So why does the maximal metabolic rate scale with a higher exponent? If doubling the number of cells doubles the metabolic rate, then each constituent cell consumes the same amount of food and oxygen as before. When the relationship is directly proportional, the exponent is 1. The closer an exponent is to 1, then the closer the animal is to retaining the same cellular metabolic power. In the case of maximum metabolic rate, this is vital. To understand why it is so important, let's think about muscle power: clearly we want to get stronger as we get bigger, not weaker. What actually happens?

The strength of any muscle depends on the number of fibres, just as the strength of a rope depends on the number of fibres. In both cases, the strength is proportional to the cross-sectional area; if you want to see how many fibres make up a rope, you had better cut the rope—it's strength depends on the diameter of the rope, not its length. On the other hand, the weight of the rope depends on its length as well as its diameter. A rope that is 1 cm in diameter and 20 metres long is the same strength, but half the weight, as a rope that is 1 cm in diameter and 40 metres long. Muscle strength is the same: it depends on the cross-sectional area, and so rises with the square of the dimensions, whereas the weight of the animal rises with the cube. This means that even if every muscle cell were to operate with the same power, the strength of the muscle as a whole could at best increase with mass to the power of 2/3 (mass067). This is why ants lift twigs hundreds of times their own weight, and grasshoppers leap high into the air, whereas we can barely lift our own weight, or leap much higher than our own height. We are weak in relation to our mass, even though the muscle cells themselves are not weaker.

When the Superman cartoons first appeared in 1937, some captions used the scaling of muscle strength with body mass to give 'a scientific explanation of

Clark Kent's amazing strength.' On Superman's home planet of Krypton, the cartoon said, the inhabitants' physical structure was millions of years advanced of our own. Size and strength scaled on a one-to-one basis, which enabled Superman to perform feats equivalent, for his size, to those of an ant or a grasshopper. Ten years earlier, J. B. S. Haldane had demonstrated the fallacy of this idea, on earth or anywhere else: 'An angel whose muscles developed no more power, weight for weight, than those of an eagle or a pigeon would require a breast projecting for about four feet to house the muscles engaged in working its wings, while to economise in weight, its legs would have to be reduced to mere stilts.'

For biological fitness, it's plainly important to be strong in proportion to weight, as well as just having brute strength. Flight, and many gymnastic feats such as swinging from trees or climbing up rocks, all depend on the strength-to-weight ratio, not on brute strength alone. Numerous factors (including the lever-length and contraction speed) mean the forces generated by muscle can actually rise with weight. But all this is useless if the cells themselves grow weaker with size. This might sound nonsensical—why would they grow weaker? Well, they would grow weaker if they were limited by the supply of oxygen and nutrients, and this would happen if the muscle cells were constrained by a fractal network. Muscle would then have two disadvantages—the individual cell would be forced to become weaker, and at the same time the muscle as a whole would be obliged to bear greater weights. A double whammy. This is the last thing we would want: there is no way out of muscle having to bear greater weights with increasing size, but surely nature can prevent the muscle cells becoming weaker with size! Yes it can, but only because fractal geometry doesn't apply.

If muscle cells don't become weaker with larger size, their metabolic rate must be directly proportional to body mass: they should scale with an exponent of 1. For every step in mass there should be an equal step in metabolic rate, because if not the muscle cells can't sustain the same power. We can predict, then, that the metabolic power of individual muscle cells should not decline with size, but rather scale with mass to an exponent of 1 or more; they should not lose their metabolic power. This is indeed what happens. Unlike organs such as the liver (wherein the activity falls sevenfold from rat to man, as we've seen) the power and metabolic rate of the skeletal muscle is similar in all mammals regardless of their size. To sustain this similar metabolic rate, the individual muscle cells must draw on a comparable capillary density, such that each capillary serves about the same number of cells in mice and elephants. Far from scaling as a fractal, the capillary network in skeletal muscles hardly changes as body size rises.

