Oxygen supply capacity breathes new life into critical oxygen partial pressure (P_crit)

The relationship between metabolic rate (MR; Box 1) and environmental oxygen has long been of interest because of its implications for human health, fisheries management, biogeography, species diversity and evolution (Tang, 1933; Lindroth, 1942; Hall, 1966; Fry and Hart, 1948; Childress, 1968; Weibel, et al., 1991; Suarez, 1998; Ern et al., 2016; Rogers et al., 2016; Ultsch and Regan, 2019; Slesinger et al., 2019). Although this relationship has taken on special significance in light of anthropogenic ocean warming and deoxygenation (Wishner et al., 2018; Breitburg et al., 2018; Claireaux and Chabot, 2019; Pörtner et al., 2017; Rubalcaba et al., 2020; Howard et al., 2020; Deutsch et al., 2020), there is no consensus on how it should be measured. A commonly employed metric is the critical oxygen partial pressure (P_crit), which is typically interpreted as a measure of hypoxia tolerance, with a lower P_crit indicating greater tolerance for low oxygen (e.g. Rogers et al., 2016; Seibel, 2011). However, as ‘hypoxia’ and ‘tolerance’ are subjective, time-dependent terms, and P_crit is inconsistently defined and measured, there is active debate about the meaning and significance of the metric (Wood, 2018; Regan et al., 2019).

Typically, P_crit is defined as the minimum P_O2 at which a given MR can be sustained and is identified by a clear decline (breakpoint) in the MR in response to declining P_O2 (Farrell and Richards, 2011; Pörtner and Grieshaber, 1993; Richards, 2011; Hall, 1966; Ultsch and Regan, 2019; Childress and Seibel, 1998). Intraspecifically, P_crit is dependent on the MR, which is itself dependent on temperature and activity levels. For comparison among species, recent protocols (Claireaux and Chabot, 2016; Reemeyer and Rees, 2019) therefore recommend measuring P_crit at the standard metabolic rate (SMR), the lowest ‘sustainable’ MR measured in a fasted and resting state. By definition, aerobic metabolism is oxygen limited at P_crit for SMR (P_crit-SMR) and aerobic scope [the difference between standard and maximum (MMR) rates] is nil. This is what early researchers referred to as the ‘oxygen level of no excess activity’ (Fry and Hart, 1948; Lindroth, 1942).

Ultsch and Regan (2019) recently argued that P_crit-SMR is the most appropriate benchmark because it is truly ‘critical’ for the animal's survival, whereas P_crit for other rates is not a lethal oxygen level. Accordingly, they defined P_crit-SMR as the P_O2 below which life cannot be sustained. However, survival is also time limited at, or even at some factor above, P_crit-SMR, because aerobic scope (and thus feeding, growth and reproduction) may be oxygen limited. Furthermore, many species rarely (or never) experience P_O2 levels near their P_crit-SMR (Deutsch et al., 2020) and, among those that do, the hypoxia is often temporary, intermittent or cyclic. Survival below P_crit-SMR is dependent on the capacity to temporarily suppress metabolic demands to match limited oxygen availability and to increase anaerobic ATP production (Seibel et al., 2014, 2018; Storey, 2015).

Lindroth (1942) argued that P_crit should be determined at the lowest P_O2 at which an organism is indefinitely viable, i.e. that for which development and reproduction are undisturbed and ‘continued prosperity’ is supported. Recent work suggests P_O2 must exceed P_crit-SMR by a factor of ∼3 to meet this benchmark. That is, a factorial aerobic scope (FAS; MMR/SMR=P_O2/P_crit; see Box 1; Seibel and Deutsch, 2020) of ∼3 corresponds to a biogeographic limit for many species (Deutsch et al., 2015, 2020). However, because, intraspecifically, P_crit is typically temperature sensitive, a FAS of 3 may be met or exceeded at low temperature even in relative hypoxia, while, at warmer temperatures, even atmospheric P_O2 may not provide sufficient metabolic scope. Thus, the ecological significance of P_crit remains unclear, regardless of the rate at which it is measured, and there is no consensus on the best way to determine P_crit.

