3.1 may be said to describe metabolic scaling. Taking logs of both sides, we get:
allometric relationships
The slowing down of metabolism at larger sizes is reflected in values of b that are less than 1. This makes them allometric relationships – that is, relationships in which a physical or (in this case) physiological property of an organism changes relative to the size of the organism, rather than changing in direct proportion to the changing size. That would be an isometric relationship, and in that case b would be equal to 1. We can see from Equation 3.2 that b is the slope of the line when metabolism is plotted against body mass on logarithmic scales.
Examples are shown in Figure 3.31 for a wide range of taxonomic groups. As mass increases, temperature‐corrected metabolic rate increases less than proportionately; b = 0.71, with, apparently, little variation between groups. We discuss actual values of b below. The intercept in these plots is logY0, such that the value of Y0 locates the relationship vertically within the plot, telling us about the absolute rate of metabolism at a given body size (and temperature). In this case, there is variation in absolute rate between the groups, despite the relationship with size being apparently the same. Such allometric relationships can be ontogenetic (changes occurring as an organism grows) or phylogenetic (changes that are apparent when related taxa of different sizes are compared).
Figure 3.31 Metabolic scaling: the relationship between metabolic rate (Y, watts) and body mass (M, g) for a variety of organisms, as indicated, on logarithmic scales. The analysis sought a single slope but allowed the intercepts of different groups to vary. For clarity, the data points shown are the averages for mass‐classes. The metabolic rates have been temperature‐corrected to ensure different studies are comparable (see Equation 3.3), but this has been omitted from the equation, top right, for clarity.
Source: After Brown et al. (2004).
rates per unit mass, and times
Note that if individual metabolic rate scales with individual mass with an exponent of b, then metabolic rate per unit mass (that is, the metabolic rate of a gram of tissue) will scale with an exponent of b–1 (simply divide both sides of Equation 3.1 by M). Similarly, the time taken to complete a process (for example to reach maturity) will scale with an exponent of 1–b, because these times are the reciprocals of rates per unit mass (the reciprocal exponent of b–1 is –(b–1) = 1–b). It may sometimes be more appropriate to examine the metabolic scaling of times or rates per unit mass than individual rates.
transport or demand?
The conventional view, as we discuss later, is that plots like Figure 3.31 arise because size imposes constraints on rates of supply (of oxygen, nutrients and so on), and of transport of materials generally within the organism, which determine metabolic rates. Metabolic rates then constrain the resources available for reproduction, growth and so on (Brown et al., 2004). An alternative viewpoint, however, is that size is closely co‐adapted with investment in reproduction and growth (and hence with the demands for these), with rates and means of transport evolving to satisfy those demands (Harrison, 2017). Size in this case is an integral part of an organism’s overall life history, which has evolved to match its environment. From this perspective, a mouse, for example, metabolises rapidly, and an elephant metabolises more slowly, to fuel their respective life histories, of which size is a part. Each has evolved transport networks sufficient to service their metabolism. We discuss life history evolution more fully (and why mice might be fast, and elephants slow) in Chapter 7. The alternative viewpoints are summarised in Figure 3.32. Bringing the two together, we may conclude that there is really no single driver of metabolic scaling. Rather, we should see life histories and their metabolic demands as being co‐adjusted with transport system design (Glazier, 2014).
Figure 3.32 Schematic representation of the two main approaches to the relationship between metabolic rate and size. In one, indicated by the red arrows, size sets limits to rates of supply of nutrients (and of transport generally, for example of waste product excretion) and these supply routes or transport networks constrain an organism’s metabolic rate. In the other approach, indicated by black arrows, an organism’s life history, of which size is an integral part, evolves to match its environment, and the metabolic rate evolves to satisfy the demands (the metabolic capacity) of an organism with such a life history. The transport network, in turn, evolves to satisfy the demands of the metabolic rate.
temperature dependence
For a more complete account of metabolic theory, we must add the effects of temperature on metabolism to the effects of size. We saw in Section 2.3 that over a biologically realistic range of temperatures, the rate, Y, of a metabolic process is expected to increase exponentially, and this is conventionally described by the Arrhenius equation:
in which Y0 is the normalisation constant, as above, E is the activation energy required for that process, k is the so‐called Boltzmann’s constant and T is the temperature in Kelvin (a scale starting at absolute zero, in which 0°C is 273 Kelvin, and increments are the same as in the centigrade scale). For our purposes, we need only note that as temperature increases,
Clearly, we can bring metabolism, size and temperature together and obtain
(3.4)
In practice, however, most studies focus on size and allow for temperature by dealing
