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# Metabolic trait diversity shapes marine biogeography

## Abstract

Climate and physiology shape biogeography, yet the range limits of species can rarely be ascribed to the quantitative traits of organisms1,2,3. Here we evaluate whether the geographical range boundaries of species coincide with ecophysiological limits to acquisition of aerobic energy4 for a global cross-section of the biodiversity of marine animals. We observe a tight correlation between the metabolic rate and the efficacy of oxygen supply, and between the temperature sensitivities of these traits, which suggests that marine animals are under strong selection for the tolerance of low O2 (hypoxia)5. The breadth of the resulting physiological tolerances of marine animals predicts a variety of geographical niches—from the tropics to high latitudes and from shallow to deep water—which better align with species distributions than do models based on either temperature or oxygen alone. For all studied species, thermal and hypoxic limits are substantially reduced by the energetic demands of ecological activity, a trait that varies similarly among marine and terrestrial taxa. Active temperature-dependent hypoxia thus links the biogeography of diverse marine species to fundamental energetic requirements that are shared across the animal kingdom.

## Main

The provisioning of energy to organisms in their natural environment is a key determinant of fitness. The energetic demands of ectothermic organisms increase with temperature and activity, and must be met by an adequate supply of oxygen (O2) and food. At a minimum, physiological survival requires that the supply of energy matches the maintenance costs of an organism in a resting state; these energy demands vary by body size, temperature and species6,7. Additional energetic costs are incurred by the growth and activity required for ecological survival, which depend on lifestyle and ecological niche and typically increase energy expenditure several-fold above resting rates8,9.

Energy provision can be limited by O2 if its availability falls short of the metabolic demands of the organism, inducing a hypoxic condition10,11,12. This is more common in aquatic environments due to the slower diffusion of O2 in water than in air13. The effects of an acute reduction in O2 on population fitness can induce considerable die-offs14,15; however, the presence of metabolic barriers in habitats under stable conditions are difficult to observe. Recent analyses suggest that the current latitude and depth limits of several marine ectothermic species coincide with an O2 pressure that is just adequate to fuel the energy demand for physiological maintenance and sustained ecological activity4,16,17. Here we evaluate the metabolic causes and biogeographical consequences of the constraints to aerobic energy by combining a mathematical model of temperature-dependent hypoxia with laboratory and field data from the broadest-available diversity of marine animal species.

## Temperature-dependent O2 tolerance

The aerobic energy balance of an organism can be represented by a Metabolic Index4 (Φ), which is defined as the ratio of O2 supply to resting demand (Fig. 1a and Methods):

$$\Phi =\frac{{\alpha }_{{\rm{S}}}}{{\alpha }_{{\rm{D}}}}{B}^{\varepsilon }{p}_{{{\rm{O}}}_{2}}\exp \,\left\{\frac{{E}_{{\rm{o}}}}{{k}_{{\rm{B}}}}\left[\frac{1}{T}-\frac{1}{{T}_{{\rm{ref}}}}\right]\right\}$$
(1)

where αD is the resting metabolic rate per unit body mass (B) at a reference temperature (Tref), and αS is the efficacy of O2 supply per unit body mass and the O2 pressure $$({p}_{{{\rm{O}}}_{2}})$$ of the ambient medium (units are described in Fig. 1 and the Methods). The ratio of αD/αS defines a first key physiological trait of an organism: its resting vulnerability to hypoxia at the reference temperature, Vh = αD/αS, which is measurable as the lowest O2 pressure (Pcrit) that can sustain resting metabolic demand (Φ = 1) (Fig. 1a). The inverse of hypoxia vulnerability is hypoxia tolerance, which is denoted Ao = 1/Vh, as defined previously4. A second key trait, Eo, is the sensitivity of hypoxia vulnerability to temperature (T), which is described by the exponential Arrhenius function (Fig. 1a) (Boltzmann constant, kB) and is equal to the difference between the temperature variation in the metaboli4c rate (Ed) and the O2 supply (Es), such that Eo = Ed − Es (Methods). The exponent ε is the allometric scaling of the supply-to-demand ratio, which is typically near zero18.

A third component of the energetic balance of an organism is the O2 needed to fuel growth and essential ecological activities. In terrestrial animals, sustained metabolic rates range from 1.5 to 7 times those at rest, a ratio termed the sustained metabolic scope (SMS)8,9. For aquatic aerobic organisms, such levels of activity would increase the resting vulnerability to hypoxia from Vh to a higher value, Vh × SMS, which requires that the minimum Φ of a given species in its environment increases above its resting minimum (which is set to 1, see above) by the same factor, denoted Φcrit. The ratio SMS will depend on the ecology and life history of each species. This ecological trait is not directly measured in marine species, but it can be estimated from the maximum metabolic rates, while Φcrit can be inferred as the lowest value of Φ that bounds the geographical distribution of a species4. If values of Φcrit match those of SMS, this strongly indicates that there is an energetic limit on marine species habitats.

To characterize the variation in these traits across diverse marine animal species, we analysed published physiological rates and thresholds (Methods), and the global geographical distributions of the species (OBIS; https://obis.org/). The dataset of 199 species includes 145 species with temperature-dependent metabolic rates and associated parameters (Ed and αD; hereafter called ‘metabolic traits’) (Extended Data Fig. 1a) and 72 species with temperature-dependent hypoxia thresholds and corresponding parameters (Eo and Vh; hereafter called ‘hypoxic traits’) (Extended Data Fig. 1b and Supplementary Table 1). The species span more than eight orders of magnitude in body size, inhabit all ocean basins and biomes (Extended Data Fig. 1c), belong to five phyla (Annelida, Arthropoda, Chordata, Cnidaria and Mollusca) and broadly but incompletely sample the metabolic, geographical and taxonomic diversity of the ocean.

## Physiological trait diversity

Resting metabolic rates normalized by temperature and body mass vary by orders of magnitude among all 145 species (Fig. 1b), but remain within the range found across the tree of life19. The critical O2 pressures also show a well-defined distribution of resting hypoxia vulnerability across species (Fig. 1b). Although metabolic rates are a direct driver of hypoxia vulnerability, the two traits are uncorrelated across species, and αD exhibits greater interspecies variation than does Vh. These observations suggest that animals with a high metabolism also have a high efficacy of O2 delivery. Indeed, the absolute metabolic rates and coefficients of O2 supply are highly correlated among species (Fig. 1c and Extended Data Table 1), which indicates that there is a strong selective pressure for tolerance to low O2 , even for species that live outside relatively small ocean regions commonly termed hypoxic zones20.

