We used meiosis II arrested mouse oocytes as a model system. MII oocytes are in a metabolic steady-state, which eases interpretations of metabolic perturbations. ATP synthesis in mouse oocytes occurs primarily through oxidative phosphorylation using pyruvate, without an appreciable contribution from glycolysis (Houghton et al., 1996), providing an excellent system to study mitochondrial metabolism. Mouse oocytes can be cultured in vitro using chemically well-defined media (Biggers and Racowsky, 2002). In our work, we used AKSOM as the culturing media (Summers, 2013). The oocytes can directly take up pyruvate supplied to them or derive it from lactate through the activity of lactate dehydrogenase (LDH) (Lane and Gardner, 2000), and they can remain in a steady-state for hours with constant metabolic fluxes. While NADH and NADPH are difficult to distinguish with fluorescence measurements due to their overlapping fluorescence spectrum, the concentration of NADH in mouse oocytes is 40 times greater than the concentration of NADPH for the whole cell (Bustamante et al., 2017) and potentially even greater in mitochondria (Zhao et al., 2011), so the autofluorescence signal from these cells (particularly from mitochondria) can be safely assumed to result from NADH.

To investigate how FLIM measurements vary with mitochondrial activities, we performed quantitative metabolic perturbations. We first continually varied the concentration of oxygen in the media, from 50±2 μM to 0.26±0.04 μM, over a course of 30 min while imaging NADH autofluorescence of oocytes with FLIM (Figure 1a, top, black curve; Video 1). NADH is present in both mitochondria and cytoplasm where it is involved in different metabolic pathways. To specifically study the response of NADH in mitochondria, we used a machine learning-based algorithm to segment mitochondria from the NADH intensity images (Berg et al., 2019; Figure 1b and Figure 1—figure supplement 1). We verified the accuracy of the segmentation with a mitochondrial labeling dye, MitoTracker Red FM, which showed a 80.6±1% (SEM) accuracy of the segmentation (Appendix 1).

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Using the segmentation mask, we obtained the intensity of NADH, $I$, in mitochondria by averaging the photon count over all mitochondrial pixels. The intensity increased with decreasing oxygen concentration (Figure 1a, top, red), as is readily seen from the raw images (Figure 1a, middle). Restoring oxygen to its original level caused a recovery of NADH intensity, indicating that the observed changes are reversible (Figure 1a; Video 1). These observations are consistent with the expectation that NADH concentration will increase at low oxygen levels due to oxygen’s role as the electron acceptor in the ETC. In addition to intensity, FLIM can be used to determine the enzyme engagement of NADH by measuring the photon arrival times, from which fluorescence lifetimes can be fitted. Enzyme-bound NADH has a much longer fluorescence lifetime than free NADH (Sharick et al., 2018), allowing bound and free NADH to be separately resolved, but the precise fluorescence lifetimes of NADH depend on a range of factors, including viscosity, pH, and the identity of the enzyme NADH binds to (Sharick et al., 2018; Ghukasyan and Heikal, 2015). To fit NADH fluorescence lifetimes, we grouped all detected photons from mitochondria to form histograms of photon arrival times from NADH autofluorescence for each time point (Figure 1a, lower). We fitted the histograms using a model in which the NADH fluorescence decay, ${\displaystyle F(\tau )}$, is described by the sum of two exponentials,

where ${\tau }_{\mathrm{l}}$ and ${\tau }_{\mathrm{s}}$ are long and short fluorescence lifetimes, corresponding to enzyme-bound NADH and free NADH, respectively, and $f$ is the fraction of enzyme-bound NADH (Sanchez et al., 2018; Sanchez et al., 2019) (Materials and methods).

We repeated the oxygen drop experiments for a total of 68 oocytes. Since the oxygen drop is much slower than the NADH redox reactions (30 min compared to a timescale of seconds), the oxygen perturbation can be safely assumed to be quasistatic, allowing the FLIM measurements to be determined as a function of oxygen levels. We averaged data from all oocytes to obtain a total of four FLIM parameters: mitochondrial NADH intensity, I, long and short fluorescence lifetimes, ${\tau }_{\mathrm{l}}$ and ${\tau }_{\mathrm{s}}$, and the fraction of enzyme-bound NADH, $f$. We determined how these parameters varied with oxygen level (Figure 1a and c). All parameters are insensitive to oxygen level until oxygen drops below ~10 μM. This observation is consistent with previous studies that showed mitochondria have a very high apparent affinity for oxygen (Chance and Williams, 1955; Gnaiger et al., 1998).

We next explored the relationship between the measured FLIM parameters and the concentration of NADH. Since bound and free NADH have different fluorescence lifetimes, and hence different molecular brightnesses, the NADH concentration is not generally proportional to NADH intensity. Assuming molecular brightness is proportional to fluorescence lifetime (Lakowicz, 2006), we derived a relation between NADH intensity, fluorescence lifetimes, and concentrations as

where $c_{\mathrm{s}}$ is a calibration factor that relates intensities and concentrations (see Appendix 1). We measured the calibration factor by titrating free NADH in vitro and acquiring FLIM data (Figure 1—figure supplement 2, Equation S4). To test the validity of this approach, we used Equations 2a,b to measure concentrations of free and bound NADH in solutions with different concentrations of purified LDH, an enzyme to which NADH can bind. The measured NADH bound concentration increases with LDH concentration while the sum of free and bound NADH concentration remains a constant and equal to the amount of NADH added to the solution (Figure 1—figure supplement 3). This result demonstrates that Equations 2a,b can be used to measure free and bound NADH concentrations from NADH intensity and lifetimes. It is well established that FLIM can be used to distinguish bound and free NADH in vivo based on the large change of fluorescence lifetime when NADH binds to enzymes (Skala et al., 2007; Heikal, 2010). Even though the exact amount that the lifetime changes depend on the specific enzyme NADH binds to (Sharick et al., 2018), enzyme-bound NADH always has a much longer fluorescence lifetime than free NADH. Therefore, the method to calculate free and bound concentrations of NADH from FLIM measurements is expected to hold in vivo. We next used this method to study NADH in mitochondria in oocytes. We applied Equations 2a,b to our FLIM data from oocytes and determined how the concentrations of free NADH, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$, and enzyme-bound NADH, ${\displaystyle \left[\mathrm{N}\mathrm{A}\mathrm{D}{\mathrm{H}}_{\mathrm{b}}\right]}$, depended on oxygen level (Figure 1c, lower right). Interestingly, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ increased as oxygen fell below ~10 μM, while $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{b}}\right]$ did not vary with oxygen level.

We next explored the impact of metabolic inhibitors on mitochondrial NADH. We first inhibited LDH by adding 9 mM of oxamate to the AKSOM media. This led to a decrease of NADH intensity in the mitochondria (Figure 2a, upper) and significant changes in all FLIM parameters (Figure 2a, lower, p<0.001). We next inhibited complex I of the ETC using 5 μM of rotenone (in the presence of 9 mM of oxamate, to reduce cytoplasmic NADH signal for better mitochondrial segmentation). This resulted in a dramatic increase of NADH intensity in the mitochondria (Figure 2b, upper) and significant changes in NADH bound ratio and long lifetime (Figure 2b, lower, p<0.001). Then we inhibited ATP synthase with 5 μM of oligomycin (in the presence of 9 mM of oxamate), which, similar to rotenone, resulted in an increase of mitochondrial NADH intensity (Figure 2c, upper) and significant changes in all FLIM parameters (Figure 2c, lower, p<0.001). Finally, we subjected the oocytes to 5 μM of FCCP (in the presence of 9 mM of oxamate), which uncouples proton translocation from ATP synthesis, and observed a decrease of mitochondrial NADH intensity (Figure 2d, upper) and significant changes in FLIM parameters (Figure 2d, lower, p<0.001). Interestingly, the direction of change of FLIM parameters under FCCP is opposite to those under rotenone and oligomycin. For each of these conditions, we used Equations 2a,b to calculate the concentrations of free NADH, ${\displaystyle \left[\mathrm{N}\mathrm{A}\mathrm{D}{\mathrm{H}}_{\mathrm{f}}\right]}$, (Figure 2e) and bound NADH, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{b}}\right]$, (Figure 2f) from the measured intensity and FLIM parameters. While rotenone and oligomycin significantly increased $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ and decreased $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{b}}\right]$, FCCP decreased $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$. It remains unclear how to relate these changes of the free and bound concentrations of NADH to the activities of mitochondrial respiration.

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We next developed a mathematical model of NADH redox reactions to relate these quantitative FLIM measurements to activities of mitochondrial metabolic pathways. NADH is a central coenzyme that binds to enzymes and facilitates redox reactions by acting as an electron carrier. There are two categories of enzymes associated with NADH redox reactions, which together form a redox cycle: oxidases that oxidize NADH to NAD⁺ and reductases that reduce NAD⁺ to NADH. The major NADH oxidase in mitochondria is complex I of ETC for mammalian cells. There are many NADH reductases in mitochondria because NADH can be reduced through different pathways depending on the energy substrate. These pathways include the TCA cycle, fatty acid catabolism via beta oxidation, amino acid catabolism such as glutaminolysis and the malate-aspartate shuttle (Salway, 2017). A comprehensive NADH redox model will include all the oxidases and reductases. For generality, we consider N oxidases and M reductases.

