Fishes in Warming Waters, the Gill-Oxygen Limitation Theory and the Debate Around Mechanistic Growth Models

Abstract

Mechanistic explanations of the impact of climate change on fish growth are currently under debate. However, critical assessments of even the most prominent theories are not always based on accurate interpretations of their underlying mechanistic models. This contribution addresses some of the major misunderstandings still causing the Gill-Oxygen Limitation Theory (GOLT) from being examined based on its actual structuring elements and assumptions, rather than erroneous perceptions. As we argue, recent critiques of the GOLT are based on implausible interpretations of respirometry data that are invoked to distinguish maintenance costs and overhead costs of growth. Discussing the current state of the debate, we emphasize the fact that fasting young and, thus, growing fish for short periods of time is not sufficient to suppress energy (i.e., oxygen) allocation to growth. In the process of dealing with these issues, several cases of apparent ‘counter-evidence’ are discussed. Highlighting the need to base critical discussions and examinations of the GOLT on its actual predictions, we recommend that testing the theory should focus on broad reviews or meta-analyses, e.g., on datasets of gill surface area and the relationship of these data to growth performance under different temperature regimes.

1. Introduction

The impact of global warming and deoxygenation on fish populations is currently the topic of heated debates among fisheries scientists, physiologists, and ecologists, and one of the central questions is the correlation between temperature and reduced final sizes in fishes and other water-breathing ectotherms [1,2,3]. In ectothermic water-breathers such as fish and aquatic mollusks or crustaceans, this “universal” phenomenon [4] manifests itself in a life-history patterns, in which early growth is faster but the size at first maturity and final body sizes are reduced at higher temperatures This pattern was already described and theorized in the early 20th century by Pütter [5] and is now widely referred to as the “temperature-size rule”, a term coined by Atkinson [6].

The consequences of ‘shrinking’ body sizes of individual fish also impact entire populations and, thus, the productivity of commercial fish stocks [7,8,9]. Changes in body size can alter trophic relationships [10] and population dynamics because fecundity often scales hyperallometrically with body mass [11]. The observed and projected ecological impacts of altered growth and life-history patterns of fish and other aquatic animals have attracted the attention of scientists in different fields and highlighted the need for mechanistic explanations [12,13,14,15,16]. However, mechanistic theories that proposed general explanations of the observed ‘shrinking’ of fish provoked criticism from physiologists, who argued against the idea that unifying mechanisms could explain observed patterns across species [17,18]. Rather, these critics argue that the mechanisms that underlie temperature-induced alterations in growth patterns are likely to be diverse and dependent on species-specific ecologies.

The two dominant theoretical proposals to explain the impact of increasing temperatures on the growth and physiological performance of fishes and other aquatic animals are the so-called OCLTT (Oxygen- and Capacity-Limited Thermal Tolerance) of Pörtner [12] and the GOLT (Gill-Oxygen Limitation Theory) developed by Pauly [15,16]. Both of these models focus on oxygen limitation as a constraint on the growth in water-breathing ectotherms (hereafter WBE) and point to the fact that these animals inhabit a respiratory medium with fundamentally different properties than air. Water not only contains 30 times less oxygen than air, it is also 55 times more viscous, and its density is 850 times higher. Moreover, diffusion is 300,000 times slower in water than it is in air [19]. Thus, both the OCLTT and the GOLT highlight the impact of limited oxygen supply on the energetic budgets of fishes. If temperatures increase, ectothermic animals need to invest more energy—and thus oxygen—in maintenance activities, which leaves a smaller fraction of the energy budget for growth. As a result, final sizes are reduced at higher temperatures. Theoretical frameworks similar to the OCLTT and the GOLT are the DEB (Dynamic Energy Budget Theory), proposed by Kooijman [20], and the OGM (Ontogenetic Growth Model) by West et al. [21], which are also based on the Pütter/von Bertalanffy growth equation, to be discussed below in more detail. Even though the DEB and OGM play a less dominant role in the current debate on climate change, their underlying models predict similar outcomes, and proponents of the DEB have recently evaluated temperature-size effects in ectotherms in more detail [22].

In this contribution, we evaluate the current debate about mechanistic models, which aim to explain temperature-induced growth patterns in fish. Our focus is on recent debates on the validity and the explanatory scope of the GOLT, which has received more critical attention than the OCLTT in the past five years [23,24,25,26,27,28]. However, a large portion of the recent critiques is, in fact, directed at the validity of mechanistic generalizations across species per se and, thus, also pertains to similar theoretical frameworks. In the following sections, we first present a short overview of the theoretical architecture of the GOLT and the so-called P-diagram [29], which serves as an explanatory tool to illustrate the relationship between the metabolic energy that is needed for maintenance activities and the available “scope for growth” of an individual organism. We then review the different arguments against the theory and against mechanistic explanations that are based on oxygen limitation as a cause of reduced final sizes in fishes and other WBE. After evaluating the datasets on which these critiques are based, we argue that a large portion of these critical assessments of the GOLT are based on assumptions that underestimate the costs of growth in juvenile or subadult fishes during short fasting periods in which the animals are subjected to respirometry experiments. The current status of physiological research calls for a more stringent assessment of overhead costs of growth in indeterminate growers to arrive at realistic results [30,31].

