Are the surface areas of the gills and body involved with changing metabolic scaling with temperature?

The metabolic rate (MR) of animals is related to body mass (M_b). MR scales with M_b according to the power equation MR=aM^b, where a is a constant and b is the scaling exponent. This relationship has attracted many researchers over the past century. Many important hypotheses regarding the scaling of MR have been proposed (Glazier, 2014a), e.g. the organ size hypothesis (Itazawa and Oikawa, 1983; Oikawa et al., 1992), the fractal resource distribution network hypothesis (West et al., 1997; Brown et al., 2004), the surface area (SA) hypothesis (Rubner, 1883; Okie, 2013) and the metabolic-level boundaries (MLB) hypothesis (Glazier, 2005, 2008, 2009, 2010).

The SA hypothesis assumes that metabolic scaling is strongly influenced by the transport of materials through external exchange surfaces, such as skin, lungs and gills (Rubner, 1883; Okie, 2013), and that MR is proportional to the surface area of an isomorphically growing organism, and it predicts that the intraspecific b-value (b_RMR) for the resting metabolic rate (RMR) is equal to 2/3 (Rubner, 1883). According to the Euclidean surface area predictions, the b-value (b_BSA) of body surface area (BSA) can be predicted from the M_b–body length (L) scaling exponent b_L. When organisms grow isomorphically throughout ontogeny, b_L is equal to 3 and b_BSA could be 2/3. However, when the growing mode of an organism is anisomorphic (elongated, flatted or thickened), b_BSA could be different than 2/3, which causes varying b_RMR among species (Okie, 2013; Hirst et al., 2014).

The MLB hypothesis proposes that metabolic level, the vertical elevation of a metabolic scaling relationship, mediates the relative influence of SA- versus volume-related metabolic processes on the body-mass scaling of metabolic rate in organisms (Glazier, 2005, 2008, 2009, 2010, 2014b). When the metabolic level is high, metabolic scaling should be chiefly influenced by fluxes of resources, wastes and/or heat across surfaces (scaling as M_b^2/3). However, when the metabolic level is low and amply met by surface-dependent processes, metabolic scaling should be more related to the energy demand required to sustain the tissues, which is directly proportional to tissue mass or volume (scaling as M_b¹) (Glazier, 2008). Therefore, organisms with higher RMR have a lower intraspecific metabolic scaling exponent, that is, b_RMR is negatively correlated with the metabolic level (Glazier, 2005, 2008, 2009, 2010). Aside from the effect of the metabolic level on b_RMR, factors influencing the relative effects of a surface- or volume-related process should also affect metabolic scaling (Glazier, 2014b), which has been supported by our recent work, which showed that a negative correlation exists between SA of red blood cells and intraspecific b_RMR among several fish species (Luo et al., 2015). It can be hypothesized that the variation in SA and its scaling may affect how metabolic level affects b_RMR.

In water-breathing fish, the gill is mainly responsible for the transport of oxygen and metabolic wastes (Perry et al., 2009). Studies have compared the scaling exponent (b_GSA) of the gill surface area (GSA) with b_RMR in several species of fish (Oikawa and Itazawa, 1985; Oikawa et al., 1999). However, whether GSA limits b_RMR remains unclear, as GSA is more related to activity metabolism but not resting metabolism (Hughes, 1984; Oikawa and Itazawa, 1985; Oikawa et al., 1999; Killen et al., 2016). Unfortunately, those previous data of b_GSA were not determined together with b_RMR in the same specimen of an experiment. As the b_GSA and b_RMR of a species may vary depending on multiple factors (e.g. temperature, dissolved oxygen, predation regime and habitat) (Ohlberger et al., 2012; Luo and Wang, 2012; Glazier et al., 2011; Mcfeeters et al., 2011; Glazier and Paul, 2017), a comparison of the b_GSA and b_RMR of the same fish species in one experiment seems to be necessary.