The temperature sensitivity of metabolic rates within species exhibits substantial variation across species (Fig. 1d). The mean value, standard deviation and range of Ed (0.69 ± 0.36 eV, 0.1–2.0 eV) are similar to the thermal acceleration of the metabolic rates of organisms that are observed across the tree of life21. The temperature sensitivity of hypoxia vulnerability also varies widely across species (Fig. 1d), but Eo has a smaller mean, standard deviation and range (0.4 ± 0.28, −0.2–1.3 eV) that includes negative values. The differences in Eo relative to Ed reflect the effect of temperature on O2 supply (ES), the positive mean value (0.29 ± 0.23 eV) of which suggests that the supply of O2 also accelerates with temperature22. The temperature effect on the supply of O2 therefore counteracts, and for species with Eo < 0, even exceeds the thermal increase in metabolic rates.

To confirm the role of the O2 supply in moderating the temperature sensitivity of the vulnerability to hypoxia, we estimated the thermal response of three processes that transport O2 from ambient fluid to body tissue: the ventilation of water past the organism, the diffusion of O2 through the boundary layer at the water–body interface and the internal transport of O2 by animals that have circulatory systems. Diffusive O2 fluxes increase with temperature in proportion to gas diffusivity (κ) and increase inversely to the decrease in kinematic viscosity (υ). The ratio of gas diffusivity to kinematic viscosity—the Schmidt number (Sc = υ/κ)—predicts a diffusive O2 flux23 for which the temperature dependence, Es, lies between 0.21 and 0.42 eV (Extended Data Fig. 3a). This range encompasses the mean value of Es that was inferred from all species for which Eo and Ed can both be estimated (Fig. 1e), but cannot account for its full interspecies range.

The ventilation of O2 to and circulation in the body may also modify the temperature sensitivity of hypoxia tolerance24,25,26,27 (Extended Data Fig. 3b). Both ventilation and circulation rates increase with temperature in cooler waters (Es = 0.55 ± 0.15 eV (mean ± s.d.)) (Fig. 1e and Extended Data Fig. 3c), but the response decreases or even reverses in warmer conditions (Es = 0.04 ± 0.18 eV (mean ± s.d.)) (Fig. 1e and Extended Data Fig. 3c). These thermal responses of the O2 supply combined with those of metabolic demand (Ed) can account for nearly the entire range of the temperature-dependence of hypoxia vulnerability (Eo). Moreover, the stronger thermal response of the ventilation and circulation rates in cool compared with warm waters is consistent with the weaker temperature sensitivity of species vulnerability to hypoxia (lower Eo) that is observed under cold relative to warm conditions (Extended Data Fig. 3d). Thus, both biological and physical responses of the O2 supply to temperature reduce the temperature sensitivity of hypoxia vulnerability, relative to that of the metabolic demand alone. The compensation of faster metabolic rates at higher temperatures by a more rapid O2 supply indicates that there is a strong selective pressure for oxygen supply to meet demand across the range of inhabited temperatures.

The variation in temperature-dependent hypoxia traits suggests that species experience distinct geographical patterns of hypoxia risk (Fig. 2). In the upper ocean, both temperature and $${p}_{{{\rm{O}}}_{2}}$$ decrease with depth, but often have opposing gradients with latitude; temperature decreases as subsurface $${p}_{{{\rm{O}}}_{2}}$$ increases away from the Equator. The resulting spatial variation in Φ depends on the strength of these gradients, and on the temperature sensitivity parameter, Eo. For species with strongly positive values of Eo, Φ decreases towards the warm low-O2 waters of the shallow tropics (Fig. 2a). However, positive Eo also weakens any vertical decrease in Φ, because the decline in ambient O2 is compensated by a slower metabolic rate, which extends the potential habitat of species into deeper waters. By contrast, for species with Eo < 0, the highest Φ is found in tropical waters, but declines rapidly with depth below the surface due to both lower O2 levels and cooler temperatures (Fig. 2c). The diversity in temperature-dependent hypoxia traits suggests that species that are limited by low Φ conditions may occupy distinct ocean habitats with global coverage, from shallow tropical waters to high-latitude and deep water, with a continuum of patterns in between (Fig. 2b).

To test whether the range of the predicted geographical habitat niches corresponds to the actual distributions of marine species, we extracted global occurrence data14 for all species in the physiological database. For the 72 species with Metabolic Index parameters (Eo, Vh), distribution data were available for most species (n = 68), and the sampling resolution of many species was sufficient to reveal clear range boundaries in depth and latitude (Fig. 2 and Extended Data Figs. 4, 5). These data include three species that have similar Vh but span the full range of Eo, from strongly positive (Eo = 0.9, northern shrimp) to slightly negative (Eo = −0.2, sea squirt) and an intermediate value (Eo = 0.2, small-spotted catshark), which predict the distinct aerobic habitat distributions of these species (Fig. 2). In all three species, range boundaries in latitude and in depth are closely aligned with a single value of Φ above which the populations are widely distributed and below which reported occurrences are rare and isolated. Geographical range boundaries across a range of depths, latitudes and longitudes also coincided with single isopleths of Φ in other well-mapped species (Extended Data Fig. 4), including species that span multiple ocean basins or different sides of the same basin (Extended Data Fig. 5).

The boundaries of the geographical ranges of species are more strongly aligned with the Metabolic Index than with either temperature or $${p}_{{{\rm{O}}}_{2}}$$ alone (Fig. 2 and Extended Data Figs. 47). This can be observed geographically: in vertical cross-sections, range boundaries follow a constant Φ value, but tend to cross multiple isotherms (Fig. 2 and Extended Data Figs. 4, 6). In mid-latitude species, range boundaries lean equatorward at shallower depths, opposite to the poleward tilt of isotherms (Fig. 2a, b and Extended Data Fig. 4a–e). At the surface, range limits can be reached as Φ declines towards the Equator, even without a gradient in $${p}_{{{\rm{O}}}_{2}}$$ (Extended Data Figs. 4a–e, 6a, c).

The alignment of range boundaries with Φ is most easily observed, however, by projecting the biogeography of species onto the temperature and $${p}_{{{\rm{O}}}_{2}}$$ state–space that they inhabit (Fig. 3). Across species from distinct phyla and multiple ocean basins, including those with sparse spatial sampling, the state-space habitat map reveals strong correlations between the temperatures and $${p}_{{{\rm{O}}}_{2}}$$ levels that bound the occurrences of species. These relationships are consistent with the expectations based on the Metabolic Index of each species, with opposite slopes for species with positive and negative Eo values, but are incongruent with habitat limitation by either a single temperature or $${p}_{{{\rm{O}}}_{2}}$$ level. The predictive ability of Φ to discriminate between inhabited and uninhabited ocean regions is better than that of temperature for 92% of species, better than $${p}_{{{\rm{O}}}_{2}}$$ for 67% of species, and better than both temperature and $${p}_{{{\rm{O}}}_{2}}$$ for 62% of species (Methods and Extended Data Fig. 7).

