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Resources and conditions are also important insofar as they determine the metabolic rates of individuals, which determine the levels of resources available to those individuals for reproduction, growth and so on, which in turn influence their life histories, their abundance and indeed all of the processes we discuss in subsequent chapters. This perspective has generated interest in a ‘metabolic theory of ecology’ (Brown et al., 2004). At the heart of metabolic theory are the effects on metabolic rate of temperature, and in particular of size (most often, body mass). We examined the effects of temperature in Section 2.3. We turn now to size.

metabolic scaling

The most fundamental point, perhaps, is that life is typically faster for small organisms than it is for large ones – metabolising at greater rates, and maturing and dying sooner. For example, per gram of body mass, a resting mouse metabolises about 20 times faster than an elephant. There are exceptions to this pattern, as we’ll see below, but the more general rule is very widespread. Putting this more formally, we can say that the rate of a metabolic process, Y, varies with size according to the equation

(3.1)

where Y₀ is referred to as the normalisation constant, reflecting the typical rate for the metabolic process concerned, M is the organism’s mass and b is the so‐called allometric exponent. Equation 3.1 may be said to describe metabolic scaling. Taking logs of both sides, we get:

(3.2)

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:

(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 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 times, and b will therefore vary. It is unwise to seek a single, universal value for b, or a single, simple basis for all metabolic scaling. The key message, picked up again in later chapters, is that the scaling of metabolic rate plays a key role in the dynamics at all levels of ecological organisation, from the individual to the whole community.