Linking scaling laws across eukaryotes Ian A. Hattona,1,2, Andy P. Dobsona,b, David Storchc,d, Eric D. Galbraithe,f,g, and Michel Loreauh a Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544; bSanta Fe Institute, Santa Fe, NM 87501; cCenter for Theoretical Study, Charles University and the Academy of Sciences of the Czech Republic, 110 00 Praha, Czech Republic; dDepartment of Ecology, Faculty of Science, Charles University, 128 44 Praha, Czech Republic; eICREA (Catalan Institution for Research and Advanced Studies), 08010 Barcelona, Spain; f Department of Mathematics, Institut de Ciència i Tecnologia Ambientals (ICTA-UAB), Universitat Autonoma de Barcelona, 08193 Barcelona, Spain; g Department of Earth and Planetary Sciences, McGill University, Montreal, QC H3A 0G4, Canada; and hCentre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS, 09200 Moulis, France Edited by Nils Chr. Stenseth, University of Oslo, Oslo, Norway, and approved September 9, 2019 (received for review January 30, 2019) Scaling laws relating body mass to species characteristics are among temperature, other environmental conditions, life stage, and re- the most universal quantitative patterns in biology. Within major gression methods (1, 6, 8–10, 22, 23). Despite these many factors, taxonomic groups, the 4 key ecological variables of metabolism, an exponent value of 3/4 has become a canonical expectation for abundance, growth, and mortality are often well described by body mass scaling, especially as additional taxonomic groups and power laws with exponents near 3/4 or related to that value, a additional species characteristics have been found to scale with commonality often attributed to biophysical constraints on metab- similar or related values (1–3, 7, 17, 18). olism. However, metabolic scaling theories remain widely debated, Most prevailing theories of metabolic scaling (with k < 1) and the links among the 4 variables have never been formally tested are based on physical constraints on the structure of a body (3, 9, across the full domain of eukaryote life, to which prevailing theory 24, 25), which in turn are thought to constrain the scaling of other applies. Here we present datasets of unprecedented scope to variables with body mass (1, 2, 7, 11, 15, 18, 20, 26). For example, examine these 4 scaling laws across all eukaryotes and link them limits on energy supply can proximally limit abundance (2, 10–12) to test whether their combinations support theoretical expectations. and the energy allocated to growth and reproduction (1, 2, 15, 18, We find that metabolism and abundance scale with body size in a 26–28). Metabolism is also known to produce harmful byproducts remarkably reciprocal fashion, with exponents near ±3/4 within that hasten senescence and shorten life span (2, 9, 29). However, groups, as expected from metabolic theory, but with exponents the many dependencies of the metabolic exponent, listed above, near ±1 across all groups. This reciprocal scaling supports “energetic suggest that metabolism is also flexible and can adjust to different equivalence” across eukaryotes, which hypothesizes that the parti- tioning of energy in space across species does not vary significantly factors, some of which have been found to be growth factors. This with body size. In contrast, growth and mortality rates scale simi- has prompted some authors to suggest that basal metabolic scaling larly both within and across groups, with exponents of ±1/4. These adjusts to growth rather than exerting fundamental control on findings are inconsistent with a metabolic basis for growth and growth and other characteristics (9, 14, 23, 27, 30–32). mortality scaling across eukaryotes. We propose that rather than One way to better understand the origin of near 3/4 scaling is limiting growth, metabolism adjusts to the needs of growth within to consider the generality of circumstances over which such major groups, and that growth dynamics may offer a viable theo- scaling holds. For example, similar scaling both within and across retical basis to biological scaling. taxonomic groups is consistent with a single underlying process, whereas different scaling regimes within or across groups are macroecology | biological scaling | metabolic theory Significance S caling laws relating body mass to a variety of species char- acteristics are among the most general quantitative patterns in biology (1–3). These scaling laws encompass such core ecolog- Metabolic scaling theory has had a profound influence on ecol- ogy, but the core links between species characteristics have not ical characteristics as metabolism (1, 2, 4–9), abundance (10–13), been formally tested across the full domain to which the theory growth (14–18), and mortality (1, 2, 19, 20). The relationship be- claims to apply. We compiled datasets spanning all eukaryotes tween a species characteristic (y) and its body mass (m) is often for the foremost