That the species habitat boundaries coincide with a lower Φ value suggests that an aerobic barrier limits the geographical ranges of marine animals (Figs. 2, 3). We determined the range-bounding value, Φcrit, for all of the species with hypoxia traits and georeferenced location data, using two independent methods that yield convergent results (Methods and Extended Data Fig. 8a, b). The average of Φcrit is approximately 3.3 (interdecile range, 1.3–6.5) (Fig. 4). For all species, waters with lower Φ values exist within their inhabited depth range, but lack confirmed sightings (Extended Data Fig. 8c).

If Φcrit is the operative habitat barrier for marine species, its values should correspond to their sustained metabolic rates relative to rest. Long-term energetic demand is not directly measured for marine organisms, but short-term experimental estimates of maximum-to-resting rate ratios (MMR/RMR) provide an empirical upper bound on SMS (Methods). We find a strong correlation between biogeographically inferred Φcrit and laboratory measured MMR/RMR values (Extended Data Fig. 9 and Extended Data Table 1), which suggests that SMS lies approximately midway between the resting and maximum rates (that is, SMS = wR + (1 − wR)(MMR/RMR); (see Methods, equation 7); wR = 0.4 ± 0.17 (mean ± s.d.), n = 14) (Extended Data Fig. 9), consistent with independent estimates of SMS from carbon isotopes in the otoliths of Atlantic cod28. Applied to the broadest compilation of MMR/RMR ratios, this scaling yields an interspecies distribution of SMS (n = 106) (Fig. 4a) that is statistically indistinguishable from that of Φcrit (Fig. 4a, Extended Data Table 1). The Φcrit values of the few sessile species that we analysed (Styela plicata, Lophelia pertusa and Crassostrea gigas) were among the lowest (Fig. 3c and Supplementary Table 1), which is consistent with their less-active lifestyles. Together, these observations provide strong evidence that Φcrit corresponds to SMS, and thus represents an energetic barrier to the geographical ranges of species.

The interpretation of Φcrit as the ratio of sustained active-to-resting metabolic rates can be further evaluated by comparing its frequency distribution across marine species to the SMS data that were directly and independently measured for terrestrial taxa4,8, including mammals, birds and reptiles (Fig. 4b). The available data reveal no significant differences between the distribution of marine Φcrit and marine and terrestrial SMS distributions (Fig. 4, Extended Data Table 1), which supports the suggestion that Φcrit is an operative limit on the geographical ranges of marine species. These results also suggest that the ratios of active-to-resting metabolic rates are a fundamental trait that represents ecological and life-history variation across the animal kingdom.

The SMS of marine taxa has important implications for empirical metrics of thermal tolerance that are widely used to infer the climate sensitivity of marine species. By elevating O2 demand, ecological and life-history activity increases the vulnerability to hypoxia from a resting threshold (Vh), to an active one, Vh × Φcrit, that is a key operative constraint on marine geographical ranges. Similarly, because hypoxia tolerance decreases with temperature for most species, ecological activity also reduces the maximum temperature at which aerobic metabolism can be sustained. Maximum temperatures for aerobic metabolism can be derived from the Metabolic Index (Fig. 1a) as the temperature at which Pcrit reaches the atmospheric O2 pressure (Patm) applied in experimental determinations of thermal tolerance29 (Methods). The distribution of this aerobic thermal limit, denoted ATmax, evaluated in a resting state (Φ = 1, $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{rest}}}$$) is highly variable among species (Fig. 5a), owing to the diversity of hypoxia traits (Eo and Vh). For all species, $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{rest}}}$$ is considerably higher than the temperatures that are encountered by the organisms in their natural habitats30,31, and for most species it is higher than temperatures found in the ocean (Fig. 5a). Similar findings have been reported based on observed critical thermal maxima, termed CTmax, measured by the loss of physiological performance in a resting state32. Indeed, the frequency distributions of $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{rest}}}$$ and CTmax are remarkably similar (Fig. 5a and Extended Data Fig. 4a–d). In four of the seven species for which both thermal tolerance metrics are known, $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{rest}}}$$ occurs at a temperature at or below CTmax (Extended Data Fig. 10). This correspondence may reflect an aerobic basis for thermal tolerance29, although the link remains controversial25,26,27,28,29,30,31,32,33,34,35,36. Whatever the underlying physiological basis for this similarity, both measures suggest that although there is a large ‘thermal safety margin’ in the face of climate warming37,38, these are derived from, and applicable to, only a state of rest.

Under the ecologically relevant energetic demand (Φ = Φcrit), the active aerobic thermal maximum, $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{act}}}$$, falls well below $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{rest}}}$$ (Fig. 5b). Indeed, calculated values of $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{act}}}$$ closely correspond to the maximum occupied environmental temperatures of individual species (Extended Data Fig. 10). Across species, the distribution of $${{\rm{AT}}}_{{\rm{\max }}}^{{\rm{act}}}$$ tracks the global volumetric frequency of ocean temperatures. Thus, species with substantial apparent thermal safety margins at rest are in fact likely to be at the limit of their active thermal tolerance in the ocean39 and will experience habitat compression even at modest levels of warming and without any depletion of O2.

## Implications

The energetic balance of organisms is a powerful framework for explaining biogeographical patterns from temperature-dependent hypoxic tolerances and constituent metabolic rates that have been well studied for decades4,5,6. Geographical range limits imposed by aerobic energy constraints apply to a greater diversity of ocean species, physiologies and habitats than previously investigated4,16, from tropical to high-latitude waters and from shallow to deep ocean niches. Our results thus extend and strengthen the hypothesis that temperature-dependent hypoxia has a major role in biogeography, by mediating how ocean temperature and O2 are experienced by organisms with diverse environmental tolerances and geographical niches. The global applicability of such constraints support their use to predict patterns of extinction caused by climate change in the geological record40 and in the future.

Sustained activity levels and the metabolic traits—the resting metabolic rate and its temperature sensitivity—that underlie aerobic energy barriers are not substantially different from the values observed in terrestrial biota. However, the hypoxia traits that shape those energetic barriers—resting hypoxia vulnerability and its temperature sensitivity—cannot be derived from metabolic traits alone because of the strong compensation by O2 supply mechanisms. Species with fast metabolisms exhibit rapid O2 supply rates (Fig. 1c and Extended Data Fig. 2), while those with high metabolic temperature sensitivities show strong thermal responses of O2 extraction (Fig. 1e and Extended Data Fig. 2). The constituent traits of active hypoxia vulnerability are also correlated: species with a lower resting hypoxia vulnerability have a higher active to resting metabolic rate ratio (Extended Data Fig. 2). These correlations act to narrow the interspecies ranges of all three key traits (Extended Data Fig. 2) and suggest that there are strong physiological trade-offs and selective pressures, the nature and causality of which remain unresolved. Whatever their mechanistic origins, these trade-offs and constraints have resulted in a breadth of temperature-dependent hypoxia tolerance and associated spatial habitat limits that allow species to collectively exploit the full range of aerobic conditions found in the modern ocean.

