## The adaptive nature of metabolic scaling and its consequences

'I make no attempt to explain the 0.75 production scaling itself. I believe that it represents a fundamental coupling between an organism and its environment, but my attempts to derive it from even more basic considerations have thus far failed'. (Charnov 1993,p. 19)

Fig. 4.7 Optimal scaling of metabolic rate, following West et al. (1999). The problem is how to pack as many metabolic 'exchange units' of constant size (circles) into a three-dimensional volume (for simplicity the figure only shows two dimensions) while minimizing transport distance along the network. Two networks are shown: for a small organism (bottom) and a large organism (top). Note that the network for the small organism resembles a small piece of the network of the large organism. This 'self-similarity' or 'fractal' assumption helps generate the three-quarter power metabolic scaling. Metabolic rate is simply determined by the number of exchange units the organism contains, and is optimized with a three-quarter power relationship to body mass.

An unexplained assumption in Charnov's model was the three-quarter scaling exponent of metabolic rate with body mass. Recently, Jim Brown and co-workers have suggested an adaptive explanation for this (but see Kozlowski etal. 2003; Kozlowski and Konarzewski 2004), and have gone on to use it to not only explore many features of life history evolution but also other biological phenomena, including both ecological and evolutionary patterns, at large and small scales. The number of uses of the explanation is so large that Jim Brown and co-workers have to deny that it is a theory of 'everything', presumably settling for a 'large bundle of important things'. Work derived from this original theory even has its own name, dubbed 'metabolic ecology'.

The three-quarter scaling is most easily understood as arising from a packaging problem (West etal.1999). Imagine that the total area provided by exchange units of constant size limits metabolic rate. These units might be alveoli in the lungs, capillaries, leaves, or root hairs. We want to know how the density of these units should scale with body mass so as to maximize metabolic rate while minimizing the delivery distance from the surface to the point of demand and while being packed into a three-dimensional volume (Figure 4.7). According to West et al., this occurs when the number of exchange units is proportional to mass to the three quarters. The maths, for those so inclined, is explained in Box 4.1.

Box 4.1 The surface area of an object is proportional to its mass to the power 2/3, because area is two-dimensional and mass is three-dimensional. The area of exchange surfaces within a volume scales with slightly greater freedom provided by two parameters, one relating to the number of exchange units (ea) and one to the distance between them (el):

In a three-dimensional shape like a body, these parameters can vary between 0 and 1. It is easily seen that a is maximized when el = 0 and sa = 1. This makes sense because when sa = 1 the area is maximized and when el = 0 the distance between exchange units is minimized.When these values are taken, the above becomes a « M3/4.

If metabolism depends directly on area of exchange units, it takes on this three quarters exponent.

 Phenomenon Assumptions Predictions Reference Scaling of plant Plants compete for Scaling of plant density Enquist population resources and grow until with size is -4/3; energy et al. (1998) density limited by them; resource supply per unit area is constant use per area is therefore size invariant Scaling of For a given mass, stem For trees of the same Enquist production and diameter depends on diameter, production is et al. (1999) life history traits wood density; independent of wood in plants proportional investment in reproduction is constant; maturity and lifespan determined by assuming determinate growth density, relative growth rate scales as -0.25, trees with dense wood live longer and mature later Above ground Plant design is limited Scaling of leaf number West structure of by branching supply (0.75), height of tree et al. (1999) plants networks; trunks and branches resist buckling while minimizing energy dissipation (0.25), flow rate per tube (0). Maximum tree height is about 100 m, limited by the required width of the supply tubes Ontogenetic Growth depends on Growth is sigmoidal, and West growth rates metabolic supply and also on the demands of maintenance at a given proportion of asymptotic mass, all organisms spend about the same proportion of metabolism on growth et al. (2001) Latitudinal Total energy flux per Number of species Allen gradient in area is body size increases with temperature et al. (2002) species richness invariant, number of in ectotherms of guilds across individuals in guild is communities constant Metabolic rates Cells within a body, Knowing metabolic rate at West in mammals, mitochondria within a one level of organization et al. (2002) cells, cell, and respiratory is sufficient to predict the mitochondria, complexes within others and molecules mitochondria are all supplied by fractal-like space-filling networks

Thus natural selection may provide an explanation for the scaling, and hence, yet another contribution to the explanation of mammalian life histories, as well as other quarter-power scaling exponents that derive from it. However, combined with other assumptions, it can be used as the starting point for the explanation of many other phenomena, most of which are supported by comparative data. Table 4.2 lists some of these. I cannot help but quote Churchill again: 'All the great things are simple, and many can be expressed in a single word'. In this case, the word is 'quarter'.