The distinction between skeletal muscle and other organs is an extreme case of a general rule—the capillary density depends on the tissue demand,, not on the limitations of a fractal supply network. If tissue demand rises, then the cells use up more oxygen. The tissue oxygen concentration falls and the cells become hypoxic—they don't have enough oxygen. What happens then? Such hypoxic cells send distress signals, chemical messengers like vascular-endothelial growth factor. The details needn't worry us, but the point is that these messengers induce the growth of new capillaries into the tissue. The process can be dangerous, as this is how tumours become infiltrated with blood vessels in cancer (the first step to metastasis, or the spreading of tumour outposts to other parts of the body). Other medical conditions involve the pathological growth of new blood vessels, such as macular degeneration of the retina, leading to one of the most common forms of adult blindness. But the growth of new vessels normally restores a physiological balance. If we start regular exercise, new capillaries start growing into the muscles to provide them with the extra oxygen they need. Likewise, when we acclimatize to high altitude in the mountains, the low atmospheric pressure of oxygen induces the growth of new capillaries. The brain may develop 50 per cent more capillaries over a few months, and lose them again on return to sea level. In all these cases— muscle, brain, and tumour—the capillary density depends on the tissue demand, and not on the fractal properties of the network. If a tissue needs more oxygen, it just asks for it—and the capillary network obliges by growing new feeder vessels.

One reason for capillary density to depend on tissue demand may be the toxicity of oxygen. Too much oxygen is dangerous, as we saw in the previous chapter, because it forms reactive free radicals. The best way to prevent such free radicals from forming is to keep tissue oxygen levels as low as possible. That this happens is nicely illustrated by the fact that tissue oxygen levels are maintained at a similar, surprisingly low level, across the entire animal kingdom, from aquatic invertebrates, such as crabs, to mammals. In all these cases, tissue oxygen levels average 3 or 4 kilopascals, which is to say about 3 to 4 per cent of atmospheric levels. If oxygen is consumed at a faster rate in energetic animals such as mammals, then it must be delivered faster: the through-flow, or flux, is faster, but the concentration of oxygen in the tissues need not, and does not, change. To sustain a faster flux, there must be a faster input, which is to say a stronger driving force. In the case of mammals, the driving force is provided by extra red blood cells and haemoglobin, which supply far more oxygen than is available in crabs. Physically active animals therefore have a high red blood cell and haemoglobin count.

Now here is the crux. The toxicity of oxygen means that tissue delivery is restricted, to keep the oxygen concentration as low as possible. This is similar in all animals, and instead a higher demand is met by a faster flux. The tissue flux needs to keep up with maximum oxygen demand, and this sets the red blood cell count and haemoglobin levels for any species. However, different tissues have different oxygen demands. Because the haemoglobin content of blood is more or less fixed for any one species, it can't change if some tissues need more or less oxygen than others. But what can change is the capillary density. A low oxygen demand can be met by a low capillary density, so restricting excess oxygen delivery. Conversely, a high tissue oxygen demand needs more capillaries. If tissue demand fluctuates, as in skeletal muscle, then the only way to keep tissue oxygen levels at a constant low level is to divert the blood flow away from the muscle capillary beds when at rest. Accordingly, skeletal muscle contributes very little to resting metabolic rate, because blood is diverted to organs like the liver instead. In contrast, skeletal muscle accounts for a large part of oxygen consumption during vigorous exercise, to the point that some organs are obliged to partially shut down their circulation.

The diversion of blood to and from the skeletal muscle capillary beds explains the higher scaling exponent of 0.88 for maximal metabolic rate: a larger proportion of the overall metabolic rate comes from the muscle cells, which scale with mass to the power of 1—in other words, each muscle cell has the same power, regardless of the size of the animal. The metabolic rate is therefore somewhere in between the resting value of mass2/3 or mass3/4 (whichever value is correct) and the value for muscle, of mass to the power of 1. It doesn't quite reach an exponent of 1 because the organs still contribute to the metabolic rate, and their exponent is lower.

So the capillary density reflects tissue demand. Because the network as a whole adjusts to tissue demands, the capillary density does actually correlate with metabolic rate—tissues that don't need a lot of oxygen are supplied with relatively few blood vessels. Interestingly, if tissue demand scales with body size—in other words, if the organs of larger animals don't need to be supplied with as much food and oxygen as those of smaller animals—then the link between capillary network and demand would give an impression that the supply network scales with body size. This can only be an impression, because the network is always controlled by the demand, and not the other way around. It seems that West and colleagues may have confounded a correlation for causality.