Finally, we will address the current status of constraint-based explanations for ontogenetic growth and allometric scaling. Notably, theories that neglect constraints to resource supply fail to explain patterns of energy allocation to growth, reproduction, and maintenance in real-life situations [32,33]. We conclude this review with a number of recommendations for future tests of the GOLT and other theoretical proposals to explain the currently observed trend of ‘shrinking’ fish. Our review focuses on the studies that have not yet been addressed in an earlier discussion of the GOLT and the counterarguments to its critiques [16]. While some of these earlier papers will be mentioned in the following sections, this is to add context and to deal with issues that were not addressed earlier.

2. ‘Shrinking’ Fishes and the GOLT

Even though the GOLT is currently mainly discussed in the context of climate warming and its impact on fishes and their ecosystems, its foundational elements were elaborated to describe the growth of tropical fish, a topic that was long neglected by European and North American fisheries scientists who mainly focused on temperate species [34,35]. The theory that emerged from this approach was then expanded to understand respiration, metabolism, growth, reproduction, food conversion, migration, and other life-history traits in aquatic organisms under changing circumstances as well as their interactions with their ecosystems and their roles within the food webs of marine and freshwater ecosystems [15,16]. Strictly speaking, the GOLT is not primarily a theory on the impact of climate on ectothermic animals but a theory of growth that seeks to explain the central life-history patterns in water-breathing ectotherms. It is important to note, however, that despite its ambition to generalize across phyla and taxa, its explanatory scope limits itself to water-breathing organisms and is, thus, not applicable to obligatory air-breathers.

The GOLT’s most fundamental building block is a modified version of Pütter’s growth equation [5], which expresses the relationship between two metabolic terms as follows:

dW/dt = HW^d − kW

(1)

Here, dW/dt is the rate of growth, W body mass, and H and k the coefficients of anabolism, or the synthesis of new materials, and breakdown metabolism. d is the scaling exponent of the anabolic term and, as the theoretical framework of the GOLT predicts (and as empirical studies show), always takes values <1 in adults (apparent exceptions will be discussed below). While earlier versions of the Pütter growth model assumed values of d = 2/3 (as did Pütter himself and also von Bertalanffy in what he called ‘growth type I’), the GOLT predicts values between ca. 0.6 and 0.9, depending on the ecology and life-history strategy of the species in question.

The value of d is equivalent to the scaling exponent of respiratory surfaces, with gills being the relevant structure in fish. Early mechanistic explanations of the Pütter growth model reasoned that the anabolic term of the equation scaled with a factor that depended on a surface, which was the reason to set the scaling exponent to 2/3 [5,36]. However, gills are highly filamentous surfaces that can grow with significantly higher slopes. While d in small fish such as guppies—on which von Bertalanffy based his observations—do indeed equal 2/3, larger or more active fish such as tuna have much higher slopes [37]. As the GOLT and its theoretical predecessors maintain, the slopes of d correlate closely with the observed scaling exponents of metabolic rate, and they shape the specific growth curve of the studied fish. As both Pütter and von Bertalanffy asserted, “growth is dependent on respiration” [36] (p. 280).

As Equation (1) states, the anabolic term (which represents the synthesis of new materials and depends on a surface) grows slower than the term of breakdown (which increases with weight). This leads, upon integration, to growth curves that are asymptotic. When d is assumed to equal 2/3, the integration takes the form

L_t = L_∞·(1 − e^{−K·(t−t₀)})

(2)

where L_t is the length at age t, L_∞ the asymptotic length, i.e., the length that the fish would reach if they were to grow indefinitely, K is a coefficient (of dimension time ⁻¹) expressing how fast the asymptote is approached, and $t_0$ the age at which the length of the organism would have been zero if it had from the onset, grown in the manner described by the equation—which does not occur in reality and, thus, $t_0$ usually takes a negative value. For weight, the equation takes the following form:

W_t = W_∞·(1 − e^{−K·(t−t₀)})^b

(3)

Here W_∞ is the weight corresponding to L_∞, K and t₀ are as defined above, and b is the exponent of a length–weight relationship of the form W = a·L^b, with b ≈ 3 in most cases, because solid bodies grow in three dimensions if they are to retain their shape. Equations (2) and (3) are often referred to as ‘special von Bertalanffy Growth Function’, or VBGF.

Alternatively, allowing d ≠ 2/3 (but maintaining d < 1) leads to

L_t = L_∞·(1 − e^{−KD·(t−t₀)}) ^1/D

(4)

for length, and

W_t = W_∞ ·(1 − e^{−KD·(t−t₀)}) ^b/D

(5)

For weight, the sole difference between the two pairs of equations being D = b (1 − d) makes the latter pair compatible with allometric scaling of gill area and respiration.