body mass scaling laws: metabolism, abun- expressed as a power law, y = cmk, where c is a constant for a dance, growth, and mortality. We show that metabolism and given variable and k is a dimensionless scaling exponent, given by abundance scaling only follow the canonical ±3/4 slopes within the slope of a straight line on a plot of log y vs. log m. Many of these some taxonomic groups, but across eukaryotes reveal reciprocal relationships are increasingly used to better understand and make near ±1 slopes, broadly supporting the “energetic equivalence large-scale predictions of the effects of the most critical global rule.” In contrast to metabolism, growth follows consistent ∼3/4 environmental problems, since they represent simple and efficient scaling within many groups and across all eukaryotes. Our predictors of fundamental variables that hold across broad taxo- findings are incompatible with a metabolic basis for growth scaling and instead point to growth dynamics as foundational to nomic groups (1, 2). Moreover, these relationships raise basic and biological scaling. persistently enigmatic problems: how are they linked, and where do they originate? Author contributions: I.A.H. designed research; I.A.H. performed research; I.A.H. analyzed Body mass scaling research largely began with the study of data; and I.A.H., A.P.D., D.S., E.D.G., and M.L. wrote the paper. basal metabolism across mammals, which found an exponent The authors declare no competing interest. near k = 3/4, known as Kleiber’s law (1, 2, 9, 21) and termed This article is a PNAS Direct Submission. “allometry” (k ≠ 1). This value did not match expectations of a This open access article is distributed under Creative Commons Attribution-NonCommercial- constant energy flux per unit tissue mass (“isometry”; k = 1) nor NoDerivatives License 4.0 (CC BY-NC-ND). of surface-volume constraints on heat dissipation over the sur- 1 Present address: Institut de Ciència i Tecnologia Ambientals (ICTA-UAB), Universitat face of geometrically similar body plans (“surface law”; k = 2/3). Autonoma de Barcelona, 08193 Barcelona, Spain. These mismatches provoked questions as to the origin of near 3/4 2 To whom correspondence may be addressed. Email: i.a.hatton@gmail.com. metabolic scaling but also the generality of the exponent, which is This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. now known to depend on many factors, including metabolic ac- 1073/pnas.1900492116/-/DCSupplemental. tivity level, taxonomic group, taxonomic level, body mass range, First published October 7, 2019. 21616–21622 | PNAS | October 22, 2019 | vol. 116 | no. 43 www.pnas.org/cgi/doi/10.1073/pnas.1900492116 consistent with multiple factors dominating under different cir- Metabolism (W). Consistent with prior work (1, 2, 7, 21), we find cumstances but inconsistent with a single universal process. that basal metabolism scales near 3/4 within some major groups In this work, we tested several of the principal assumptions (Kleiber’s law), but when viewed across 20 orders of magnitude, it and/or predictions of metabolic theories in ecology by linking is clear that distinct shifts in elevation occur between major and scaling relationships for different characteristics, with a view minor groups (e.g., ectotherms and mammals). These shifts are toward the ultimate origin of ∼3/4 scaling. These theories as- such that across all taxa, metabolism scales near isometrically, with sume that metabolism is the “master rate,” thus limited solely an exponent near 1 (k = 0.96; Fig. 1A) (4, 5). These shifts between by physical factors related to body mass and so should apply to groups are partially reduced by correcting metabolism to a standard all eukaryotes. The most comprehensive test of current theory temperature (2, 5, 6), but even correcting endotherm metabolism thus calls for data over the full eukaryote domain. Here we first still results in slopes k > 0.92 across the full eukaryote domain (SI consider the data for each characteristic separately and then Appendix, Fig. S4). This implies that mass-specific basal metabolism examine the links between them. is strongly bounded across eukaryotes (Fig. 2A), consistent with previous studies on smaller but equally broad datasets (4, 5). Scaling Across Eukaryotes We compiled data across eukaryotes for 4 core ecological Abundance (Individuals/m2). Abundance (i.e., population density of a variables, comprising 22,761 estimates drawn from nearly 2,800 species) data are drawn from different ecosystems globally, with published sources and meta-analyses (Fig. 1 and Table 1). These each estimate representing a snapshot for a species or an average data are of varying quality, with unequal representation across the value of several such points (SI Appendix). Consistent with prior size spectrum. While data for mammals and birds are generally work (1, 10, 11, 13), we find that abundance scales with body