The gill-oxygen limitation theory (GOLT) and its critics

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Science Advances  06 Jan 2021:
Vol. 7, no. 2, eabc6050
DOI: 10.1126/sciadv.abc6050


The gill-oxygen limitation theory (GOLT) provides mechanisms for key aspects of the biology (food conversion efficiency, growth and its response to temperature, the timing of maturation, and others) of water-breathing ectotherms (WBEs). The GOLT’s basic tenet is that the surface area of the gills or other respiratory surfaces of WBE cannot, as two-dimensional structures, supply them with sufficient oxygen to keep up with the growth of their three-dimensional bodies. Thus, a lower relative oxygen supply induces sexual maturation, and later a slowing and cessation of growth, along with an increase of physiological processes relying on glycolytic enzymes and a declining role of oxidative enzymes. Because the “dimensional tension” underlying this argument is widely misunderstood, emphasis is given to a detailed refutation of objections to the GOLT. This theory still needs to be put on a solid quantitative basis, which will occur after the misconceptions surrounding it are put to rest.


The need for a theory

To make sense of scientific data and their patterns, robust theories are required, which can provide an interpretative context for new findings, or which cannot, in which case the new findings are either problematic or very interesting (1). However, a situation can emerge where the practitioners of a given scientific discipline have forgotten Darwin’s dictum “odd it is that anyone should not see that all observation must be for or against some view if it is to be of any service!” (2). They not only publish articles that do not test anything but also, in the process, appear to have become utterly theory adverse and argue that the organisms or processes they study are so unique that only their ad hoc hypotheses can explain the data they generate.

This attitude is very problematic at a time when we, as a scientific community, are challenged to devise novel ways to protect marine and freshwater biodiversity threatened by overfishing, pollution, and habitat modification (3, 4) and by global changes with its attendant ills, ocean and freshwater warming (59), and acidification and deoxygenation (1012).

This is why the gill-oxygen limitation theory (GOLT) is being reintroduced here, and the case made for it to be seriously (re-)examined. Despite it being counterintuitive to the air-breathing mammals that we are, the GOLT is coherent in its content and the range of phenomena that it claims to explain. A critical examination should replace dismissals based on untenable arguments, which have created the strange situation wherein the GOLT has become controversial, e.g., in internal deliberations of the Intergovernmental Panel for Climate Change, even before it has become widely known.

To counter the tendency to discredit proposed hypotheses (rather than test them), here, after a brief presentation of the key tenets of the GOLT, a detailed presentation of the objections to the theory will be provided. Many of these objections do not pass simple tests of scholarship (e.g., they cite things that were not stated), strong evidence, or logic. Thus, the intention of the paper is primarily to clear the field of frivolous arguments such that a serious debate can begin.

This contribution is also an attempt to change the minds of aquatic biologists about notions most think are obvious, but which are incompatible with the fact that, for water-breathing ectotherms (WBE; i.e., most fish and aquatic invertebrates), life is shaped more by the distribution and concentration of dissolved oxygen (13) and the temperature of the water surrounding them than by the availability of food, which is more important for endotherms (birds and mammals). The tendency to project our mammalian biases onto WBE has resulted in a misunderstanding of many features and life histories of fish and marine invertebrates.

These are strong claims, especially because the framework of an alternative vision of the lives of WBE, i.e., the GOLT (14, 15), has, to date, not found many adherents. Recent extensive (16) and shorter elaborations (17, 18) of the GOLT exist. Each parsimoniously explains several biological features of, behaviors of, and experimental results with WBE that mostly have no other (simple) explanations (Table 1).

Table 1 Some physiological and related differences between young and older WBE.

Here, item (1) is the cause of all others. “Relative” stands for “per unit weight.”

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The GOLT presents a unifying theory—based solely upon first principles and their corollaries—that explains growth and related phenomena in both marine and freshwater fishes and aquatic invertebrates. It should replace several ad hoc hypotheses common in ichthyology, limnology, and marine biology.

How the GOLT defines growth

The GOLT builds on concepts developed by von Bertalanffy (1924), who built on earlier work by Pütter (25), and whose main feature is that organic growth (dw/dt) can be seen as the difference between two processes, i.e.dw/dt=Hwdkw(1)where the two terms on the right are usually called anabolism and catabolism, respectively, and where d < 1. Here, an increase of body mass (dw/dt) is the difference between body mass that is newly (Hwd) synthesized and the body mass (kw) that is degraded (see below). As simple as Eq. 1 seems to be, considerable confusion exists regarding the definition of the two terms on the right.

In the GOLT, anabolism refers to the synthesis of body tissues (including gonad material); the process requires an amino acid pool to provide building blocks for proteins and adenosine triphosphate (ATP) to provide the “energy” required for synthesis. Here, ingested food is not energy; rather, food is oxidized (i.e., “burnt”) to generate ATP, which may be considered to be energy (26).

Thus, the process of anabolism requires oxygen, which must enter the body through some permeable surface. Therefore, in WBE, the parameter d in Eq. 1, is equivalent to the exponent (dG) of a relationship linking gill (or another respiratory) surface area (G) to body weight of the form G = a·WdG, which determines how much anabolism can occur. For this reason, the oxygen consumption of WBE scales with body weight with a factor (dO2) that should be and is near dG (27). As we shall see below, dG—and hence dO2 as well—drops below 1 once fish have grown past a certain body mass (and past metamorphosis in teleosts).

In contrast, catabolism, as defined in the GOLT, is directly proportional to body weight because it consists of the spontaneous denaturation of the proteins and other molecules contributing to that weight. Protein molecules can fulfill their function (e.g., as enzymes) only if they keep their native quaternary structure, usually maintained by weak H-bonds (28). In the long term, they cannot maintain that structure because they are constantly subjected to Brownian motion (2931). Thus, all such molecules have half-lives that become shorter when temperature increases (28, 32).