Details on the integration and properties of what may be called the generalized VBGF can be found in Pauly [16,35]. While seasonal versions of the VBGF exist [15], they are not considered here as the relevant issue is that, within species, empirical estimates of asymptotic size (L_∞ and W_∞) tend to be lower in populations exposed to higher temperatures (at least at temperatures above 4–5 °C, below which the cold denaturation of proteins must be expected to play a role; see [38]). This can be demonstrated by using the growth parameters of thousands of fish species in FishBase (www.fishbase.org).

The difference, or ‘dimensional tension’, between gill surface area (GSA) and body mass results in a decreasing capacity to take up oxygen in large adults compared to juveniles and, thus, in the asymptotic growth that is a characteristic of all versions of the VBGF. The GOLT interprets this changing relationship between respiratory surfaces and the three-dimensional bodies that need to be supplied with oxygen as the cause for the steady decline in growth rates. Even though gills are filamentous organs whose surfaces can increase with slopes of >2/3, the growth of their respiratory surfaces is not proportional to the mass increase in a growing body, even though oxygen demand may indeed decrease with body mass due to the hydrodynamic advantage of large size [39], the reduced number of potential predators [40], the greater ease at capturing larger, energy-dense prey [41], and the allometric scaling of internal organs that may consume more oxygen than other tissues.

The relationship between GSA and body weight at different sizes can be presented in the so-called P-diagram ([29], Figure 1). In this diagram, the difference between the oxygen demand of maintenance metabolism (Q_maint) defined as the backward-extrapolated metabolism of an old/large and, hence, non-growing WBE at W_max, and the supply that results from the relationship between gill surface area and body weight is then visualized as ‘scope for growth’ ([15]; Figure 1A).

Aside from explaining the reasons behind the cessation of growth at the maximum or asymptotic weight (W_∞ or W_max), the schematic versions also illustrate why fish and other WBE tend to be smaller when subjected to higher temperatures (Figure 2).

The concept of the P-diagram has recently been criticized by McKenzie [24], who also claimed that the GOLT lacks a proper definition of maintenance metabolism. However, maintenance metabolism, as defined in the GOLT (see text above and Figure 1A), is a variable that can only be defined retroactively and which includes all the energetic expenditures of a fish with the exception of growth [15] (p. 159). In contrast to standard metabolic rate (SMR), which is usually defined as the metabolic rate of a resting and fasting animal (thus expected to devote none of its energy to growth), maintenance metabolic rate cannot be directly measured in growing fish because they will hardly ever be in a state where all of their metabolic energy will be invested exclusively in maintenance. We return to this issue further below.

In his critique of the P-diagram, McKenzie suggested abandoning the maintenance metabolic rate as a parameter and to look at the difference between standard and maximum metabolic rate instead, which could then be defined as “aerobic metabolic scope”. However, this is not the same as scope for growth, as described in Figure 1; indeed, aerobic metabolic scope does not allow for any inferences on growth. Even if aerobic scope were to remain the same over a wide range of body sizes, it would still not correspond to ‘scope for growth’ (see Figure 1A). In other words, aerobic scope can remain self-similar over a wide range of body sizes, but only because energy allocation to growth decreases with size, and the energy that would otherwise be invested in the production of new tissues is increasingly used for their maintenance.

While maintenance metabolic rate is indeed not a parameter that can be measured directly but only inferred retrospectively, the problem with estimates and measurements of standard metabolic rate (SMR) is that it remains unclear which energetic expenditures they include. Most of the recent critiques of the GOLT are based on the explicit or implicit assumption that SMR does not include significant overhead costs of growth, and that realistic values can be established if the animals have been fasted for a sufficient period of time [23,27,44]. If this were the case, it would be possible to compare the scaling exponents of gill surface area and of standard metabolic rate and infer whether surface area is indeed a limiting factor. However, as we show in the following, this view is not supported by current physiological research, and as newer studies show, the overhead costs of growth remain after 24 h and other commonly used fasting periods [30,45]. As long as this problem, illustrated in

Table 1. Definitions of SMR and related terms in the dataset used to support the argument of Scheuffele et al. [23]. P = parameter; F = fasting period: h = hours; O2C = O₂ consumption rate; RMR = routine metabolic rate; and N = not stated.

#	P	F	Source
1	SMR	N	Al-Kadhomiy [46]
2	SMR	24 h	Beauregard et al. [47]
3	SMR	24 h	Blasco et al. [48]
4	RMR	24 h	Bowden et al. [49]
5	N	N	Clark et al. [50]
6	RMR	45–72 h	Clark et al. [51]
7	SMR	20 h	Cutts et al. [52]
8	BMR	N	Degani et al. [53]
9	SMR (a)	48 h	Duthie [54]
10	SMR	24 h	Herrmann & Enders [55]
11	O2C	24 h	Ken-Ichi [56]
12	RMR	24 + 48 h	Li et al. [57]
13	RMR	48 h	Li et al. [58]
14	O2C	24 h	Miyashita [59]
15	RMR	36 h	Huang et al. [60]
16	RMR	?? (b)	Hughes [61]
17	SMR	24 h	Seppänen et al. [62]
--	--	--	--

^(a) SMR/Standard O₂ consumption; ^(b) Fish were held “generally without feeding” (unspecified).

and Table S1 in Supplementary Materials, is not resolved, a retrospectively inferred parameter, and, thus, the P-diagram, may be the best way to arrive at reasonable estimates of energy allocation to growth and to other activities.