In addition to the gills, the skin of fish may play an exchange role (Hughes and Al-Kadhomiy, 1988; Glover et al., 2013). Approximately 10–30% of resting oxygen uptake is via the skin of adult water-breathing fish (Kirsch and Nonnotte, 1977; Steffensen et al., 1981; Feder and Burggren, 1985; Takeda, 1990). It can be assumed that the exchange capacity of the fish body may be linked to its total surface area (TSA), including not only GSA, but also body surface area (BSA). In the larvae of Atlantic salmon (Salmo salar), the scaling exponent of TSA (b_TSA, 0.95) was virtually identical to b_RMR (0.94) (Wells and Pinder, 1996a,b), implying a close link between the two parameters. Therefore, according to the MLB hypothesis, it can be hypothesized that the increase in metabolic level may result in a reduction of b_RMR to values tending to be close to b_TSA.

Temperature is one important ecological factor influencing the metabolic level of fish (Jobling, 1994; Clarke and Johnston, 1999; Peck et al., 2003, 2005; Ohlberger et al., 2012; Luo and Wang, 2012). The metabolic level of fish increases as temperature increases, which reduces the intraspecific metabolic scaling exponent, as b_RMR is negatively correlated with metabolic level (Glazier, 2005, 2008, 2009, 2010). The negative effects of higher temperatures on b_RMR have been observed in many fish species (Xie and Sun, 1990; Killen et al., 2010; Luo and Wang, 2012). However, the increase in temperature may change the exchange SA of fish. For example, the functional respiratory area of gills may be enlarged by thermal plastic remodeling in the goldfish [Carassius auratus (Linnaeus 1758)] (Sollid et al., 2005; Sollid and Nilsson, 2006). Thus, the potential surface constraints of the gill exchange surface can be changed by temperature. The determination of respiratory SA, accompanied with the metabolic level of fish, may supply strong evidence for the effect of temperature on b_RMR.

In the present study, RMR and the exchange SA parameters, including BSA, GSA and TSA, of the goldfish were assessed at two temperatures, 15°C and 25°C. Then, the scaling exponents of those parameters were analyzed. Our earlier studies have suggested that previous theories can only partially explain the scaling exponent of RMR of the goldfish (Huang et al., 2013; Li et al., 2016). It remains unclear which exchange surface constricts the metabolic scaling exponent of the goldfish and how temperature affects its metabolic scaling. This study aimed to determine the influence of increasing the metabolic level at a higher temperature on the metabolic scaling of the goldfish and determine the link between metabolic scaling exponents and SA parameters of both the gills and the body.

where ΔC_DO is the difference in the dissolved oxygen concentration (mg O₂ l⁻¹) between the outlets of an experimental chamber and the control chamber, and v is the water flow rate through the chamber (l h⁻¹). The dissolved oxygen concentration was measured at the outlet of the chamber using an oxygen meter (HQ30, Hach Company, Loveland CO, USA). The water flow rate was determined by collecting the outflow water into a beaker over a given period of time. The water flow rate was adjusted to ensure that the oxygen concentration in the outlet water was higher than 7 mg l⁻¹ to avoid physiological stress (Zhang et al., 2014).