mass extensive, data are particularly limited for unicellular protists and near −3/4 within some major groups (Damuth’s law) (2, 10, 11, 13), less complete for some groups of invertebrates. Data often exhibit but that across eukaryotes, the exponent is closer to −1 (k = −0.95; considerable residual variability, limiting the accuracy with which Fig. 1B). Surprisingly, we find that the slopes within groups and the exponent values can be determined within particular taxonomic shifts in elevation between groups are largely reciprocal to basal groups. At the largest scales, however, estimated exponent values metabolism (Fig. 1A). ECOLOGY are typically robust to different assumptions, measurement tech- Species abundance is known to vary greatly through time or niques, and regression methods, as well as more specific con- along an environmental gradient, and thus it is not surprising that siderations (Methods and SI Appendix). These relationships the abundance-mass relationship reveals large residual variation. offer a comprehensive view to distinguish the scaling across Contributing to this residual variation is the trophic level of a and within major groups, which helps delimit the generality of species, with, for example, carnivorous mammals approximately any underlying process. 10-fold less abundant than herbivorous mammals (Box 1), which A B 104 1012 W = 0.001m0.96 N = 0.38m 0.95 Basal metabolism (watts) W R2 = 0.96 R2 = 0.81 Abundance (ind./m2) N 108 1 n = 3965 n = 3051 k= 4 104 10 k= 8 10 1 - 1 - - - -- - - 12 4 - - -- -- 10 10 k k - - -- 16 -- 8 - 10 10 1 C D 108 P = 2.7m0.74 Z = 0.74m 0.24 4 Maximum growth (g/yr) P 10 R2 = 0.97 R2 = 0.60 104 Mortality rate (1/yr) Z n = 2729 n = 3798 1 1 k=0 4 10 1 - - 0 -- 8 - - - -- - 10 - -- -- - 4 k k 10 1 - 12 k= 10 12 8 4 10 10 10 1 104 108 10 12 10 8 10 4 1 104 108 Mammal Protist Body mass (g) m Plant Ectotherm Fig. 1. Scaling of basic variables with body mass. (A) Basal metabolism, (B) population abundance, (C) maximum growth, and (D) mortality rate. Points (n) in all plots (A–D) are separate species values, except for plants represented by multiple points for the same species. Ectotherms and protists were also not aggregated into species values in D, due to limited mortality data among smaller sized species. For illustration, we have split eukaryotes into mammals, protists, plants, and ectotherm vertebrates and invertebrates. More resolved groups down to taxonomic orders are detailed in SI Appendix, Figs. S1–S3 and Tables S1–S4. Birds (gray points) often have similar rates to mammals and thus are difficult to see in the plots (SI Appendix). We show bacteria (also gray points) for reference, where available, but limit our discussion to eukaryotes. Scaling exponents k and 95% CIs are shown in the insets for major groups (these exclude birds and bacteria). Black empty circles are humans, with ranges shown in B for cities and hunter-gatherer communities (not included in the analysis). Hatton et al. PNAS | October 22, 2019 | vol. 116 | no. 43 | 21617 Table 1. Scaling of basic, transformed and combined variables. Mortality (1/yr). In general, mortality rate is the inverse of life span (years), but life span is commonly defined in different ways. Expected Observed Basic variables Units Relation Scaling Scaling C.I. Maximum life span is often measured in captivity under ideal conditions and represents an intrinsic physiological potential, often i. Metabolism W watts W~m 0.75 0.96 ± 0.006 quite distinct from the extrinsic ecological reality. In contrast, av- ii. Abundance N ind./m2 N~m 0.75 0.95 ± 0.016 erage field life span is measured over a population in the wild and iii. Growth P g/yr P~m 0.75 0.74 ± 0.005 represents a more realistic but more variable measure. Estimating mortality in the wild is more difficult for species such as trees and iv. Mortality rate Z 1/yr Z~m 1 0.25 0.24 ± 0.006 fish that produce large numbers of offspring, most of which die in Transformed variables early ontogeny. Our data include both measures of mortality and, following previous studies, are normalized to field mortality rates i. Specific metabolism watts/g W/m ~ m 1 0.25 0.04 ± 0.006 when possible (20). As might be expected from the challenges in ii. Population biomass g/m2 Nm ~ m1 0.25 0.05 ± 0.016 measuring mortality, there is significant dispersion in the rela- iii. Growth rate 1/yr P/m ~ m 1 0.25 0.26 ± 0.005 tionship. However, consistent with prior work (1, 2, 19, 20), we find that mortality scales near −1/4, both across and within many major iv. Lifespan yrs 1/Z ~ m1 0.25 0.24 ± 0.006 groups. Data are limited among smaller-sized species, so that we Combined variables are less confident of the scaling within some groups, such as pro- tists and invertebrates (Fig. 1D). H1. Population metabolism watts/m2 WN ~ m0 0 0.01 ± 0.013 H2. Lifetime growth unitless P/m/Z ~ m0 0 0.01 ± 0.007 Testing Equivalence Hypotheses H3. Growth efficiency