For convenience, we introduced a reduced notation to describe models of the enzyme kinetics of these oxidases and reductases. We began by illustrating our reduced notation using reversible Michaelis-Menten kinetics as an example (Keleti, 1986; Miller and Alberty, 2002; Smith, 1992). The conventional, full notation for these kinetics (Figure 3a, left) explicitly displays all chemical species that are modeled in this reaction scheme — free NADH, ${\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}$, free enzyme, ${\mathrm{O}\mathrm{x}}_i$, free NAD⁺, ${\mathrm{N}\mathrm{A}\mathrm{D}}_{\mathrm{f}}^{+}$, and NADH bound to the enzyme — as well as the forward and reverse reaction rates — $k_{-1}$, $k_1$, $k_{-2}$, and $k_2$ Our reduced notation for reversible Michaelis-Menten kinetics (Figure 3a, right) is an alternative way of representing the same mathematical model. In this reduced notation, only free NADH, free NAD⁺, and NADH bound to the enzyme are explicitly shown, while the free enzyme concentration is only represented as entering through the effective binding rates $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{b}}$ and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{`}\mathrm{b}}$. The conventional, full notation, and the reduced notation are alternative ways of representing the same mathematical model, but the reduced notation is convenient to use in the derivation that follows (see Appendix 2).

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We next introduced a generalized enzyme kinetics using our reduced notation (Figure 3b), which contains not only free NADH, free NAD⁺, and NADH bound to the enzyme, but also NAD⁺ bound to the enzyme, and the reaction rates for oxidation and reduction of the bound coenzymes. In this reduced notation, all the binding and unbinding rates, and the reaction rates, can be functions of metabolite concentrations, protein concentrations, and other factors such as pH and membrane potential. As in the reversible Michaelis-Menten kinetics example, these rates can depend on the concentration of the free enzyme itself. This dependency on free enzyme concentration can be nonlinear, as could occur if the enzyme oligomerizes. Furthermore, the rates may depend on the concentration of free NADH, free NAD⁺, and the enzyme complexes. Thus, while the reduced notation for the generalized enzyme might appear to describe a first-order reaction, it can actually be used to represent reactions of any order, with arbitrary, nonlinear dependencies on the concentration of the enzyme itself, as well as arbitrary, nonlinear dependencies on other factors. In order to model the dynamics of enzymes described by such generalized kinetics, it is necessary to specify the functional form of all the rates, as well as specify mathematical models for all the variables that enter these rates (i.e., free enzyme concentration, membrane potential, pH, etc.) (Appendix 2). However, in what follows, we will derive results that hold true, irrespective of the functional form of the rates or the presence of additional, implicit variables. Thus, remarkably, these quantitative predictions are valid for enzyme kinetics of any order, with arbitrary nonlinearities in the rates.

To begin our derivation, we note that under this generalized enzyme kinetics (Figure 3b), the net flux through the ith oxidase at steady-state is:

where $\left[\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\bullet {\mathrm{O}\mathrm{x}}_i\right]$, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}}^{+}\bullet {\mathrm{O}\mathrm{x}}_i\right]$, and $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ are the concentrations of the ith oxidase-bound NADH, NAD⁺, and free NADH, respectively. $r_{{\mathrm{o}\mathrm{x}}_i}^{+}$ and $r_{{\mathrm{o}\mathrm{x}}_i}^{-}$ are the forward and reverse oxidation rates. $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{b}}$ and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{u}}$ are the binding and unbinding rates. The second equality in Equation 3 results from the steady-state condition, where the net binding and unbinding flux equals the net oxidation flux.

We next considered a redox cycle between NADH and NAD⁺ with multiple oxidases and reductases. To account for all possible NADH redox pathways, we developed a detailed NADH redox model with N oxidases and M reductases described by the generalized enzyme kinetics (Figure 4a and Figure 4—figure supplement 1). In this model, NADH and NAD⁺ can bind and unbind to each oxidase and reductase. Once bound, NADH can be reversibly oxidized to NAD⁺ by the oxidases, and NAD⁺ can be reversibly reduced to NADH by the reductases, forming a redox cycle. The functional dependencies of the binding and unbinding rates, and the reaction rates, can be different for each oxidase and reductase, and each of these rates can be nonlinear functions of free enzyme concentrations, NADH concentration, and other factors such as pH and membrane potential. Modeling the dynamics of this redox cycle requires specifying the precise number of oxidases and reductases, the functional forms of the rates, and mathematical models for all the variables the rates implicitly depend on. However, we will show that quantitative predictions regarding the interpretation of FLIM measurements can be made that generally hold, independent of these modeling choices.

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FLIM cannot resolve the association of NADH with individual enzymes in cells, but rather, provides quantitative information on the global states of bound and free NADH. Thus, to facilitate comparison to FLIM experiments, we coarse-grained the detailed redox model by mapping all N oxidases into a single effective oxidase and all M reductases into a single effective reductase (Figure 4b and Appendix 3). This coarse-graining is mathematically exact and involves no approximations or assumptions.

In the coarse-grained redox model, NADH can be bound to the effective oxidase, $\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\bullet \mathrm{O}\mathrm{x}$, bound to the effective reductase, $\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\bullet \mathrm{R}\mathrm{e}$, or can be free, ${\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}$. Hence, the concentration of NADH bound to all enzymes is, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{b}}\right]=\left[\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\bullet \mathrm{O}\mathrm{x}\right]+[\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\bullet \mathrm{R}\mathrm{e}]$, and the total concentration of NADH is, $\left[\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\right]=\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{b}}\right]+\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$. The kinetics of the effective oxidase and reductase are represented by the coarse-grained forward, $r_{\mathrm{o}\mathrm{x}}^{+}$, and reverse, $r_{\mathrm{o}\mathrm{x}}^{-}$, oxidation rates, and the forward, $r_{\mathrm{r}\mathrm{e}}^{+}$, and reverse, $r_{\mathrm{r}\mathrm{e}}^{-}$, reduction rates. The global flux through all the oxidases in the detailed redox model equals the global flux through the coarse-grained oxidase, which at steady-state is:

where $k_{\mathrm{o}\mathrm{x}}^{\mathrm{b}}$ is the rate that free NADH binds the effective oxidase, $k_{\mathrm{o}\mathrm{x}}^{\mathrm{u}}$ is the rate that NADH unbinds the effective oxidase, and the last equality results because the coarse-grained redox loop is a linear pathway so the global oxidative flux must equal the global binding and unbinding flux at steady-state. The conservation of global flux explicitly relates the effective binding and unbinding rates and the reaction rates of the coarse-grained model to those of the detailed model (Appendix 3, Figure 4—figure supplement 1). The binding and unbinding kinetics of NADH and NAD⁺ to the effective oxidase and reductase are described by eight coarse-grained binding and unbinding rates (Figure 4b). The coarse-grained reaction rates and binding and unbinding rates can be arbitrary functions of metabolite concentrations, enzyme concentrations, and other factors (i.e., pH, membrane potential, etc.). These effective rates can even be functions of $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}}_{\mathrm{f}}^{+}\right]$, and the concentration of other variables, and thus can include reactions of arbitrary order. Hence, this coarse-grained model is a generic model of NADH redox reactions. Fully specifying this model would require explicitly choosing the functional form of all the rates and incorporating additional equations to describe the dynamics of all the implicit variables that the rates depend on (Appendix 2). We next demonstrate that quantitative predictions regarding the interpretation of FLIM measurements of NADH can be made that are valid irrespective of the form of the rates or the presence of implicit variables.

At steady-state, the model can be further coarse-grained, without approximation, to consider only free NADH, with a turnover rate of ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$, and free NAD⁺, with a turnover rate of ${\displaystyle {\tilde{r}}_{\mathrm{r}\mathrm{e}}}$ (Figure 4c). Our key prediction is that the steady-state global oxidative flux of NADH is (Appendix 4):

where

and

This prediction results from the steady-state assumption where the net binding and unbinding flux of NADH from the oxidase balances the net oxidative flux through the oxidase (Equation 4 and Appendix 4). The turnover rate of free NADH, ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$, is proportional to the difference between the NADH bound ratio $\beta$, that is, the ratio between bound and free NADH concentrations, and the equilibrium NADH bound ratio, ${\beta }_{\mathrm{e}\mathrm{q}}$ (i.e., what the bound ratio would be if the global oxidative flux is zero). ${\displaystyle {\beta }_{\mathrm{e}\mathrm{q}}}$ and the prefactor ${\displaystyle \alpha }$ are independent of the reaction rates of the oxidase and reductase and can be explicitly related to the binding and unbinding rates of the coarse-grained model (Appendix 4, Equation S43 and S45).

In mitochondria, the major NADH oxidation pathway is the ETC. Thus, Equations 5a-c predict that there is a direct connection between quantities that can be measured by FLIM of NADH in mitochondria (i.e., $\beta$ and $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$) and the flux through the ETC (i.e., $J_{\mathrm{o}\mathrm{x}}$). Equations 5a-c suggest a procedure for using FLIM to infer flux through the ETC: if a condition can be found under which there is no net flux through the ETC, then ${\beta }_{\mathrm{e}\mathrm{q}}$ can be measured with FLIM. Once ${\beta }_{\mathrm{e}\mathrm{q}}$ is known, then subsequent FLIM measurements of $\beta$ allows ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$, and hence $J_{\mathrm{o}\mathrm{x}}$, to be inferred (up to a constant of proportionality ${\displaystyle \alpha }$) (Appendix 5).