The traditional brokenstick method (Yeager and Ultsch, 1989) depends on a relatively constant MR as P_O2 declines (i.e. it must be ‘regulated’) and a discontinuity in the measured rate must be readily apparent at P_crit, where the rate begins to conform to available P_O2. In the brokenstick method, the intersection of two regressions, through the oxygen-limited and oxygen-independent portions of the trial, is taken as P_crit (Yeager and Ultsch, 1989). This methodology imparts a great deal of importance to the constancy or degree of oxygen independence of the regulated rate at high P_O2 values as variation in slope will affect the determined P_crit value. A more consistent method uses only a single regression (MR versus P_O2) through the oxygen-limited portion of the curve and solves for SMR, which must be established at higher P_O2. This limiting low oxygen (LLO) method is not influenced by variation in rate throughout the trial (Reemeyer and Rees, 2019). However, SMR estimates may vary with respirometry or statistical methodology (Chabot et al., 2021) and other variables, which will influence P_crit determination (e.g. Negrete and Esbaugh, 2019; Regan and Richards, 2017). What constitutes SMR is also unclear for the majority of species and many measurements to date may be more appropriately described as routine metabolic rate (RMR), for which activity and feeding history are unknown. Additionally, P_O2 below P_crit-SMR is in the realm of ‘incipient lethal oxygen levels’ (Claireaux and Chabot, 2016), where MR does not necessarily decline in direct proportion to P_O2. In that case, the slope of that portion of the trial will vary unpredictably, resulting in variable estimates for P_crit-SMR. Some studies have forced the oxygen-limited regression through the origin, increasing the consistency of P_crit-SMR determination, but without a stated physiological justification (McArley et al., 2019).

Others have recommended statistical approaches to identify an inflection point in the data that represents P_crit (Muggeo, 2003) or to statistically describe the entire relationship between MR and P_O2, without implication of any physiological mechanism or evidence that oxygen is driving the apparent relationship (non-linear regression, regulation index, or Michaelis–Menten analysis; Mueller and Seymour, 2011; Cobbs and Alexander, 2018; Wood, 2018). However, P_crit is effectively lost in such mathematical descriptions of the entire respirometry trial and it is unclear what information can be extracted from a description of the entire trial, much of which may not have any functional relationship to available oxygen.

For an individual organism, activity elevates MR and a higher rate will become oxygen limited (i.e. reach its P_crit for that MR) at higher oxygen pressures (Fry and Hart, 1948; Claireaux and Chabot, 2016). The relationship between MR and its corresponding P_crit is often assumed to be curvilinear between P_crit-SMR and P_crit-max (the P_crit for the maximum metabolic rate, MMR). Claireaux and Chabot (2016) described this relationship schematically as the limiting oxygen level (LOL) curve. They note that, for any rate lower than MMR, ventilatory rates and cardiac output increase as environmental P_O2 declines until the LOL curve is reached. Further decline in P_O2 requires that extraneous metabolic costs (e.g. activity) be reduced. While this description is physiologically consistent, the authors did not explain or justify the curvilinear shape of the LOL curve and their direct measurements do not unambiguously support the assumed shape. Unfortunately, this assumption masks the quantifiable oxygen dependency of MR described by Seibel and Deutsch (2020) and reinforces the idea that P_crit-SMR is an independent metric that reflects environmental hypoxia tolerance.

Here, we define P_crit as the P_O2 at which physiological oxygen supply mechanisms are operating at maximum capacity (α, see Box 1; Deutsch et al., 2015, 2020; Seibel and Deutsch, 2020; Kielland et al., 2019; Lindroth, 1942). P_crit values are a MR-specific measure of the oxygen supply capacity, α, that can be mathematically defined as P_crit=MR/α (see detailed derivation below and definitions in Box 1). α is a species- and temperature-specific constant that describes the linear dependency of P_crit on MR and, equivalently, the oxygen dependency of the MMR (MMR=P_O2×α; Seibel and Deutsch, 2020). We show here that α can be determined directly without an obvious breakpoint in, nor a standardized level of, MR.

