## 4.1.5. Allometric relationships

Author: A. Jan Hendriks

Reviewers: Nico van den Brink, Nico van Straalen

Learning objectives:

You should be able to

- explain why allometrics is important in risk assessment across chemicals and species
- summarize how biological characteristics such as consumption, lifespan and abundance scale size
- describe how toxicological quantities such as uptake rates and lethal concentrations scale to size

Keywords: body size, biological properties, scaling, cross-species extrapolation, size-related uptake kinetics

Introduction

Globally more than 100,000,000 chemicals have been registered. In the European Union more than 100,000 compounds are awaiting risk assessment to protect ecosystem and human health, while 1,500,000 contaminated sites potentially require clean-up. Likewise, 8,000,000 species, of which 10,000 are endangered, need protection worldwide, with one lost per hour (Hendriks, 2013). Because of financial, practical and ethical (animal welfare) constraints, empirical studies alone cannot cover so many substances and species, let alone their combinations. Consequently, the traditional approach of ecotoxicological testing is gradually supplemented or replaced by modelling approaches. Environmental chemists and toxicologists for long have developed relationships allowing extrapolation across chemicals. Nowadays, so-called Quantitative Structure Activity Relationships (QSARs) provide accumulation and toxicity estimates for compounds based on their physical-chemical properties. For instance bioaccumulation factors and median lethal concentrations have been related to molecular size and octanol-water partitioning, characteristic properties of a chemical that are usually available from its industrial production process.

In analogy with the QSAR approach in environmental chemistry, the question may be asked whether it is possible to predict toxicological, physiological and ecological characteristics of species from biological traits, especially traits that are easily measured, such as body size. This approach has gone under the name “Quantitative Species Sensitivity Relationships” (QSSR) (Notenboom, 1995).

Among the various traits available, body-size is of particular interest. It is easily measured and a large part of the variability between organisms can be explained from body size, with r2 > 0.5. Not surprisingly, body size also plays an important role in toxicology and pharmacology. For instance, toxic endpoints, such as LC50s, are often expressed per kg body weight. Recommended daily intake values assume a “standard” body weight, often 60 kg. Yet, adult humans can differ in body weight by a factor of 3 and the difference between mouse and human is even larger. Here it will be explored how body-size relationships, which have been studied in comparative biology for a long time, affect the extrapolation in toxicology and can be used to extrapolate between species.

Fundamentals of scaling in biology

Do you expect a 104 kg elephant to eat 104 times more than a 1 kg rabbit per day? Or less, or more? On being asked, most people intuitively come up with the right answer. Indeed, daily consumption by the proboscid is less than 104 times that of the rodent. Consequently, the amount of food or water used per kilogram of body weight of the elephant is less than that of the rabbit. Yet, how much less exactly? And why should sustaining 1 kg of rabbit tissue require more energy than 1 kg of elephant flesh in the first place?

A century of research (Peters, 1983) has demonstrated that many biological characteristics Y scale to size X according to a power function:

Y = a Xb

where the independent variable X represents body mass, and the dependent variable Y can virtually be any characteristic of interest ranging, e.g., from gill area of fish to density of insects in a community.

Plotted in a graph, the equation produces a curved line, increasing super-linearly if b > 1 and sub-linearly if b < 1. If b=1, Y and X are directly proportional and the relationship is called isometric. As curved lines are difficult to interpret, the equation is often simplified by taking the logarithm of the left and right parts. The formula then becomes:

log Y = log a + b log X

When log Y is plotted against log X, a straight line results with slope b and intercept log a. If data are plotted in this way, the slope parameter b may be estimated by simple linear regression.

