Response to ‘Coming up for air’

We (Seibel et al., 2021) proposed a novel method to quantify the oxygen supply capacity (α), which defines the maximum metabolic rate (MMR) that can be achieved at a given oxygen pressure. In their Correspondence article, Farrell et al. (2021) suggest that our method lacks ‘empirical realism’. The core issue is whether the MMR is linearly related to P_O2 and whether that line intercepts the origin as required for the constancy of α, from which its predictive power derives. However, Farrell et al. (2021) overlooked the published derivation and validation of α, which is based on decades of empirical physiology and which demonstrates convincingly that α is constant from rest to maximum exertion (Deutsch et al., 2015; Seibel and Deutsch, 2020).

For a given species, α, calculated as the standard metabolic rate (SMR) divided by its critical P_O2 (P_crit-SMR), accurately predicts MMR measured at any P_O2 up to its critical P_O2 (P_crit-max; see Fig. 2D in Seibel and Deutsch, 2020). Thus, α is equal to both SMR/P_crit-SMR and MMR/P_crit-max and is effectively the slope of a straight line between those two endpoints that intercepts the origin (Fig. 1). Beyond P_crit-max, in hyperoxia, MMR plateaus (see Fig. 2B in Seibel and Deutsch, 2020). A non-linear relationship that begins and ends at identical α values would be highly improbable. The interpretation of some of this same data as curvilinear (Farrell et al., 2021) stems, at least in part, from the inclusion of measurements made at oxygen pressures beyond P_crit-max (i.e. in the hyperoxic plateau). Thus, α does not typically change with oxygen or activity (Fig. 1) and can be used to predict MMR and aerobic scope.

The anatomy of the oxygen cascade does change between rest and activity, and some individual steps in the cascade appear non-linearly related to P_O2. However, these non-linear steps do not preclude a linear relationship between MMR and inspired P_O2 (Seibel and Deutsch, 2020). Regardless of the underlying mechanism, Fig. 1 shows that MMR is linearly related to P_O2 for most species analyzed to date. Although there may be species and conditions for which α changes with activity or oxygen, such examples would not diminish the utility of α. Instead, they may reveal how α is acted upon by natural selection. For example, α may vary with swimming in species that ram ventilate, with depth in species that migrate vertically across oxygen and temperature gradients, or with size and life stage as circulatory systems develop.

Methodological issues may also influence the apparent shape of the MMR–P_O2 relationship. MMR measurement relies on a fish's willingness to swim during exercise protocols (Slesinger et al., 2019), which may vary with exertion and oxygen. Furthermore, because MMR declines in direct proportion to P_O2, supposed ‘normoxic’ MMR measurements may be underestimates. Seibel et al. (2021) recommended that, when feasible, MMR be measured across a range of P_O2 values, including hyperoxia, to minimize our shared concern (Farrell et al., 2021) of extrapolating from ‘one experimental data point … that has associated error’. Also, when feasible, MMR should be elicited by more than one protocol (e.g. chase and flume). Our method provides much-needed consistency for determining oxygen supply capacity directly and provides a means of predicting MMR and aerobic scope.

Farrell et al. (2021) are correct that MMR can't continue upward indefinitely with P_O2. Instead, it plateaus at P_crit-max. A realistic estimate of P_crit-max, rather than aerobic scope as suggested by Farrell et al. (2021), is required to avoid over-estimates of MMR. P_crit-max has been directly measured for numerous species but, more importantly, it can be approximated as the highest P_O2 that persists in a species' natural environment (i.e. air-saturation for most shallow, coastal species; Seibel and Deutsch, 2020).

Farrell et al. (2021) are concerned that α is a ‘black box’ that hides all relevant physiological mechanisms. However, this is true of all integrative physiological metrics, including SMR and aerobic scope, and does not diminish the utility of these metrics or the importance of the underlying mechanistic physiology. Rather, these ‘black boxes’ act to reduce inherent physiological complexity in order to understand relationships at different scales. The α is informative despite the complexity of the oxygen cascade, just as aerobic scope is informative despite incomplete knowledge of the specific oxygen requirements of the underlying processes (Farrell, 2016).

Farrell et al. (2021) state that we dismissed approaches that aim to understand the entire MR versus P_O2 response curve and that, having done so, we fail to recognize that our ‘measure of P_crit-SMR may not actually relate to SMR’. This indicates an important misunderstanding of our method. We are not measuring P_crit-SMR. We are measuring the oxygen supply capacity (α). We did compare P_crit-SMR derived from α with that using other methods simply as a demonstration of its relative accuracy and precision. However, our method concerns only α and does not require that P_crit be measured at SMR because the P_crit measured for any rate provides the same information (α) from which the P_crit for any other rate (including SMR) can be calculated. For example, SMR could be independently determined and P_crit-SMR can then be calculated as SMR/α, even if α was determined at MMR. The physiological changes that occur at P_O2 above P_crit are not relevant to the measurement of α, and statistical descriptions of the entire MR versus P_O2 trial apparently seek to answer a different question.

Furthermore, we do not ‘discourage examining data that depart from [our] model’, nor do we simply ‘draw a straight line through two data points’. We determined oxygen supply for every P_O2 bin in 50 published trials. The highest value in each trial is the oxygen supply capacity (α) for that individual or species. The metabolic rate divided by α is P_crit for that rate, whether standard, routine or maximum. Any P_O2 (up to P_crit-max) multiplied by α is the maximum metabolic rate that could be achieved at that P_O2. Our theory, rather than ‘trying to enforce itself onto real data’, is derived from real data and provides a powerful tool for understanding how metabolism changes with oxygen and temperature. Our paper ‘breathed new life into P_crit’ by clarifying its biological significance as a measure of oxygen supply capacity.