The data needed to calibrate the Metabolic Index and diagnose the relative role of O2 supply and demand can be derived from standard respirometry data, but currently the number of sampled species comprises only a small fraction of the total marine biodiversity. They include few species without circulatory systems; species without a clear Pcrit (‘oxyconformers’); or species pairs with well-characterized predator–prey or other ecological relationships that may modulate the physiological response to climate change. A systematic and concerted effort to expand data on Metabolic Index parameters across a wider variety of marine biota, especially those with rich biogeographical data, and populations that may adapt hypoxia traits over regional scales or between generations, will be key to further evaluating the role of temperature-dependent hypoxia in shaping marine biogeography, ecological interactions and habitat loss in a warming climate.

## Methods

### Derivation of the  Metabolic Index

The Metabolic Index is defined as per a previous study4 as the ratio of the rates of the O2 supply to and demand by an organism. In general, both rates are dependent on temperature (T) and body mass (B). Following standard metabolic scaling, the O2 demand can be written:

$$D={\alpha }_{{\rm{D}}}{B}^{\delta }\exp \,\left\{\frac{-{E}_{{\rm{d}}}}{{k}_{{\rm{B}}}}\left(\frac{1}{T}-\frac{1}{{T}_{{\rm{ref}}}}\right)\right\},$$
(2)

where the rate coefficient (αD) has units of O2 per unit body mass per time (we use μmol O2 g−3/4 h−1). It is scaled by the exponential Arrhenius function of absolute temperature, which captures the temperature dependence often described by a Q10 factor42. When estimating parameters, the body mass is normalized to the median experimental body mass so that it is non-dimensional. Thus, when T = Tref, an organism of median body mass has a resting metabolic rate of D = αD.

The supply of O2 to the body may also scale with body size, temperature and ambient O2 pressure $$({p}_{{{\rm{O}}}_{2}})$$, such that:

$$S={\hat{\alpha }}_{{\rm{S}}}(T){B}^{\sigma }{p}_{{{\rm{O}}}_{2}}$$
(3)

The function $${\hat{\alpha }}_{{\rm{s}}}(T)$$ represents the efficacy of the O2 supply. It is a rate coefficient (in μmol O2 g−3/4 h−1atm−1), but becomes an absolute mass-normalized rate (μmol O2 g−3/4 h−1) only when multiplied by the ambient O2 pressure (we use units of atm). The exponent, σ, for the allometric scaling of the O2 supply with body mass is typically very similar to that of O2 demand18, although the two may differ.

The temperature dependence of $${\hat{\alpha }}_{{\rm{s}}}(T)$$ may be complicated, as it reflects the combined effect of multiple steps in the O2 supply chain, including ventilation and circulation rates that are under biological control, as well as diffusive O2 flux across the water–body boundary. Because diffusive gas fluxes are governed by physical and chemical kinetics, their temperature dependence follows the known scaling of gas exchange across a diffusive boundary layer43. Standard gas exchange models are well approximated by an Arrhenius function (Extended Data Fig. 3a):

$${\hat{\alpha }}_{{\rm{S}}}(T)={\alpha }_{{\rm{S}}}\exp \,\left\{\frac{-{E}_{{\rm{s}}}}{{k}_{{\rm{B}}}}\left(\frac{1}{T}-\frac{1}{{T}_{{\rm{ref}}}}\right)\right\}$$
(4)

where the scalar coefficient αS has the same units as the function $${\hat{\alpha }}_{{\rm{s}}}(T)$$, but is a constant that does not depend on temperature. The same equation can be applied to ventilation rates and circulation rates, although in contrast to diffusion, for biological rates a single Es value will not necessarily apply over the entire temperature range of a species (Extended Data Fig. 3b). Even so, equation (4) provides a flexible formula for biological fluxes that vary nonlinearly with temperature over a finite temperature range.

Inserting equations (2)–(4) into the definition of the Metabolic Index, we get:

$$\varPhi =\frac{S(B,T,{p}_{{{\rm{O}}}_{2}})}{D(B,T)}=\frac{1}{{V}_{{\rm{h}}}}{B}^{\varepsilon }{p}_{{{\rm{O}}}_{2}}\exp \,\left\{\frac{{E}_{{\rm{o}}}}{{k}_{{\rm{B}}}}\left[\frac{1}{T}-\frac{1}{{T}_{{\rm{ref}}}}\right]\right\}$$
(5)

where

$$\varepsilon =\sigma -\delta ,$$
(6a)
$${E}_{{\rm{o}}}={E}_{{\rm{d}}}-{E}_{{\rm{s}}}$$
(6b)

and

$${V}_{h}={\alpha }_{D}/{\alpha }_{s}$$
(6c)

The defining formula (equation (5)) is identical to equation (1) in the main text, and to the previously described formula given by ref. 4. It differs in form from that described previously4 because it is normalized to a reference temperature (Tref) such that when T = Tref (here specified at 15 °C), the coefficient (αS/αD = 1/Vh, which is denoted Ao in the previous study4) is the inverse of Pcrit at that reference temperature. We have also chosen a more intuitive annotation for the allometric exponents (σ, for ‘supply’ and δ, for ‘demand’). The only substantial difference in this formulation is that the contributions of the O2 supply and demand to the temperature sensitivity of hypoxia tolerance (that is, Eo) are made explicit, rather than being accounted for implicitly (for example, see supplementary figure 2 of the previous study4). This allows the net temperature dependence of the tolerance of hypoxia to be partitioned into supply and demand effects using equations (6a)–(6c).

### Data compilation and parameter estimation

The physiological parameters of the Metabolic Index (Φ) are derived from laboratory measurements of hypoxic thresholds (Pcrit) and resting metabolic rates (D) at multiple temperatures. The measurements are taken from published literature, adding to previous compilations7,40,44. The original studies and parameter values are listed in Supplementary Table 1, and yield 145 species with metabolic rate parameters, and 72 species with hypoxia parameters (including four based on lethal thresholds (LC50)). The species with Pcrit data range over 8 orders of magnitude in body mass, from 5 phyla (Annelida, Arthropoda, Chordata, Cnidaria and Mollusca), including 31 malacostracans, 26 fishes, 9 molluscs, 2 copepods, and 1 species each for ascidians, thaliasceans, scleractinian corals and annelid worms.

Metabolic traits (δ, αD, Ed) are derived from fitting equation (2) with mass-normalized resting metabolic rates (μmol O2 h−1g−3/4) that have been experimentally determined at multiple temperatures. Hypoxia traits (ε, Vh and Eo) are derived by substituting paired experimental temperatures and Pcrit data (atm) in equation (5) (as variables T and $${p}_{{{\rm{O}}}_{2}}$$), and solving for the parameters that give Φ = 1, the condition in which the physical O2 supply and resting metabolic demand are balanced. Parameters describing the net O2 supply (αS and Es) were estimated from equations (6a)–(6c), that is, αS = αD/Vh and Es = Ed − Eo, for the subset of species for which Pcrit and metabolic rates are both available at multiple temperatures. The temperature dependence of the net O2 supply is compared to independent estimates based on the individual steps in the O2 supply chain: diffusion, ventilation and circulation (Extended Data Fig. 3). With species for which body mass varied by less than a factor of 2, we set δ = 3/4 and ε = 0, values that typify most species, including those investigated here.