Across wide size ranges, slope b often turns out to be a multitude of ¼ or, occasionally, ⅓. Rates [kg∙d-1] of consumption, growth, reproduction, survival and what not, increase with mass to the power ¾ , while rate constants, sometimes called specific rates [kg∙kg-1∙d-1], decrease with mass to the power –¼. So, while the elephant is 104 kg heavier than the 1 kg rabbit, it eats only (104)¾ = 103 times more each day. Vice versa, 1 kg of proboscid apparently requires a consumption of (104)-¼ kg∙kg-1∙d-1, i.e., 10 times less. Variables with a time dimension [d] like lifespan or predator-prey oscillation periods scale inversely to rate constants and thus change with body mass to the power ¼. So, an elephant becomes (104)¼ = 10 times older than a rabbit. Abundance, i.e., the number of individuals per surface area [m-2] decreases with body mass to the power –¾. Areas, such as gill surface or home range, scale inversely to abundance, typically as body mass to the power ¾.

Now, why would sustaining 1 kg of elephant require 10 times less food than 1 kg of rabbit? Biologists, pharmacologists and toxicologists first attributed this difference to area-volume relationships. If objects of the same shape but different size are compared, the volume increases with length to the power 3 and the surface increases with length to the power 2. For a sphere with radius r, for example, area A and volume V increase as A ~ r2 and V ~ r3, so area scales to volume as A ~ r2 ~ (V⅓)2 ~ V⅔. So, larger animals have relatively smaller surfaces, as long as the shape of the organism remains the same. Since many biological processes, such as oxygen and food uptake or heat loss deal with surfaces, metabolism was, for long, thought to slow down like geometric structures, i.e., with multitudes of ⅓. Yet, empirical regressions, e.g. the “mouse-elephant curve” developed by Max Kleiber in the early 1930s show a universal slope of ¼ (Peters, 1983. This became known as the “Kleiber’s law”. While the data leave little doubt that this is the case, it is not at all clear why it should be ¼ and not ⅓. Several explanations for the ¼ slope have been proposed but the debate on the exact value as well as the underlying mechanism continues.

Application of scaling in toxicology

Since chemical substances are carried by flows of air and water, and inside the organism by sap and blood, toxicokinetics and toxicodynamics are also expected to scale to size. Indeed, data confirm that uptake and elimination rate constants decrease with size, with an exponent of about –¼ (Figure 1). Slopes vary around this value, the more so for regressions that cover small size ranges and physiologically different organisms. The intercept is determined by resistances in unstirred water layers and membranes through which the substances pass, as well as by delays in the flows by which they are carried. The resistances mainly depend on the affinity and molecular size of the chemicals, reflected by, e.g., the octanol-water partition coefficient Kow for organic chemicals or atomic mass for metals. The upper boundary of the intercept is set by the delays imposed by consumption and, subsequently, egestion and excretion. The lower end is determined by growth dilution. Both uptake and elimination scale to mass with the same exponent so that their ratio, reflecting the bioconcentration or biomagnification factor in equilibrium, is independent of body-size.

Figure 1. Regressions of elimination rate constants [kg∙kg-1∙d-1 = d-1] as a function of organism mass m [kg], ranging from algae to mammals, and organic chemicals' octanol-water partition ratio Kow and metal mass within the limits set by consumption and production (redrawn by author, based on Hendriks et al. 2001).

Scaling of rate constants for uptake and elimination, such as in Figure 1, implies that small organisms reach a given internal concentration faster than large ones. Vice versa, lethal concentrations in water or food needed to reach the same internal level after equal (short-term) exposure duration are lower in smaller compared to larger organisms. Thus, the apparent "sensitivity" of daphnids can, at least partially, be attributed to their small body-size. This emphasizes the need to understand simple scaling relationships before developing to more elaborate explanations.

Using Figure 1, one can, within strict conditions not elaborated here, theoretically relate median lethal concentrations LC50 [μg L-1] to the Kow of the chemical and the size of the organism, with r2 > 0.8 (Hendriks, 1995; Table 1). Complicated responses like susceptibility to toxicants can be predicted only from Kow and body size, which illustrates the generality and power of allometric scaling. Of course, the regressions describe the general trends and in individual cases the deviations can be large. Still, considering the challenges of risk assessment as outlined above, and in the absence of specific data, the predictions in Table 1 can be considered as a reasonable first approximation.