Once the determination of RMR was finished, the fish were killed by an overdose of anesthetic. M_b was weighed to the nearest 0.1 g, and L was measured to the nearest 0.1 cm. For the fish at 25°C, M_b ranged from 4.24 to 299.6 g (mean±s.d.=63.32±64.98 g) and L ranged from 7.13 to 23.78 g (mean±s.d.=14.37±5.31 cm). For the fish at 15°C, M_b ranged from 4.42 to 223.2 g (mean±s.d.=68.32±68.99 g) and L ranged from 7.47 to 24.03 g (mean±s.d.=15.01±5.38 cm). Then, all fins were dissected from the body of each individual, and a piece of plastic paper was stuck to the fish body, equal in shape and size, to fix body regions (Jaworski and Holm, 1992; Tucker et al., 2002; O'Shea et al., 2006). The fins and the plastic paper were laid out and photographed to measure the SA using the computer-linked image-analysis program Image-Pro Plus 6.0 (Media Cybernetics, Rockville, MD, USA). Next, the gill mass (M_g) was measured, and the first four gill arches were dissected from the left gill of the fish and put into Bouin's fluid. The gills were kept in the fixative for 2 days before transfer to 70% alcohol just before measuring GSA. GSA was calculated using a standardized gill morphology assessment (Hughes, 1966, 1984) with a microscope (Aigo Digital Technology Co., Beijing, China). The number of filaments on each gill arch was counted. Then, each arch was divided into six equal parts, and one middle filament of each part was chosen to measure the filament length and the secondary lamellae frequency on one side of the filament. Lastly, the bilateral areas of the secondary lamellas of the filaments from the second arch were determined as an averaged lamellae area. Total filament number (FN), total length of all gill filaments (TFL, mm), mean filament length (FL, mm), mean lamellae frequency (LF, mm⁻¹) and mean lamella area (LA, mm²) were obtained. The following formula was employed to calculate the GSA (mm²) of an individual: GSA=4×TFL×LF×LA. The final numbers of fish specimens were 30 for 25°C and 28 for 15°C.

The data were calculated using Microsoft Excel 2003 (Microsoft Corporation, Redmond, WA, USA), and statistical analysis was conducted using SAS (PROC MODEL, SAS v.9.4; SAS Institute, Cary, NC, USA). The original data were used for size allometric analysis using nonlinear regression and were fitted to three sets of four models, involving two straight lines and two power functions: assumed additive, normal, homoscedastic error or additive, normal, heteroscedastic error, and multiplicative, lognormal, heteroscedastic error (Lolli et al., 2017; Packard, 2017). Ordinary least squares regression was used for regression procedures. The SAS programming code used in the present study is modified from the code of Packard (2017) and is available online (see Script 1). Parameters (a, b, c, d and y0) were estimated based on the Marquardt procedure. The treatments ware set as 0 for fish at 15°C and 1 for those at 25°C. For each model, the effect of temperature on each variable was tested by the coefficient of the temperature term (c), and the effect of temperature on scaling exponent was tested by the coefficient of the interaction term (d) (Script 1). If the value for d was not significant, then the model with no interaction term was used for analysis. If the value for c was not significant, then the model with no temperature term was used for analysis. Akaike’s information criterion (AIC) was used to examine the quality of models (Burnham et al., 2011; Lolli et al., 2017; Packard, 2017). The model with the smallest AIC value was referred to as the best model and the AIC difference (ΔAIC) from the best model was used to rank the models (Burnham et al., 2011; Lolli et al., 2017; Packard, 2017). Regression parameters of scaling slopes were reported with 95% confidence intervals (CI) in graphs. Each scaling slope was compared with 2/3 using 95% CI. P-values less than 0.05 were considered statistically significant.

The regressions between M_b and most parameters including RMR, TSA, GSA, BSA, TFL, FN, LF and LA had the lowest ΔAIC values using models with lognormal, heteroscedastic error (model 9, 12 in Table S1). Most of those relationships were described with the best quality by the power function (with intercept) with lognormal, heteroscedastic error (model 12). The power functions (no intercept) with lognormal, heteroscedastic error (model 9) also worked satisfactorily, with low ΔAIC values (mostly lower than 2) for most relationships and with the best quality for the relationship between RMR and M_b. The regressions between GSA and M_g also had lognormal, heteroscedastic error and could be described by the power functions and straight lines with (or without) intercept satisfactorily. Thus, power functions with no intercept (model 9) were adopted and displayed in log–log graphs for ease of comparison with traditionally reported scaling functions.