Equations 5a-c are valid irrespective of the functional forms of the rate laws, which may have nonlinear dependencies on metabolite concentrations, enzyme concentrations, and other factors. While Equation 5a seems to imply first-order kinetics in $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$, the rates can also be arbitrary functions of $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$, so Equations 5a–c hold for kinetics of any order. Equations 5a-c are also applicable if the rates depend on additional variables that have their own dynamical equations (as long as the system is at steady-state): as an example, Appendix 7 shows that Equations 5a-c result when the N oxidases and M reductases are each described by reversible Michaelis-Menten kinetics, a model in which the rates depend on the concentration of free enzymes (which is a dynamical variable). More generally, if detailed biophysical models of the NADH oxidases are available, then parameters of these models can be explicitly mapped to the coarse-grained parameters of the NADH redox model. Appendix 7 and Appendix 7—table 1 contain mappings between the coarse-grained model and a number of previously proposed detailed biophysical models of NADH oxidation in the ETC (Beard, 2005; Korzeniewski and Zoladz, 2001; Hill, 1977; Jin and Bethke, 2002; Chang et al., 2011). However, since Equations 5a-c are valid for a broad set of models, they can be used for flux inference without the need to specify the functional form of the rates or the variables they depend on. This is because the rates are coarse-grained into two effective parameters $\alpha$ and ${\beta }_{\mathrm{e}\mathrm{q}}$, which can be experimentally determined. This generality results from the steady-state assumption and the topology of the reactions resulting in the net binding and unbinding flux of NADH from the oxidase balancing the net oxidative flux. Thus, Equations 5a–c provide a general procedure to infer the ETC flux from FLIM measurements of NADH in mitochondria.

We applied this procedure to analyze our oxygen drop experiments (Figure 1) by assuming that there was no net flux through the ETC at the lowest oxygen level achieved for each oocyte (implying that the measured value of $\beta$ at that oxygen concentration corresponds to ${\beta }_{\mathrm{e}\mathrm{q}}$ for that oocyte). We also assumed that $\alpha$ and ${\beta }_{\mathrm{e}\mathrm{q}}$ do not change with oxygen levels, which is reasonable since, as noted above, they are independent of the reaction rates of the oxidase and reductase. The measured value of ${\beta }_{\mathrm{e}\mathrm{q}}$ allowed us to obtain a prediction for $J_{\mathrm{o}\mathrm{x}}$ as a function of oxygen concentration for the oocytes (Figure 5a). To test these predictions, we directly determined $J_{\mathrm{o}\mathrm{x}}$ as a function of oxygen concentration by measuring the OCR of the oocytes using a nanorespirometer (Lopes et al., 2005) (Materials and methods). The direct measurements of $J_{\mathrm{o}\mathrm{x}}$ from OCR quantitatively agree with the predictions of $J_{\mathrm{o}\mathrm{x}}$ from FLIM for all oxygen concentrations (Figure 5a), strongly arguing for the validity of the model and the inference procedure. This agreement supports the assumption that $\alpha$ and ${\beta }_{\mathrm{e}\mathrm{q}}$ are independent of oxygen levels.

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So far, we have inferred the ETC flux up to a constant of proportionality ${\displaystyle \alpha }$, allowing the relative changes of ETC flux to be inferred from FLIM of NADH. ${\displaystyle \alpha }$ cannot be determined by FLIM alone. If an absolute measurement of the ETC flux can be obtained at one condition, then ${\displaystyle \alpha }$ can be calibrated to predict absolute ETC fluxes for other conditions. OCR measurement provides a means to calibrate ${\displaystyle \alpha }$ (Appendix 5, Equation S48). We used oocytes cultured in AKSOM media at 50±2 μM oxygen as a reference state, which, from our OCR measurements yielded ${\displaystyle J_{\mathrm{o}\mathrm{x}}=56.6\pm 2\mu \mathrm{M}/s}$ (SEM) and hence a constant of proportionality of $\alpha =5.4\pm 0.2{\mathrm{}\mathrm{s}}^{-1}$. Using this value of $\alpha$, we can predict absolute values of $J_{\mathrm{o}\mathrm{x}}$ under various perturbations assuming $\alpha$ remains a constant. We note that $J_{\mathrm{o}\mathrm{x}}$ is a flux density with units of concentration per second, an intensive quantity that does not depend on the mitochondrial volume. Multiplying $J_{\mathrm{o}\mathrm{x}}$ by the volume of mitochondria in an oocyte gives the total ETC flux, proportional to OCR, in that oocyte. In all subsequent discussions, ETC flux refers to flux density unless otherwise noted.

We next applied the inference procedure and a constant of $\alpha =5.4\pm 0.2{\mathrm{}\mathrm{s}}^{-1}$ to analyze the experiments of oxamate, FCCP, rotenone and oligomycin perturbations (Figure 2). We dropped oxygen levels to determine ${\beta }_{\mathrm{e}\mathrm{q}}$ in the presence of oxamate (Figure 5—figure supplement 1h) and applied Equations 5a-c to infer the impact of oxamate on $J_{\mathrm{o}\mathrm{x}}$ at 50 μM oxygen (i.e., control levels of oxygen). Surprisingly, while the addition of oxamate greatly impacts FLIM parameters, including a 29±2% (SEM) decrease in intensity and a 10±3% increase in bound ratio (Figure 2a), this procedure revealed that the predicted ETC flux with oxamate (${\displaystyle J_{\mathrm{o}\mathrm{x}}=55.2\pm 3.2\mu \mathrm{M}/s}$) is the same as that without oxamate (${\displaystyle J_{\mathrm{o}\mathrm{x}}=55.4\pm 1.9\mu \mathrm{M}/s}$) (Figure 5b; p=0.95), which was confirmed by direct measurements of oocytes’ OCR that yielded ${\displaystyle J_{\mathrm{o}\mathrm{x}}=55.4\pm 1.5\mu \mathrm{M}/s}$ and ${\displaystyle J_{\mathrm{o}\mathrm{x}}=54.9\pm 0.7\mu \mathrm{M}/s}$ before and after the addition of oxamate, respectively (Figure 5b; p=0.85). We next analyzed the FCCP experiment. We obtained ${\beta }_{\mathrm{e}\mathrm{q}}$ by dropping oxygen in the presence of FCCP (Figure 5—figure supplement 1h) and applied Equations 5a-c to infer the impact of FCCP on $J_{\mathrm{o}\mathrm{x}}$ at 50 μM oxygen. We predicted that FCCP increased the flux to ${\displaystyle J_{\mathrm{o}\mathrm{x}}=67.7\pm 1.5\mu \mathrm{M}/s}$, which was confirmed by the directly measured ${\displaystyle J_{\mathrm{o}\mathrm{x}}=74.0\pm 3.7\mu \mathrm{M}/s}$ from OCR (Figure 5b; p=0.30). Following the same FLIM based inference procedures, we predicted that the addition of rotenone and oligomycin reduced the fluxes to ${\displaystyle J_{\mathrm{o}\mathrm{x}}=16.7\pm 1.4\mu \mathrm{M}/s}$ and ${\displaystyle J_{\mathrm{o}\mathrm{x}}=15.7\pm 1.3\mu \mathrm{M}/s}$, respectively, which was again confirmed by corresponding direct measurements of OCR that yielded ${\displaystyle J_{\mathrm{o}\mathrm{x}}=11.1\pm 0.4\mu \mathrm{M}/s}$ and ${\displaystyle J_{\mathrm{o}\mathrm{x}}=22.3\pm 0.6\mu \mathrm{M}/s}$ (Figure 5b; p=0.31 and p=0.17). The quantitative agreement between predicted fluxes from FLIM and directly measured fluxes from OCR under a variety of conditions (i.e., varying oxygen tension, sodium oxamate, FCCP, rotenone, and oligomycin) demonstrates that Equations 5a-c can be successfully used to infer flux through the ETC in mouse oocytes. This agreement also supports the assumption that $\alpha$ is a constant across these different perturbations.

The work described above used the relation ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}=\alpha \left(\beta -{\beta }_{\mathrm{e}\mathrm{q}}\right)}$ to predict the flux through the ETC from FLIM measurements. We next show that the model also predicts a relationship between ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ and the fluorescence lifetime of enzyme-bound NADH, ${\tau }_{\mathrm{l}}$, in mitochondria. This provides a second means to use the model to infer ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$, and hence $J_{\mathrm{o}\mathrm{x}}$, from FLIM of NADH. Specifically, we assumed that NADH bound to the oxidases have a different average lifetime, ${\tau }_{\mathrm{o}\mathrm{x}}$, than NADH bound to the reductases, ${\tau }_{\mathrm{r}\mathrm{e}}$, which is reasonable because NADH bound to different enzymes do exhibit different fluorescence lifetimes (Sharick et al., 2018). This assumption implies that the experimentally measured long lifetime of NADH in mitochondria, ${\tau }_{\mathrm{l}}$, is a weighted sum of these two lifetimes,

Using the coarse-grained NADH redox model at steady-state, Equation 6 leads to a non-trivial prediction that ${\tau }_{\mathrm{l}}$ is linearly related to $1/\beta$ (Appendix 5):

where the slope $A$ and offset $B$ can be explicitly related to ${\tau }_{\mathrm{o}\mathrm{x}}$, ${\tau }_{\mathrm{r}\mathrm{e}}$, and the coarse-grained binding and unbinding rates. Such a linear relationship is indeed observed in individual oocytes subject to oxygen drops (Figure 6a and Figure 6—figure supplement 1), supporting the assumptions of the model. Combining Equations 7 and 5b leads to a predicted relationship between ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ and NADH long fluorescence lifetime (Appendix 5):

where ${\tau }_{\mathrm{e}\mathrm{q}}$ is the equilibrium NADH long fluorescence lifetime, that is, the value of the long lifetime when the global oxidative flux is zero. This provides a second means to infer ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ from FLIM measurements: dropping oxygen and plotting the relationship between ${\tau }_{\mathrm{l}}$ and $1/\beta$ provides a means to measure A and B from Equation 7, while ${\tau }_{\mathrm{e}\mathrm{q}}$ can be obtained from the NADH long fluorescence lifetime obtained at the lowest oxygen level. Once A, B, and ${\tau }_{\mathrm{e}\mathrm{q}}$ are known, ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ can be inferred solely from NADH long fluorescence lifetime ${\tau }_{\mathrm{l}}$, using Equation 8.