Spontaneous loss of quaternary structure by protein molecules occurs throughout the body and requires no energy (beyond the kinetic energy of Brownian motion). Thus, catabolism as defined in the GOLT requires no oxygen either. It is therefore weight proportional even if denaturation proceeds at different rates in different molecule types, because the ratios between molecule types would not change much in the course of ontogeny (at least past the larval stage).

Integrating the differential equation in Eq. 1 is straightforward, and when d in that equation is set equal to 2/3, this yields the von Bertalanffy growth function (VBGF), which for length has the formLt=L(1eK(tt0))(2)where Lt is the mean length (however measured) at age t of the WBE in question, L is their asymptotic length, i.e., the mean length they would attain after an infinitely long time, K is a growth coefficient (of dimension time−1), and t0 is the (usually negative) age they would have had at a length of zero if they had always grown in the manner predicted by the equation (which they usually have not, as the growth rate of fish larvae and early juveniles is usually more rapid than predicted by the VBGF) (33).

Combining this equation with a length-weight relationship of the form W = a·Lb leads to a version of the VBGF that can express growth in weight, i.e.Wt=W(1eK(tt0))b(3)where W is the mean weight attained after an infinitely long time and all other parameters are as defined previously.

When d ≠ 2/3, but still <1, the integration of Eq. 1 yields what may be called the generalized VBGF; for length, this isLt=L(1eKD(tt0))1/D(4)where D = 3(1 − d). Note that here, the exponent of the length-weight is equal to 3, as is (nearly) the case in the overwhelming majority of fish (34) (see also for fishes and for invertebrate species).

For weight, the generalized VBGF isWt=W(1eKD(tt0))b/D(5)where D = b(1 − d), which makes Eq. 5 more versatile than Eq. 3.

When d = 0.75 (and thus, D = 0.75), Eq. 4 is equivalent to what was called a “general model” of growth (35), which, however, is not general because the value of d does vary between taxa (14, 16). Note that whether one uses the standard VBGF or its generalized versions [including versions that account for ubiquitous seasonal growth oscillations (16)], a reasonably good fit to length/age data pairs is obtained, including estimates of asymptotic lengths (L) that are close to observed maximum lengths (Lmax). Important exceptions are tuna and other large WBE with relatively high values of d (i.e., 0.90 ≤ d ≤ 0.95, and hence, 0.3 ≥ D ≥ 0.15). In such cases, the estimates of L that are obtained are much higher than Lmax (compare Fig. 1A with Fig. 1B), with W being also overestimated (Fig. 1C).

Fig. 1 Different forms of the von Bertalanffy growth function.

The VBGF fitted to bluefin tuna (T. thynnus) length-age data pairs (137). (A) Standard VBGF (Eq. 2), which assumes d = 2/3, and hence D = 1 (which can thus be omitted). (B) Same length-at-age data, fitted by Eq. 4, with b = 3 and D = 0.3, corresponding to d = 0.9 (46). (C) Two versions of the generalized VBGF for weight (Eq. 5), with D = 1 and D = 0.3, with weights converted from lengths using W = 0.0182·L3 (from FishBase;, where W is in g and (fork) length is in cm. Note the position of Lm [from (138)] relative to Lmax, L and Wm relative to Wmax and W, and that the weights at inflection points of the growth curves (Wi) are much higher than Wm, i.e., that bluefin tuna growth is still accelerating when they reach maturity.

The scope of the GOLT

If one can agree with the above definitions and constraints, the various predictions of the GOLT (Table 1) follow logically while being empirically verified. Most natural scientists other than some fish physiologists, once informed of the points above, tend to accept the elements of Table 1 as straightforward corollaries. This is important, given two massive challenges related to the respiration of WBE in an age of global warming, i.e., the accelerating deoxygenation of the oceans and freshwater bodies, and the increasing role of aquaculture in supplying global seafood markets.

Studying the effect of temperature increases and deoxygenation requires a robust theory of why WBE, particularly old/large (and hugely fecund) individuals, are as sensitive as they are to such changes (3639). Similarly, for the insights of physiologists to be able to assist in increasing aquaculture production, the theory that guides them has to be compatible with the fact that large sums are spent by in the aquaculture industry to aerate the ponds in which WBE are raised (40, 41).


The terms of the debate

So, what are the objections? They are presented here in a series of tables briefly stating the objections and their sources and providing a brief refutation, along with a reference to one or several articles presenting the evidence cited as refutation.

Each table addresses a different class of arguments, i.e., (i) the gill surface area of WBE either does (or could) grow as required to keep up with a growing volume, i.e., with body weight; (ii) some WBE contradict key tenets of the GOLT (dG > 1, or large size in tropical waters); (iii) identifying the cause of the decline in metabolic rates with increasing weight; (iv) different definitions of “anabolism” and “catabolism”; and (v) miscellaneous discipline-related and/or philosophical objections.

Gill lamellae versus book pages

The first group of objections (Table 2) is also the most important. In fact, if any of these objections were tenable, then the GOLT would be eviscerated. These objections refer to gills functioning as a surface, and thus being limited by the geometric constraint that they cannot keep up with the three-dimensional (3D) growth of the bodies that they supply with oxygen (Fig. 2).

Table 2 Arguments raised against the GOLT: Claims by Lefevre and associates.

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Fig. 2 Schematic representation of water flows across the gills of a fish.

