= Modeled mitochondrial oxygen traces
{title2=$q=-V\,dc_{\mathrm{O}_2}/dt$}
The following original model chooses piecewise constant <oxygen> consumption rates in a closed, well-mixed $2\,\mathrm{mL}$ suspension. It assumes no gas exchange, negligible addition <volume (thermodynamics)>, a corrected sensor and sufficient <oxygen> throughout. The curves are invented teaching examples, not measured traces, fitted data or a reconstruction of Figure 6.
\Image[/mitochondrial-modeled-oxygen-and-control-readouts.png]
{title=MODELED <oxygen> traces: a faster decline need not mean more <ATP>}
{description=Original piecewise linear <concentration> curve and corresponding consumption-rate plot. All rates and times are chosen. The finite first <ADP> pulse is stipulated to be used by minute 2; the second supplies excess <ADP>. <Oligomycin> remains present after <FCCP>. Substrate additions and intermediate chain interventions from the published protocol are omitted. The residual after <antimycin A> is shown as background, not <ATP> production.}
Start at $c(0)=220\,\mathrm{\mu mol\,L^{-1}}$. On successive one-minute intervals, choose positive slope magnitudes $5,25,5,25,6,32,1\,\mathrm{\mu mol\,L^{-1}\,min^{-1}}$. Integration gives the endpoint <concentrations> $220,215,190,185,160,154,122,121\,\mathrm{\mu mol\,L^{-1}}$. This is just the recurrence $c_{i+1}=c_i-r_i\Delta t$, with $\Delta t=1\,\mathrm{min}$; the step plot shows $r_i=-dc/dt$. The trace remains above zero by construction.
During the second <ADP> interval, the total consumption rate is
$$
q=-V\frac{dc_{\mathrm{O}_2}}{dt}
=(0.002\,\mathrm L)(25\,\mathrm{\mu mol\,L^{-1}\,min^{-1}})
=50\,\mathrm{nmol\,min^{-1}}.
$$
With a stipulated mitochondrial <protein> mass of $0.20\,\mathrm{mg}$, that is $250\,\mathrm{nmol\,min^{-1}\,mg^{-1}}$. Using a denominator twice as large would halve this reported rate without changing the raw <oxygen> trace. Comparisons therefore need both loading and the chosen denominator.
For a second bookkeeping check, the modeled post-<antimycin A> slope of $1\,\mathrm{\mu mol\,L^{-1}\,min^{-1}}$ corresponds to $2\,\mathrm{nmol\,min^{-1}}$ background. If that background is additive and unchanged in earlier windows, subtraction gives $48$, $10$ and $62\,\mathrm{nmol\,min^{-1}}$ for excess <ADP>, <oligomycin> and <FCCP>, respectively. The <oligomycin>-sensitive difference is $48-10=38\,\mathrm{nmol\,min^{-1}}$. It is a difference in <oxygen> consumption under the chosen interventions, not an <ATP> production rate. Converting it into one requires additional coupling assumptions or an independent <ATP> measurement; the <FCCP> interval shows why slope alone cannot supply that conversion.
Across the whole seven-minute example, consumption is $V(c(0)-c(7))=198\,\mathrm{nmol}$, also the sum of the seven interval consumptions. This equality checks the rate-to-amount accounting independently of how the intervals are named. Actual records require attention to calibration, mixing transients, background stability and decreasing <oxygen> availability; a line that steepens or flattens alone does not identify the cause.
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