This definition provides a strong theoretical justification for the concept of P_crit and eliminates most, if not all, of the recently described pitfalls associated with its measurement (Wood, 2018; Reemeyer and Rees, 2019; Ultsch and Regan, 2019; Regan et al., 2019). This new definition calls into question widely held beliefs about the ecological and evolutionary significance of P_crit. As stated by Fry and Hart (1948), ‘the worth of such data to the ecologist must ultimately depend on proof that they have real significance as values limiting the activity of the organism in nature’. Many decades later, such proof is still not readily available and most evidence is merely correlative. Strong comparative approaches have been employed to demonstrate that P_crit sometimes reflects physiological adaptations to low oxygen (Regan et al., 2019; Wishner et al., 2018; Childress and Seibel, 1998; Mandic et al., 2009). However, a comparatively low P_crit value is often assumed to reflect adaptation to environmental hypoxia, even where oxygen levels never approach P_crit or in the absence of direct environmental P_O2 measurements. We argue here that P_crit is not a measure of hypoxia tolerance, per se, but rather an indirect way of measuring the oxygen supply capacity, itself adapted to supply oxygen for species-specific metabolic demands and aerobic scope across temperatures and at the prevailing oxygen pressure.

For an aerobic organism to obtain sufficient energy for survival, oxygen supply must meet oxygen demand. Oxygen demand is the aerobic MR, most commonly measured at rest (SMR) or at maximum exertion (MMR). The total amount of oxygen available for cellular respiration at any given time is a function of the ambient environmental P_O2 and the physiological oxygen supply (α₀; Seibel and Deutsch, 2020; Deutsch et al., 2015). Physiological oxygen supply, α₀, comprises each step in the oxygen cascade (Weibel et al., 1991) from ventilation and blood–oxygen binding to cardiac output and circulation. We define and calculate α₀ as the rate of oxygen consumption (MR) per unit of available environmental oxygen pressure (α₀=MR/P_O2; µmol O₂ g⁻¹ h⁻¹ kPa⁻¹). When an individual has maximized these physiological mechanisms of oxygen delivery, the physiological oxygen supply, α₀, has reached the physiological oxygen supply capacity, α, and MR=MMR at that P_O2.

To determine α, the rate of oxygen consumption (MR) is monitored using established respirometry techniques as P_O2 declines. Each MR measurement period within a trial is divided by the corresponding P_O2 to provide a value for the rate of oxygen supply (α₀) at that point in the trial. For any given MR, oxygen supply increases toward its maximum capacity (α) as P_O2 declines and the highest α₀ value is taken as the maximum capacity, α. To be more conservative, here we designated the mean of the three highest α₀ values as α. The α-line (MMR=P_O2×α) defines the limiting P_O2 for any MR between SMR and MMR. The oxygen supply capacity is conserved across MR (i.e. the α-line is linear, with slope, α, and intercept at the origin), as was demonstrated previously (Seibel and Deutsch, 2020) and is further supported here.

When MR is maintained at SMR throughout a trial, as is often the goal in traditional P_crit trials, α₀ will gradually increase throughout the run and may only reach α briefly at P_crit-SMR. Thus, α may be represented by few points before α₀ declines again as the subject either suppresses metabolism or experiences physiological failure. A decline in α₀ below P_crit-SMR is frequently observed, which diminishes the accuracy of traditional methods, such as the LLO or the brokenstick method, that rely on the slope of the that portion of the curve (Fig. 2). If, in contrast, the MR is directly proportional to P_O2 below a given P_crit, then α₀ will plateau when it reaches capacity, α. In this case, multiple α₀ values can be averaged to determine α (see lines 1 and 2 in Fig. 1A,B). A final consideration is that α₀ may increase following the brief plateau if substantial error exists in P_O2 measurement (i.e. sensor calibration error; see Fig. 1). This occurs because P_O2 measurement error leads to a measurable rate of oxygen consumption (the numerator) even as P_O2 (the denominator) apparently, but erroneously, approaches anoxia. This is sometimes apparent as a positive y-intercept in the MR versus P_O2 trace. This is recognizable (Fig. 1) using the α-method but would represent undiagnosed error using traditional techniques. Once diagnosed, P_O2 measurement error may be corrected. No correction has been applied to any of the data in the present manuscript.