We analysed the parameters of the Metabolic Index in two complementary ways. First, we compare the interspecies frequency distributions of each parameter, which emphasizes the diversity of traits and their relationships across marine biota, and enables comparisons between traits that are not all measured in all species. Second, we examine the intraspecies relationships between traits whenever multiple traits from the same species are available. Such analyses provide a more direct test of physiological mechanisms, but are taxonomically restricted and more sensitive to random errors in the experimental determination of parameters.

We use MATLAB’s nonlinear fitting routine (fitnlm.m) to solve for species traits (parameters) that minimize the squared residual errors. We report the central estimate of each parameter, the Pearson correlation coefficient (r2) and the P value based on two-sided Student’s t-tests, and the number of raw observations in Supplementary Table 1. With species parameters obtained from equations (1), (2), (6), relationships between traits are subsequently analysed using a standard linear least squares MATLAB routine (regress.m). Regression parameters, their 95% confidence intervals, correlation coefficients (r2), the P value based on two-sided t-tests and the number of raw observations for each relationship are reported in Extended Data Table 1.

To maximize the number of species analysed, we include those for which experiments were conducted at as few as 2 temperatures. However, all reported relationships among traits were confirmed using the subset of species (n = 14) for which regressions of metabolic rates and Pcrit data against temperature were statistically significant (P < 0.05, two-tailed t-test) (Extended Data Fig. 2).

### Determination and validation of Φcrit

The limiting value of the Metabolic Index in each species habitat (Φcrit), was estimated by pairing species location data with hydrographic conditions at those locations. Occurrence data were downloaded from the Ocean Biodiversity Information System (OBIS; http://iobis.org) in September 2019. Of the 72 species with hypoxia traits, OBIS contains georeferenced presence data for 68. To estimate the hydrographic conditions at each specimen location, we used monthly climatological temperature and O2 fields at a resolution of 1° latitude and longitude and at 33 depths from the World Ocean Atlas45,46. For analysis of temperature at the sea surface (z = 0 m) (Fig. 5), we include the diurnal temperature range from satellite remote sensing (data downloaded from https://www.ghrsst.org/ghrsst-data-services/products/) to estimate the globally resolved peak daytime surface temperatures.

Species occurrences were paired to hydrographic data by binning them to the World Ocean Atlas grid for every month based on the locations provided in OBIS. Hydrographic conditions were determined at the central depth of the minimum and maximum depths reported by OBIS, or from either depth alone if only one metric was provided. Occurrences were discarded if the range of conditions within that depth range differed from the central estimate by more than 2 °C for temperature or 20% for O2. For occurrences that did not have depth information altogether, we assigned a minimum depth at the sea surface and maximum depth at the seafloor47. In cases in which even this maximum uncertainty in depths satisfied the error tolerance (2 °C for temperature and 20% for O2) the location data were retained. The Metabolic Index (that is, equation (5)) was computed based on species-specific traits and the paired hydrographic data for the occupied sites of each species.

Of the more than 1.5 million OBIS occurrences used here, only 0.1% mapped to climatological conditions in which Φ falls below 1. This environmental condition is physiologically unsustainable, yet may arise from transient species movements, or a mismatch between the climatological temperature and O2 fields used to compute Φ and the true in situ hydrographic conditions at the time occurrence data were recorded. Only three species in our dataset had more than 5% of OBIS occurrences for which Φ < 1, and two of them (Sergia tenuiremis and Sergia fulgens) are known to be vertical migrators. Because of the likelihood that these occurrences do not reflect viable long-term habitats, but instead are being used as a temporary refuge that requires metabolic suppression, we report the Φcrit values that do not include such locations. Of the three species with more than 5% of locations that had Φ < 1, the removal of those points affected the estimate Φcrit by <0.3 for two of them (S. fulgens and M. pammelas), and thus has a negligible effect on our results. We report the Φcrit estimates both with and without the inclusion of rare locations for which Φ < 1 (Supplementary Table 1).

We evaluated the Φ value that best defines the boundary of the geographical range of each species in two independent methods, which use identical data but differ in the degree of data aggregation over space and time. The first equates Φcrit with the lower tail in the frequency distribution of Φ across all occupied sites in OBIS, for each species. The second computes the Φ value that maximizes its predictive skill in segregating inhabited and uninhabited grid cells globally, using a machine-learning technique. The two methods, which are described below, give highly consistent results (Extended Data Fig. 8a), but the first approach is presented in the main text (Fig. 4), owing to its conceptual and computational simplicity.

#### Occurrence histogram

The ecological parameter, Φcrit, is estimated from the cumulative distribution function as the value of Φ above which the most of the occurrences of each species are found (5th and 10th percentiles). The two values yield similar Φcrit values, and their range encompasses the Φcrit derived from a machine-learning algorithm (see ‘The F1-score’; Extended Data Fig. 8a), but can be applied objectively to species for which the three-dimensional distribution is too complex or sparsely sampled to identify a clear boundary to the geographical range. We present the median of Φcrit in our primary results, but include both values in Supplementary Table 1.

As sampling density decreases, the lowest observed Φ value may not reflect the true minimum within a species habitat. However, we found that the distribution of Φcrit for all species was similar regardless of sampling intensity (Extended Data Fig. 8b), and not biased towards higher values of Φcrit (Fig. 4). We therefore did not restrict the analysis based on the number of occurrences.

#### The F1-score

We evaluated the ability of Φ to separate the ocean into inhabited and uninhabited portions for each species, using a standard statistical categorization metric, the F1-score48,49. The F1-score is computed based on the presence and absence of a species on a regular grid (latitude, longitude, depth and month), for which the environmental conditions fall above and below a threshold value, which we varied. The value of the environmental threshold that yields the maximum F1-score is the one that best segregates global grid cells into inhabited and uninhabited conditions for the environmental parameter of interest. Φcrit is estimated as the Φ value that optimizes the predictive skill of categorizing habitat (maximum F1-score).