g/yr/watt P/W ~ m0 0 0.19 ± 0.009 By variously combining any 2 of these 4 scaling relationships through multiplication or division, we can obtain combined H4. Lifetime metabolism watts/g/yr W/m/Z ~ m0 0 0.22 ± 0.009 variables that have previously been proposed to be invariant with Each basic variable scales with body mass m raised to a power, often body mass, termed “equivalence relationships” (Table 1). By expected to be α = 3/4. Observed scaling exponents are for all eukaryote “equivalence,” we do not mean to imply that everything is equal data shown in Figs. 1 and 2 (with 95% CIs). Exponents that differ signif- and residual variation is small, but simply that there is no significant icantly from expectations (α ≉ 3/4) are boxed in red, while matches (∼3/4) trend with body mass. These equivalence relationships form core are boxed in green. Basic variables can be transformed and combined to assumptions and/or predictions of the metabolic theory of ecology yield 4 hypothesized equivalence relationships with expected mass ex- ponents near 0 (H1 to H4). H1 is supported even though W and N do not (2), but so far have only been validated within major groups. We themselves match expectations, because both basic variables remain in- treat these as hypotheses that can be formally tested across eukary- verse to one another. H3 and H4 are rejected because α ≉ 3/4 for metab- otes, where the null expectation is a mass exponent of 0 (k = 0). We olism (W ) across eukaryotes. have combined variables using taxonomic information down to the species level when available, to preserve as much of the residual variation as possible (Methods and SI Appendix). This allows us to themselves are at least 10-fold less abundant than plants (SI Appendix, Fig. S8). Further accounting for their very high den- sities, the plant data in Fig. 1B are for monoculture stands of Transformed variables Combined variables trees rather than natural, more diverse assemblages (SI Appen- dix). Birds exhibit a very weak abundance-mass relationship with A Specific metabolism watts/g E Pop. metabolism watts/m2 0.1 high residual variation, which is not fully understood (12). Much 1 4 k = 0.04 10 of the remaining residual variation reflects species population 4 10 fluctuations rather than systematic variation between species, B Population biomass g/m2 suggesting that residual variation across species is only marginally k = 0.01 8 104 10 greater than that within species (SI Appendix, Fig. S5). Both the scaling and residual variation in the abundance-mass F Lifetime growth Unitless 1 2 relationship can also be linked to several other well-known abun- 1 10 dance patterns. In Box 1, we transform the abundance data in Fig. k = 0.01 4 10 1B (N ∼ mk) by applying simple functions to N and m, which allows k = 0.05 predictions for other types of relationships and distributions widely C Growth rate 1/yr G Growth efficiency g/yr/watt 108 studied in ecology. Overall, the abundance data reveal broad 104 consistency with other patterns, but also highlight several critical 104 mismatches calling for further attention (SI Appendix, section 8 1 k = 0.26 k = 0.19 and Fig. S8). D Lifespan yrs H Lifetime metabolism watts/g/yr Growth (g/yr). To ensure that data are comparable across very 1 2 k = 0.24 k = 0.22 1 10 different groups, measures of growth were integrated over the 4 entire life cycle and are equivalent to the maximum population 10 2 10 growth rate (intrinsic growth rate, or rmax) multiplied by individual 12 8 4 10 10 10 1 104 108 10 12 10 8 10 4 1 104 08 10 adult body mass (2, 15, 16, 33). Alternative measures of growth that apply only to particular groups or life stages tend to converge Body mass (g) Mammal Protist Plant Ectotherm on similar measures, so that our confidence in the growth re- lationship across eukaryotes and within most major groups is rel- Fig. 2. Scaling of transformed and combined variables with body mass. Data are the same as in Fig. 1 but transformed (A–D) and combined (E–H) as atively high (SI Appendix). Consistent with prior work (2, 15–17), per Table 1, using taxonomic information down to the species level, where maximum growth exhibits ∼3/4 scaling within major groups, available. The dashed lines show within-group relationships, while the solid similar to within-group metabolic scaling. However, in contrast to line shows the cross-group relationship. Colors of the exponent k correspond metabolism, robust ∼3/4 growth scaling is also preserved across to colors of boxes in Table 1. The x- and y-axes have equal order of mag- groups (15–17) (Fig. 1C). nitude spacing. Boxplots are shown for each variable. 