metabolic rates (see, for example, Figure 3.31). Here, too, therefore, we return to Equation 3.2 and the value of b.
a basis for metabolic scaling: SA and RTN theories
What should the allometric exponent be? As explained before, most answers to this question have focused on constraints on rates of transport. There have been two main types of theory: surface area theories (SA) and resource‐transport network theories (RTN), both with histories stretching back to the 1800s (Glazier (2014); and see Glazier (2005) for a much fuller subdivision of theories). SA theories argue that the rate of any metabolic process is limited by the rate at which resources for that process can be transported in, or at which the heat or waste products generated by the process can be transported out. This transport occurs across a surface, either within the organism or between the organism and its environment, the extent of which increases with the square (power 2) of linear size – as too, therefore, does the metabolic rate. However, assuming no change in shape, mass itself increases with the cube (power 3) of linear size. Hence, the metabolic rate, rather than keeping up with this increase in mass (where b would be 1) lags behind, scaling with mass with an exponent (b) of
RTN theories, on the other hand, focus on the geometries of transport networks that would optimise the flow of nutrients being dispersed from a centralised hub to target tissues within an organism, or the flow of waste products carried away in an equivalent manner in the opposite direction. Derivations based on networks assumed to be of this type are more complex than the simple area‐to‐volume arguments applied above. However, we can ignore these details and note simply that initial attempts to derive a metabolic scaling rule based on such networks led to a b value of
a universal b?
Attempts like these to derive an ‘expected’ value for b have often been motivated by a wish to discover fundamental organising principles governing the world around us –universal rules linking metabolism to size – a single, common value of b (Brown et al., 2004). Others have suggested that such generalisations may be oversimplified (Glazier, 2010, 2014). There need be no conflict between these two viewpoints. It can be valuable to have a single, simple theory that goes a long way towards explaining the patterns we see in nature. But it is also valuable to have a more complex, multifaceted theory that explains even more, including apparent exceptions to the simple rule. Similarly, when we examine data for these relationships, it can be valuable to focus on the general trend and fit a single line to the data, even if there is considerable variation around that general trend. But it is also valuable to treat that variation not as noise but as something requiring an explanation in its own right – for which a more complex model may be required.
A review of the data, overall, argues against a universal value for b. The analysis in Figure 3.31 suggested that a single value between 0.67 and 0.75 was appropriate for multicellular animals (metazoa), unicellular organisms and plants. However, a more detailed look suggests that metazoa do indeed have an exponent of around 0.75, but for unicellular eukaryotes (protists) the value is close to 1 (isometry) and for prokaryotes significantly greater than 1 (Figure 3.33) (DeLong et al., 2010). DeLong et al. hypothesise, with some empirical support, that the prokaryote value greater than one reflects an increase in genome size (and hence metabolic complexity) as organism mass increases; and that the protist value of one reflects a linear increase with size in ATP‐synthesising (energy‐generating) sites bound to membranes, which are surfaces. The metazoan value then reflects more conventional body surface or transport network constraints (DeLong et al., 2010).
Figure 3.33 Relationships between metabolic rate and body mass for heterotrophic prokaryotes, protists and metazoans, plotted on logarithmic scales. The black lines and closed points are for active metabolic rates and the grey lines and open points for resting rates. In each case, the fitted slopes (± SE) are shown. All are significant (P < 0.05).
Source: After DeLong et al. (2010).
As another example, the allometric exponent in plants appears to be consistently different between, on the one hand, seedlings and the smallest plants, and on the other, larger saplings and adult plants – close to 1 for plants with masses less than 1 g and converging to 0.75 as masses exceed around 100 g (Figure 3.34), though the particular mass values should not be taken too literally. In this case, the authors hypothesise that for larger plants, the photosynthetic machinery is distributed across surfaces (principally of leaves), whereas for smaller plants most or all of the tissue (and hence a volume) is photosynthetically active (Mori et al., 2010). A curvilinear relationship has also been proposed for mammals, but this time with the opposite curvature, starting at 0.57 and rising to 0.87 (Kolokotrones et al., 2010).
Figure 3.34 The allometric exponent of metabolism in plants decreases with plant size. (a) The relationship between temperature‐adjusted respiration rate and above‐ground plant mass across a wide range of masses on logarithmic scales. A curvilinear power function was fitted to the data, the changing slope of which is shown in (b).
Source: After Mori et al. (2010).
Note, to add a further perspective, that alongside SA and RTN theories, there is an equally long tradition of emphasising body composition as a driver of metabolic rate, with some organisms having a much higher proportion of structural, low‐metabolising tissue than others (see Glazier, 2014); and other studies again have emphasised the importance of changing shape (which the simpler theories assume remains constant) and show that the shifting patterns of metabolic rates with shape support the SA but not the RTN theories of metabolic scaling (Hirst et al., 2016). However, particular values of b, and the truth or otherwise of the hypotheses proposed to explain them, are less important than the more general point that an organism’s rate of metabolism reflects a whole host of constraints and demands, and different factors will therefore dominate in their effects in different organisms, and at different