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We next used the lifetime method (Equation 8) and the bound ratio method (Equation 5b) to separately infer ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ in oocytes subject to a wide variety of conditions (varying oxygen levels, with oxamate, FCCP, rotenone, and oligomycin). We obtained A, B, ${\beta }_{\mathrm{e}\mathrm{q}}$, and ${\tau }_{\mathrm{e}\mathrm{q}}$ for these different conditions (Figure 5—figure supplement 1 and Figure 6—figure supplement 1), and used the two different methods to provide two independent measures of ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ (assuming $\alpha$ is constant across all conditions). The predictions of ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ from these two methods quantitatively agree under all conditions (Figure 6b, p=0.73), which is a strong self-consistency check that further supports the use of the model to infer ETC flux from FLIM measurements of NADH.

After thoroughly testing the NADH redox model and the inference procedure in mouse oocytes, we next investigated if it can be used in other cell types. We chose human tissue culture cells for this purpose, since they are widely used as model systems to study metabolic dysfunctions in human diseases including cancer (Vander Heiden et al., 2009) and neuropathology (Lin and Beal, 2006).

While mouse oocytes have a negligible level of NADPH compared to NADH (Bustamante et al., 2017), the concentrations of NADH and NADPH are similar in tissue culture cells (10–100 μM averaged over the whole cell) (Lu et al., 2018; Park et al., 2016; Blacker et al., 2014). Since NADPH and NADH have overlapping fluorescent spectra (Patterson et al., 2000), the presence of NADPH may complicate the interpretation of FLIM experiments. Thus, we investigated the impact of background fluorescence, such as from NADPH, on the flux inference procedure. If the background fluorescence does not change with the perturbations under study, then it can be treated as an additive offset that systematically makes the measured concentrations of free and bound NADH different from their actual values. In this case, a derivation in Appendix 5 demonstrates that the background fluorescence can be incorporated into the equilibrium bound ratio ${\beta }_{\mathrm{e}\mathrm{q}}$ and does not impact the flux inference procedures. In other words, if the modified ${\beta }_{\mathrm{e}\mathrm{q}}$ can be reliably determined, then the measured concentrations of free and bound fluorescent species can be used in place of the true values of NADH in Equations 5a-c to infer the ETC flux. An alternative possibility is that the background fluorescence does change with the perturbations under study, but in a manner that is proportional to the change in NADH. In this case, the background fluorescence can be incorporated into the equilibrium bound ratio $\alpha$ and, once more, does not impact the flux inference procedures (Appendix 5). If the background fluorescence changes in some more complicated manner, then the inference procedure may no longer be valid. Thus, depending on the behavior of NADPH, it either might or might not interfere with the inference procedure: no impact if NADPH is either constant or proportional to changes in NADH, a possible impact otherwise. Therefore, the validity of the inference procedure in the presence of significant NADPH fluorescence must be established empirically.

We next tested the inference procedures experimentally in hTERT-RPE1 (hTERT-immortalized retinal pigment epithelial cell line) tissue culture cells. We started by exploring the impact of metabolic perturbations on mitochondrial NAD(P)H: the combined signal from NADH and NADPH (which are indistinguishable) from mitochondria. We first cultured the cells in DMEM with 10 mM galactose (Materials and methods). We then inhibited complex I of the ETC by adding 8 μM of rotenone to the media. This resulted in a significant increase of mitochondrial NAD(P)H intensity (Figure 7a, upper). We segmented mitochondria using a machine learning-based algorithm from the intensity images of NAD(P)H, and fitted the fluorescence decay curves of mitochondrial NAD(P)H to obtain changes in FLIM parameters (Materials and methods). All FLIM parameters displayed significant changes (Figure 7a, lower, p<0.001, and Figure 7—figure supplement 1). We next uncoupled proton translocation from ATP synthesis by adding 3.5 μM CCCP to the media. This led to a decrease of NAD(P)H intensity in the mitochondria (Figure 7b, upper) and significant changes in NAD(P)H bound ratio and short lifetime, but in opposite directions as compared to rotenone perturbation (Figure 7b, lower, p<0.01, and Figure 7—figure supplement 1). Finally, we perturbed the nutrient conditions by culturing the cells in DMEM with 10 mM glucose. FLIM imaging revealed an increase of mitochondrial NAD(P)H intensity (Figure 7c, upper) and significant changes in all FLIM parameters as compared to the galactose condition (Figure 7c, lower, p<0.001, and Figure 7—figure supplement 1).

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Inference of the ETC flux from FLIM measurements requires a measurement of ${\beta }_{\mathrm{e}\mathrm{q}}$. Since rotenone is known to drastically decrease the OCR of hTERT-RPE1 cells to near zero (MacVicar and Lane, 2014), we used the NAD(P)H bound ratio measured in the presence of rotenone as ${\beta }_{\mathrm{e}\mathrm{q}}$. Different values of ${\beta }_{\mathrm{e}\mathrm{q}}$ were obtained for glucose and galactose conditions by adding 8 μM of rotenone to each condition (Figure 7—figure supplement 1). We next calculated the concentrations of free NAD(P)H, ${\displaystyle \left[\mathrm{N}\mathrm{A}\mathrm{D}(\mathrm{P}){\mathrm{H}}_{\mathrm{f}}\right]}$, from the FLIM parameters using Equation 2a. ${\displaystyle \left[\mathrm{N}\mathrm{A}\mathrm{D}(\mathrm{P}){\mathrm{H}}_{\mathrm{f}}\right]}$ displayed significant changes for all perturbations (Figure 7d). Using Equation 5b and assuming ${\displaystyle \alpha }$ is a constant, we calculated the NAD(P)H turnover rate, ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$, from the FLIM measurements and ${\beta }_{\mathrm{e}\mathrm{q}}$. ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ changed significantly for all perturbations (Figure 7e). Multiplying ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ and ${\displaystyle \left[\mathrm{N}\mathrm{A}\mathrm{D}(\mathrm{P}){\mathrm{H}}_{\mathrm{f}}\right]}$, we obtained the predicted ETC flux, $J_{\mathrm{o}\mathrm{x}}$, which increased under FCCP, decreased under glucose and reduced to zero under rotenone (Figure 7f).

To test the model predictions, we compared the predicted ETC flux with previous direct OCR measurements of the same cell type that we used, under the same conditions (MacVicar and Lane, 2014). Remarkably, the predicted changes in ETC fluxes are in quantitative agreement with the directly measured OCR across all conditions as estimated from Figure 1A of MacVicar and Lane, 2014: CCCP is predicted to increase the ETC flux by 14±3% (SEM), in agreement with the 18±21% increase from OCR measurement (p=0.80); Glucose is predicted to decrease ETC flux by 33±3%, in agreement with the 46±9% decrease from OCR measurements (p=0.30), shifting metabolism from oxidative phosphorylation to aerobic glycolysis. Since we used $\beta$ from rotenone treatment as ${\beta }_{\mathrm{e}\mathrm{q}}$, the predicted decrease in ETC flux after the addition of rotenone is 101±2%, which is in agreement with the 82±2% decrease from OCR measurement (p=0.28). This quantitative agreement between predicted ETC fluxes and measured OCR across all perturbations demonstrated the applicability of the NADH redox model and the flux inference procedures to tissue culture cells, even though they contain substantial levels of NADPH.

Having established the validity of the NADH redox model and the associated flux inference procedures, we next applied it to study energy metabolism in mouse oocytes. We began by investigating the processes that determine the ETC flux in MII mouse oocytes. Mitochondrial-based energy metabolism can be viewed as primarily consisting of three coupled cycles: the NADH/NAD⁺ redox cycle (which our NADH redox model describes), the proton pumping/dissipation cycle, and the ATP/ADP production/consumption cycle (Figure 8a). At the most upstream portion of this pathway, the reduction of NAD⁺ to NADH is powered by a supply of nutrients, while at the most downstream portion, energy-demanding cellular processes hydrolyze ATP to ADP. To test whether nutrient supply and energy demand set ETC flux, we investigated the effect of perturbing these processes. To perturb supply, we first varied the concentration of pyruvate in the media from 181 μM (which is standard for AKSOM) to either 18.1 μM or 1.81 mM, and observed significant changes in NADH intensity and FLIM parameters (Figure 8b, left), demonstrating that the NADH metabolic state is altered. To perturb demand, we began by adding 10 μM nocodazole to the media, which disassembled the meiotic spindle, an energy user, and resulted in significant changes in NADH FLIM parameters (Figure 8b, center). Similarly, the addition of 10 μM latrunculin A disassembled the actin cortex and also produced significant changes in NADH FLIM parameters (Figure 8b, right).