21618 | www.pnas.org/cgi/doi/10.1073/pnas.1900492116 Hatton et al. in population metabolism (10–13), this has not been previously Box 1. Linking abundance patterns. tested across all eukaryotes. We find that the exponents and the Abundance can be viewed in many ways in ecology. The data from shifts in elevation between groups of each variable are reciprocal. Fig. 1 B can be transformed by applying simple functions to N and/ Specifically, the residual variation in each of the metabolism- or m to make predictions for four other well-known ecological mass and abundance-mass relationships (Fig. 1 A and B) is abundance patterns. These predictions are compared to the known partly compensatory, so that when a species (or group) is above patterns (A to D) to highlight consistency and identify mismatches. the line in 1 variable, it tends to be below the line in the other variable and vice versa. As a result, the product of these 2 var- 104 108 1012 Carnivore Abundance-mass scaling N~m 0.95 iables shows an equivalence in energy use at the population level Herbivore Different major groups are that appears to hold across 20 orders of magnitude in body mass Protist known to exhibit other known Plant N abundance relations and (combining Fig. 2 A and B gives E). 1 Invert. distributions. These groups 4 H2: Lifetime Growth. Lifetime growth (dimensionless) estimates the 10 Vertebrate are sub-sampled from the m 8 maximum number of offspring produced over the average lifetime 10 Bird original data in Fig. 1 B. 10 12 10 8 10 4 1 104 108 of an individual. This hypothesis, also known as “lifetime re- Sample productive effort” (33) or the “equal fitness paradigm” (17), Function Prediction Known pattern suggests that populations are broadly near a steady state in Inverse of N A ~ m0.73 A. Home range scaling abundance, such that reproductive rates multiplied over an av- 108 erage life span in the wild should be nearly constant across Mammal home range area, A species (2, 17, 33). Combining reproduction and survival in this 106 A is known to scale with mass, way has been proposed as a key component of fitness (17, 33). m, but as A ~ m1 (1, 43, 44). 104 1/N Data are limited for field estimates of reproductive rates, and Suggests encounter rates 100 m and grouping scale with size. our use of maximum reproductive growth will tend to over- 100 104 106 estimate actual lifetime growth in the wild (SI Appendix). ECOLOGY Moreover, data remain limited among smaller-sized classes for Sum of N in f(m) ~ m 0.88 B. Size spectra scaling fully evaluating this hypothesis among protists and invertebrates. bins of m 108 Size-frequency distributions Despite these limitations, and consistent with prior work (2, 17, f(m) in aquatic communities are 33), we find broad support for this equivalence relationship 104 power law, but k 1 (1, 46). across eukaryotes (combining Fig. 2 C and D gives F). 1 Suggests diversity scales 4 m negatively with size class. H3: Growth Efficiency. Growth efficiency (g/yr/W) estimates the 10 12 8 4 10 10 10 1 104 maximum amount of mass produced per unit energy of basal Mean & var. 2Nm ~ metabolism. This hypothesis, also known as the “cost of growth,” Nm1.92 C. Mean-variance scaling 108 of Nm in suggests that an approximately constant fraction of metabolism is bins of m Population variation in space 104 and time scales with the allocated to growth (27, 28, 32), which forms a basic assumption 2Nm in many growth models, going back to that of Bertalanffy (18, 25, 1 mean from k = 1.5 to 2 (47). 26, 34). Our use of basal metabolism for estimating growth effi- 4 10 Consistent with variations Nm ciency may underestimate the energy needed to fuel maximum 8 in population biomass (Nm). 10 10 4 0.01 1 100 10 4 growth, so that this measure represents a possible upper limit of growth efficiency (27, 28). Consistent with prior work (27, 28), we Number of f(N) ~ (lnN; , ) D. Species abundance find that within major groups, growth efficiency is largely equiva- 0.4 species in bins of N Community-level species lent but that across groups, this hypothesis is not supported, with 0.3 f(N) abundance distributions are clear shifts between groups, such that mammals and birds are 0.2 often lognormal (48). several orders of magnitude less efficient than unicells in converting Consistent with mammal energy into new biomass (combining Fig. 2 A and C gives G). 0.1 N species abundances globally. 0.0 10 10 10 8 10 6 10 4 10 2 H4: Lifetime Metabolism. Lifetime metabolism (W/g/yr) estimates the amount of basal energy used per gram of tissue over the Abundance patterns in A and B relate to the scaling of the N ~ m relation, whereas C and D relate to the structure of residual variation maximum lifetime of an individual. This hypothesis, also known as in N ~ m. While the functional forms of all four known patterns are the “rate of living” hypothesis (and characterized as “live fast, die recovered, there are mismatches of predicted and known scaling in young”), is based on oxygen radicals produced as byproducts of A and B, which suggest other factors (see SI Appendix and Fig. S8). metabolism that are known to accelerate senescence and thus reduce life span (2, 9, 29). Whereas H2 applies to average