The F1-score is calculated as the harmonic mean of precision and recall, with equal weighting given to both measures. Precision measures the probability that the presence of the species in waters for which Φ ≥ Φcrit is a true positive (TP; specimen reported in the space in which they are predicted to occur) rather than a false positive (FP; specimen reported in a space predicted to be below the Φ threshold). Recall is the probability that a specimen is actually reported where Φ > Φcrit (that is, how likely is a true positive relative to a false negative (FN); missing observations above Φcrit). In terms of these variables, the F1-score can be expressed as:

$${F}_{1}={\left(\frac{{{\rm{recall}}}^{-1}+{{\rm{precision}}}^{-1}}{2}\right)}^{-1}=\,\frac{2{\rm{TP}}}{2{\rm{TP}}+{\rm{FN}}+{\rm{FP}}}$$
(7)

This metric does not give weight to true absence data (species known to be not present), which are infrequently and inconsistently reported in marine species data. It is thus well suited to categorization problems based on OBIS data. A model with perfect precision and recall would have F1 = 1. The absolute F1-score cannot be meaningfully compared between species, as it depends on the total number of grid cells included, as well as the total number of occupied sites. However, when applied to the same species and geographical region, the variations in F1-scores between different values of the same environmental parameter, or between different environmental parameters (for example, Φ versus T), are a meaningful metric for the relative skill of a given parameter and its threshold value. Optimal F1-scores were used to compare the predictive skill of different environmental parameters.

#### Comparison to SMS

To determine whether Φcrit is consistent with independent estimates of the ratio of active-to-resting metabolic rates, we compared the frequency distributions of both metrics. An appropriate direct comparison of the habitat constraint (Φcrit) to metabolic rate ratios would be based on active metabolic rates sustained over the time scales of population maintenance (termed ‘SusMR’)8. Such rates are not measured in marine species. However, maximum rates of metabolism are commonly measured in laboratory experiments. Long-term sustained metabolic rates can be expressed as a weighted average of the maximum rates (MMR) obtained under extreme exertion and the minimum rates that apply in a state of rest (RMR):

$${\rm{SusMR}}\,=\,{w}_{{\rm{R}}}\times {\rm{RMR}}+(1-{w}_{{\rm{R}}})\times {\rm{MMR}}$$

where wR represents the effective weight of the resting state in the time–mean sustained metabolic rate. Dividing both sides by RMR, and noting the definition of SMS (SMS = SusMR/RMR), yields the equation in the main text:

$${\rm{SMS}}={w}_{{\rm{R}}}+(1-{w}_{{\rm{R}}})({\rm{MMR}}/{\rm{RMR}}),$$
(8)

which can be rearranged, substituting the definition of factorial aerobic scope (FAS = MMR/RMR) to estimate the weighting factor, wR:

$${w}_{{\rm{R}}}=\frac{{\rm{FAS}}-{\rm{SMS}}}{{\rm{FAS}}-1}.$$
(9)

Carbon isotope analyses of the otoliths of Atlantic Cod22 suggest that SMS ≈ 2, whereas FAS50 ranges from 3.3 to 3.8, yielding a range of wR from 0.39 to 0.47. We estimated the weighting of resting metabolic rates (that is, wR) using the measured ratios of MMR/RMR and Φcrit for the species in our dataset (Supplementary Table 1), and find a mean and interspecies variation (s.d.) (wR = 0.40 ± 0.17, n = 14; Extended Data Fig. 9 and Supplementary Table 1) that is consistent with the direct geochemical estimate for cod. Extending the mean value from these 14 species to a broader group of species with measured FAS41 (n = 106) but no SMS or Φcrit, we find a distribution of SMS that is statistically indistinguishable from the overall distribution of Φcrit (Extended Data Table 1). Interspecies variation in the estimates of wR probably reflects both real biological differences in activity levels and the substantial methodological uncertainties that originate from both laboratory rates (MMR and RMR) and biogeographically derived Φcrit values. Regardless of the precise values of wR and their uncertainties, the fact that they are all positive (FAS values are at or above Φcrit) and that Φcrit is significantly correlated with laboratory-derived SMS and MMR/RMR measurements (Extended Data Table 1 and Extended Data Fig. 9) indicates that aerobic energy availability is a habitat constraint.

#### Estimation of ATmax

The maximum temperature at which aerobic respiration can be sustained is estimated by extrapolating the empirical relationship between Pcrit and temperature to the mean atmospheric $${p}_{{{\rm{O}}}_{2}}$$ (Patm), at which CTmax experiments are carried out. We thus find the solution to the equation:

$${A}_{{\rm{o}}}{P}_{{\rm{a}}{\rm{t}}{\rm{m}}}\,\exp \,\left\{\frac{{E}_{{\rm{o}}}}{{k}_{{\rm{B}}}}\,[\frac{1}{{{\rm{A}}{\rm{T}}}_{max}}-\frac{1}{{T}_{{\rm{r}}{\rm{e}}{\rm{f}}}}]\right\}=\{\begin{array}{cc}1 & {{\rm{r}}{\rm{e}}{\rm{s}}{\rm{t}}{\rm{i}}{\rm{n}}{\rm{g}}{\rm{A}}{\rm{T}}}_{max}\\ {\varPhi }_{{\rm{c}}{\rm{r}}{\rm{i}}{\rm{t}}} & {{\rm{a}}{\rm{c}}{\rm{t}}{\rm{i}}{\rm{v}}{\rm{e}}{\rm{A}}{\rm{T}}}_{max}\end{array}$$
(10)

Because the Pcrit data are all below Patm, the solutions to this equation (ATmax) are necessarily extrapolated beyond the experimental range of temperatures over which Eo is estimated. If Eo was constant across the full range of temperatures, this extrapolation would only be influenced by the random errors in Pcrit measurements, but would not incur a systematic bias across all species, yielding a histogram of ATmax with a robust mean value. However, the available data indicate that Eo increases systematically (albeit slightly) with temperature (Extended Data Fig. 3). We correct for this bias in the extrapolation of Pcrit curves to the aerobic thermal maximum, by including an empirically derived linear increase in Eo with temperature, as discussed next.

The slope of the relationship (denoted by the derivative of Eo with respect to temperature, dEo/dT) is estimated in multiple ways, to evaluate the uncertainty in these extrapolations. First, we use the intraspecies difference in Eo among species for which it can be separately estimated both above and below Tref, as discussed in the main text and shown in Extended Data Fig. 3. This yields a mean intraspecies dEo/dT = 0.036 eV/°C (0.55 eV/15 °C, where 15 °C is the difference between the two temperature bins 0–15 °C and 15–30 °C). Second, we consider the differences in Eo between colder waters (T < Tref) and warmer waters (T > Tref) for all species. This estimate, dEo/dT = 0.013 eV/°C (0.2 eV/15 °C; Extended Data Fig. 3) gives a lower value because it includes interspecies variation. Finally, as a third method for estimating the potential variation in Eo with temperature, we directly fit the Pcrit curves (equation (5) for all species with more than 2 temperatures, including a linear relationship between Eo and temperature. We discard any fits that predict a Pcrit that declines towards zero at high temperatures (T 30 °C), as this would imply an unrealistic (infinite) tolerance for hypoxia at high temperatures. As a second check on the curve fits, we compare the Akaike information criterion (AIC) for the model with a linear increase in Eo to our standard model with a constant Eo. We only retain those curve fits in which the AIC did not decrease, indicating that the additional parameter did not reduce the information content of the model despite the additional parameter. Across species, this yields a mean value of dEo/dT = 0.022 eV/°C, which falls in between the previous two values. We apply this interspecies mean value as a default value for all species (Fig. 5), since it yields results that are not biased relative to values derived from species-specific dEo/dT wherever both are available (Extended Data Fig. 10). The range of dEo/dT estimates is used to generate the error bars of the estimates of ATmax plotted in Extended Data Fig. 10.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this paper.