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We next performed additional perturbations of nutrient supply, inhibiting the conversion of lactate to pyruvate by LDH (with 9 mM oxamate) and inhibiting the malate-aspartate shuttle (with 11 mM AOA). We performed additional perturbations of energy demand by inhibiting protein synthesis (with 1 mM cycloheximide) and ion homeostasis, by varying extracellular potassium concentrations from 0 mM to 15 mM, inhibiting the Na⁺/K⁺ pump (with 2 mM ouabain), and adding an ionophore (10 μM gramicidin). All perturbations resulted in significant changes in NADH FLIM parameters (Figure 8—figure supplement 1), showing that NADH metabolic state is generally impacted by varying nutrient supply and cellular energy demand. We next used the NADH redox model and the measured FLIM parameters to infer the concentration and effective turnover rate of free NADH for these perturbations. The free NADH concentrations, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$, and turnover rates, ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$, displayed large variations across the perturbations, ranging from 33.5±1 μM (SEM) to 56.0±2.8 μM and from 1.0±0.05 s^-1 to 1.65±0.09 s^-1, respectively (Figure 8c). Surprisingly, the changes in $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ and ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$ were highly anti-correlated such that the data points primarily fell within a region where the inferred ETC flux, ${\displaystyle J_{\mathrm{o}\mathrm{x}}={\tilde{r}}_{\mathrm{o}\mathrm{x}}\left[\mathrm{N}\mathrm{A}\mathrm{D}{\mathrm{H}}_{\mathrm{f}}\right]}$, is a constant 55.5 μM·s^-1 (Figure 8c, solid line, shaded region indicates 5% error). Indeed, ANOVA tests confirmed that perturbing nutrient supplies and cellular energy demand lead to no significant change in either the inferred ETC flux (Figure 8d, p=0.20) or directly measured OCR (Figure 8e, p=0.07). Thus, while nutrient supply and cellular energy demand strongly affect mitochondrial NADH redox metabolism, they do not impact ETC flux. In contrast, ETC flux is impacted by perturbing proton leak and ATP synthesis (Figure 5). Taken together, this suggests that the ETC flux in mouse oocytes is set by the intrinsic properties of their mitochondria, which can adjust their NADH redox metabolism to maintain a constant flux when nutrient supplies and cellular energy demand are varied. The mechanistic basis of this homeostasis of ETC flux is unclear and will be an exciting topic for future research.

Our results presented so far were performed by averaging together FLIM measurements from all mitochondria within an oocyte. However, FLIM data is acquired with optical resolution, enabling detailed subcellular measurements. To see if there are spatial variations in FLIM measurements within individual oocytes, we computed the mean NADH fluorescence decay time for each mitochondrial pixel. The mean NADH fluorescence decay time displays a clear spatial gradient, with higher values closer to the oocyte center (Figure 9a).

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To quantify this gradient in more detail, we partitioned mouse oocytes into equally spaced concentric regions (Figure 9b) and fitted the fluorescence decay curves from mitochondrial pixels within each region to obtain FLIM parameters as a function of distance from the oocyte center. NADH intensity, bound ratio, and long lifetime in mitochondria all display significant spatial gradient within oocytes (Figure 9c). Next, using Equations 5a-c and ${\beta }_{\mathrm{e}\mathrm{q}}$ obtained at the lowest oxygen level, and confirming that ${\beta }_{\mathrm{e}\mathrm{q}}$ is uniform within the oocyte with complete inhibition of ETC using high concentration of rotenone (Figure 9—figure supplement 1), we predicted the ETC flux, ${\displaystyle J_{\mathrm{o}\mathrm{x}}}$, as a function of distance from the oocyte’s center. The ETC flux displayed a strong spatial gradient within oocytes, with a higher flux closer to the cell periphery (Figure 9d). Note that, as described above, $J_{\mathrm{o}\mathrm{x}}$ is actually a flux density with units of concentration per second. Thus, the measured flux gradient is not merely a reflection of variations in mitochondrial density, but instead indicates the existence of subcellular spatial heterogeneities in mitochondrial activities.

To investigate the origin of this flux gradient, we inhibited ATP synthase using 5 μM of oligomycin and repeated measurements of subcellular spatial variations in inferred fluxes. After inhibition, $J_{\mathrm{o}\mathrm{x}}$ decreased at all locations throughout the oocytes and displayed an even more dramatic flux gradient (Figure 9e). If oligomycin completely blocks ATP synthase, then the remaining flux must be the result of proton leak. If it is further assumed that proton leak remains the same with and without oligomycin, then the flux due to ATP synthase in control oocytes can be determined by subtracting the flux after oligomycin inhibition (i.e., the proton leak) from the flux before inhibition. Performing this procedure throughout oocytes indicates that proton leak greatly increases in mitochondria near the periphery of oocytes, where ATP production decreases (Figure 9f). This implies that the subcellular gradient in ETC flux is primarily caused by a gradient in proton leak and that mitochondria near the periphery of oocytes are less active in ATP production than those in the middle of the oocyte.

We hypothesized that a gradient in proton leak would result in a gradient of mitochondrial membrane potential, with lower membrane potential closer to the cell periphery where proton leak is the greatest. To test this, we measured mitochondrial membrane potential using the membrane potential-sensitive dye TMRM, which preferentially accumulates in mitochondria with higher membrane potential (Al-Zubaidi et al., 2019). We observed a strong spatial gradient of the intensity of TMRM in mitochondria within oocytes, with dimmer mitochondria near the cell periphery (Figure 9g and h), indicating that mitochondria near the periphery of the oocyte have a lower membrane potential. This result is robust to locally normalizing TMRM intensity by mitochondrial mass using a membrane potential insensitive dye (Mitotracker Red FM), or using an alternative membrane potential-sensitive dye, JC-1 (Figure 9—figure supplement 2). The predicted flux of proton leak and mitochondrial TMRM intensity shows a strong negative correlation (Figure 9i), confirming our hypothesis.

Taken together, these results show that MII mouse oocytes contain subcellular spatial heterogeneities of mitochondrial metabolic activities. The observation that proton leak is responsible for the gradient of ETC flux suggests that the flux heterogeneity is a result of intrinsic mitochondrial heterogeneity. This is consistent with our conclusion from the homeostasis of ETC flux (Figure 8) that it is the intrinsic rates of mitochondrial respiration, not energy demand or supply, that controls the ETC flux. The causes and consequences of the subcellular spatial variation in mitochondrial activity remain unclear and are an exciting topic for future research.

Despite extensive studies and applications of FLIM in metabolic research (Bird et al., 2005; Skala et al., 2007; Heikal, 2010; Sharick et al., 2018; Sanchez et al., 2018; Liu et al., 2018; Sanchez et al., 2019; Ma et al., 2019), it remains a challenge to relate FLIM measurements to the activities of the underlying metabolic pathways in cells. We overcame this challenge by developing a coarse-grained NADH redox model that leads to quantitative predictions for the relationship between FLIM measurements and the flux through the ETC. The model was constructed by explicitly coarse-graining a detailed NADH redox model with an arbitrary number of oxidases and reductases that represent all the possible enzymes involved in NADH redox reactions. The reactions in the detailed NADH redox model can be of arbitrary order and depend on implicit variables (i.e., free enzyme concentration, membrane potential, pH, etc.), which obey their own dynamical equations. The dynamics of the redox model will, of course, depend on the precise number of oxidases and reductases, the functional forms of the rates, and specific mathematical models for all the variables the rates implicitly depend on. However, the quantitative predictions relating FLIM measurements and ETC flux are independent of these modeling choices. Coarse-graining the detailed NADH redox model reduces all oxidases to an effective oxidase and all reductases to an effective reductase. The kinetic rates of the coarse-grained model can be related to those of the detailed model by keeping the global fluxes through the oxidases and the reductases the same in both models. The coarse-grained model predicts that the flux through the ETC is a product of the turnover rate and the concentration of free NADH (Equation 5a). The turnover rate is proportional to the difference between the nonequilibrium and the equilibrium NADH bound ratio (Equation 5b), which are measurable by FLIM of NADH (Equation 5c). Thus, this model provides a generic framework to relate FLIM measurements of NADH to the flux through the ETC in mitochondria.

The central assumption required for the validity of Equations 5a-c is that the redox reactions, and binding and unbinding processes, can be approximated as being at steady-state (i.e., undergoing only quasistatic changes over perturbations or development). At steady-state, the net binding and unbinding flux balances the oxidative flux of NADH. Therefore, the measurement of binding and unbinding state of NADH from FLIM allows the inference of the ETC flux, irrespective of the detailed behaviors of the oxidative reactions.

Remarkably, all the binding and unbinding rates of the NADH redox model are coarse-grained into two effective parameters: ${\displaystyle \alpha }$ and ${\beta }_{\mathrm{e}\mathrm{q}}$, which can be experimentally measured. We determined the value of ${\displaystyle \alpha }$ from an OCR measurement (Appendix 5, Equation S48), and we determined the value of ${\beta }_{\mathrm{e}\mathrm{q}}$ from FLIM of NADH at low oxygen levels or from rotenone perturbation (Appendix 5, Figure 5—figure supplement 1h). In MII mouse oocytes, ${\displaystyle \alpha }$ do not significantly vary in response to oxygen or drug and nutrient perturbations. This is demonstrated by the agreement between the predicted ETC flux and the measured OCR with a constant ${\displaystyle \alpha }$ of 5.4±0.2 s^-1 across a variety of conditions (Figure 5). ${\displaystyle \alpha }$ is predicted to depend only on the coarse-grained unbinding rates of NADH from the enzymes (Equation S43), so the observed constancy of ${\displaystyle \alpha }$ implies that the perturbations in this study primarily impacted the reduction/oxidation reaction rates (and not the unbinding rates). In other scenarios, such as when the concentrations of enzymes change, the coarse-grained unbinding rates might change, so ${\displaystyle \alpha }$ might not be a constant. In contrast, ${\beta }_{\mathrm{e}\mathrm{q}}$ does vary with drug and nutrient perturbations, but not with oxygen level, allowing ${\beta }_{\mathrm{e}\mathrm{q}}$ to be obtained at the lowest oxygen level for different drug and nutrient conditions (Figure 5—figure supplement 1h and Figure 8—figure supplement 1). Using these two parameters, we inferred the effective turnover rate of free NADH, ${\displaystyle {\tilde{r}}_{\mathrm{o}\mathrm{x}}}$, from FLIM measurements of NADH. By multiplying this turnover rate with the concentration of free NADH, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ (also obtained from FLIM measurements using Equation 2a), we inferred the ETC flux from Equation 5a. Thus, all the complex behaviors of the binding and unbinding and reaction rates are captured by the variations in FLIM parameters of NADH, and our coarse-grained model provides a generic way to interpret these variations.

