Power, Sex, Suicide
Page 23
Questioning the universal constant
There are various reasons to question whether the fractal model is really true, but one of the most important is the validity of the exponent itself—the slope of the line connecting the metabolic rate to the mass. The great merit of the fractal model is that it derives the relationship between metabolic rate and mass from first principles. By considering only the fractal geometry of branching supply networks in three-dimensional bodies, the model predicts that the metabolic rate of animals, plants, fungi, algae, and single celled organisms should all be proportional to their mass to the power of ¾, or mass0.75. On the other hand, if the steady accumulation of empirical data shows that the exponent is not 0.75, then the fractal model has a problem. It comes up with an answer that is found empirically not to be true. The empirical failings of a theory may inculcate a fantastic new theory—the failings of the Newtonian universe ushered in relativity—but they also lead, of course, to the demise of the original model. In our case here, fractal geometry can only explain the power laws of biology if the power laws really exist—if the exponent really is a constant, the value of 0.75 genuinely universal.
I mentioned that Alfred Heusner and others have for decades contested the validity of the 3/4 exponent, arguing that Max Rubner’s original 2/3 scaling was in fact more accurate. The matter came to a head in 2001 when the physicists Peter Dodds, Dan Rothman, and Joshua Weitz, then all at MIT in Cambridge, Massachusetts, re-examined ‘the 3/4 law’ of metabolism. They went back to the original data sets of Kleiber and Brody, as well as other seminal publications, to examine how robust the data really were.
As so often happens in science, the apparently solid foundations of a field turned to rubble on closer inspection. Although Kleiber’s and Brody’s data did indeed support an exponent of 3/4 (or in fact, of 0.73 and 0.72, respectively) their data sets were quite small, Kleiber’s containing only 13 mammals. Later data sets, comprising several hundred species, generally failed to support the 3/4 exponent when re-analysed. Birds, for example, scale with an exponent close to 2/3, as do small mammals. Curiously, larger mammals seem to deviate upwards towards a higher exponent. This is in fact the basis of the 3/4 exponent. If a single straight line is drawn through the entire data set, spanning five or six orders of magnitude, then the slope is indeed approximately 3/4. But drawing a single line already makes an assumption that there is a universal scaling law. What if there is not? Then two separate lines, each with a different slope, better approximate the data, so large mammals are simply different from small mammals, for whatever reason.3
This may seem a little messy, but are there any strong empirical reasons to favour a nice crisp universal constant? Hardly. When plotted out reptiles have a steeper slope of about 0.88. Marsupials have a lower slope of 0.60. The frequently cited 1960 data set of A. M. Hemmingsen, which included single-celled organisms (making the 3/4 rule look truly universal) turned out to be a mirage, reforming itself around whichever group of organisms were selected, with slopes varying between 0.60 and 0.75. Dodds, Rothman, and Weitz concurred an earlier re-evaluation, that ‘a 3/4 power scaling rule… for unicellular organisms generally is not at all persuasive.’ They also found that aquatic invertebrates and algae scale with slopes of between 0.30 and 1.0. In short, a single universal constant cannot be supported within any individual phyla, and can only be perceived if we draw a single line through all phyla, incorporating many orders of magnitude. In this case, even though individual phyla don’t support the universal constant, the slope of the line is about 0.75.
West and his collaborators argue that it is precisely this higher level of magnification that reveals the universal importance of fractal supply networks—the non-conformity of individual phyla is just irrelevant ‘noise’, like Galileo’s air resistance. They may be right, but one must at least entertain the possibility that the ‘universal’ scaling law is a statistical artefact produced by drawing a single straight line through different groups, none of which conforms individually to the overall ‘rule’. We might still favour a universal law if there was a good theoretical basis for believing it to exist—but it seems the fractal model is also questionable on theoretical grounds.
The limits of network limitation
There are some circumstances when it is clear that supply networks do constrain function. For example, the network of microtubules within individual cells are highly efficient at distributing molecules on a small scale, but probably set an upper limit to the size of the cell, beyond which a dedicated cardiovascular system is required to meet demand. Similarly, the system of blind-ending hollow tubes, known as trachea, which deliver oxygen to the individual cells of insects, impose quite a low limit to the maximum size that insects can attain, for which we can be eternally grateful. Interestingly, the high concentration of oxygen in the air during the Carboniferous period may have raised the bar, facilitating the evolution of dragonflies as big as seagulls, which I discussed in Oxygen. The supply system can also influence the lower limits to size. For example, the cardiovascular system of shrews almost certainly nears the lower size limit of mammals: if the aorta gets much smaller, the energy of the pulse is dissipated, and the drag caused by blood viscosity overcomes smooth flow.