Table 1. Lethal concentrations and doses as a function of test animal body-mass

Species | Endpoint | Unit | b (95% CI) | r2 | nc | ns | Source
Guppy | LC50 | mg∙L-1 | 0.66 (0.51-0.80) | 0.98 | 6 | 1 | 1
Mammals | LD10≈MTD | mg∙animal-1 | 0.73 | 0.69‑0.77 | 27 | 5 | 2
Birds | Oral LD50 | mg∙animal-1 | 1.19 (0.67-0.82) | 0.76 | 194 | 3…37 | 3
Mammals | Oral LD50 | mg∙animal-1 | 0.94 (1.18-1.20) | 0.89 | 167 | 3…16 | 4
Mammals | Oral LD50 | mg∙animal-1 | 1.01 (1.00-1.01) |  | >5000 | 2…8 | 5

MTD = maximum threshold dose, repeated dosing. LD50 single dose, b = slope of regression line, nc = number of chemicals, ns = number of species. Sources: 1 Anderson & Weber (1975), 2 Travis & White (1987), 4 Sample & Arenal (1999), 5 Burzala-Kowalczyk & Jongbloed (2011).

Allometry is also important when dealing with other levels of biological organisation. Leaf or gill area, the number of eggs in ovaries, the number of cell types and many other cellular and organ characteristics scale to body-size as well. Likewise, intrinsic rates of increase (r) of populations and the production-biomass ratios (P/B) of communities can also be obtained from the (average) species mass. Even the area needed by animals in laboratory assays scales to size, i.e., by m¾, approximately the same slope noted for home ranges of individuals in the field.

Future perspectives

Since almost any physiological and ecological process in toxicokinetics and toxicodynamics depends on species size, allometric models are gaining interest. Such an approach allows one to quantitatively attribute outliers (like apparently "sensitive" daphnids) to simple biological traits, rather than detailed chemical-toxicological mechanisms.

Scaling has been used in risk assessment at the molecular level for a long time. The molecular size of a compound is often a descriptor in QSARs for accumulation and toxicity. If not immediately evident as molecular mass, volume or area often pops up as an indicator of steric properties. Scaling does not only apply to bioaccumulation and toxicity from molecular to community levels, size dependence is also observed in other sections of the environmental cause-effect chain. Emissions of substances, e.g., scale non-linearly to the size of engines and cities. Concentrations of chemicals in rivers depend on water discharge, which in itself is an allometric function of catchment size. Hence, understanding the principles of cross-disciplinary scaling is likely to pay off in protecting many species against many chemicals.

References

Anderson, P.D., Weber, L.J. (1975). Toxic response as a quantitative function of body size. Toxicology and Applied Pharmacology 33, 471-483.

Burzala-Kowalczyk, L., Jongbloed, G. (2011). Allometric scaling: Analysis of LD50 data. Risk Analysis 31, 523-532.

Hendriks, A.J. (1995). Modelling response of species to microcontaminants: Comparative ecotoxicology by (sub)lethal body burdens as a function of species size and octanol-water partitioning of chemicals. Ecotoxicology and Environmental Safety 32, 103-130.

Hendriks, A.J. (2013). How to deal with 100,000+ substances, sites, and species: Overarching principles in environmental risk assessment. Environmental Science and Technology 47, 3546−3547.

Hendriks, A.J., Van der Linde, A., Cornelissen, G., Sijm, D.T.H.M. (2001). The power of size: 1. Rate constants and equilibrium ratios for accumulation of organic substances. Environmental Toxicology and Chemistry 20, 1399-1420.

Notenboom, J., Vaal, M.A., Hoekstra, J.A. (1995). Using comparative ecotoxicology to develop quantitative species sensitivity relationships (QSSR). Environmental Science and Pollution Research 2, 242-243.

Peters, R.H. (1983). The Ecological Implications of Body Size. Cambridge University Press, Cambridge.

Sample, B.E., Arenal, C.A. (1999) Allometric models for interspecies extrapolation of wildlife toxicity data. Bulletin of Environmental Contamination and Toxicology 62, 653-66.

Travis, C.C., White, R.K. (1988) Interspecific scaling of toxicity data. Risk Analysis 8, 119-125.