We make the following specific recommendations for use of the α-method. α should be determined during exercise when possible because α₀ will be maintained near capacity (α₀≈α) for a greater proportion of the trial and/or α may be reached multiple times throughout the trial (Fig. 1). If measured at MMR, α₀ will be near α for the entire trial, in which case, α may be determined as the mean of all α₀ values. A further advantage of measurement in an active state is that α can be determined at higher P_O2, well above potentially lethal oxygen levels. If α is measured near SMR, we recommend averaging no more than the three the highest α₀ values to determine α. This diminishes the underestimating effect of averaging sub-maximal oxygen supply values above and below P_crit-SMR.

As with traditional measures of MR and P_crit, values of α may be influenced by the precision of the measurement. However, because α is not dependent on the MR, it is not dependent on the behavior of the animal in the respiration chamber, the technique used to measure the rate (e.g. closed versus intermittent-flow respirometry) or the duration of the trial (Fig. 3). In some cases, extended trial duration could result in physiological acclimation of, or impairment to, the oxygen supply cascade in response to declining oxygen or accumulated metabolic waste, respectively, resulting in variation in α relative to shorter trials (Reemeyer and Rees, 2020; Regan and Richards, 2017; Snyder et al., 2016; Sollid et al., 2005). However, differences in methodology between studies that only influence the MR will not affect the oxygen supply capacity.

We used several approaches to test the applicability, generality and precision of the α-method. We first extracted and re-evaluated available literature data for 40 species in five phyla (Table S1) using studies that provided a ‘representative’ trace showing the relationship between the oxygen consumption rate and P_O2. Data were extracted from these respirometry curves using WebPlotDigitizer 4.2 (https://automeris.io/WebPlotDigitizer/index.html). The oxygen supply capacity was determined as described above and compared with α calculated using literature values of P_crit and the corresponding MR.

Many of the ‘representative’ curves provided in the literature may be the best of many trials, rather than being truly representative. Thus, they may not provide the most rigorous test of this method. Accordingly, we also tested full datasets (Table S2) for zooplankton (Wishner et al., 2018), shrimp (A.L.B. unpublished), squid (M.A.B. and B.A.S., unpublished) and fishes (Slesinger et al., 2019; A.A., unpubllished) to assess inter-individual variance in α. Oxygen consumption for each species was measured using published methods, based on accepted practices (Clark et al., 2013; Chabot et al., 2016) using chambers of appropriate size and type for each species (see references in Table S2). All experiments were carried out with IACUC approval (protocol # W IS00007992, W IS00004975).

To assist in the adoption of this new method, calculation of α has been incorporated into the R package ‘respirometry’ with an optional parameter available to define the MR of interest for P_crit estimation (http://cran.r-project.org/package=respirometry). The ‘calc_pcrit’ and ‘plot_pcrit’ functions now return side-by-side comparisons of P_crit calculated using a brokenstick regression (Yeager and Ultsch, 1989), non-linear regression (Marshall et al., 2013), the LLO method (Reemeyer and Rees, 2019), the sub-prediction interval method (Birk et al., 2019), and the α-method presented here. Additionally, the function ‘calc_alpha’ determines α when the user inputs MR and oxygen data. This software is freely available on CRAN.