3. Fish Growth, Respiration and Temperature

A number of recent studies have argued that oxygen limitation cannot be expected to impact growth if either resting or active metabolic rates scale with a similar exponent as gill surface area [23,27,28,44]. As these authors argue, the fact that gill surface area scales isometrically with oxygen consumption implies that the capacity to meet oxygen demand is independent of body size. Hence, they assert that changing relationships between respiratory surfaces and body mass in the course of ontogeny cannot explain declining growth rates at greater size or lower final size at higher temperatures.

This argument is often invoked as a criticism of mechanistic growth models that attribute the asymptotic form of growth curves in fish to the different scaling exponents of gill surface area and body weight [18]. While Lefevre et al. [63] initially seemed to argue against the impact of surface–volume relationships on oxygen uptake capacity per se, later studies revisited this argument and examined the relationship between the slopes of metabolic rate and gill surface area [18,23,27]. As these authors argued, it is not body size itself that should be regarded as the relevant parameter in this context but rather metabolic rate, which also scales (with negative allometry) with body mass, as the extensive literature on metabolic scaling consistently shows (see, e.g., [64]). Thus, as long as metabolic rate and gill surface area scale with the same exponents, critics of the GOLT suggest that oxygen limitation can be excluded as a factor that could impact growth.

For this reasoning to hold, it would be necessary to actually exclude the metabolic costs of growth from the respiratory measurements that underlie the model in question (as attempted by depriving fish of food for 24–36 h, as illustrated in Table 1) and to assume that none of the measured oxygen intake is diverted toward growth. Only then could SMR be interpreted as done by these authors.

In contrast to this line of reasoning, the similarity between the slopes of oxygen uptake and gill surface area is a constitutive element of the GOLT (see, e.g., [15], pp. 36–37 and the review of De Jager and Dekkers [65] on the gill surface area and respiration of fishes). The notion on which the proposed tests of the GOLT by Scheuffele et al. [23] and Lonthair et al. [27] is based on is a parameter, which these authors believe to be derived from the GOLT itself, the parameter b_S, which is defined as the difference between the scaling exponents of standard metabolic rate (b_SMR) and that of gill surface area (b_GSA). Thus:

$$b_S={b_{GSA}-b}_{SMR}$$

(6)

As both Scheuffele et al. [23] and Lonthair et al. [27] assert, for the GOLT to apply, b_S should be ≤0. Therefore, the authors conclude that values of b_S > 0 would refute this theory since the allometric slopes of the GSA would make it increase faster than SMR with increasing body size. However, the GOLT always assumed b_GSA and b_MR to take similar values; this is explicitly stated in Pauly [15] (p. 36–37), where b_GSA and b_MR are referred to as ‘d_G’ and ‘d_Q,’ respectively, and are used as a substitute for one another throughout his book.

Values of b_S < 0 would be in contradiction of Fick’s Law of Diffusion, which states that respiration is proportional to respiratory area [66], while values of b_S > 0 would not make evolutionary sense, as no fish would invest energy in producing gills of a size in excess of what it needs, which would make it more vulnerable to parasites accessing its bloodstream through the thin epithelia that separate it from the surrounding water. Thus, the GOLT considers deviations from b_S ≈ 0 to be caused by methodological issues of no theoretical importance.

The reasoning of Lefevre et al. [18,63], Scheuffele et al. [23], and Lonthair et al. [27] seems to assume that the GOLT, and its statement that b_GSA < 1, implies that fish growth continues until vital body functions would be close to collapse; i.e., that fish are supposed to grow their way into asphyxia. What b_GSA ≈ b_MR < 1 implies, in reality, is that growing fish have less and less surplus oxygen to allocate to growth until at the maximum size they can reach in a given environment (and temperature). They can do everything they normally do (forage, digest their food, escape predators, etc.) except grow. In other words, what the GOLT claims is not that old fish lose their aerobic scope and collapse because they cannot maintain their normal activities but rather because they do not have surplus oxygen to devote to further growth. Unfortunately, Lonthair et al. [28] did not address this issue in their response to Pauly and Müller [67], which leaves open the question of what the proportionality between b_GSA and b_MR might actually imply.

The change in energy allocation from growth to increasing ‘maintenance’ (not reproduction; see [68]) is unrelated to Equation (6), which assumes that the GOLT predicts a critical shortage of available oxygen for even basic maintenance functions in the course of a fish’s growth trajectory. Thus, Equation (6) is biologically meaningless and will only generate values that can be predicted to be identical across species, i.e., values ≈ 0. This also implies that Figure 3, from Scheuffele et al. [23], which summarizes their vision of the relationship between metabolism and growth, depicts a biologically unrealistic scenario. Moreover, Figure 3 implies that fish would grow forever if not constrained by some external factor that is independent of their physiology. One such factor is food availability, explicitly mentioned, e.g., by Lefevre et al. [18], who state that “many studies focus on food limitation as the causative factor for declining fish sizes in the field”. However, these studies did not demonstrate why food availability declines rapidly, for each fish species, in each of the world’s ecosystems, and at precisely the sizes that happen to mimic well-established temperature–size relationships, i.e., patterns caused by oxygen stress. Also, note that, unlike food, oxygen can only be stored in the organism for very short periods of time (in myoglobin).