RMR of the goldfish was higher at 25°C than at 15°C (c=0.994, P<0.0001) and the b_RMR at 15°C (0.898) was significantly larger than that at 25°C (0.815; Fig. 1), as demonstrated by the significant interaction between temperature and M_b (d=−0.0826, P=0.0418; model 9 in Table S1).

TSA of the goldfish scaled with M_b by a b_TSA of 0.837 at 25°C, larger than that at 15°C (0.789; Fig. 2), as demonstrated by the significant d value (d=0.0475, P=0.0332; model 9 in Table S1). b_GSA and b_BSA were not significantly different between temperatures, and the common values (b_GSA: 0.889; b_BSA: 0.673) were used for both temperatures. b_GSA and b_TSA were significantly larger than 2/3 at both temperatures. At 25°C, b_RMR was different to both b_BSA and b_GSA, but not to b_TSA, whereas at 15°C, b_RMR was different to both b_BSA and b_TSA, but not to b_GSA (Figs 1 and 2). The results showed the negative effects of temperature on both TSA and GSA, as demonstrated by the small but significant c values (c=−0.244, P=0.0044 for TSA; c=−0.101, P=0.0286 for GSA) (model 9 in Table S1). BSA showed no significant difference between temperatures.

GSA scaled with M_g by the common b value of 1.019 at both temperatures (Fig. 3; model 9 in Table S1). TFL, FL and LF scaled with M_b and showed no significant differences in both scaling exponents and intercepts between temperatures (Table S1). FN and LA also scaled with M_b by common scaling exponents at both temperatures. However, the significant c values for FN (−0.0438) and LA (−0.0863) showed negative effects of temperature (model 9 in Table S1).

Our study presented a positive association between temperature and RMR, but a negative association between temperature and b_RMR in the goldfish, which can be explained by the MLB hypothesis that the increase in metabolic level intensifies the relative importance of surface area processes on fluxes of resources, wastes and/or heat (Glazier, 2005, 2008, 2009, 2010), and/or by changes in relative exchange SA (Luo et al., 2015). The negative effect of metabolic level on b_RMR in our study supports the MLB hypothesis. In addition, the results showed a small but significant reduction in the values of TSA and GSA of the goldfish from 15°C to 25°C, which suggests that the plasticity in exchange surfaces at the individual level may also contribute to the reduced b_RMR at the higher temperature. Whether the exchange surfaces at the cellular level change with temperature needs to be further studied.

The oxygen uptake of fish occurs via its whole surface, including not only the gills but also the skin (Feder and Burggren, 1985; Glover et al., 2013). Actually, a considerable proportion of oxygen transfers occur via the fish body surface, e.g. 12% in the carp (Cyprinus carpio), 13% in the trout (Salmo gairdnerii), 23% in the tench (Tinca tinca) and 27% in the plaice (Pleuronectes platessa) (Kirsch and Nonnotte, 1977; Steffensen et al., 1981; Takeda, 1990). The present results showed that the b_RMR was larger than the b_TSA at 15°C, and the two tended to be equal at 25°C, which agrees with our prediction. It can be explained, according to the MLB hypothesis (Glazier, 2005, 2008, 2009, 2010; Hirst et al., 2014), that the low metabolic demand and thus the weak constraints of TSA allow a larger b_RMR at a lower temperature, whereas at a higher temperature, the enhanced constraints of TSA on RMR result in a lower b_RMR, tending to be close to b_TSA. An alternative explanation for the present results can be deduced according to the viscosity hypothesis proposed by Verberk and Atkinson (2013). The viscosity of water increases with decreasing temperature (Jumars et al., 1993; Von Herbing, 2002). According to the equation in Jumars et al. (1993), the viscosity of freshwater is 0.89×10⁶ m² s⁻¹ at 25°C and 1.14×10⁶ m² s⁻¹ at 15°C, suggesting an approximately 28% increase in the viscosity of the water cooling down from 25°C to 15°C in our study. The higher viscosity of cold water results in thicker boundary layers and larger drag, and thus may impede respiratory ventilation and oxygen uptake of aquatic organisms (Verberk and Atkinson, 2013). For very small organisms (<1 mg), oxygen uptake can be met by diffusion rather than convection of water, and the effect of water viscosity on oxygen uptake is not important; however, for organisms heavier than 1 mg, oxygen uptake switches to be convection dependent and is affected by water viscosity more seriously in smaller individuals because of smaller Reynolds number (Makarieva et al., 2008; Verberk and Atkinson, 2013). Therefore, within the organisms beyond the critical size of 1 mg, smaller individuals may exhibit a greater depression of MR than that of larger individuals at lower temperature, thus resulting in a negative association between temperature and b_RMR.