Published MR and P_crit, as well as the directly determined and calculated α values, are presented in Table S1 for 40 species of diverse aquatic and terrestrial animals. The individual trials used are presented pictorially in Table S1. The MR, P_crit and α values were converted to common units (µmol O₂ g⁻¹ h⁻¹ and kPa) for ease of comparison. The physiological oxygen supply capacity, α, was determined using the α-method and calculated as the published MR divided by its corresponding P_crit (α=MR/P_crit). The α values determined using these two distinct methods are correlated (Fig. 4; y=1.96x−0.10; Pearson's r=0.96, P<0.001, n=40). However, the α-method resulted in higher values on average (mean difference=4.4%), which reflects its dependence on the highest α₀ values, rather than a regression through all values as in other P_crit methods. In the species analyzed here, α ranged from 0.09 µmol O₂ g⁻¹ h⁻¹ kPa⁻¹ for the deep-sea anglerfish, Melanocetus johnsoni, at 5°C (Cowles and Childress, 1995), to 4.11 µmol O₂ g⁻¹ h⁻¹ kPa⁻¹ at 28°C for larval zebrafish, Danio rerio (Mandic et al., 2020; Table S1). These values are within the range previously reported for a similar diversity of species and are correlated with the MR at which they were determined (α=0.195×SMR+0.317; Pearson's r=0.98, P<0.001; Fig. 4; Deutsch et al., 2020; Seibel and Deutsch, 2020). The oxygen supply capacity is not significantly correlated with P_crit (Pearson's r=−0.10, P=1) but it is elevated, relative to oxygen demand, in the few species included here that inhabit persistently hypoxic environments, such as pronounced oxygen minimum zones (OMZs; Tables S1 and S2).

In our dataset, α₀ increased in response to declining P_O2 in all cases and peaked or reached a plateau in all but two cases (see Table S1). As P_O2 continued to decline below P_crit-SMR, α₀ either was maintained at capacity (α₀=α), with MR conforming directly to available P_O2 for the remainder of the trial (line 1; Fig. 1), or declined, indicating metabolic suppression or physiological failure below P_crit-SMR (line 2; Fig. 1). In all cases, the mean of the highest three resulting α₀ values was designated as the oxygen supply capacity, α (Table S1), which was used to generate the α-line (see Box 1). In two cases (see Table S1), a continuous increase in α₀, with the highest value occurring at the lowest P_O2, provided no clear peak or plateau, which likely indicates P_O2 measurement error. For these two cases, the highest α₀ was substantially higher than the mean of the highest three values. Unlike when using traditional P_crit methods, these potential P_O2 measurement errors are identifiable using the α-method. The highest α₀ for the remaining species was only slightly higher than, and directly correlated with, the mean of the three highest values (y=1.1x−0.03; R²=0.99, n=40).

The oxygen supply capacity, α, was insensitive to differences in MR that resulted from differences in respirometry methods, such as closed versus intermittent flow respirometry, trial duration or organismal stress (Fig. 3). However, within a species, higher temperature typically results in higher oxygen supply capacity (see Chitala ornata and Lates calcarifer; Table S2). This occurs because α supports maximum oxygen demand, which is elevated by temperature. Maximum or active MRs have been measured during declining P_O2 for several species. Below P_crit-max, the decline in MMR with P_O2 was described well by the α-line (MMR=P_O2×α) and MMR and P_crit-SMR trials provided similar α values (Fig. 5).

To assess intraspecific variability and to ensure that our analysis was not biased by the use of idealized representative curves from the literature, we also analyzed several existing respirometry datasets (Table S2): Euphausia mucronata, a midwater krill from the Eastern Tropical Pacific OMZ (C.J.W. and B.A.S., unpublished), Farfantepenaeus duorarum, an estuarine pink shrimp (A.L.B. unpublished), the Atlantic spiny dogfish, Squalus acanthias (A.A., unpublished), and two oceanic squids, Illex illecebrosus and Sthenoteuthis oualaniensis (M.A.B., unpublished). Among these species, α varied from 3.35±2.64 in E. mucronata to 0.41±0.13 in S. acanthias. The α₀ response types (described above) varied among individuals of each species, but typically displayed a clear peak or plateau, facilitating unambiguous identification of α. The α values were similar for each species whether derived using the α-method or from P_crit and SMR determined using the LLO method (Reemeyer and Rees, 2019).

P_crit, regardless of how it is determined, is a rate-specific measure of α and it is α, rather than P_crit per se, that provides relevant physiological information. Because α is not specific to a particular MR, P_crit need not be measured at SMR or any other specific MR. It is also not necessary for an individual animal to maintain a consistent MR throughout a trial or for a MR versus P_O2 curve to have a clear breakpoint. The oxygen supply capacity, whether directly determined or extracted from a P_crit value, defines the α-line that describes the oxygen limit for every MR from SMR to MMR. This definition of P_crit is consistent with current physiological theory and the α-method provides a simple, repeatable and precise alternative to existing methods of determining P_crit. Moreover, this definition, with its emphasis on oxygen supply capacity, clarifies the physiological and evolutionary significance of P_crit.

