Quantitative mismatch between empirical temperature-size rule slopes and predictions based on oxygen limitation

Model

At a given temperature i there should be a maximum body mass, Mmax_i, for which the maximum temperature-dependent surface-specific flux of oxygen, fmax_i (with unit mass O₂ area⁻¹ time⁻¹) allows for oxygen uptake to match consumption, and where a further increase in size would lead to an oxygen deficit. This can be expressed as:

where the left side of the equation gives oxygen uptake and the right side represents oxygen demand. Amax_i is the maximum surface area used for oxygen uptake. Thus, the exact area of the organism that should be considered here will depend on the type of organism (i.e. gill surface area [e.g. fish] or other specific areas of the body surface where oxygen uptake occurs [e.g. ventral body region of Daphnia]). β is the allometric scaling exponent describing the relationship between body mass and oxygen consumption, and k_i is the parameter describing temperature-dependent oxygen consumption (with unit mass O₂ body mass⁻¹ time⁻¹). The relationship between A and M can be expressed as A = α∙M^c, where the constant α gives the mass specific surface area used for oxygen uptake (with units area mass⁻¹) when M = 1. The constant c is the allometric scaling exponent describing the relationship between body mass and area over which oxygen can diffuse. Thus, since maximum body size will only be limited by oxygen availability when oxygen demand increases faster than supply with increasing body size, the model is only valid for c < β.

Substituting Amax with α∙Mmax^c and rearranging Eq. (1) yields:

It should be emphasized that fmax is the maximum surface specific flux of oxygen which will be reached at the maximum body size. Thus, the surface specific flux of oxygen increases with increasing body size up to fmax, which is when oxygen becomes limiting. With this increase in body size, mass specific metabolism decreases with increasing body mass according to the value of the scaling exponent β. Thus, systems for delivery of oxygen to cells once it has entered the body will not become constraining with the increasing surface specific flux of oxygen as individual size increases.

By using Eq. (2) on log-scale we can express the linear proportional change in maximum body mass with an increase in temperature from j to i as:

As can be seen from this, for a given difference between β and c, the predicted response in maximum body mass to a change in temperature depends on the corresponding relative changes in fmax and oxygen consumption. If the proportional change in these two are equal, then no response in maximum body mass is predicted. To evaluate the strength of temperature effects on maximum body mass, Eq. (3) is used to calculate the slope of the change in log maximum body mass with increasing temperature by dividing the right hand size by i–j (i.e. Δlog Mmax °C⁻¹). From these slopes, the percentage change per degree increase in temperature is obtained as 100% (e^slope − 1).

Estimating model parameters

For isometric growth, the allometric scaling exponent c describing the relationship between body mass and area over which oxygen can diffuse has a value of 2/3. However, many organisms change their body shape throughout ontogeny, resulting in scaling exponents different from 2/3. Using Euclidian geometry, boundary values for this scaling exponent in organisms that lack gills and thus obtain oxygen directly through the body surface can be calculated from the scaling exponent of the body length-mass relationship²². For D. magna we estimated the scaling exponent of the body length-mass relationship to be 2.72^23,24, which results in boundary values (possible minimum and maximum values) for the surface area-body mass scaling exponent c of 0.684 and 0.735 (see Ref.²² for equations). Thus, these values were used in separate calculations of the predicted body mass changes.

Two assumptions are applied to predict body mass changes based on empirical measurements of fmax; (1) that the amount of body area available for oxygen uptake for a given body mass, and hence the constant a, is independent of temperature, and (2) that fmax depends only on temperature and is independent of body size. We describe below how, for our application of the model, assumption (1) can be relaxed, and we also confirm the validity of assumption (2).

Temperature-specific estimates of k and fmax were obtained using the same approach and data as Kielland et al.²¹, and we repeat the methods of that study in brief here. Individuals of a single clone of D. magna were acclimated to 17, 22 and 28 °C over three generations to ensure complete intra- and inter-generational plasticity. D. magna from the population used in the study show a monotonic increase in fitness within this temperature range²⁵, thus these data conform to the suggested criteria for evaluation of the temperature-size relationship²⁶. Measurements of oxygen consumption (V̇O₂) and critical dissolved oxygen thresholds (cO_2crit, i.e. oxygen level above which mass-specific oxygen consumption, V̇O₂*, remains unconstrained, and below which consumption declines) were then conducted on individuals at their respective acclimation temperatures (n = 77, 86 and 84 individuals at 17, 22 and 28 °C, respectively). To provide data on metabolic rates that as closely as possible resemble those experienced in the wild, animals were not starved, and they were allowed to perform their spontaneous swimming activity during measurements. Temperature-specific estimates of k were obtained directly from oxygen consumption data (see “Statistics” section). At a given temperature, fmax is proportional to the product of how available oxygen is in the environment (i.e. concentration cO₂) and the maximum efficiency by which the animal can obtain it (i.e. maximum area-specific oxygen diffusion into the body per unit oxygen available). The area-specific (and hence mass-specific) oxygen diffusion per unit oxygen available in the environment is at its maximum at cO_2crit. Thus, for a given individual, V̇O₂*/cO_2crit provides a measure of the maximum efficiency with which it can obtain oxygen at a given temperature²¹. This measure is identical to what has more recently been termed the oxygen supply capacity (or α)²⁷. For each of the three experimental temperatures we multiplied these efficiencies with the corresponding temperature-specific oxygen concentrations at saturation to obtain estimates of temperature-specific values of fmax.