Many previous theories have attributed metabolic scaling to the limits of the exchange SA of organisms, which was assumed to scale by M_b^2/3 (Rubner, 1883; Okie, 2013). In the present study, the value of b_BSA of the goldfish (0.673) was not significantly different to 2/3, but the value of b_GSA (0.889) was larger than 2/3, independent of temperature. Greater than 2/3 scaling in respiratory SA is common, as the gill has high fractal dimensions (Okie, 2013; Weibel et al., 1991), e.g. 0.997 in three tunny species (Muir and Hughes, 1969), 0.85 in Atlantic mackerel Scomber scombrus (Hughes and Gray, 1972), 0.900 in Oncorhynchus mykiss (Jager and Dekkers, 1975), 0.794 in C. carpio (Oikawa and Itazawa, 1985), 0.813 in P. major (Oikawa et al., 1999) and 0.768 in Sander lucioperca (Satora and Wegner, 2012). In addition, the TSA of the goldfish also scaled with body mass by exponents larger than 2/3 at both temperatures. Similarly, previous studies have proposed that the scaling exponent of SA may be different to 2/3 (Glazier, 2010, 2014b). Those results suggest that the exchange SA may scale with body mass by different exponents, depending on the objective SA, and that the presupposed scaling value of 2/3 for the SA needs to be modified.

Our results showed a small, negative change in the GSA of the goldfish from 15°C to 25°C, different to a previous report (Sollid et al., 2005), which could be due to the relative warmer temperature range in our study. The remarkable increase of the GSA of the goldfish occurred within a lower temperature range (from 7.5°C to 15°C), and no further obvious changes were observed above 15°C (Sollid et al., 2005). The ontogenetic changes in GSA can be attributed to changes in different gill constituent variables depending on species, such as an increase in secondary lamellae area (Muir and Hughes, 1969; Mazon et al., 1998; Hartl et al., 2000) and an increase in filament length or lamellae frequency (Severi et al., 1997; Mazon et al., 1998; Timmerman and Chapman, 2004; Karakatsouli et al., 2006). LA and FL scaled with M_b of the goldfish by exponents larger than that of LF, which suggests that the increase in GSA with M_b is mainly attributed to the increase in LA and FL. In addition, GSA scaled isometrically with M_g (scaling exponent: 1.019) at both temperatures. Compared with GSA, M_g is a variable that can be determined more easily. Therefore, it suggests a potential method to estimate mass scaling of GSA conveniently by determining the mass scaling of M_g.

In conclusion, the present study supplied comparable data of several SA parameters and RMR, as well as the mass scaling exponents of the goldfish under different temperatures. The results verified the negative effect of metabolic level on b_RMR and showed that the increase in metabolic level with temperature resulted in a reduction of b_RMR to values tending to be close to b_TSA, which supports the MLB hypothesis. An alternative explanation is that the higher viscosity of cold water impedes respiratory ventilation and oxygen uptake and reduces MR more in smaller individuals than in larger individuals at lower temperature, thus resulting in a negative association between temperature and b_RMR.

We thank Mr Bo Zhang for his help in fish collection. We sincerely thank the anonymous reviewers for their helpful suggestions and comments.