P_crit is a measure of α, from which the maximum metabolic rate achievable at any P_O2 can be determined. Seibel and Deutsch (2020) showed that, for most shallow, coastal species, α is nearly identical whether determined as SMR/P_crit-SMR or MMR/21 kPa. Interestingly, when using the α-method, α is often slightly (mean 4.4%) higher than that determined as SMR/P_crit-SMR (Table S1). A higher α predicts a higher MMR, suggesting that some previous MMR measurements may have been slight underestimates. MMR trials typically start at P_O2 near air saturation and allow oxygen to decline by as much as 5–10%. In fact, Clark et al. (2013) specifically recommended maintaining oxygen saturation above 80%. However, if, as we suggest, MMR declines linearly with P_O2 below P_crit-max, MMR values will be underestimated in proportion to the measurement P_O2. Thus, the reported match between α values determined in active and resting states may reflect off-setting discrepancies in both MMR and P_crit-SMR determination. We suggest that future studies should maintain P_O2 at or above air saturation during MMR measurement.

Selection for elevated metabolic capacity (i.e. athleticism), increased aerobic scope or metabolic performance in persistent hypoxia may elevate the oxygen supply capacity (Fig. 6A,B). In order to distinguish between these selective pressures, P_crit-max (the P_O2 above which no further increase in metabolism is possible) must be determined (Fig. 6). A P_crit-max less than air saturation suggests adaptation to persistent hypoxia. Most terrestrial or shallow-living aquatic species have a P_crit-max near atmospheric P_O2 (21 kPa; Seibel and Deutsch, 2020). For such species, additional environmental oxygen (hyperoxia) should not elevate metabolism. Among these ‘normoxic’ species, most of the variation in P_crit reflects the differing temperature sensitivities of MMR and SMR and, thus, the FAS (Fig. 6C).

Inserting MMR and SMR into Eqn 1 and rearranging shows that FAS×P_crit-SMR=P_crit-max, meaning that, for species with similar P_crit-max, P_crit-SMR is inversely correlated with FAS (Fig. 6C; Seibel and Deutsch, 2020). FAS explained 95% of the variation in P_crit-SMR among 39 taxonomically diverse, normoxic species analyzed by Seibel and Deutsch (2020). No variation could be attributed to measured differences in environmental P_O2. Thus, P_crit-max, rather than P_crit-SMR, provides a useful measure of hypoxia tolerance, while hypoxic effects, as noted by Ern et al. (2016), are evident at any P_O2 less than P_crit-max (air saturation for most coastal species). Below P_crit-max, MMR and aerobic scope are diminished in all species and the decrement is quantifiable using the α-line. The precise decrement in aerobic scope that results in reduced fitness is unknown, but recent work suggests that a FAS of ∼3 defines biogeographical limits for many species (Deutsch et al., 2020) and may provide the hypoxic benchmark above which species are ‘indefinitely viable’ as prescribed by Lindroth (1942). Although a lower P_crit-SMR allows SMR to be sustained to that lower P_O2, it is rare for most species to experience hypoxia less than about 3 times P_crit-SMR (Deutsch et al., 2020). For those that do, hypoxia is typically intermittent or cyclic and selects for the ability to temporarily suppress metabolism rather than, or in addition to, enhanced oxygen supply capacity (Seibel, 2011; Mandic et al., 2009; McArley et al., 2019; Storey, 2015).

In contrast, adaptation to persistent hypoxia (longer than a diel or tidal cycle) results in a relatively high α for a given MR, which improves active (not resting) performance in low oxygen (Seibel and Deutsch, 2020). As a result, P_crit is reduced at all metabolic levels and both P_crit and P_crit-max are low relative to those of similar species adapted to higher oxygen environments (Childress and Seibel, 1998; Seibel, 2011; Wishner et al., 2018; Mandic et al., 2009; Richards, 2011; Regan et al., 2019). For example, for Gnathophausia ingens, a deep-sea lophogastrid crustacean living permanently in the OMZ, α is ∼1.39 µmol g⁻¹ h⁻¹ kPa⁻¹, which is among the highest values measured here and its P_crit-max is extremely low (∼2.5 kPa; Fig. 5; Childress, 1968). Gnathophausiaingens lives persistently below 600 m in the California Current OMZ and rarely, if ever, experiences oxygen levels higher than its P_crit-max.

