatively similar predicted and measured ROS production. The model predicts also that a pulse of succinate is associated with decrease of ΔΨ. Such counterintuitive decrease of ΔΨ induced by increase of substrate for respiration is shown in . The value of ΔΨ decrease, induced by the same pulse of succinate, can be different, depending, for instance, on membrane leak, as illustrated by two curves in . Measurements of ΔΨ using safranine O fluorescence revealed that the mean ΔΨ at low succinate (0.2 mM) is greater than at high succinate (2 mM) (), thus validating the paradoxical prediction of the model.
Decrease in ΔΨ on transition from the oxidized to the reduced state.
Interactions between N2 centers and quinones in complex I.
The succinate threshold for a switch to the reduced state depends on the parameters of pyruvate transport and TCA reactions; here we do not investigate the quantitative details with respect to bistability, but emphasize only the qualitative similarity of predicted and measured behavior. With regards to the considered in the previous sections normal “working” steady state, the predicted levels of free radicals are robust with respect to the model parameters, as the next section shows.
Sensitivity to parameters and robustness of the model predictions
The sensitivity of simulations to variations in model parameters is shown in Table S1
for each type of experimental data presented in (dynamics of NAD+
reduction, maximal and state 4 respiratory fluxes). The sensitivity is also listed for simulated levels of free radicals shown in . The results indicate that significant changes in some parameters hardly affect the simulations (e.g. kqp_FS
). Evidently, the data do not restrict the parameter values and they could not be defined unambiguously. However, changes in these parameters within the range, for which fitting remains good, do not affect the predictions in terms of free radical levels. The parameters shown in red highly affect the simulations. However, it is possible that different combinations of such parameters could fit the measured data equally well because of mutual compensatory changes. In this case, despite the high sensitivity, the parameters can have a wide range of values for which a good fit is obtained. Confidence intervals rather than sensitivity are used to characterize the robustness of parameter determination.
Different sets in the global space of parameters that fit the experimental data could be identified using our stochastic algorithm for minimization of the objective function χ2
(sum of squares of deviations from measured data normalized by standard deviations). The algorithm identified confidence intervals for parameters based on fixed thresholds of χ2 
shows the 99% confidence intervals for the free parameters. The ranges for which the values give a good fit to the data are large. Thus, even though the measurements cover various modes of respiratory chain operation, the data do not restrict the parameters sufficiently to define them unambiguously. Various sets over a wide range of parameters can describe the data equally well. However, the situation is different for free radical levels predicted for the simulated experimental conditions. lists intervals for predicted free radical levels simulated using the parameters sets that fit the data with χ2 that is below the threshold. The confidence intervals for free radical levels are generally much narrower, so the predicted values are more robust. Although the intervals for SQ at Qo sites in succinate-fueled mitochondria are relatively large, they are clearly almost the same for both conditions (with or without rotenone). This result agrees with data indicating that the SQ content at Qo practically shows no dependence on the presence of rotenone (). The levels of all free radicals in complex I under the conditions for the first two simulations are very robust, despite the high parameter variability. If the parameters give a good fit, the model predicts similar levels of complex I radicals. Although the intervals are relatively large under the third condition (pyruvate/malate supply), it is evident that they are much lower than the intervals for the condition of succinate supply, as well as the levels of radicals in complex III.