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:

(3.3)

      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, decreases, increases, and hence the metabolic rate increases exponentially.

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 not with simple metabolic rates but temperature‐compensated 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 or 0.67.

      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 or 0.75 (West et al., 1997), while subsequent elaborations confirmed this value if the velocity of flow itself scales with mass, but suggested a value closer to 0.67 if velocity does not vary significantly with mass (Banavar et al., 2010). A value of 0.75 is attractive in that it conforms with an empirical estimate derived long ago by Kleiber (1932) from an analysis of metabolic rates in a number of birds and mammals. This had given rise to the so‐called ‘Kleiber’s law’, but the law had lacked a convincing theoretical underpinning before West et al.’s study.

      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