To understand whether variation in P_crit and α reflects adaptation for metabolic capacity, aerobic scope or hypoxia, P_crit-max must also be known (Fig. 6A). In the absence of such data, one can tentatively infer the adaptive value of α if it is assumed that FAS for the species of interest falls within the typical range (∼3 to 6; Seibel and Deutsch, 2020; Killen et al., 2016; Peterson et al., 1989). If true, P_crit-max will be 3–6 times P_crit-SMR, which can be calculated as SMR/α. An estimated P_crit-max value substantially less than 21 kPa suggests that oxygen supply mechanisms may be adapted for hypoxia. For G. ingens, mentioned above, P_crit-max, estimated as 6×SMR/α, is only 2.42 kPa (Table S1), which is very near its independently measured P_crit-max (Fig. 5). In contrast, an assumed P_crit-max of 21 kPa would provide an unlikely FAS of 57. Thus, G. ingens high oxygen supply capacity facilitates high FAS (∼6) in extreme hypoxia (<2.5 kPa) (Childress and Seibel, 1998). Higher oxygen values (>2.5 kPa) will not enable higher MR or aerobic scope in G. ingens. The difference between P_crit-max and P_crit-SMR is less than 2 kPa for G. ingens, consistent with the very low and narrow oxygen range experienced in its deep-sea habitat (Childress and Seibel, 1998). Similarly, the goldfish maintains high FAS across a wide temperature range and in hypoxia by elevating α and reducing both P_crit and P_crit-max (Fig. 6C).

The α-method results in estimates of α that are similar to those determined from P_crit using other methods. This is especially true for the LLO (SMR extension) method (Reemeyer and Rees, 2019) because the variation in the MR for which P_crit is being determined is reduced or eliminated. The LLO method is further improved if the oxygen-limited portion of the curve is forced through the origin (McArley et al., 2019). If α is determined from a P_crit that is measured by the intersection of curves above and below P_crit-SMR, large errors may occur if the low-P_O2 portion of the curve does not extend through the origin (Fig. 2). Below P_crit-SMR, oxygen transport or oxidative metabolism may be failing or shutting down. Thus, the low-P_O2 portion of the curve cannot be informative of performance in a non-lethal oxygen range. For ∼30% of the trials in Table S1, MR is not directly proportional (does not conform) to P_O2 below P_crit-SMR as evidenced by a decline in α₀ following a peak at α. Using the α-method, MR measurements below P_crit-SMR are not diagnostic and, thus, are not relevant. However, P_crit is often defined as the transition between oxyconformation (below P_crit) and oxyregulation (above P_crit) (e.g. Rogers et al., 2016). The definitions of P_crit and α in the present study require reassessment of the concepts of oxygen consumption regulation and conformation (Gnaiger, 1993; Pörtner and Grieshaber, 1993).

