The recent paper in Journal of Experimental Biology by Seibel et al. (2021) claims that a new term, the oxygen supply capacity (α), breathes new life into critical oxygen partial pressure (Pcrit). Despite the authors' good intentions, their idea does not resuscitate the 88 year old concept of Pcrit (Tang, 1933). It would be wonderful if the α-line were prescriptive and biologically meaningful across species and environmental conditions, but we suspect that it is not. We argue for what the data have told us for years: the relationship between any metabolic rate (MR) and PO2 is not adequately described by straight lines.
We raise several problems with their overly simple linear model: maximum metabolic rate (MMR)=PO2×α. This linear line is anchored at the origin, must pass through any critical PO2 (Pcrit), has a single α value and passes through MMR.
While the authors claim the α-line can ‘predict the maximum achievable MR at any PO2’, this claim is misleading for three reasons. Foremost, the indefinitely long line cannot predict the maximum aerobic MR (i.e. 
O2,max) of the animal without knowing its metabolic scope. Second, the line is based on only one experimental data point (Pcrit) that has associated error. If Pcrit is based on standard metabolic rate (SMR) data, the point lies near the origin and therefore has a large leverage effect on the slope of the line. Small absolute errors in its position can create large absolute errors in MMR, in proportion to the animal's metabolic scope. Finally, real data for several species raise doubts that the line is straight.
Fry and Hart (1948) had a clear conceptual framework for investigating hypoxia tolerance in fish: ‘… the simplest and most direct methods of obtaining ecologically significant values of oxygen uptake are to measure the maximum rate over a series of varying oxygen tensions down to the asphyxia level, and to measure the standard rate over a range of tensions at which oxygen would not be the limiting factor even for the maximum rate’. This empirical approach is what Fred Fry envisaged for his ‘Limiting Oxygen Concentration’ (LOC) curve (Fry, 1947). In fact, reliable empirical data for MMR versus PO2 with groups of fish exist (Claireaux and Lagardère, 1999; Claireaux et al., 2000) and these LOC curves, based on best-fit statistical relationships, were non-linear. Still needed are empirical data using single fish.
Unfortunately, the ‘oxygen supply’ concept (α0) is not new, other than its name. Its units are identical to those used for ‘conductance’, a term defined and long-used by respiratory physiologists (Piiper et al., 1971) to represent the effectiveness of gas transport in each step down the oxygen cascade. ‘Oxygen supply capacity’ (α) would be the maximum conductance of the entire system from the environment to the mitochondrion, but its use places all levels of oxygen cascade into a black box, where we lose sight of what determines conductance at each level. Maximum conductance is influenced by individual conductances at all levels of the oxygen cascade, from ventilation of the gas exchange surface, to diffusion through the surface, oxygen binding to respiratory proteins and dissolved in plasma, convection by the cardiovascular system and diffusion through tissues to mitochondria where oxygen is finally consumed. Fundamentally, most of these conductances are non-linear with respect to PO2. Oxygen binding in blood, for example, is sigmoidal or hyperbolic, and mitochondrial kinetics are hyperbolic. Furthermore, the tissues supported at SMR (oxygen-sensitive internal organs) are different from those supported at MMR (aerobic red muscle), so the ‘anatomy’ of the oxygen cascade changes between rest and activity. Thus, we are not convinced that the mechanisms in the oxygen cascade remain identical at Pcrit-SMR and Pcrit-MMR, an apparent requirement of the theory.
Sadly, Seibel et al. (2021) do not address the very reasonable criticisms of Pcrit theory and methods (e.g. Wood, 2018; Cobbs and Alexander, 2018; Marshall et al., 2013). Also, they discourage examining data that depart from their model: ‘Using the α-method, MR measurements below Pcrit-SMR are not diagnostic and, thus, are not relevant’. They dismiss attempts of other approaches to understand responses as ‘purely descriptive’ and imply that approaches to understand the entire MR versus PO2 response curve ‘may not have any functional relationship to available oxygen’. In particular, the regulation index (RI), a metric that evaluates the empirical ability of animals to regulate MR in relation to PO2 (Mueller and Seymour, 2011), is dismissed as being devoid of underlying physiological mechanisms. Having discarded a need to consider the entire responses, they fail to recognize that their measure of Pcrit-SMR may not actually relate to SMR. For example, some species show a progressive MR decrease at high PO2 (e.g. Fig. 2B of Seibel et al., 2021, showing data from Mueller and Seymour, 2011), so the MR at Pcrit-SMR would appear lower than SMR. The whole response could represent metabolic suppression in response to hypoxia. In fact, well laid out and rigorously tested methodological and analytical procedures demonstrate the real possibility of metabolic suppression and show that Pcrit-SMR can be interpolated from empirical data for MR versus PO2 (Chabot et al., 2016, 2021; Claireaux and Chabot, 2016) rather than by selecting three points with the highest α values. Absent from Seibel et al. (2021) is the crucial validation that estimated Pcrit values do indeed match those measured using reliable SMR determinations.
In conclusion, we feel Seibel et al. (2021) presented a theory trying to enforce itself onto real data. They cite 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’) to justify their theory, but the equation of Seibel et al. (2021) lacks empirical realism. Indeed, forcing an equation relating MMR and PO2 through the origin does not represent empirical data (Claireaux and Lagardère, 1999; Claireaux et al., 2000) in a biologically meaningful manner. Still, you can always draw a straight line through two data points, in this case a biologically meaningless intercept at the origin and a potentially subjective Pcrit-SMR. Besides, MMR cannot be derived from such a line without prior knowledge of metabolic scope in normoxia.