eco- logical lifetimes, this hypothesis relates to the maximum physio- logical lifetime. While prior work has shown mixed support for this characterize the scaling and coefficients both within and across major hypothesis (9, 29), data remain limited for fully rejecting this hy- groups for each of 4 equivalence hypotheses, H1 to H4 (Table 1 and pothesis within groups (SI Appendix, Fig. S7), but across groups, Fig. 2 E–H). our data do not support this hypothesis (combining Fig. 2 A and D 2 gives H). H1: Population Metabolism. Population metabolism (W/m ) esti- In summary, our data support 2 previously proposed equiva- mates the total amount of basal energy used by a population per lence relationships connected to the population energetics (H1) unit area. This hypothesis, also known as the “energetic equiva- and steady-state dynamics (H2) of abundance but are inconsistent lence rule,” suggests an energetic basis of abundance, whereby with 2 other equivalence hypotheses often used to argue for a energy partitioning among species in space exhibits no trend with metabolic basis for growth and mortality scaling (H3 and H4). This body mass (2, 10, 12, 13). Although prior work has shown that raises the question of the ultimate basis for the scaling of these within major groups, the product of the reciprocal ±3/4 scal- variables and in particular the origin of k ∼ 3/4 growth scaling ing of basal metabolism and abundance gives an equivalence across eukaryotes. Hatton et al. PNAS | October 22, 2019 | vol. 116 | no. 43 | 21619 Linking Growth and Metabolism Several authors have suggested that basal metabolic scaling is Many of the theories for body mass scaling are based on structural rather underpinned by growth scaling and not the reverse (9, 14, dimensions of the body thought to limit the flux of raw materials or 23, 27, 30–32). The idea that metabolism adjusts to growth factors products of metabolism (3, 9, 24, 25). The flux of energy needed to rather than growth being limited by structural constraints on me- support all life processes is in turn assumed to dominate the tabolism has been proposed for a variety of different reasons. In SI scaling of other characteristics (1, 2, 7, 11, 15, 18, 20, 26). While Appendix, section 9, we summarize some of the different lines of our results are indeed consistent with a metabolic basis for evidence for this proposal. In addition, we list growth phenomena abundance both across and within groups (H1), it is only within at different levels of organization and stages of development, be groups that metabolism can account for the allometric scaling of they normal, abnormal, or experimentally induced, that are known other variables, such as growth and mortality (H3 and H4). to have downstream effects on metabolism. Extreme limits on Basal metabolism exhibits 2 scaling regimes: an allometric metabolism can always limit growth, but except in some cancers regime within groups (k < 1, and typically k ∼ 3/4) and a near- (e.g., the Warburg effect) (37), normal or experimental changes in isometric regime across groups (k ∼ 1). These 2 scaling regimes metabolism rarely cause downstream changes in growth. Growth is often seen to be regulated upstream of metabolism, which adjusts are also observed in mammals in different activity states. Basal to fuel protein synthesis and cell replication and ensures coregulation metabolism across mammals scales near 3/4, as has long been in both variables (SI Appendix, Fig. S8). Despite the many indica- known (1, 2, 7, 21), while maximum (35) and minimum (torpor) tions that metabolism adjusts to growth, so far no quantitative (36) metabolic rates scale as k ∼ 1 (Fig. 3B). Mass-specific meta- theory for ∼3/4 growth scaling has been proposed that encompasses bolic rates across eukaryotes and across activity levels in mammals the numerous instances in which such scaling is observed. are thus limited over the same 3 orders of magnitude. This iso- metric regime suggests the existence of strict metabolic boundaries Outlook. We believe there may be a universal process underlying the above which eukaryote cells may be damaged by metabolites or ubiquitous tendency for growth to scale near 3/4. This single growth unable to pack mitochondria and below which tissues cease to scaling regime suggests something fundamental in the dynamics of maintain function or become unviable (4, 5, 8, 25). Within this how mass changes over time across very different kinds of living broad 3 orders of magnitude scope, basal metabolism exhibits a systems. The idea that metabolism adjusts to fuel growth within tendency to scale near 3/4 within groups but is systematically groups and adheres to metabolic boundaries across groups ap- staggered in elevation across groups so as to remain within near- pears to provide a more parsimonious explanation for the link isometric metabolic boundaries (Fig. 3C). These 2 distinct scaling between the growth and metabolism relationships shown in Fig. regimes are suggestive of multiple underlying processes. 