4. Do Fish Stop Allocating Oxygen to Growth When in a Respirometer?

To test the conclusions of Lefevre et al. [63] and Scheuffele et al. [23] and their argument that oxygen limitation cannot impact fish growth as long as the slopes of metabolic rate and gill surface area are similar, it is worthwhile to take a closer look at the studies cited to make this claim (Table 1 and Table S1). For the argument of these authors to hold, it would be necessary to completely exclude the costs of growth from the measurements of metabolic rates.

However, as a comparative analysis of the definitions and methodologies of the 37 studies cited in Lefevre ([44], see Table S1), Lefevre et al. [63] and the 17 studies reported in Scheuffele et al. ([23]; see Table 1) shows, the experimental setups are too divergent for consistent and realistic inferences of SMR in growing fish, if this parameter is understood as the metabolic rate excluding activities, digestion, and especially growth. Also, most of these studies were not designed to distinguish between the metabolic activities of juvenile and adult animals and the resulting differences in energy allocation. However, as numerous studies have shown, juvenile and subadult animals do not suddenly stop allocating energy to growth when in a respirometer. In fact, respirometry on smaller and younger individuals always includes energetic overhead costs of growth [30,31,45]. The issue seems to be that the studies from which these data were extracted were not based on experiments originally designed to compare SMR in juveniles (with high growth rates) with adult animals (with lower growth rates). However, in the context of the issue raised by the GOLT, an evaluation of the declining oxygen supply to larger individuals is only possible if reduced growth costs are considered.

As Rosenfeld et al. [30] reminded us, fish physiologists follow different procedures and definitions than researchers who work on endotherms: “[b]ecause of the strong influence of growth on SMR, physiologists studying maintenance metabolism in birds and mammals generally work on adults; in contrast, fish physiologists commonly measure SMR on actively growing juveniles in laboratory experiments”.

In the hope of arriving at realistic values, fish physiologists often use fish that have been deprived of food for 24 h before respirometry and then estimate SMR based on the lowest measured value. Yet, it remains unclear for how long fish must be food-restricted to exclude their metabolism contributing to growth, which is the reason why SMR can only be reliably measured in non-growing, i.e., large, adults (see, e.g., McNab [69]). Similarly, more recent studies define SMR as “the metabolic rate of an adult, inactive, unstressed, postprandial ectotherm”, i.e., for animals where no growth expenditures are expected to be included in measured metabolic rates [32].

While many studies rely on fasting periods of 24 h before placing their fish in respirometers, this is probably not sufficient to exclude the overhead costs of growth. As Rosenfeld et al. [30] demonstrate, the measured values of SMR depend heavily on the food rations consumed in the days and weeks before fasting. In their study on juvenile salmonids, animals fed heavily before a 35 h fasting period showed no less than 67% overhead costs of growth. Therefore, Rosenfeld et al. [30] recommended putting the fish on maintenance ratios for extended periods before respirometry. Even in fish fed only moderate rations for 2–3 weeks before the experiment, overhead growth expenditure fluctuated between 19 and 33%.

If we examine the studies upon which criticism of the GOLT is based in the light of the recommendations of Rosenfeld et al. [30] and Chabot et al. [31], we see that almost none of the research setups that underlie the studies in Lefevre [44] and Scheuffele et al. [23] conform to the recommended standards. In the 37 studies cited in [44], many authors did not indicate fasting periods; some only fasted the fish for 10 h (Table S1). The only longer fasting period was maintained in a study on Antarctic fish, where the metabolic effects of specific dynamic action, or SDA, can last for 2 weeks or longer. In a more recent study by Scheuffele et al. [23], the average fasting period was 36.6 h, and some datasets did not indicate whether the fish were food-deprived (Table 1).

Only one study [70] indicated how pre-fasting maintenance rations were calculated. As the experiments of Rosenfeld et al. [30] show, information like this is necessary to infer if SMR data are realistic, i.e., represent metabolic rates that exclude growth expenditures. Only one study followed the suggestion of Chabot et al. [31] to report feeding conditions in the weeks prior to respirometry and pre-respirometry fasting [71]. Another study reported that the fish were on maintenance rations 5–10 days before the experiment [70]. Overall, this suggests that the data cited by Lefevre [44], Lefevre et al. [63], and Scheuffele et al. [23] do not provide evidence against the GOLT unless the applied methodology can clearly account for costs of growth.

5. Allometric Scaling of Gill Surface Area at Different Life Stages

It is essential to the GOLT that, past the larval and early juvenile life stages of fish (which exclusively breathe water), gill surface area does not increase proportional to body mass. While the larvae in most fish species are still capable of cutaneous respiration and typically show exponential growth in both length and weight, d in late juveniles and adult fish becomes <1 once they fully rely on respiration through their gills, which is a testable hypothesis. However, a recent paper on this question claimed to have identified a fish species with d > 1 in the adult stage and argued that this finding refuted the GOLT [26].