Note that once water has flown between lamellae (which extracted most of the O2 it contained), there is no point for this water to flow through another set of lamellae. Hence, gills function as a surface, although their arrangement in 3D space may suggest otherwise.

Some authors believe that this is a simple problem, i.e., that if their gill surface area is too small, WBE can simply enlarge it, i.e., grow bigger gills. However, these authors do not perceive the underlying geometric problem. So far, only one contribution (42) has tackled this problem head-on and advanced the following two fundamental arguments:

1) The surface area of gills is similar to the surface area of the pages of a book, which can increase in proportion to its volume. Similarly, gill surface area, which can be seen as equivalent to book pages, can always keep up with body weight.

2) The oxygen requirement of fish declines as they grow older/larger, and thus, these fish do not need to maintain the gill surface area/body weight ratio occurring in young/small fish.

The raison d’être of books is to be read, i.e., opened. The transfer of knowledge from their pages through the readers’ eyes to the brain is analogous to the transfer of oxygen from the water flowing past the gills to the blood of a WBE. In our 3D world, only one layer of paper can be read at a time. Similarly, despite appearances to the contrary, in fish, only one “layer” of gill lamellae is in the path of a flow of water across gills (Fig. 2).

Given the efficiency of gills at extracting oxygen from flowing water, there would be little to be gained by putting subsequent layers (i.e., “pages”) of lamellae behind the first one. Hence, gill surface area, as complex as it may appear, functions like a sieve, perpendicular to the water flowing through the gill chamber. This implies that in a WBE, gill surface area cannot grow in three dimensions and thus cannot keep up with the 3D body that it supplies with oxygen. Note that the improvements of gas exchange performance that may be achieved by changing the pattern of perfusion of the lamellae or reducing blood residence time by altering branchial blood flow would not overcome the “dimensional tension” occurring when a surface limits the rate of a process required to by a growing volume (17).

The disconcerting argument in (42), that the GOLT assumes gills to behave like spheres, becomes even more disconcerting when one notes that the argument that WBE are able to maintain the same gill surface area/body weight ratios as their weight increases is backed up with only two very questionable references. The first is a contribution on the gill area of spangled perch, Leiopotherapon unicolor, with a scaling factor dG = 1.04 (43) and, second, the bivalve, Solemya velum, with reported scaling factors of 1 between gill surface area and gill mass and dG = 0.85 between gill surface area and body weight (44) (see also Table 3).

Table 3 Arguments raised against the GOLT: Issues regarding gill surface areas.

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The choice of (43) is unfortunate, as there are several reviews, jointly covering more than 150 species, showing that typical scaling factor for fish gill surface area ranges from dG = 0.6 to 0.9 (27, 4548). For “medium-sized” fish (200 g), it was shown (27), on the basis of data from well over 200 papers covering 121 fish species, that the mean value of dG = 0.811 is very close to the mean estimate of the scaling factor of metabolic rate versus body weight, dO2 = 0.826, which confirmed the results of an earlier comprehensive review (49); thus, these two values were averaged to obtain a robust estimate of dG = dO2 = 0.82, assuming that dG causes dO2. This was based on Fick’s law of diffusion, which states that the total amount of oxygen that can diffuse into the circulatory system of a WBE isQ=dP·U·G·WBD1(6)where Q is the oxygen uptake (ml hour−1). Here, dP is the difference between the oxygen partial pressure on either side of the membrane (in atm); U is Krogh’s diffusion constant, that is, the amount of oxygen (in ml) that diffuses through an area of 1 mm2 in 1 min for a given type of tissue (or material) when the pressure gradient is 1 atm of oxygen per μ (μm), and G is the surface area of the gills (total area of the secondary lamellae). Last, WBD is the water-blood distance or the “water-capillary distance” (50), i.e., the thickness of the tissue between water and blood in μ (27) that cannot be reduced much without risks to the structural integrity of the gill lamellae.

Note also that Eq. 6 also applies to WBE that lack gills and blood, as part of a closed system for distributing oxygen to the tissues (51). One such example is provided by the arrow worms (Chaetognatha), in which, in the absence of gills, the body integument serves as respiratory organ (52) and whose thickness cannot be reduced as they grow.

However, the authors of (42) argue that, while fish could grow gill surface area such that dG = 1 is maintained, they do not need to do so because they suggest that “the activity of oxidative enzymes falls with body mass in fishes” and cited (53) as source. This argument is problematic for two reasons: the first obvious, the second less so: (i) It begs the question why growing fish—if not forced by a declining relative O2 supply—should have evolved to reduce their O2 consumption and shift from relying on oxidative to glycolytic enzymes, the latter catalyzing metabolic processes that are far less efficient than the former. (ii) It attributes to fish biological features (i.e., gill lamellae) that they could multiply and use but somehow choose not to, which makes its claims about fish physiology unfalsifiable. Thus, the authors of (42) can assert that dG < 1 in the overwhelming majority of fish species so far studied does not refute their claim that gill surface areas can grow according to dG = 1. At the same time, single (and questionable) cases with dG = 1 (43) “confirm” that gill surface area can keep up with body weight.

Argument (i) mistakes cause and effect (54) (Table 4). The physiologists who documented that the preponderance of oxidative enzymes in the tissues of small/young fish is replaced, in large/old fish by a preponderance of glycolytic enzymes, were well aware that this shift contradicts standard hypotheses about fish physiology (53, 5557). Thus, the authors of (57) titled the report of their findings: “A violation of the metabolism-size scaling paradigm: Activities of glycolytic enzymes in muscle increase in larger-size fish.” Note, however, that this feature not only is compatible with the GOLT but also is one of its consequences. Argument (ii) evidently points to Popper’s “decision criterion” that claims that cannot be falsified in principle are not part of science (1).