Within such limits, does the supply network limit the rate of delivery of oxygen and nutrients, as specified by the fractal model? Not really. The trouble is that the fractal model links the resting metabolic rate with body size. The resting metabolic rate is defined as oxygen consumption at rest, while sitting quietly, well fed but not actively digesting a meal (the ‘post-absorptive state’). It is therefore quite an artificial term—we don’t spend much time resting in this state, still less do animals living in the wild. At rest, our metabolism cannot be limited by the delivery of oxygen and nutrients. If it was, we would not be able to break into a run, or indeed sustain any activity beyond sitting quietly. We wouldn’t even have the reserves of stamina required to digest our food. In contrast, though, the maximum metabolic rate—defined as the limit of aerobic performance—is unquestionably limited by the rate of oxygen delivery. We are swiftly left gasping for breath, and accumulate lactic acid because our muscles must turn to fermentation to keep up with demand.
If the maximum metabolic rate also scaled with an exponent of 0.75, then the fractal model would hold, as the fractal geometry would predict the maximal aerobic scope, which is to say the range of aerobic capacity between resting and maximum exertion. This might happen if the maximal and resting metabolic rates were connected in some way, such that (evolutionarily) one could not rise unless the other did. This is not implausible. There certainly is a connection between the resting and maximal metabolic rates: in general, the higher the maximal metabolic rate, the higher the resting metabolic rate. For many years, the ‘aerobic scope’ (the increase in oxygen consumption from resting to maximal metabolic rate) was said to be fixed at 5 to 10-fold; in other words, all animals consume around 10 times more oxygen when at full stretch than at rest. If true, then both the resting and the maximal metabolic rates would scale with size to the power of 0.75. The entire respiratory apparatus would function as an indivisible unit, the scaling of which could be predicted by fractal geometry.
So does the maximum metabolic rate scale with an exponent of 0.75? It’s hard to say for sure, as the scatter of data is confoundingly high. Some animals are more athletic than others, even within a species. Athletes have a greater aerobic scope than couch potatoes. While most of us can raise oxygen consumption 10-fold during exercise, some Olympic athletes have a 20-fold scope. Athletic dogs like greyhounds have a 30-fold scope, horses 50-fold; and the pronghorn antelope holds the mammalian record with a 65-fold scope. Athletic animals raise their aerobic scope by making specific adaptations to the respiratory and cardiovascular systems: relative to their body size, they have a greater lung volume, larger heart, more haemoglobin in red cells, a higher capillary density, and suchlike. These adaptations do not rule out the possibility that aerobic
scope might be linked to size as well, but they do make it hard to disentangle size from the muddle of other factors.
Despite the scatter, there has long been a suspicion that the maximum metabolic rate does scale with size, albeit with an exponent that seemed greater than 0.75. Then in 1999, Charles Bishop, at the University of Wales in Bangor, developed a method of correcting for the athletic prowess of a species, to reveal the underlying influence of body size. Bishop noted that the average mammalian heart takes up about 1 per cent of body volume, while the average haemoglobin concentration is about 15 grams per 100 ml of blood. As we have seen, athletic mammals have larger hearts and a higher haemoglobin concentration. If these two factors are corrected against (to ‘normalize’ data to a standard), 95 per cent of the scatter is eliminated. Log maximum metabolic rate can then be plotted against log size to give a straight line. The slope of this line is 0.88—roughly, for every four steps in metabolic rate there are five steps in mass. Critically, this exponent of 0.88 is well above that for resting metabolic rate. What does that mean? It means that maximum metabolic rate and mass are closer to being directly proportional—we are closer to the expectation that for every step in mass we have a similar step in metabolic rate. If we double the body mass—double the number of cells—then we very nearly double the maximum metabolic rate. The discrepancy is less than we find for the resting metabolic rate. This means that the aerobic scope rises with body size—the larger the animal, the greater the difference between resting and maximum metabolic rate; in other words, larger animals can in general draw on greater reserves of stamina and power.
This is all fascinating, but the most important point, for our purposes, is that the slope of 0.88 for maximal metabolic rate does not tally with the prediction (0.75) of the fractal model—and the difference is statistically highly significant. On this count, too, it seems that the fractal model does not correspond to the data.
Just ask for more
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 (mass0.67). 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 t
issue 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.