Let us see how one of these arguments works for a life history trait: growth rate (West et al. 2001). The aim is to predict how animals grow from basic assumptions about how metabolic rate scales with body mass. First, assume that growth is fuelled by metabolism, and occurs by cell division. Then assume that during growth a proportion of metabolism is devoted to making new tissue and a certain amount to maintenance. The addition of new mass depends on supply, proportional to mass to the three quarters (how the units of exchange scale), and on demand, which is directly proportional to mass (the number of cells). Thus, as growth occurs, and mass increases, demand begins to outstrip supply, ultimately limiting growth (Figure 4.8).

This is beguilingly simple. If this simple model of growth is correct, it can be shown that all animals should fit a single sigmoidal curve equation, and this equation contains no less than three quarter-power exponents resulting from the original metabolic assumptions. The values of some of the equations' parameters vary from species-to-species and this is what gives different organisms apparently different growth profiles. Data for several vertebrates do indeed fit the predicted curves, although in practice they cover different parts of it: at birth a cow is already expending half its energy in maintenance, and quickly approaches asymptotic size. A cod, however, only spends half its energy on maintenance after 6 years of age. The cod can

 \ Cow \ \ / Cod \ --------- --------------j Fig. 4.8 The sigmoid curve of animal growth and development, following West etal. (2001). The curve is asymptotic simply because demand increasingly outstrips supply as size increases. The underlying equation of the line is the same for all species. However, because some of the parameters take species-specific values, the curves would appear to differ for different species when plotted on the same axes shown. Cod live most of their lives on the left part of their curve maturing well before the asymptote (indeterminate growth), and cows mostly on the right, maturing closer to it (determinate growth). Compare this with Figure 4.6. Time Fig. 4.8 The sigmoid curve of animal growth and development, following West etal. (2001). The curve is asymptotic simply because demand increasingly outstrips supply as size increases. The underlying equation of the line is the same for all species. However, because some of the parameters take species-specific values, the curves would appear to differ for different species when plotted on the same axes shown. Cod live most of their lives on the left part of their curve maturing well before the asymptote (indeterminate growth), and cows mostly on the right, maturing closer to it (determinate growth). Compare this with Figure 4.6. never really expect to reach asymptotic size, and in turn devotes a substantial proportion of total lifetime energy to growth (about 40%). A cow, however, expends only about 1% of its lifetime energy use to growth, and only 10% of its pre-maturation energy use. Thus, according to the model determinate and indeterminate growth simply reflect whether an organism reaches asymptotic size before death. The model, even more so than Charnov's model, emphasizes the similarities among organisms rather than their differences,something that applies to most of the models that rely on common scaling assumptions. The model is mechanistic rather than evolutionary, and merely invokes natural selection for the original metabolic scaling assumption. Rather nicely, Charnov (2001), following on from the above work, has incorporated these new growth assumptions into a modified theory of mammalian life history evolution. The new model is able to retain the original quarter-power scaling predictions of his original model (Charnov 1991) but now makes more realistic assumptions about growth and metabolism that themselves derive from first principles. Life history theory, therefore, traditionally concentrating on individual traits, moving into invariant relationships and a more whole organism view, has come full circle again. Throughout however, the big message the field brings is that natural selection has played a fundamental role in generating both the basic similarities that organisms share, and the differences that make each unique. The field has made exciting claims in recent years about how evolution shapes us all, and there is more excitement to come. The following chapters consider traits that do not find traditional placement in life history evolution,but might easily have done so.

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