Table 4 Arguments raised against the GOLT: Mistaking cause and effect.

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Regarding the bivalve S. velum, the fact that a value of d = 1 is reported (44) for the scaling factor between gill surface area and gill mass misses the point. What matters here is the scaling factor between gill surface area and the weight of the entire body, which at dG = 0.85 is relatively high for a bivalve, but well under 1, as required by the GOLT. Other bivalves have values of dG and dO2 ranging between 0.5 and 0.8 (58) and thus suffer from “ontogenic anaerobiosis” (59). An exception may be the giant clams of the family Tridacnidae, which are phototrophic and thus produce their own oxygen (60).

Note that despite the argument that gill surface area grows under the constraints of a surface, it does not mean that this growth should be proportional to length squared, i.e., isometric growth, with dG = 2/3, although von Bertalanffy (24) thought so. He erroneously referred to instances of 2/3 < d < 1 as growth that is “intermediate between surface and weight proportionality” (Table 5).

Table 5 Arguments raised against the GOLT: Different definitions of anabolism and catabolism.

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However, values of dG ≈ 0.67 seem to occur only in very small fish such as the guppies which von Bertalanffy (22, 24) used to illustrate his theory of growth. An even lower estimate of dG = 0.60 was obtained for Mistichthys luzonensis, in which an adult does not reach more than 25 mm and a weight of about 0.05 g (61, 62). The other extreme appears to occur in bluefin tuna (Thunnus thynnus), where a well-documented value of dG = 0.90 for adult specimens has been published (46).

The matter with exceptions, or exceptions matter

As stated above, the GOLT is falsifiable, i.e., it would be refuted if well-founded estimates of dG ≈ 1 or worse dG > 1 were shown, for example, via a meta-analysis, to routinely occur in the adult stages of WBE (and excluding air-breathing taxa). So far, credible estimates of dG >> 1 have been found to occur only in teleost larvae (6365), which also breathe through their integument and fins (66), while estimates of dG ≈ 1 have been reported from juvenile fish transiting from the high dG values in larvae to the values of dG < 1 typical of the adults (64, 65).

There will be a tendency for published estimates of dG to be on the high side when, as is often the case, only the small representatives of a species are studied. In the case of spangled perch (43), the published estimate of dG = 1.04 pertained to juvenile fish reaching at most 30% of the maximum weight typically attained by that species (see Table 3). This was similar to the specimens of icefish (Chaenocephalus aceratus) (67), for which dG ≈ 1 [after correction from 1.09 due to the inappropriate use of a “type II” regression (68)]. More cases of this sort have been documented (69).

Thus, while this may appear as special pleading, in view of their theoretical importance, it may be recommended that dG values should preferably pertain to adults (i.e., larger than a third of the maximum weight typically reached by the species in question). In the future, it would be fair to expect criticisms of the GOLT to take account of existing meta-analyses, rather than search for isolated estimates that differ from the results of meta-analyses but seem to support one’s point.


Fish growth versus reproduction

One of the main issues in ichthyology, though it is not often perceived as such, is the relationship between growth and reproduction. The majority of authors writing on this topic repeat the usual belief that the relationship between growth and reproduction is explained by stating that “the growth of fish slows down upon reaching maturity because their energy is redirected from growth to reproduction,” or a variant of this phrase (7077). This notion implies a “biphasic growth” with a rapid growth phase before the length at first maturity is reached, and a slower phase thereafter, as illustrated by Fig. 3A.

Fig. 3 Two views of the relationships between size at first maturity and maximum size.

(A) Traditional view, where “linear” growth slows down when length at first maturity (Lm; black star) is reached, with growth then continuing at a reduced pace, depending on circumstances [i.e., a, b, or c; redrawn from (77)]. (B) More appropriate, but uncommon, view, with growth expressed as change in body weight (in line with Eq. 1). This shows not only that weight at first maturity in females and males (Wm; black star) is reached when growth is still accelerating (i.e., Wm < Wi, the inflexion of the curve) but also that females grow faster and reach larger weights than the males despite investing more in reproduction (see also text and Table 6). Graph based on length growth parameters, a length-weight relationship, and length at first maturity for Alaska pollock (Gadus chalcogrammus) in FishBase (, which contains hundreds of similar datasets.

What is not realized, however, is that this phrase, like all statements about complex phenomena, is a hypothesis. Moreover, this hypothesis is contradicted by four sets of observations: (i) Fish kept in aquaria and that never mature and spawn reach maximum sizes that are similar to those of reproducing conspecifics in the wild. (ii) In most fish species, the females are larger than the males, although they devote more energy to reproduction. (iii) In most fish species, growth in weight is more rapid after maturity is reached than before. (iv) Mean length at first maturity, in fish, correlates tightly with the maximum length that can be reached in a given environment.

Regarding item (i), popular aquarium fish such as clown loach (Chromobotia macracanthus) do not breed in captivity but still approach a common maximum length of about 16 cm (78). Similarly, most saltwater aquarium fish such as damsels or butterfly fish do not breed in tanks but again reach a common maximum length similar to the one in the wild (79); many of the saltwater or freshwater fish kept by home aquarists never mature and spawn. However, although they are fed ad libitum, they stop growing at some point. In addition, triploid (and thus sterile) fish exhibit growth patterns largely similar to those of their diploid brethren (80). This should suffice to kill the notion that it is reproduction that causes growth to cease. However, it has become a zombie idea: It does not die.

Similarly, regarding item (ii), in over 80% of fish families where females and males look alike, it is the females that eventually reach larger sizes (Fig. 3B), even if this growth dimorphism can become attenuated in certain circumstances (81). This strong female dimorphism should lead to a rethink of the notion that the cost of reproduction causes growth to decline. However, some authors, when confronted with this evidence, have doubled down and suggested that males have the higher reproductive cost.