Over the entire size range of the species they studied, horn shark (Heterodontus francisci), the authors found a scaling exponent that stayed solidly within the range predicted by the GOLT, with d = 0.877 ± 0.065 (Figure 4A). However, by using a segmented regression, with a separate low slope (‘d’ = 0.564) for the smallest three individuals, Prinzing et al. [26] obtained a convenient slope of ‘d’ = 1.012 ± 0.113 for the remaining 16 fish (Figure 4B). It is unclear which goal the segmented regression in this paper was intended to serve other than to use specific segments of the overall slopes as an argument against the GOLT, especially given the lack of correspondence between the apparent hockey stick form and the metabolic slopes at the same size ranges. However, what matters to our model is not the slope at specific ontogenetic stages of an organism but on a size range that reflects the asymptotic growth curves that are commonly found in fish.

It is important to note, however, that in fish, and specifically shark species for which changing slopes of GSA scaling have been reported, e.g., in the thresher shark Alopias vulpinus [72], it is at the high end of the size range that the d parameter takes lower values when the gills can be assumed to be so large as to have difficulties to increase their lamellar area further. The higher values reported for the higher size ranges are unlikely to reflect the further growth of gills for this species. This strengthens the assumption that a segmented regression primarily served to shape the debate, and it is surprising that Lonthair et al. [28] cite Prinzing et al. [26] as one of two studies supporting their claim that “there are several robust datasets in the literature of GSA scaling close or even above [d] = 1”.

6. “Gills Are Folded Surfaces, Not Spheres”

The notion that fish gills can grow with scaling exponents equal to or even larger than unity (i.e., d ≥ 1) is related to one of the more idiosyncratic points used as an argument against the GOLT. As Lefevre et al. [18,63] repeatedly argued, surface–volume relationships would not apply to the relationship between gills and the body they have to supply with oxygen. As these authors state, “gills are folded surfaces, not spheres”, which is obviously true. However, they argue that this geometrical fact would invalidate the assumption of declining surface–volume relationships in growing bodies and that “the scaling of surface area to volume is not constrained by spherical geometry”. This statement reveals an interesting type of confusion about scaling properties between organs and entire bodies. As outlined above, gills typically have higher scaling exponents than expected in the early 20th century literature on this topic (i.e., d > 2/3 in species that can reach lengths exceeding 5–10 cm). However, the argument that “gills are folded surfaces, not spheres” does not pertain to the problem in question: if gills increased their surfaces at the same rate as spheres, their slope relative body size would not be ≈2/3 but significantly lower (depending on the species and size range, it could be as low as <0.2). This argument would only apply to fish growth if gills would have to support themselves and their own oxygen consumption. Since they have to supply an entire organism with oxygen, hyperallometric slopes are needed to reach even exponents of ≈2/3.

7. The Improbable Existence of Large Tropical Fish

The GOLT implies that larger fish are generally constrained in their further growth by the oxygen uptake capacity of their gills to the body’s interior. This applies particularly to large fish species in the tropics, where the high temperatures increase oxygen demand. The constraints to sufficient oxygen delivery from the gills have resulted in several evolutionary developments that enabled tropical fish to take up oxygen from the air. Most of the large fish species of tropical freshwater ecosystems, such as the lower Mekong River and the Amazonian floodplains, are indeed air-breathers, for example, the Amazonian osteoglossid Arapaima gigas, whose anabolism is constrained by neither the size of its gills nor by the oxygen content of the water in which it occurs and which has shown highly atypical growth patterns, to which no version of the VBGF can be fitted [15,73].

However, not all tropical freshwater ‘giants’ breathe air, and a notable example is the giant freshwater stingray (Urogymnus polylepis), which has recently been revealed to be the largest known freshwater fish extant [74]. Even though there are reasonable explanations for why the aquatic megafauna of the Mekong is so extraordinary—and the giant fish species typically occur in the vicinity of deep holes where they can cool themselves—critics of the GOLT have used the existence of large tropical species as arguments against the theory. As outlined elsewhere [15,16], large tropical fish have developed strategies to keep their body temperatures at levels that allow them to reduce their maintenance metabolism efficiently. However, the apparent problem of large tropical fishes reveals a more fundamental point of the GOLT, and the invoked counterexample allows us to address it even more concisely.

A fundamental implication of the theory is that inter- and intraspecific (and, thus, ontogenetic) metabolic scaling depends on two different mechanisms. Thus, the decrease in the metabolic rates in growing fish of the same species differs from lower metabolic rates in large species compared to smaller ones. As outlined elsewhere, “the GOLT does not claim to explain why minnow and anchovies are small while carp and tuna grow to much larger sizes. We have the theory of evolution for that. What the GOLT explains is why a minnow and a large tuna have gills which supply them, as they grow, with a declining oxygen per unit weight, and that this reduced oxygen supply reduces their growth rate until finally, they stop growing” [75]. While this may seem trivial, it is, in fact, of crucial importance for the GOLT’s mechanistic model (assuming that no other factors impact individual sizes within a given population before maximum possible sizes are reached, e.g., food limitation or size-based predation). Decreasing oxygen consumption rates within species and individuals are caused by slower growth rates at greater body sizes (not decreasing maintenance costs, as assumed by Lefevre et al. [18] or White and Marshall [32]). This is not the case in differences between species, where scaling patterns can be explained as the result of evolutionary selection. Since large animals cannot consume and metabolize the same amounts of energy as smaller ones, the evolution of their sizes depends on selection for low maintenance costs. In short, a whale shark with the metabolic rates of a guppy would be a physical, chemical, and physiological impossibility. A large fish with such high maintenance costs would not only collapse due to oxygen limitation but also a lack of food resources that would be sufficient to maintain its energetic demand—it is only its relatively lower metabolic rates that allow it to exist and reach its immense size.