While we found that ${\beta }_{\mathrm{e}\mathrm{q}}$ is smaller than $\beta$ in mouse oocytes, this does not generically have to be true. Thus, if a perturbation is observed to decrease the NADH bound ratio $\beta ,$ it does not necessarily imply a decrease of the ETC flux. Similarly, a decrease of NADH long lifetime is not necessarily associated with an increase of the ETC flux. Therefore, measurements of $\alpha$ and ${\beta }_{\mathrm{e}\mathrm{q}}$ are required to use Equations 5a-c to infer ETC flux from FLIM measurements of NADH.

In this section, we clarify the underlying assumptions and limitations of the model to facilitate the accurate interpretation of FLIM measurements of NADH in different biological contexts.

To use the coarse-grained NADH model, segmentation needs to be performed to separate the mitochondrial NADH signal from the cytoplasmic NADH signal, because they encode different metabolic fluxes. In mouse oocytes, the segmentation can be reliably performed based on NADH images due to the higher NADH intensity in mitochondria than cytoplasm. Mitochondrial movements are also slow in MII oocytes (Video 1); hence, long exposure times can be used to obtain high contrast NADH images. For cells where NADH contrast is low, such as in yeast cells (Papagiannakis et al., 2017; Shaw and Nunnari, 2002), MitoTracker dye (Appendix 1 Figure 1—figure supplement 1) or mitochondrial associated fluorescent proteins (Westermann and Neupert, 2000) will likely be needed for reliable segmentation of mitochondria.

One of the most important assumptions that enables the coarse-grained model to be used to predict fluxes is that the NADH redox cycle can be well approximated as being at steady-state, that is, the rate of change of NADH concentrations is much slower than the kinetic rates, including the binding/unbinding rates and the reaction rates. This is true for mouse oocytes, where the NADH intensity does not significantly change over the course of hours. This assumption also holds for slow processes such as the cell cycle (Papagiannakis et al., 2017), which occurs on the timescale of hours compared to timescales of seconds for the kinetic rates. This claim is supported by the success of the model on human tissue culture cells. The steady-state approximation could fail for rapid dynamics of NADH, such as the transient overshoot of NADH in neurons induced by acute external stimulus (Díaz-García et al., 2021), but this needs to be tested experimentally.

While NADH and NADPH share the same fluorescence spectrum, NADH concentration is 40 times greater than the concentration of NADPH for the whole mouse oocytes and presumably even higher for mitochondria (Bustamante et al., 2017). NADPH concentration can be comparable to that of NADH for other cell types such as tissue culture cells (Park et al., 2016). However, we have shown that the presence of NADPH signal and other background fluorescence signals only affect the equilibrium bound ratio ${\beta }_{\mathrm{e}\mathrm{q}}$ or the prefactor ${\displaystyle \alpha }$, and hence does not affect the flux inference procedure if ${\beta }_{\mathrm{e}\mathrm{q}}$ can be reliably determined and ${\displaystyle \alpha }$ remains a constant (Appendix 5). This was validated in tissue culture cells by comparing predicted ETC flux (Figure 7) with previous OCR measurements (MacVicar and Lane, 2014).

Finally, when relating NADH FLIM measurements to the ETC flux, we did not explicitly consider the contribution to the flux through FADH₂. This is a valid approximation when the FADH₂ oxidative flux is much smaller than the NADH oxidative flux, as is often the case since pyruvate dehydrogenase plus the TCA cycle yields four NADH molecules but only one FADH₂ molecule per cycle. Alternatively, if the FADH₂ flux is proportional to the NADH flux, then a rescaled value of ${\displaystyle \alpha }$ can be used in Equation 5b to effectively account for both fluxes. The proportionality of FADH₂ flux and NADH flux is expected when NADH and FADH₂ are produced from the same redox cycle with fixed stoichiometry, such as the pyruvate dehydrogenase and TCA cycle. This proportionality will break down if significant amounts of NADH and FADH₂ are produced in independent cycles where the stoichiometry varies, for example, when the glycerol phosphate shuttle acts as a reductase in mitochondria for FADH₂ but not for NADH.

Given these underlying assumptions, the model needs to be tested before being applied to other biological systems. The present study provides an example for such tests in mouse oocytes and human tissue culture cells by comparing the predicted ETC flux from FLIM with direct measurements of OCR across a wide range of perturbations.

Cells transduce energy from nutrients to power various cellular processes. The ETC flux represents the total rate of energy conversion by mitochondria. Despite the detailed knowledge of the biochemistry of mitochondrial metabolism, it is still unclear what cellular processes determine ETC flux or how cells partition energetic fluxes to different cellular processes, including biosynthesis, ion pumping, and cytoskeleton assemblies. Energetic costs of specific cellular processes have been estimated from theoretical calculations (Stouthamer, 1973) or through inhibition experiments (Mookerjee et al., 2017). The latter typically involves measurements of the change of metabolic fluxes, such as OCR, upon inhibition of specific cellular processes, and interpreting this change as the energetic cost of the inhibited process. This interpretation is valid if metabolic flux is determined by the energy demand of different cellular processes in an additive manner. This assumption has not been thoroughly tested. Using the NADH redox model, we discovered a homeostasis of ETC flux in mouse oocytes where perturbing energy demand and supply do not impact ETC flux despite significantly changing NADH metabolic state. On the other hand, perturbing ATP synthesis and proton leak greatly impacted the ETC flux. From these results, we concluded that it is the intrinsic rates of mitochondrial respiration, rather than energy supply or demand, that controls the ETC flux in mouse oocytes. While NADH metabolic state significantly changed in response to perturbing energy demand and supply, indicating cell metabolism was indeed impacted, it is unclear if these perturbations also influenced ATP, ADP, or AMP levels. Future work, including direct measurements of ATP, ADP, and AMP levels, will be required to uncover the mechanism of flux homeostasis. More broadly, our work demonstrates that it is a prerequisite to understand the regulation of ETC fluxes in order to correctly interpret the changes of ETC flux upon inhibiting subcellular processes.

The mechanism of the homeostasis of ETC flux is unclear. One possibility is the presence of flux buffering pathways, where the change of ATP fluxes induced by process inhibition is offset by the opposing change of fluxes through the buffering pathways. Enzymes such as adenylate kinase are known to buffer concentrations of adenine nucleotide (De la Fuente et al., 2014), but it is unclear if they also buffer fluxes. Another possibility is a global coupling of cellular processes, where the change of ATP consumption by one process is offset by the change of others. Changes in proton leak could also compensate for changes in ATP production. Additional work will be required to distinguish between these (and other) possibilities.

FLIM data is obtained with optical resolution, enabling subcellular measurements of NADH metabolic state. Interpreting these measurements using the NADH redox model enables inference of metabolic fluxes with subcellular resolution. Using this method, we discovered a subcellular spatial gradient of ETC flux in mouse oocytes, where the ETC flux is higher in mitochondria closer to the cell periphery. We found that this flux gradient is primarily a result of a spatially heterogeneous mitochondrial proton leak. It will be an exciting aim for future research to uncover the causes and consequences of the subcellular spatial variation in mitochondrial activity.

Frozen MII mouse oocytes (Strain B6C3F1) were purchased from EmbryoTech. Oocytes were thawed and cultured in droplets of AKSOM media purchased from MilliporeSigma in plastic petri dish. Mineral oil from VitroLife was applied to cover the droplets to prevent evaporation of the media. Oocytes were then equilibrated in an incubator at 37°C, with 5% CO₂ and air saturated oxygen before imaging. For imaging, oocytes were transferred to a 2-μl media droplet in a 35-mm glass bottom FluoroDish from WPI covered with 400–500 μl of oil. The glass bottom dish was placed in an Ibidi chamber with temperature and gas control during imaging. Temperature was maintained at 37°C via heated chamber and objective heater. CO₂ was maintained at 5% using gas tanks from Airgas.

The hTERT-RPE1 cell line is an established wild-type cell line received from the Cheeseman lab that has been validated based on behavior and properties. The hTERT-RPE1 cell line was maintained and tested for mycoplasma contamination in the Needleman lab on a regular basis (Southern Biotech).

Cell lines were maintained at 37°C and 5% CO₂. Cells were grown in Dulbecco’s modified Eagle’s medium (DMEM) (11966025, Gibco) supplemented with 10% fetal bovine serum (FBS), 0.5 mM sodium pyruvate, 5 mM HEPES, 1% penicillin and streptomycin, and either 10 mM glucose or 10 mM galactose. Cells were passaged in glucose or galactose at least three times before imaging. Cells were plated on 35 mm glass bottom FluoroDishes from WPI for imaging. Right before imaging, the media was replaced with 1 ml of phenol red-free DMEM (A1443001, Gibco) supplemented with 0.5 mM sodium pyruvate, 4 mM L-glutamine, 10 mM HEPES, and either 10 mM glucose or 10 mM galactose.

Our FLIM system consists of a two-photon confocal microscope with a 40× 1.25 NA water immersion Nikon objective, Becker and Hickle Time Correlated Single Photon Counting (TCSPC) acquisition system and a pulsed MaiTai DeepSee Ti:Sapphire laser from Spectra-Physics. NADH autofluorescence was obtained at 750 nm excitation wavelength with a 460/50 nm emission filter. Laser power at the objective was maintained at 3 mW. The scanning area was 512 by 512 pixels with a pixel size of 420 nm. Acquisition time was 30 s per frame. Oocytes were imaged with optical sectioning across their equators. A histogram of NADH fluorescence decay times was obtained at each pixel of the image.