## Data availability

The data used in this study are described in the Methods. The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The MATLAB code is available at GitHub (https://github.com/cadeutsch/Metabolic-Index-Traits).

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## Acknowledgements

We thank T. Brey for contributing data, E. Howard for statistical advice, H. Frenzel for computational support and W. Verberk, M. Pinsky and A. Bates for insightful suggestions that improved the clarity of presentation. This work was made possible by grants from the Gordon and Betty Moore Foundation (GBMF#3775) and the National Science Foundation (OCE-1419323 and OCE-1458967) and the National Oceanic and Atmospheric Administration (NA18NOS4780167).

## Author information

Authors

### Contributions

C.D and J.L.P. analysed the data and wrote the paper with input from B.S.

### Corresponding author

Correspondence to Curtis Deutsch.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Amanda Bates, Malin Pinsky and Wilco C. E. P. Verberk for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Species metabolic rates and hypoxia tolerances from laboratory studies.

a, b, Measured metabolic rates (μmol g−1 h−1) (a) and critical O2 pressures (Pcrit) (b) versus temperature (°C) in published laboratory experiments (circles). For clarity, metabolic rates are shown only for the subset of species with Pcrit data. c, Location data from OBIS for all species with Pcrit measured at multiple temperatures, yielding calibrated Metabolic Index parameters. The number of species with occurrences in the Pacific, Atlantic and Indian Oceans are labelled. Maps of the occurrences of individual species are available at https://obis.org/.

### Extended Data Fig. 2 Correlations and diversity in traits that govern geographical range boundaries.

ac, The key traits that make up resting hypoxia vulnerability (Vh = αD/αS) (a), its temperature sensitivity (Eo = Ed − Es) (b) and the elevated hypoxia vulnerability under activity (Vh × Φcrit) (c) all exhibit significant correlations (standard linear regression, two-tailed t-test, P < 0.05) between their constitutive parameters, regardless of whether we use all 72 species (dashed lines) or the subset of 14 species (dotted lines) for which the traits themselves were derived from statistically significant fits to equations (1) and (2) (see Methods and Extended Data Table 1). As in Fig. 1, points and error bars (centred dot, if shorter than marker) are mean ± s.e.m. for species with more than two independent experimental temperatures. See Supplementary Table 1 for the number of independent temperature experiments used for each species. The number of species used in each correlation is n = 48 (a, b) and n = 56 (c). See Extended Data Table 1 for statistics on two-sided t-tests of trait correlations. df, Observed diversity in resting hypoxia vulnerability (Vh) (d), its temperature sensitivity (Eo) (e) and active hypoxia vulnerability (Vh × Φcrit) (f), is measured as the interquartile range (IQR) among all species (red bars). We also quantified the diversity in species traits in the absence of observed correlations in the underlying metabolic traits. The correlations are removed by replacing species variation in the indicated parameter with the interspecies mean value. The diversity of the resulting trait is recomputed from the IQR (blue bars). Specifically, we replace the variable αS (a) with its mean value to derive a new distribution and IQR of Vh (d, blue bar); replace the variable Es (b) with its mean value to derive a new distribution and IQR of Eo (e, blue bar) and replace the variable Φcrit (c) with its mean value to derive a new distribution and IQR of the active hypoxia tolerance (Vh × Φcrit; f, blue bar). For all three central traits, the correlation and putative trade-offs among the underlying constitutive parameters act to reduce the interspecies diversity of the trait that governs habitat range limits.

### Extended Data Fig. 3 Temperature sensitivity of processes that govern the O2 supply.

a, The rate of diffusive flux across the boundary layer increases with temperature in proportion to Scn, where the Schmidt number (Sc) is the ratio of seawater viscosity (υ) to O2 diffusivity (κ). Typical values of the exponent, n, are −1/2, −2/3 and −1, depending on the underlying model of boundary layer renewal23. In all cases, the empirically derived curves (solid) are well approximated by an Arrhenius function (dashed) with corresponding activation energy parameters (that is, Es) ranging from 0.21 eV (for n = −1/2; blue) to 0.27 eV (for n = −2/3; green) and 0.42 eV (for n = −1; red). b, Experimental measurements of rates of the ventilation (solid) and circulation (dashed) of animals. Rates at multiple temperatures are from published studies of six species, including three annelids24 (Nereis virens, blue; Nereis succinea, brown; and Nereis diversicolor, gold; all with n = 6 independent experiments) and three chordates (Lepornis macrochirus26, red; Oncorhynchus mykiss51, green; Leiopotherupon unicolor25, cyan; all with n = 7 independent experiments). c, For each species, the temperature sensitivity of each rate is determined by fitting to an Arrhenius function above and below 20 °C, the approximate thermal midpoint of all data. Histograms of activation energy in each temperature range (insets) are significantly different (two-sample Kolmogorov–Smirnov test; P = 5× 10−4) for warm conditions (Es = 0.04 ± 0.18 (mean ± s.d.)) and cool waters (Es = 0.55 ± 0.15 (mean ± s.d.)). d, Distributions of Eo computed from experimental data at temperatures at or above 15 °C (red bars) are higher than for the same parameter computed using only temperatures at or below 15 °C (blue bars). For species for which at least two Pcrit values were available both above and below Tref, the difference between Eo for warm and cold temperatures (green bars) is always greater than zero, and has a mean value (0.55 eV) similar to the change in temperature dependence of ventilation and circulation rates across cold and warm temperatures (green bars in c).