One such case is (82) (see also Table 6, number 6.2); it was suggested (83), in an effort to refute the claim above (84), that males had the higher reproductive effort. This was backed with a graph from an unpublished thesis that did not even compare male and female reproductive output (see Table 6, no. 6.2). In reality, females are, by definition, the sex with the higher reproductive output, which also can be shown empirically in almost all groups of animals reproducing sexually [reviewed in (85)]. There are a few exceptions (e.g., parental care by male seahorse), but they are not pertinent here.

Table 6 Arguments raised against the GOLT: Spawning versus growth and vice versa.

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Regarding item (iii), Figs. 1C and 3B show that, in fish, the ratio of weight at first maturity (Wm) to asymptotic weight (W) can be much lower than the corresponding ratio for length (Lm/L), which is frequently not realized because the overwhelming majority of growth curves drawn reflect growth in length (Fig. 3A).

From length growth curves, one can get the impression that spawning strongly affects growth, hence the name “reproductive load” for the Lm/L ratio (86). However, growth is a process primarily involving mass (see Eq. 1), as reflected in weight growth curves. Weight growth curves have marked inflection points, where growth rate (dw/dt) is highest (at Wi), and thus, the question may be asked whether Wm > Wi or, on the contrary, Wi > Wm. Taking the second derivative of generalized VBGF for weight growth (Eq. 5) and setting it equal to zero allow us to identify Wi, where the growth rate changes from increasing to decreasingWi=W·(1(D/b))b/D(7)

As may be seen from Figs. 1C and 3B, the weight at the inflection point of these curves is higher than the mean weight at first maturity of the population in question (i.e., Wi > Wm). This result, which can easily be reproduced for multiple species of (large) fish (Table 7), implies that as fish reach maturity, their growth in weight is still accelerating, which refutes the reproductive load hypothesis.

Table 7 Theoretical versus empirical predictions of weight at fish maturity.

The “theoretical” predictions of Wm based on the GOLT (Eq. 9) match the empirical estimates based on Eq. 10 (90); the relationship of Wm to the inflexion (Wi; Eq. 7) of weight growth curves is also as predicted (see text).

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The question thus arises: If the reproductive load concept does not hold, i.e., if reproduction does not cause growth to decline, what then is the relationship between reproduction and growth in fish and, by extension, in other WBE?

Equation 1 with d < 1 implies that the heavier fish get, the less O2 per unit weight they will get, which should imply—other things being equal—more frequent occurrences of respiratory stress and hypercapnia. All we need to assume, therefore, is the existence of a threshold weight (Wm) at which the high frequency of respiratory stress or hypercapnia events triggers the hormonal cascade that leads to maturation (87). Thus, one can defineA=(W1/b/W1/b)D(8)from whichWm=W·(1/A)b/D(9)with A being the ratio of gill surface area (or O2 supply) at W over the gill surface area (or O2 supply) at Wm (16, 87).

A first estimate of A = 1.365 was published in 1984 (see Fig. 4C) (87), whose 95% confidence interval is 1.218 to 1.534, as estimated using the Fieller method (88) (see applied to the data of table S1. These data covered 56 pairs of Lm and L in 34 different fish species ranging from guppies to tuna and raised to the power of 3/(1 − d), which here substitutes for weights.

Fig. 4 Growing fish mature when their relative gill surface area reaches a threshold.

(A) In the ontogeny of fish, when their relative gill surface area declines, their oxygen supply declines as well; when the latter reaches 1.3 to 1.4 times the oxygen supply required for maintenance and routine activities, i.e., as fish increasingly get “out of breath” (and suffer from hypercapnia), the hormonal cascade is initiated that leads to gonad maturation and spawning. (B) If the same fish are in a stressful, e.g., warmer environment, causing oxygen demand to be elevated, the same 1.3 to 1.4 threshold will cause them to mature and spawn at smaller sizes. (C) Plot, whose 56 points represent the 34 fish species, ranging from guppies to tuna (87) (see the Supplementary Materials) used to estimate the average threshold value of 1.36 (with 95% confidence interval of 1.218 to 1.534). (D) Same plot but for different populations of redband trout (Oncorhynchus mykiss). (E) Ditto for Yellowstone cutthroat trout (Oncorhynchus clarkii). (F) Ditto for mountain whitefish (Prosopium williamsoni).

Because A−1 = 0.733, combining with Eq. 7 and rearranging (see the Supplementary Materials) lead to the conclusion that d > 0.733 implies Wi > Wm, d ≈ 0.733 implies Wi ≈ Wm, and d < 0.733 implies Wi < Wm. Thus, in small fishes, which usually had small values of d (e.g., 0.6 in the diminutive goby M. luzonensis) (16, 62), Wm > Wi, while the opposite, Wm < Wi, applies to larger fishes (e.g., bluefin tuna; see Figs. 1C and 3B).

This also aligns with the empirical relationships between Lm and L (with and without additional variables) in 265 fish species in FishBase (, covering 88 families and 27 orders, with an average scaling factor of ≈0.9 emerging (89). The simplest of these relationships waslog(Lm)=0.898·log(L)0.0782(10)when Lm and L are in cm.

Equation 10 implies that fish with an asymptotic length of 10 cm reach maturity at a length of 6.6 cm, while fish with asymptotic lengths of 100 and 1000 cm reach maturity at 52 and 412 cm, respectively. These values, when converted to weights, are well within the confidence interval of the value of Wm predicted by Eq. 7 (Table 7).