Oxygen level was measured in the Ibidi chamber with an electrode-based oxygen sensor (GasLab). Since the oil layer covering the media droplet was very thin, the oxygen level in the droplet was assumed to be in instant equilibration with the chamber.

To separate mitochondrial NADH signal from cytoplasmic signal, we performed machine learning-based segmentation algorithms on NADH intensity images. We used the freeware Ilastik (Berg et al., 2019), which implements a supervised learning algorithm for pixel classification. The classifiers were trained to separate mitochondrial pixels from cytoplasmic pixels with a greater than 80% accuracy, as tested by MitoTracker Red FM (Appendix 1, Figure 1—figure supplement 1). We grouped photons from all mitochondrial pixels to obtain a histogram of NADH decay times for each oocyte and for each image of tissue culture cells. To extract the FLIM parameters of NADH bound fraction $f,$ long lifetime ${\tau }_{\mathrm{l}}$ and short lifetime ${\tau }_{\mathrm{s}}$, we fitted the histogram with $G=\mathrm{IRF*}{(C}_{1}F+C_2)$, where $\mathrm{*}$ indicates a convolution, and $\mathrm{I}\mathrm{R}\mathrm{F}$ is the instrument response function of the FLIM system, measured using a urea crystal. ${\displaystyle F\left(\tau \right)=f\cdot \exp \left(-\frac{\tau }{{\tau }_{\mathrm{l}}}\right)+\left(1-f\right)\cdot \exp \left(-\frac{\tau }{{\tau }_{\mathrm{s}}}\right)}$ is the two-exponential model for the NADH fluorescence decay. $C_1$ is the amplitude of the decay and $C_2$ is the background noise. The fitting was performed with a custom MATLAB code using a Levenberg-Marquardt algorithm (Yoo, 2018). To obtain the intensity, $I$, of mitochondrial NADH, we first measured the average number of photons per mitochondrial pixel, and divided it by the pixel area, 0.185 μm², and pixel scanning time 4.09 μs. The flux of ETC is inferred using Equations 5a-c for each oocyte and for tissue culture cells in a single image. Heatmaps of mean NADH fluorescence decay times were obtained by computing NADH fluorescence decay time of each mitochondrial pixel and averaging over neighboring mitochondrial pixels weighted by a Gaussian kernel with a standard deviation of 20 pixels. All FLIM measurements were taken from distinct individual oocytes and distinct images of tissue culture cells. Error bars in all figures of FLIM represent standard error of the mean across different individual oocytes or across different images for tissue culture cells. Number of oocytes is reported with n. Number of images for tissue culture cells is reported with N.

FLIM curves were independently fit for each individual oocyte. The reported error bars in this manuscript are standard errors of the mean (SEMs) across these measurements, which depends on the level of variation (the standard deviation) between the oocytes. Two sources of variation in FLIM measurements across the oocytes are: (1) true biological variations between oocytes and (2) fitting errors in the FLIM analysis. To estimate the error of fitting, we performed bootstrapping with randomly drawn points with substitution from each fluorescence decay curve for 53 oocytes. There are ~66,000 photons per oocyte, from which we generated 10 bootstrapped decay curves per oocyte to estimate the fitting error. The fitting error is computed as the variance and covariance of the fitted parameters across bootstrapped decay curves and averaged over 53 oocytes.

At high oxygen level in the AKSOM condition, the bootstrapping yields a variance of 2.2×10^-4, 4.6×10^-3 ns², and 6.0×10^-4 ns² for bound fraction, long lifetime, and short lifetime, respectively. The cell-to-cell variances obtained from a single fit per oocyte are 4.4×10^-4, 9.5×10^-3 ns², and 1.6×10^-3 ns² for bound fraction, long lifetime, and short lifetime, respectively. Hence the bootstrapping error accounts for 50%, 49%, and 40% of the cell-to-cell variance in bound fraction, long lifetime, and short lifetime, respectively. The bootstrapping yields a covariance of -1.0×10^-3 ns between bound fraction and long lifetime, which only accounts for ~20% of the covariance between these two variables during oxygen drop experiment. The inferred mean flux for oocytes at high oxygen levels in AKSOM is ${\displaystyle ⟨J_{\mathrm{o}\mathrm{x}}⟩=56.6\mu \mathrm{M}\cdot {\mathrm{s}}^{-1}}$. Propagating the error of fitting in all parameters from the bootstrapping analysis to the inferred flux gives a standard error of the mean in $J_{\mathrm{o}\mathrm{x}}$ of 1.1 μM·s^-1. The standard error of the mean in $J_{\mathrm{o}\mathrm{x}}$ obtained from a single fit per oocyte was 2.0 μM·s^-1. Thus, fitting errors account for ∼50% of the standard error of the mean in $J_{\mathrm{o}\mathrm{x}}$ .

Oxygen drop experiments for oocytes were performed by mixing nitrogen-balanced 5% O₂ gas with 0% O₂ gas at different ratios to create a continuous oxygen drop profile. CO₂ was maintained at 5%. Oocytes were imaged for 10 min at 5% O₂, 30 min during the continuous drop from 5% O₂ to approximately 0% O₂, and 20 min after quickly returning to 5% O₂. Oxygen levels were simultaneously monitored with an electrode-based oxygen sensor in the Ibidi chamber. 5% O₂ corresponds to ~50 μM of oxygen concentration in the culturing media. All the drug perturbations for oocytes were performed by equilibrating oocytes in the AKSOM media containing the corresponding drug for 15–30 min before the oxygen drop experiments. Pyruvate and potassium perturbations were performed by making KSOM media following Cold Spring Harbor Laboratory protocols with varying concentrations of sodium pyruvate and potassium, respectively. For oligomycin, FCCP, rotenone and pyruvate perturbations, 9 mM of sodium oxamate was also added to the media to suppress cytoplasmic NADH signal for better mitochondrial segmentation. The addition of the oxamate does not change the ETC flux of the mitochondria (Figure 5b).

For hTERT-RPE1 cells, drug perturbations were performed by replacing the media with drug-containing media through pipetting. Cells were imaged for 20–30 min immediately after drug perturbations.

All drugs were purchased from Sigma-Aldrich. Temperature was maintained at 37°C. CO₂ was maintained at 5%.

The OCR of the oocytes was measured using the nanorespirometer from Unisense (Lopes et al., 2005). A batch of 10–15 oocytes was placed at the bottom of a glass capillary with a diameter of 0.68 mm and a height of 3 mm. The capillary well is filled with AKSOM media or drug-containing media for metabolic perturbations. After an equilibration time of ~2 hr, a steady-state linear oxygen gradient is established in the capillary well due to the balance of oocyte respiration and oxygen diffusion. A motor-controlled electrode-based oxygen sensor (Unisense) is used to measure the oxygen gradient. The OCR is calculated as the product of the oxygen gradient, diffusivity of oxygen in the media, taken to be 3.37×10^-5 cm²/s, and the cross-sectional area of the capillary well, which was 0.36 mm². The entire system was enclosed in a custom-built chamber with temperature and gas control. Temperature was maintained at 37°C. Oxygen level was continuously varied during oxygen drop experiments by slowly mixing 20% O₂ with 0% O₂ from gas tanks, and maintained at the air saturation level for drug and pyruvate perturbations. OCR was measured on a group of 10–15 oocytes at a time. Single-oocyte OCR was obtained by dividing the measured OCR by the number of oocytes in the group. Error bars in all figures of OCR represent standard error of the mean across different groups of oocytes normalized by the number of oocytes in each group. Number of oocytes is reported with n. Number of groups is reported with N.

For the comparison between inferred ETC flux and measured ETC flux of the oocytes, two-sample t-test was performed on the vectors of inferred single-cell ETC flux (with n elements, where n is the number of oocytes) and the batch OCR measurements (with N elements, where N is the number of batch groups). For the comparison between inferred ETC flux and measured ETC flux of the tissue culture cells, two-sample t-test was performed on the vectors of inferred relative change of ETC flux (with n elements, where n is the number of images) and the relative change of OCR estimated from Figure 1A of MacVicar and Lane, 2014 (with N elements, where N is the estimated number of OCR data points).

The spatial distribution of mitochondrial membrane potential within oocytes was measured with a potential-sensitive dye TMRM (Sigma-Aldrich). Oocytes were cultured in AKSOM with 100 nM TMRM for 30 min before imaging. TMRM signal was obtained at 830 nm excitation wavelength with 560/40 nm emission filter. Mitochondrial TMRM intensity in different regions of the oocyte was computed by dividing the total number of photons from that region by the number of pixels in the same region. Heatmaps of mitochondrial TMRM intensity were obtained by computing photon counts for each mitochondrial pixel and averaging over neighboring mitochondrial pixels weighted by a Gaussian kernel with a standard deviation of 20 pixels. To normalize TMRM intensity by mitochondrial mass, we cultured oocytes in AKSOM with 100 nM MitoTracker Red FM and 25 nM TMRM for 30 min before imaging. We also cultured oocytes in AKSOM with 1 μg/ml JC-1 dye for 3 hr before imaging.

Mitochondrial membrane potential of hTERT-RPE1 cells was measured with TMRM. The cells were cultured in DMEM with 100 nM TMRM for 15–30 min before imaging. To measure membrane potential under drug perturbations, the original media was pipetted out and replaced with media containing both 100 nM TMRM and the drug. The cells were imaged for 20–30 min immediately after drug perturbations. TMRM intensity ratio was obtained by normalizing the mitochondrial TMRM intensity by the cytoplasmic TMRM intensity.