### Extended Data Fig. 4 Spatial distributions of the Metabolic Index, temperature and $${{\boldsymbol{p}}}_{{{\rm{O}}}_{2}}$$ compared to occurrences of species that occupy diverse latitude and depth ranges.

ae, Species inhabit mid to high latitudes. fh, Species are found in tropical waters. Fields of Φ (colours), temperature and O2 pressure are zonally averaged over the longitudinal range of each species. a, Cyclopterus lumpus (95 °W–35 °E). b, Tautogolabrus adspersus (50–80 °W). c, Gadus morhua (75 °W–40 °E). d, Zoarces viviparus (10 °W–30 °E). e, Gadus ogac (110 °W–40 °E). f, Penaeus aztecus (40–120 °W). g, Funchalia villosa (100 °W–40 °E). h, Gennadas valens (100 °W–47 °E). Observed species occurrences are plotted (blue dots). A single lower limit of Φ bounding each species range is contoured (Φcrit; black lines) alongside isotherms of temperature (white lines; °C) and isopleths of $${p}_{{{\rm{O}}}_{2}}$$ (grey lines; atm). Published upper thermal limits (CTmax) are contoured in green where available, based on maximum monthly ocean temperatures (°C). Green asterisks denote species for which CTmax occurs above all mapped maximum monthly temperatures. For most species, Φcrit more skilfully categorizes occupied habitat than either upper temperature limits or lower $${p}_{{{\rm{O}}}_{2}}$$ considered individually. This skill is shown by the ratio of F1-scores of Φ relative to temperature or to $${p}_{{{\rm{O}}}_{2}}$$ (in parentheses, respectively) from the full four-dimensional species distribution. For G. morhua, the monthly range of Φcrit is also mapped (dashed black lines). For G. ogac, mapped occurrences, Φ and water properties are from the Atlantic Ocean only. Land regions are shaded in grey.

### Extended Data Fig. 5 Maps of the Metabolic Index, temperature and $${{\boldsymbol{p}}}_{{{\rm{O}}}_{2}}$$ compared to species distributions.

af, Mapped variables are averaged from the surface to the 95th percentile depth of each species. ac, P. borealis (0–450 m). df, Stenobrachius leucopsarus (0–225 m). A single lower limit of Φ (Φcrit; black lines) is consistent with habitat range limits found in the Pacific and Atlantic Oceans for P. borealis (a) and different sides of the Pacific Ocean for S. leucopsarus (d). By contrast, no single maximum temperature or minimum $${p}_{{{\rm{O}}}_{2}}$$ is consistent with each species’ range limit across those regions. The increased skill of Φcrit is encapsulated by the higher F1-scores of Φ relative to temperature (b, e) or to $${p}_{{{\rm{O}}}_{2}}$$ (c, f) (in parentheses, respectively) from the full four-dimensional species distribution. Occurrence data for each species are shown (blue dots).

### Extended Data Fig. 6 Spatial distributions of the P. borealis, S. canicula and S. plicata compared with the Metabolic Index, temperature and $${{\boldsymbol{p}}}_{{{\rm{O}}}_{2}}$$.

ai, Spatial distributions of the species shown in Fig. 2 (P. borealis, S. canicula and S. plicata) were compared with the Metabolic Index (a, d, g), temperature (b, e, h) and $${p}_{{{\rm{O}}}_{2}}$$ (c, f, i). ac, P. borealis. df, S. canicula. gi, S. plicata. A single lower limit of Φ bounding each species range is contoured (Φcrit; black lines). For all species, Φcrit more skilfully categorizes occupied habitat than either temperature limits or lower $${p}_{{{\rm{O}}}_{2}}$$ considered individually. This skill is shown by the higher ratio of F1-scores of Φ relative to temperature or to $${p}_{{{\rm{O}}}_{2}}$$ (in parentheses, respectively) from the full four-dimensional species distribution. Occurrence data for each species are shown (blue dots). Regions for zonal averaging are as in Fig. 2.

### Extended Data Fig. 7 Predictive skill of the Metabolic Index in delineating the species geographical range, compared with temperature or $${{\boldsymbol{p}}}_{{{\bf{O}}}_{{\bf{2}}}}$$ alone.

The F1-score measures the ability of each environmental variable at a given threshold value to categorize the ocean into inhabited and uninhabited regions. The maximum F1-score for Φ is then compared with the maximum value for temperature or $${p}_{{{\rm{O}}}_{2}}$$ thresholds and plotted on a log10 scale such that positive (negative) values indicate a stronger (weaker) predictive skill for Φ. Printed numbers (n) on the graph indicate the number of species that fall within each quadrant (for numbers in quadrants) or to each side of the axes (for numbers on axes). For example, Φ outperforms T in 44 species, and underperforms in only 4, and outperforms both T and $${p}_{{{\rm{O}}}_{2}}$$ in 30 species. A maximum F1-score was found for 48 species, while 8 additional species had no clear maximum in F1-score.

### Extended Data Fig. 8 Critical value of the Metabolic Index at the limit of species geographical range (Φcrit).

a, The values of Φcrit are compared using two independent methods. The first (y axis) is determined from the peak in the F1-score for categorization into occupied and unoccupied sites (see Methods). The second (x axis) is determined from the bottom percentile (5–10%) of Φ values inhabited by the species. b, The histogram of Φcrit is not sensitive to the number of occurrence observations. c, For all species, waters with lower Φ values than Φcrit exist within the depth range of these species, but lack confirmed specimens. The dashed line indicates the 1:1 line.

### Extended Data Fig. 9 Relationship between Φcrit and the ratio of maximum-to-resting metabolic rates (MMR/RMR), among all species with empirical estimates of both parameters.

Blue dots, species with empirical estimates of both parameters (Supplementary Table 1). Lines of constant wR (see equation (9) are shown for a sustained metabolic rate that is equal to the resting rate (SMS = 1; wR = 1; blue line), the maximum metabolic rate (SMS = FAS; wR = 0; red line) or the mean apparent species value (wR = 0.4; green line) in which sustained the metabolic rate is approximately midway between minimal (resting) and maximal rates. Independent geochemical estimate of SMS based on carbon isotopes in the otoliths of Atlantic cod28 are shown as a triangle.

### Extended Data Fig. 10 Relationship across species between thermal tolerance of species measured in laboratory studies and predicted from the Metabolic Index.

Critical thermal maxima (CTmax), reported in previous studies, were measured at the onset of loss of physiological function in a resting state. The ATmax are predicted from the Metabolic Index, as the temperature at which the O2 threshold for metabolic rate in either resting state (Pcrit) or under sustained activity levels (Pcrit × Φcrit), reaches the mean atmospheric O2 pressure, Patm (see Methods, equation (10)). The extrapolation of Pcrit curves to the atmospheric pressure is based on n = 3 independent estimates of the linear increase in Eo with temperature (see Methods). Points for each species are the mean, and error bars show the range among the resulting n = 3 estimates of ATmax. Data are shown for all species in Fig. 5, for which all parameters are available (n = 7). For four out of seven species, the ATmax is reached before the CTmax, even in a resting state. For all seven species, the active ATmax is reached at a lower temperature than the CTmax, and is comparable to the maximum temperature that the species inhabits at atmospheric pressure.

## Supplementary information

### Supplementary Table 1

Table containing species metabolic and hypoxia traits, and model statistics.

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Deutsch, C., Penn, J.L. & Seibel, B. Metabolic trait diversity shapes marine biogeography. Nature 585, 557–562 (2020). https://doi.org/10.1038/s41586-020-2721-y

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• ### A committed fourfold increase in ocean oxygen loss

• Andreas Oschlies

Nature Communications (2021)