What follows from this is that evolutionary roads to large body sizes are more unlikely to occur and more difficult to navigate in tropical waters since they rely on more refined mechanisms that allow for a reduction in metabolic rates, e.g., by performing regular cooling dives as shown by large marine tropical species [15,16]. However, the evolution of large water-breathing ectotherms is, in itself, not in contradiction to the GOLT’s underlying model; while the theory has a good explanation for the strikingly close correlation between latitudes and average fish sizes, the existence of whale sharks or large sunfish is not a real challenge—by contrast, their rareness, along with their unique adaptations to life at low latitude, neatly illustrates the GOLT’s predictive value.

8. Fish Growth Under Hyperoxia

Another critique of the GOLT that was recently published, claimed to test “its predictions” by exposing fish (Galaxia maculatus) to high (150%) levels of hyperoxia [25]. The hypothesis behind this study was that if fish grown in hyperoxia do not grow faster than in normoxia, this observation would demonstrate that oxygen is not a limiting factor to fish growth. This experimental setup is a surprising choice, however, given the fact that it has been known for nearly a century that at such high levels of hyperoxia, oxygen can act as a stressor and, thus, increase maintenance costs, which would leave less oxygen available for growth (see, e.g., the review by McArley et al. [76]).

Much more realistic results can be produced in experiments where mild hypoxia is used to quantify the effect of oxygen on growth and reproduction (see e.g., [29,77]), and they confirmed the GOLT’s predictions. Other experiments that would effectively test the GOLT could focus on the food conversion efficiency of fish (i.e., growth increment/food intake), which ought to decline with size [78], a feature that the GOLT assigns to a declining oxygen availability for growth because of the higher maintenance demand in larger individuals. This would be useful as the GOLT is incomplete because it does not deal sufficiently with food and feeding-related issues [16].

Skeeles et al. [25], however, assume that hypoxia exposure would be an imperfect test for the limiting role of oxygen for growth. As they argue, “depriving fish of normal oxygen levels may trigger a range of physiological processes (e.g., gill remodelling) that divert resources away from somatic growth, leading to a flawed understanding of the role of oxygen in the [temperature-size rule] under normoxic conditions” [25]. While gill remodeling as a response to hypoxia is indeed observed in many freshwater species (and it illustrates that the gills are a crucial bottleneck in the oxygen cascade), this argument does not only lack support from empirical data but its underlying assumptions are inconsistent. Suppose an organism is short of a crucial resource like oxygen. In that case, it is inappropriate to consider only the processes that are involved in rescuing the organism from the stressful situation but not the stressful situation itself. As experimental studies on the impact of increasing temperatures on fish show, stressful conditions activate mechanisms that reduce metabolic costs instead of adding even higher energy expenditures [79]. Thus, this argument is not based on current physiological knowledge, illustrating that ectotherms can utilize plastic responses to reduce metabolic costs. However, these reduction strategies do indeed come at the expense of growth, which is what the GOLT essentially relies on.

The choice of the species used for the experiment by Skeeles and Clark [25], the inanga (Galaxia maculatus), is also surprising given the fact that it is known to be an oxyconformer, which is rare in bony fish. Even more curiously, the authors mention none of the numerous papers that reported on the lack of a capacity for oxyregulation in G. maculatus (e.g., [80,81] or the many other studies by Urbina on the respiratory behaviour of inanga). While this species would have allowed for a case study aimed at an identification of the limits of generalizing mechanistic theories, its oxyconforming behavior is far from representative for WBE.

9. Avoiding Mechanistic Overstretch

While both the GOLT and the OCLTT understand themselves as generalizing mechanistic theories, their scope is defined within explanatory constraints. Stating that surface/volume relationships imply a changing balance of energy allocation to growth and maintenance is not a deterministic statement that will predict the specific adaptation to temperature or oxygen availability to the life histories of all known fish species. Strictly speaking, it only defines the phenotypic space of possibilities that are available to them in order to cope with the challenge of oxygen limitation.

Despite the critiques of generalizing mechanistic accounts such as the GOLT or the OCLTT [17], another trend towards mechanistic hypotheses seems to take shape. Instead of looking at the constraints, another strand of studies aims at the theorization of the various physiological control mechanisms that may explain growth reductions before oxygen limitation becomes a threat [3,82,83,84]. While most of the mechanisms described in these studies may indeed occur and explain growth reductions in specific species and situations, they are unlikely to underlie universal and interspecies patterns.