The difference in estimated fmax across temperatures includes two potential mechanisms. First, there may be effects of temperature on how efficiently individuals obtain oxygen from the environment per area of the body that allows for oxygen uptake. This includes both plasticity in biological characteristics (e.g. oxygen carriers, membrane permeability) and physical characteristics of the water (e.g. diffusivity, viscosity and resulting boundary layers surrounding respiratory surfaces). However, the method used does not allow for quantifying the actual area of the body used for oxygen uptake. Thus, a second effect of temperature on fmax in those data may be due to plasticity in the shape of the organism (i.e. proportion of the body surface allowing for oxygen diffusion), and hence the constant α in the expression describing the relationship between mass and area given above. Thus, although the model (Eq. (2)) does not explicitly consider potential temperature effects on the relationship between mass and surface area used for oxygen uptake, any such effects are included when using the estimated temperature effects on fmax to make predictions about the strength of the temperature-size relationship.

One assumption of our application of the model described above is that fmax is independent of body size. This was not tested by Kielland et al.²¹. Thus, we tested for an effect of body mass on fmax using their data²⁸. We calculated fmax for each individual as described above, and fitted an lme model (package nlme²⁹) with fmax as a function of temperature (fixed factor) and body mass (mg, covariate), and with run as a random effect. The estimated effect of body mass on fmax was weakly negative and non-significant (slope ± SE − 0.29 ± 0.21, p = 0.168). Thus, this assumption appears to be valid for our application of the model.

We also use Eq. (3) to predict the strength of the temperature-size relationship in the absence of phenotypic plasticity. Under this scenario the temperature dependence of maximum oxygen diffusion can be calculated by the OSI approach²⁰. According to this, maximum oxygen diffusion will change proportionally with the product of diffusivity and oxygen concentration. Thus,

where DO₂ is the diffusivity of oxygen (m² s⁻¹, increasing with temperature²⁰) and is calculated as a temperature dependent product of viscosity and diffusivity in water^30,31. pO₂ is the ambient oxygen partial pressure, αO₂ is the solubility of oxygen in the water, and cO₂ is the oxygen concentration at saturation (mg O₂ l⁻¹, decreasing with temperature³²). According to this, OSI increases with increasing temperature²⁰. For the experimental temperatures used by Kielland et al.²¹, OSI has values of 0.06461, 0.06723 and 0.07037 µg O₂ h⁻¹ m⁻¹ at 17, 22 and 28 °C, respectively.

Statistics

All statistical analyses were carried out in the statistical software R v. 3.3.3³³. We make separate predictions about the temperature-size slope for the two temperature intervals (17–22 and 22–28 °C). To incorporate empirical uncertainty in temperature responses of k and fmax we used a bootstrapping-procedure to estimate means and 95% confidence intervals (i.e. 2.5 and 97.5 percentiles) for the temperature-size slopes. Each bootstrap replicate was sampled with replacement, with sample sizes equal to the number of observations from each of the 12 runs obtained by Kielland et al.²¹. For each replicate sample we calculated fmax for each individual based on their V̇O₂*, cO_2crit, and the oxygen content at 100% saturation at their respective temperatures. We then fitted an lme model (package nlme²⁹) with fmax as a function of temperature (fixed factor), and with run as a random factor. From this model we extracted the estimated temperature-specific values of fmax. We then obtained the temperature-specific oxygen consumption parameter k from the same replicate sample using an lme model containing log (oxygen consumption) as the dependent variable, temperature as a fixed factor, log (body mass) as a covariate, and run as a random factor. The intercepts from this model (i.e. for a Daphnia of 1 mg) were back-transformed and used as estimates of k. Finally, all the above parameter estimates were applied together with the estimated allometric scaling exponent²¹ (β = 0.801) in Eq. (3) to predict the temperature-size slope for that replicate. A total of 10,000 replicates were run to estimate mean values and 95% confidence intervals. The temperature-size slope predictions were calculated separately for the two boundary values of the surface area-body mass scaling exponents (c = 0.684 and 0.735). To produce equivalent estimates of predicted slopes when fmax ∝ OSI, this bootstrap procedure was repeated while setting the temperature-specific values of fmax equal to the calculated OSI values (see above).

Source: Ecology - nature.com