Unfortunately, Eq. 9 does not work below 1.3 cm, i.e., it predicts Lm > L. It can be hypothesized that all such small fish are semelparous, i.e., will spawn only once before they die, as documented in minute gobies (90). Equation 10 does not work either with large semelparous species such as Pacific salmon (Oncorhynchus spp.), whose reproductive strategy, however, is a derived trait connected with their diadromous life history (91).

Last, regarding item (iv) above, there is the huge environmental plasticity of fish, which can manifest itself both in individuals used for aquarium experiments (92, 93) and in the wild. Regarding the latter, it was noted that “tropical fishes living near the limit of their tolerance for low temperature grow to larger size at such temperatures” (94). In such cases, i.e., when the maximum length (Lmax) or the computed asymptotic length (L) changes, the mean length at first maturity (Lm) also changes in the same direction such that the ratio Lm/Lmax or Lm/L remains approximately constant.

The GOLT provides an explanation for the near constancy of Lm/Lmax or Lm/L by postulating that spawning is induced by the same mechanism that also causes growth to decline (i.e., asymptotic growth). As fish grow in weight, their gills, whose surface area has grown with the scaling factor d < 1, deliver less O2 per unit of body weight (Fig. 4A).

Thus, growing fish will gradually experience more respiratory stress and hypercapnia, and a level of either is finally reached that initiates the hormonal cascade leading to maturation (95, 96). Gonadal products are elaborated, often by using fat accumulated in the summer and fall as a fuel (97). When, in spring, the gonadal products are released, the gill surface area/body weight ratio increases again, and summer growth can resume, etc.

With time, however, the fish grow heavier despite generating an increasing reproductive output, and the ratio of gill surface area/body weight declining further (Fig. 4, A and B). Thus, growth gradually ceases, but life (and reproduction) does not need to, as exemplified by adult whitefish (Coregonus spp., Salmonidae) that can live a decade or more after they have ceased to grow (98). The same occurs in a number of coral reef fishes, for example, in the families Acanthuridae and Scaridae (99, 100).

The threshold gill surface area, and hence the relative metabolic rate at which spawning is initiated, is similar among different fish families (see Fig. 4C and fig. S2) because such a critical threshold would be conserved through evolutionary time. Thus, when the growth of teleosts causes their metabolic rate to drop to about 1.3 to 1.4 times their maintenance metabolic rate (i.e., something that fish can monitor in real time), then sexual maturation is initiated. Figure 4 (D to F) provides further examples of this generalization [see also (101)].

Temperature and maximum sizes

The major critique (42) of a contribution based on the GOLT that predicted that ocean warming would reduce the maximum size of fish (7) proposed no alternative explanation as to why fish should remain smaller at higher temperatures. Thus, another contribution (102) is examined here, in some detail, as its authors attempt to answer the question whether “oxygen limitation in warming waters is a valid mechanism to explain decreased body size in aquatic ectotherms” (Fig. 5A).

Fig. 5 Fish, at higher temperatures, tend to grow fast toward smaller maximum sizes.

(A) “Observed phenomenon” that needs to be explained [adapted from an insert in figure 1 of (102)]. (B) Simplified version of figure 1 in (25). (C) Atlantic cod (G. morhua) has wide geographic and temperature ranges; in Eastern Iceland (1° to 10°C), they reach much larger sizes than in French waters (8° to 18°C), based on data in (138, 139).

Answering this question would also solve the riddle posed earlier by an author (92) who was surprised by his observation, based on guppies raised at different temperatures that “[t]he results indicate that the differences in growth rate established in young fish do not persist throughout life. Initially slow-growing fishes may surpass initially fast-growing fishes, and finally reach a greater length-at-age,” as reported and illustrated earlier (Figure 5B) (25) and well documented in the literature, for example, for Atlantic cod (Gadus morhua) (Fig. 5C).

Six potential explanations were presented and discussed by these authors (102), as documented in their figure 1, from which all quotes below are extracted. These potential explanations are then summarized, illustrated (Fig. 6, A to F), and commented upon. All but the first of these potential explanations can be viewed as alternatives to the GOLT:

Fig. 6 Simplicity versus scope in explaining why higher temperatures lead to smaller sizes.

The six explanatory models are adapted from (102) and were presented in two columns, as “intrinsic mechanisms” (A to D) and “extrinsic mechanisms” (E and F). Here, they are arranged according to the perceived complexity of the mechanism(s) they require (abscissa) and their generality or “scope” (ordinate).

1) The GOLT [or “GOL hypothesis” in (102)]. The GOLT, based on the inherent properties of gills as 2D surface that must remain exposed to an oxygen-laden water flow, assumes that they will provide decreasing amount of oxygen per unit weight to the bodies of growing WBE (Fig. 6A). Hence, increased temperatures, which increase oxygen demand, will force them to remain smaller [Fig. 5C; see also (7)]. However, the fish kept at higher temperature may, at first, experience a more rapid growth than those kept at low temperature, which also explains the above quote [from (92)]. Note also that many inferences on the growth of fish and other WBE are based on juveniles, whose growth is usually accelerated by temperature increases, and not on adults, whose growth is often depressed by increased temperature (Table 1). The preference of researchers for working with juvenile fish is understandable (they require smaller aquaria, require less food, etc.), but it can lead to confusion, as illustrated by one of the few aquatic biologists who raised fish (albeit small ones) under different temperatures from larvae to adults (92) and who penned the quote above.