3 and is supported by multiple lines of evidence (SI Appendix, Growth, on the other hand, appears to exhibit a single scaling section 9 and Fig. S8). We propose that basal metabolism in part regime (k ∼ 3/4) both within and across species, with no systematic reflects ontogenetic adjustments and evolutionary adaptations to shifts in elevation, which is consistent with a single underlying efficiently fuel growth and turnover within groups, and that groups process. Similar growth scaling is also observed in mammals at are staggered to remain within energetic limits. This could account different life stages. Maximum reproductive growth (mammals in for why basal metabolism and maximum growth within groups have red in Fig. 1C) (15, 16) scales very similarly to maximum onto- such similar scaling (Fig. 3A), but it does not solve the problem of genetic (14) and prenatal (19) growth (Fig. 3B). We also observe how this scaling arises. similar ∼3/4 scaling in the growth of whole communities across We are not prepared to speculate as to what general process large biomass gradients in grasslands, forests, lakes, and oceans might underpin growth scaling, but in many of the instances in (16). These community-level growth relationships represent a which such scaling is observed, we are essentially seeking an higher level of organization that cannot be accounted for by any explanation for the relationship between mass m and its de- α strictly individual-level constraint. This single scaling regime, de- dt = cm , where α ∼3/4 and c is a constant (33). For α < 1, rivative, dm scribing growth across and within groups and across life stages and this relationship implies a continually diminishing growth rate levels of organization, eludes any single metabolic explanation and and, more specifically, a characteristic scale-free form of sub- instead may point to a more general underlying process. exponential growth. Integrating this relationship, we see how A Eukaryote scaling B Mammal scaling bounds C Causal pathways 104 --- ---- Activity level 104 Carnivore Metabolism - -- - - - - - log Growth & Metabolism Basal metabolism Growth and metabolism Growth and metabolism 1 (watts) Herbivore (watts) k= Maximum -- 1 -- k= - Protist Basal 4 1 - 1 - Torpor 10 Plant - - k - Invertebrate 8 1 Life-stage - 10 k= - 4 Vertebrate Ontogenetic -- - - 10 - 12 -- -- - 10 Max. growth Bird Max. growth Reproductive k= (g/sec.) (g/sec.) k - Prenatal 16 All taxa 8 10 10 10 12 10 8 10 4 1 104 108 k 1 1 100 104 106 108 k 1 Body mass (g) Body mass (g) log Body mass Fig. 3. Growth and metabolism size class scaling. In A and B, original data from Fig. 1 A and C are binned into logarithmic body size classes to highlight the principal relationships and give equal representation to different size classes. Exponents and 95% CIs are shown to the right of each plot for the nonbinned data, with filled circles for metabolism (above) and empty circles for growth (below). (A) Scaling relationships for basal metabolism (W) and maximum growth (g/s) from Fig. 1 A and C (with additional taxonomic groups) reveal scaling similarities for most groups but a systematic divergence at larger body mass across groups. (B) Mammal scaling relationships for metabolism across activity levels and maximum growth across life stages shows the boundaries in which these characteristics vary. (C) The prevailing view is that metabolism determines growth scaling (orange arrows of causality) on the basis that within a given major group, both variables scale as k ∼ 3/4 (parallel thick lines). The metabolic view, however, cannot explain why growth follows the same universal k ∼ 3/4 scaling both within and across groups, given that metabolism often shows shifts in elevation between groups. A more parsimonious view is that across groups, metabolism adheres to isometric bounds (k ∼ 1), but that within major groups, metabolism adjusts to growth scaling of k ∼ 3/4 (blue arrows of causality). 21620 | www.pnas.org/cgi/doi/10.1073/pnas.1900492116 Hatton et al. 1 are provided in SI Appendix. Further analysis is possible using the raw data mass changes over time and approaches mðtÞ ∼ t1−α. Thus, this and source code available at https://zenodo.org/record/3145281. solution predicts how the timing of life history events will scale Metabolism measurements were normalized to 20 °C for all taxa except with body mass as t ∼ m1−α, consistent with the timing of ges- mammals and birds, using both published values of Q10 and the Arrhenius tation, maturity, and life span (1, 2, 20, 38) (Fig. 2D). Un- factor with standard activation energies, revealing only slight differences derstanding the origin of this simple growth relationship may among methods that do not alter our conclusions (SI Appendix, Fig. S4). All shed light on a number of other linked characteristics