The MASROS (‘Maintain aerobic scope and regulate oxygen supply’) framework, first proposed by Atkinson et al. [84], may represent the earliest account of such explanations of temperature-induced growth reductions. Similar to the ‘Aerobic scope protection’ (ASP) hypothesis proposed by Jutfelt et al. [81], which states that fishes reduce the costs of specific dynamic action (SDA) by consuming smaller meals at higher temperatures, it highlights internal control mechanisms that reduce fish sizes. Both accounts are compatible with the GOLT, as is the ‘ghosts of an oxygen limited past’ of Verberk et al. [3], which proposes an influence of the evolutionary past and an adaptation of growth to specific thermal environments in such a way that critical sizes are not reached. Such a mechanism seems plausible, and the GOLT does not exclude the possibility that inherited growth patterns could be adaptive, even though such a mechanism would raise the question why other ‘pasts’ are not inscribed into the evolutionary makeup of organisms that no longer experience oxygen limitation at present. A similar, yet far less plausible, account was recently presented by Johansen’s et al. PASLED (‘Protect aerobic scope and limit energy demand’) hypothesis, which seems to be a variant of the MASROS framework but without an clear explanation of why higher temperatures actually become limiting to growth.

The most important differences between these proposals and the GOLT (or the OCLTT) is that the two latter theories focus on the constraints within whose boundaries biological organisms can function, or on “the constraints of what can be alive”, as Wouters [85] has put it. Adaptive responses, as described by the MASROS, ASP, PASLED, or ‘ghosts of an oxygen-limited past’, are likely not universal, as there is not one single evolutionary ‘solution’ to a challenge like oxygen limitation. The GOLT does not incorporate such forms of ‘mechanistic overstretch’ into its theoretical framework. Instead of presenting a deterministic account of interspecific evolutionary responses to oxygen limitation, it outlines the space of possibilities in which adaptation is possible. As Hordijk [86] has reminded us, evolution is not guided by ‘entailing’ laws. However, while it is characterized by a seemingly unbounded diversity of biological phenomena, it is also not entirely ‘free’ but constrained by morphospaces, whose boundaries are set by geometrical and physical properties. Looking for specific adaptive mechanisms within the phenospace of what is biologically possible may be captivating, but it is unlikely to bring about generalizable insights that explain interspecific or even universal patterns and, thus, result in a parsimonious theory.

While the above-mentioned mechanistic hypotheses are compatible with constraint-based theories and may have merit in identifying the detailed adaptive responses to oxygen limitation in specific taxa, this may not be the case for the recent attempt to explain metabolic scaling patterns as adaptationist optimization strategies, for which “the invocation of physical constraints is unnecessary”, as White and Marshall [32] argued. While it is clear that both metabolic scaling and organismic growth patterns are optimized by natural selection, theories with an exclusive focus on optimization fail to explain within which boundaries and with respect to which constraints biological traits are optimized. Such versions of adaptationism have been the target of the biting sarcasm of Gould and Lewontin [87], whose arguments do not need to be repeated here, but losing sight of the ‘constraints of what can be alive’ that frame the phenotypically available space of possible adaptions is likely to result in unrealistic predictions on how organisms behave under stress or quickly changing circumstances. While the GOLT acknowledges the importance of adaptationism as a mode of biological explanation, it is built on the principle that optimization can only be appreciated in relation to the constraints presented by the relevant physical and physiological phenospace.

10. Conclusions: The Current Debate and the Future of Mechanistic Models

As the recent debate around mechanistic growth theories illustrates, the question of how published data on metabolic rates should be integrated into a mechanistic growth model depends on how their interpretation is linked to the respective relevant parameters. As shown here, the currently available evidence fails to support the notion that the oxygen uptake capacity that would be available to support further growth is independent of body size. Even though the direct cost of growth cannot immediately be measured during respirometry, empirical evidence shows that it impacts respirometry data in growing animals [30,45]. Moreover, the difficulty in isolating the overhead costs of growth from measured metabolic rates does not imply that the GOLT may be untestable (as subtly suggested by Wootton et al. [80]). What it shows, rather, is that empirical data only allow for biologically meaningful interpretations if they are analyzed and evaluated within the framework of coherent scientific models that account for the different energetic expenditures.

Testing the GOLT, and other mechanistic growth theories, may benefit most from confrontations of the patterns they predict with the result of meta-analyses, comprehensive reviews, and/or global analyses. While using single-species studies may result in valuable insights, their primary value lies in the identification of the limits of the theory’s applicability. Thus, for example, the study of Galaxia maculatus mentioned above, which identified hyperoxia as a limit to the positive role of oxygen in the growth of fish, would have been helpful if it had been framed as testing when and where ambient oxygen, for fish, turns from an asset to a liability.

The question of how climate warming impacts the physiology and, eventually, the population structure and ecosystems of fishes may be one of the most pressing current issues in all the different fields in which this group of animals is studied. Solving this question does not only require more research but also the development of convincing models that allow for a better and more general understanding of the complex phenomena we are currently observing.