2) “Different temperature dependence of DNA replication (development) results in smaller cells and faster division at warmer temperatures.” Fish that remain smaller at higher temperatures have, to the author’s knowledge, never been shown to have smaller cells, and if they did, this would be the reason for their smaller size in warm water only if they had the same number of cells, as do, e.g., tardigrades and small nematodes. This, as well, has never been demonstrated. Hypothesis (2) (Fig. 6B) is probably another case of cause and effect being inverted (Table 4), as often happens when things correlate (103). Some of the largest fish, e.g., tuna, have very small cells, while the much smaller lungfish have large cells (104106). It seems that in fish at least, cell size is linked with DNA content and activity level but not with size (107). On the other hand, the higher cellular turnover implied by “faster cell division at warmer temperatures” would be associated with a higher rate of protein denaturation, which is a central tenet of the GOLT (see above).

3) “Decreasing growth efficiency at higher temperatures means that less energy is converted to growth.” This is not an explanation because it shifts that which must be explained from “reduced growth when temperature is high” to “decreased growth efficiency” (Fig. 6C), which is a restatement of the issue at hand. The GOLT explains decreased growth efficiency [i.e., K1, growth increment/food ingested (108, 109)] by pointing out that when WBE are exposed to higher temperatures, more of their oxygen supply is diverted to basal metabolism, leaving less available to assimilate food. Hence, the amino acid pool of fish spills over and “is excreted by the gills and kidney as incompletely oxidized nitrogenous compound”—the latter point from (110), which cites (111115) [see also (116)].

4) “Higher size-specific allocation to reproduction at higher temperatures […] leaves less energy for growth.” This argument (Fig. 6D), for which no supporting evidence was presented, is not pertinent in any case because the effects of temperature on fish growth manifest themselves well before size at first maturity is reached (see Fig. 5C).

5) “Faster increases in energy demand (metabolism, activity cost, etc.) compared with food availability leaves [less] energy for growth and reproduction in […] warmer environments.” This is a complex hypothesis, implying that tropical ecosystems make less food available to consumers than colder ecosystem (Fig. 6E), which would be hard to test. Fortunately, there is no need to because experiments can be and have been conducted in vitro where food is provided ad libitum and where fish kept at cooler temperatures grow to be larger than those at higher temperatures (92, 117, 118). The only reason this point is perhaps not obvious is that laboratory growth experiments are difficult to run with large/old fish and thus are mostly conducted with juvenile fish, with the initial growth acceleration due to higher temperatures leaving the strongest impression. Only when small, short-lived fishes are monitored over their entire life spans does the phenomenon appear, which was found so puzzling (92).

6) “Increased predation mortality at higher temperatures drives an evolutionary response of higher net energy allocation to reproduction versus growth.” This is hypothesis (4) in another guise (Fig. 6F). Evoking a complex “evolutionary response” is not an explanation of anything because, as was said so elegantly, “nothing in biology makes sense except in the light of evolution” (119). The point, rather, is to identify the mechanism in question. However, it will be quite difficult, given that, as stated for (4), fish grown under experimental conditions and without opportunity to spawn remain smaller at higher temperatures (92, 117, 118). The critique of items (2) to (6) is serious: Proposed hypotheses should be able to withstand a confrontation with common sense observations. Moreover, several of the hypotheses in Fig. 6 were only complex restatements of the issue at hand.

In contrast, the GOLT proposes a mechanism for the reduced body size of fish and invertebrates under global warming that is simpler than what needs to be explained and that is based on consensual knowledge, including that gills cannot be perceived as trans-dimensional Escher-like objects (42). In addition, the GOLT makes numerous predictions pertaining to domains that, at first glance, appear to be unrelated to temperature affecting the size of fish.

This is because the constraints on the surface area of gills are real: Their surface area was optimized in the course of evolution to allow their owners to reach first maturity relatively fast, after which growth can gradually slow down.

The notion that gill surface area cannot be limiting because lamellae can be added as required (42) is false because gills function similarly to a sieve, i.e., must be perpendicular to the water that flows through them. This means that they can grow in height and in breadth, but not in depth: They cannot grow in the third dimension, and thus, 3D bodies must experience a declining oxygen supply as they grow. Moreover, gills are a favorite site for parasite infestation, and fish and aquatic invertebrates have good reasons to keep them as small as possible (102, 120). Thus, gill surface area is not limiting to young/small fish, but they are to big adults.

The GOLT offers a coherent framework for exploring these phenomena and a vast number of related observations. This is not the case for just-so hypotheses.


The nature of explanations

The physicist Wolfgang Pauli is supposed to have said, “God made the bulk; surfaces were invented by the devil” (121). When thinking about the explanations provided by the GOLT, which is concerned primarily with the “tension” between volumes and surfaces (and the arguments denying such tensions that have been advanced against the GOLT’s explanations), it is appropriate to recall what is meant by an “explanation.” Rather than pedantry, some reflections are required to define the terms of the debate and the criteria that are applied below.

To become widely accepted, scientific explanations, in addition to being (obviously) congruent with the facts at hand, should be consilient with related disciplines, parsimonious, independent from the observer, and “productive,” i.e., make unexpected predictions.

The first of these is the notion that the different scientific disciplines, while autonomous in their investigations of the phenomena upon which they focus, cannot accept explanations that violate constraints established by other scientific disciplines (122) or by logic, geometry, and mathematics. Thus, biological organisms must comply with physical laws, and the processes comprising their metabolism must comply with constraints studied by chemists.

An explanation consists, therefore, of “mapping” a phenomenon observed by the practitioners of a given discipline onto constraints, rules, or “laws” that are parts of an underlying discipline. An example is “Bergmann’s rule” (123), which explains why high-latitude mammals and birds tend to have bigger bodies and shorter appendages (ears, limbs, and tails) than their congeners in more temperate climes. It is built on the idea that, while they generate heat in their bodies (a volume, which tends to grow according to length cubed), mammals and birds radiate (i.e., lose) heat through their body surface (proportional to length squared). Hence, increasing body weight and reducing the size of appendages through evolutionary time will reduce heat loss, by reducing body surface per unit volume. Bergmann’s rule relies on consilience, specifically on facts of geometry and physics, to make a case concerning the biology of homeotherms.