Since the molecular brightness of NADH depends on the fluorescence lifetime of NADH, which changes drastically upon binding enzymes, the NADH concentration is not linearly proportional to NADH intensity. FLIM provides an accurate way of measuring NADH concentrations by simultaneously measuring fluorescence intensity and lifetime. We now derive the NADH intensity-concentration relation from the FLIM measurements. Assuming molecular brightness is proportional to the fluorescence lifetime, and therefore that free and bound NADH have different contributions to the measured intensity, we have

where $I$ is the intensity of NADH and $c_{\mathrm{s}}$ is a calibration factor that depends on the laser power. From Equation S1, we obtained the concentrations of free and bound NADH:

where $f$ is the fraction of bound NADH.

To get the calibration factor $c_{\mathrm{s}}$, we titrated NADH in AKSOM solutions and fitted the calibration curve using:

where ${\tau }_{\mathrm{s}\mathrm{o}\mathrm{l}}$ is the lifetime of NADH in solution. ${\tau }_{\mathrm{s}\mathrm{o}\mathrm{l}}$ was directly measured by FLIM (Figure 1—figure supplement 2), allowing us to obtain $c_{\mathrm{s}}$ from the fit (Figure 1—figure supplement 2).

To test if absolute concentrations of free and bound NADH can be accurately measured from FLIM of NADH, we prepared solutions with known total concentration of NADH, and titrated the concentration of LDH, an enzyme to which NADH can bind. We prepared the solutions with 50 mM TRIS buffer, 150 mM NaCl at pH 7.6 and 37°C. We added a total concentration of 50 μM NADH to the solution and titrated LDH concentrations at 0, 1.4, and 3.5 μM. We first performed single exponential fitting of the NADH decay curve at 0 μM LDH, where all NADH are free, to obtain the NADH short lifetime (Figure 1—figure supplement 3d). From the NADH intensity (Figure 1—figure supplement 3a), we obtained the calibration factor $c_{\mathrm{s}}$ using Equation S4 with ${\displaystyle \left[\mathrm{N}\mathrm{A}\mathrm{D}{\mathrm{H}}_{\mathrm{s}\mathrm{o}\mathrm{l}}\right]=50\mu M}$. We then fixed the short lifetime and performed two-exponential fitting of the NADH decay curve at LDH concentrations of 1.4 μM and 3.5 μM to obtain the bound ratio (Figure 1—figure supplement 3b) and long lifetime (Figure 1—figure supplement 3c). As expected, NADH bound ratio increases with LDH concentrations, as there is more enzyme for NADH to bind. Finally, we calculated free NADH concentration $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ and bound NADH concentration $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{b}}\right]$ using Equations S2 and S3 from the FLIM parameters. Remarkably, the free and bound concentrations of NADH both change with LDH concentrations but the total concentration remains at 50 μM (Figure 1—figure supplement 3e). This result shows that Equations S2-S3 can be used to accurately measure the concentrations of free and bound NADH from FLIM measurements of NADH.

The kinetic equations of the reversible Michaelis-Menten kinetics (Figure 3a, left) for the ith oxidase are

where $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ is the concentration of free NADH, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}}_{\mathrm{f}}^{+}\right]$ is the concentration of free NAD⁺, $\left[{\mathrm{O}\mathrm{x}}_i\right]$ is the concentration of free oxidase, and $\left[\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\bullet {\mathrm{o}\mathrm{x}}_i\right]$ is the concentration of the NADH-oxidase complex. $k_{-1}$, $k_1$, $k_{-2}$, and $k_2$ are the forward and reverse reaction rates.

In the reduced notation as introduced in Figure 3a (right), the same enzyme kinetics is described by

where $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{b}}$ and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{u}}$ are the effective binding and unbinding rates of NADH to the oxidase, and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{`}\mathrm{b}}$ and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{`}\mathrm{u}}$ are the effective binding and unbinding rates of NAD⁺ to the oxidase. In this reduced notation, the concentration of the free oxidase ${\displaystyle \left[\mathrm{O}{\mathrm{x}}_i\right]}$ is absorbed into the effective binding rates: that is, $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{u}}=k_{-1}$, $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{b}}=k_1\left[{\mathrm{O}\mathrm{x}}_i\right]$, $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{`}\mathrm{u}}=k_2$, and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{`}\mathrm{b}}=k_{-2}\left[{\mathrm{O}\mathrm{x}}_i\right]$. Hence ${\displaystyle \left[\mathrm{O}{\mathrm{x}}_i\right]}$ becomes an implicit variable whose behavior is not evident from the reduced notation diagram (Figure 3a, right). Modeling the full dynamics of a reversible Michalis-Menten enzyme requires specifying the equation for ${\displaystyle \left[\mathrm{O}{\mathrm{x}}_i\right]}$:

The reduced notation (Figure 3a, right; Equations S9–S12) and the full notation (Figure 3a, left; Equations S5-S8) are mathematically identical and describe the exact same kinetics.

The reduced notation for the ith oxidase displaying generalized enzyme kinetics (Figure 3b) refers to the following class of mathematical models:

where $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ is the concentration of free NADH, $\left[{\mathrm{N}\mathrm{A}\mathrm{D}}_{\mathrm{f}}^{+}\right]$ is the concentration of free NAD⁺, $\left[{\mathrm{O}\mathrm{x}}_i\right]$ is the concentration of free oxidase, $\left[\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\bullet {\mathrm{O}\mathrm{x}}_i\right]$ is the concentration of the NADH-oxidase complex, and $\left[{\mathrm{N}\mathrm{A}\mathrm{D}}^{+}\bullet {\mathrm{O}\mathrm{x}}_i\right]$ is the concentration of the NAD⁺-oxidase complex. $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{b}}$ and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{u}}$ are the effective binding and unbinding rates of NADH to the oxidase, $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{`}\mathrm{b}}$ and $k_{{\mathrm{o}\mathrm{x}}_i}^{\mathrm{`}\mathrm{u}}$ are the effective binding and unbinding rates of NAD⁺ to the oxidase, and $r_{{\mathrm{o}\mathrm{x}}_i}^{+}$ and $r_{{\mathrm{o}\mathrm{x}}_i}^{-}$ are the forward and reverse oxidation rates. These rates can be arbitrary functions of implicit variables, such as the concentration of free oxidase, $\left[{\mathrm{O}\mathrm{x}}_i\right]$, the mitochondrial membrane potential, ${\displaystyle \mathrm{\Delta }{G}_{\mathrm{H}}}$, $\mathrm{p}\mathrm{H}$, and other factors:

Thus, while Equations S13-S16 superficially appear to be linear and first order, they can actually refer to nonlinear reactions of any order because the rate can depend on $\left[{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}_{\mathrm{f}}\right]$ and other variables (Equations S17a–f).

The implicit variables that these rates depend on can each be governed by their own dynamics that are arbitrary functions of other variables:

Describing the dynamics of the enzyme requires specifying the implicit variables that the rates depend on, the functional form of these dependencies, and the additional equations for the dynamics of the implicit variables (Equations S18). However, we will show that the predicted relationship between FLIM measurements of NADH and fluxes does not depend on these modeling choices. Thus, the reduced notation is convenient for deriving these relations for a broad class of models.

We consider an NADH redox loop consisting of M reductases and N oxidases (Figure 4—figure supplement 1), each of which is described by the generalized enzyme kinetics (Figure 3b; Equations S13-S16). We coarse-grain this detailed NADH redox model by coarse-graining all oxidases into a single effective oxidase and all reductases into a single effective reductase (Figure 4b). We relate the kinetic rates of the coarse-grained model to those of the detailed model by keeping the global binding and unbinding fluxes and the global reaction fluxes through the oxidases and reductases the same as the detailed model.

We first coarse-grain the oxidases and reductases:

We require the global binding and unbinding fluxes of NADH to the effective oxidase and reductase to be equal to the sum of their binding and unbinding fluxes to all of the individual oxidases and reductases:

which leads to

We require the global forward and reverse reaction flux through the effective oxidase and reductase to be equal to the sum of the reaction fluxes through all of the individual oxidases and reductases:

which leads to

By applying the same procedure to ${\mathrm{N}\mathrm{A}\mathrm{D}}^{+}$, we can obtain the effective reduction rates $r_{\mathrm{r}\mathrm{e}}^{+}$, $r_{\mathrm{o}\mathrm{x}}^{-}$ and the effective binding and unbinding rates of ${\mathrm{N}\mathrm{A}\mathrm{D}}^{+}$ : $k_{\mathrm{o}\mathrm{x}}^{\mathrm{`}\mathrm{b}}$, $k_{\mathrm{r}\mathrm{e}}^{\mathrm{`}\mathrm{b}}$, $k_{\mathrm{o}\mathrm{x}}^{\mathrm{`}\mathrm{u}}$, $k_{\mathrm{r}\mathrm{e}}^{\mathrm{`}\mathrm{u}}$. We omit the derivation here because these rates are not needed to infer ETC flux. We hence explicitly related the kinetic rates of the coarse-grained model (Figure 4b) to those of the detailed model (Figure 4—figure supplement 1). We note that under the generalized enzyme kinetics (Figure 3b; Equations S13-S16), all kinetic rates are considered to be general functions of enzyme concentrations, metabolite concentrations, and other factors, and thus, all of the rates in the coarse-grained model can also depend on all of those factors. These implicit variables can obey their own dynamical equations (Equation S18). The coarse-graining presented here is mathematically exact and independent of both the functional forms of these rates and the functional form of the dynamic equations of the implicit variables.