and abundance data were gathered over a relatively large area and originally represents a critical area for further research. reported in aerial units (e.g., m2), including aquatic species, normalized over multiple depths in the water column. The mammal growth shown in Fig. 3B Conclusion was calculated for each life stage as follows: (i) maximum reproductive growth is from population time series data or, more commonly, calculated from life The data that we report here provide a comprehensive view of history measurements, integrated over the entire life cycle; (ii) maximum on- the most general boundaries within which life exists. Although togenetic growth is obtained from near the inflection point of somatic growth the data are of varying quality, gathered over many decades curves of different mammal species (14); and (iii) prenatal growth is calculated using different methods and for different objectives, the pat- as birth mass divided by gestation period (19) (SI Appendix). terns exhibit remarkable regularity and surprising connections We used ordinary least squares (OLS; type I regression) to calculate regression between core species characteristics. statistics, which is the standard approach in bivariate regression when the Population abundance is known to be highly variable, but when dependent variable is measured with greater error than the independent viewed at the largest scales reveals a reciprocal pattern with me- variable (1) (SI Appendix). Exponents from reduced major axis (type II regres- tabolism, mirroring both the slopes and shifts in elevation sion) are similar to OLS for all cross-taxa regressions, although differences are within and across major groups. This supports an equivalence apparent among more resolved groups, and can be obtained from SI Ap- pendix, Tables S1–S4. The binning of data shown in Fig. 3 was achieved by in population-level energy use, consistent with a fundamental en- taking the geometric mean value in each logarithmic size class, which allows ergetic basis to abundance that spans all eukaryotes (H1). The equal representation for different size classes when data are not evenly dis- maintenance of steady state in abundance is necessarily mediated by tributed across the size range (7). the dynamics of birth and death, which also results in an equiva- We combined variables to test equivalence hypotheses (Table 1) using several lence in lifetime growth (H2). These relationships are intriguing taxonomic levels to ensure that residual variability in combined variables is ECOLOGY given that so many presumably adaptive traits scale with body size, largely preserved. When direct species matches could not be made in both and yet across eukaryotes, their combined influence has a relatively datasets, we combined measurements from the more-dispersed variable neutral net effect on population energy use (H1) and individual with order-level regression predictions from the less-dispersed variable, lifetime growth potential (H2). These 2 broadly supported equiva- which preserves the majority of residual variation while ensuring that re- lence relationships, in connection with other known abundance gression equations are comparatively reliable. In the relatively few cases patterns (Box 1), suggest the possibility for a more synthetic un- where limited taxonomy precluded order-level regressions, we then com- bined estimates with regression predictions for major groups, and we derstanding of the different aspects of abundance in ecology. linked all variables using the same 3 taxonomic levels for all hypotheses in The widely held view that a metabolic basis underpins the al- Table 1. More specific treatments for particular groups and paired variables lometric scaling of growth and mortality is not supported across are described in SI Appendix. eukaryotes (H3 and H4). Most importantly, the consistent ∼3/4 scaling of growth found across groups cannot be explained by a ACKNOWLEDGMENTS. We thank B. Drossel, M. Smerlak, A. Machac, single set of metabolic constraints, given that metabolic scaling K. Scherrer, J. R. Burger, J. M. Grady, and 2 reviewers for commenting on exhibits different scaling regimes within and across groups. In an earlier version of the manuscript and A. M. Makarieva, W. K. W. Li, and contrast to metabolic scaling, the striking similarities in growth A. Belgrano for sharing data. Funding was provided in part by the James McDonnell Foundation and the NSF (Award 1115838, to A.P.D.), the Czech scaling within and across groups, and also across life-stages and Science Foundation (16-26369S, to D.S.) and the European Research levels of organization, are indicative of a single generating process, Council (ERC) under the European Union’s Horizon 2020 Research and In- begging further understanding. We propose considering the dy- novation Programme (Agreement 682602, to E.D.G.). 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