The key feature of this type of explanation is that it avoids infinite regress: An observation is explained once it is mapped onto a more basic framework, i.e., there is no need to map the basic framework onto an even more basic one. Thus, in the example above, there is no need for biologists to explain why heat loss is proportional to a surface, although it has been, for a while at least, a legitimate research question for physicists (124).

Parsimony is the requirement that an explanation should be “small” relative to the “size” of what needs to be explained (125) Parsimony is another term for “Ockham’s razor,” the rule that among competing hypotheses, the simplest one is (generally) to be favored (126).

The third requirement of a scientific explanation is that it must be nonlocal, i.e., it must not favor a privileged observer or standpoint (127). For example, we should not project our mammalian preoccupation with the food that we require to maintain our elevated temperature onto WBE, which require far less food, but to which the extraction of oxygen from their surrounding medium is a challenge that air-breathers often find difficult to imagine. Last, a successful hypothesis should not only explain the facts at hand and map them parsimoniously onto the fabric of a more basic discipline but also make successful predictions, i.e., make sense of facts that it was not designed to explain.

Some real issues with the GOLT

While the GOLT can obviously deal with objections that are beside the point (Table 8), there are several areas in which this theory is really deficient. One of these is that the GOLT is still largely a qualitative theory, frequently unable to make quantitative predictions. For example, while the GOLT met the challenge posed by whale shark—the largest extant fish—occurring in warm tropical waters by evoking their yo-yo type “cooling dives” (Table 3, no. 3.2), it cannot, at present, provide quantitative constraints for a model that could predict the duration of such dives as a function of whale shark size and depth-temperature gradients. Such a model could be tested using the data on whale sharks occurring in the Persian Gulf [which is both warm and shallow (128)] and the Red Sea [whose deeper waters are very hot and briny (129)].

Table 8 Arguments raised against the GOLT: Miscellaneous, mainly normative arguments.

View this table:

At present, it can only be stated that the GOLT is not refuted by the presence of whale shark in these extreme environments, although cooling dives are not possible, because these whale sharks are juveniles and young adults below 10 m in the Gulf and below 7 m in the Red Sea (130133). Fully grown whale sharks, those assumed to require frequent cooling dives, are reported to exceed 18 m (134), which makes them over 11 times heavier than at 8 m.

Another example is the experiment explicitly conducted as an explicit test of the GOLT (93), which predicts that fish raised in (mildly) hypoxic conditions should reach maturity at a smaller size than fish raised in normoxia. The GOLT passed this test, while alternative hypotheses did not (93). However, this prediction concerned only the direction of the response, and not its strength.

Another deficiency of the GOLT is assuming that the WBEs in question always get enough food to grow (i.e., the converse of most studies that deal with food limitation, but tacitly assume that the oxygen needed to turn ingested food into energy is always available, and at no cost). This issue is obviously related to the investment required to produce ova and sperms by mature WBE, whose reproduction can be understood only by considering seasonal growth oscillations, a topic not considered here [but see (16)].

Clearly, the GOLT will have to be assimilated into a bioenergetics model or vice versa. However, the intellectual effort this represents will only be undertaken if oxygen supply to the bodies of WBE is perceived as the constraint that it is, and hence this contribution.

The GOLT and evolution

How a further elaboration of the GOLT would look cannot be anticipated, at least not by the author. However, such elaboration, if successful, may influence the way we view the evolutionary process. It may lead to a realization that evolution has two ways of handling challenges, depending on their nature. In the first, the challenge is met head-on by an adaptation (for example, when grazers neutralize a toxic substance in the leaves of a plant or when a parasite gradually becomes a symbiont). In the second, the challenge (e.g., gravity, oxygen requirements, and heat buildup) cannot be overcome by a metabolic or behavioral trick. In the latter case, all that can occur is what may be called a set of “accommodations.”

The dimensional tension (17) between the gill (or other respiratory) surface of a WBE and its body weight results in the accommodations that are made explicit by the GOLT, which should not be perceived as adaptations. When the challenge posed by geometric or physical constraints cannot be accommodated, the corresponding region of morphological space remains unoccupied. This is why neither the huge spiders stalking Frodo and Sam in Lord of the Rings nor even beetle-shaped insects of more than 18 cm (135) can exist. As for fish, this is why the megatooth (Megalodon) could not reach more than twice the length of the great white shark (i.e., 20 m), implying a weight nearly 10 times greater, as claimed in a Discovery Channel “documentary” film. The GOLT requires that these two types of evolutionary challenges be recognized and distinguished, lest colleagues continue to believe that if fish suffer from warmer temperature and deoxygenation, they will just grow larger gill.


Supplementary material for this article is available at

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: I thank the numerous colleagues who, over the years, have variously challenged me to come up with a better framing for my “oxygen story,” notably R. Froese, W. Cheung, and J. Bigman. Included also are the critics, to whom I respond in Tables 2 to 8. They found this story at least worthy of criticism, rather than being simply ignored. I thank K. Meyer, Idaho Department of Fish and Game for Fig. 4 (D to F), C. “Elsa” Liang for assistance with the equations establishing the relationship between Wm and Wi, and E. Liu and E. Chu for drafting Figs. 1 to 6. Funding: The author acknowledges that he received no funding in support of this research. Competing interests: The author declares that he has no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the author.
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