Several methods have been developed to quantify the degree of regulation, loosely defined as the ability to maintain a constant MR across a range of oxygen levels, from the respiratory response of organisms over the complete range of measured oxygen pressures (e.g. regulation index: Mueller and Seymour, 2011; Tang, 1933; non-linear regression: Marshall et al., 2013; and best-fit approaches: Cobbs and Alexander, 2018; Muggeo, 2003). Proponents of these methods argue that a great deal of information is lost by distilling a respirometry trial down to a single critical P_O2 (Marshall et al., 2013; Wood, 2018). However, the oxygen supply capacity, which we argue is the important information provided by P_crit, is lost in mathematical descriptions of the entire trial and it is unclear what information is gained from such analyses. Unless the relationship between MR and P_O2 above the α-line is causal, there is no particular reason to describe it at all. At P_O2 above the α-line, MR may correlate with a number of covariables, including time in captivity, time since feeding, accumulation of metabolic waste, diel cycles, stress and activity. Some of these variables may be controlled for during experiments, but even so, as Ultsch and Regan (2019) point out, it is impossible to know whether MR above P_crit is supporting the same maintenance processes that comprise a MR at its P_crit (Ultsch and Regan, 2019). For example, vision is energetically expensive and it may be diminished at oxygen values between P_crit-max and P_crit-SMR (McCormick et al., 2019). Thus, apparent oxyregulation is not necessarily oxygen independence and a lack of apparent regulation (i.e. conformation) is not necessarily oxygen dependence. Rather, what is typically referred to as regulation are the physiological adjustments that provide additional oxygen to meet the concurrent metabolic demands as environmental P_O2 declines, regardless of the constancy or level of those demands. Only at maximum exertion for a given P_O2 is oxygen supply operating at capacity and, as a result, MMR conforms linearly to environmental oxygen availability between P_crit-SMR and P_crit-max (Fig. 5).

True conformation (a continuous decline in MR in direct proportion to P_O2 throughout a respirometry trial), in contrast, implies that an individual is operating continuously at its maximum oxygen supply capacity or that it completely lacks aerobic scope (i.e. MMR equals SMR regardless of oxygen availability). For example, Nautilus pompilius was described as an oxyconformer (Boutilier et al., 1996), but Fig. 3C shows that α₀ increases as P_O2 declines, suggesting some level of active regulation. Staples et al. (2000) later showed that N. pompilius does in fact regulate and that the previously reported elevated rate and oxyconformation may have been due to surgery-induced stress. Regardless, the α we determined from the two N. pompilius studies is similar (Fig. 3C). True conformation is seemingly rare (Ultsch and Regan, 2019), even among animals that lack complex circulatory systems (Rutherford and Thuesen, 2005).

An additional benefit of measuring α is that it can be measured at MRs higher than SMR and at oxygen pressures well above lethal limits. Measuring P_crit-SMR using any traditional method requires exposing animals to potentially lethal oxygen levels below P_crit-SMR. If measured in an active state, as recommended here, an organism can simply lower oxygen demand by reducing activity as oxygen becomes limiting. If desired, the SMR can be measured independently at high P_O2 to determine aerobic scope or to calculate P_crit-SMR. Thus, the α-method, applied to MRs above SMR, as recommended here, will alleviate the effects of measurement on animal welfare.

Thus, direct determination of α has several advantages over P_crit and non-linear descriptions of respirometry data. (1) The α-method provides a direct and unambiguous measure of oxygen supply capacity. Using the α-method, any two independent researchers will arrive at the same α value from the same dataset. The ambiguity in other P_crit methods arises because they are influenced by which MR measurement periods are included in the oxyconforming regression (below P_crit) and how the MR itself is calculated. (2) The value of α obtained using the α-method is more accurate than other P_crit methods as evidenced by the ability to predict limiting oxygen levels for any MR or to predict the maximum achievable MR at any P_O2. The accuracy derives from the fact that the α-line runs through the origin and that α is consistently defined as the highest α₀ value (or mean of highest 3). (3) The α-method does not depend on respirometry methodology to the extent that other methods do, which means that any two researchers are likely to arrive at similar values for the same species even if using very different respirometry methods.

P_crit is a rate-specific measure of the oxygen supply capacity (α), rather than hypoxia tolerance. Variation in P_crit-SMR reflects variation in FAS, while P_crit-max indicates the P_O2 below which that aerobic scope becomes oxygen limited. We suggest that α is the important physiological information provided by traditional P_crit measurements and that this information is lost in non-linear analyses. We describe a method to directly determine α, which has several advantages over traditional P_crit determination, including (1) less ambiguity and greater accuracy, (2) fewer constraints on respirometry methods and the analysis of respirometry data, and (3) greater predictive power and ecological and physiological insight. The oxygen supply capacity enables testing of previously obscured hypotheses regarding aerobic scope and its response to environmental change.

We thank E. V. Thuesen for helpful suggestions on an early draft and anonymous reviewers for their constructive comments.