4
SEARCHING FOR SIMILARITY

UNFORTUNATELY, NOT EVERYONE kept up with developments in the biology of body size.

Elephants are intelligent, long-lived animals with complex social and emotional lives. We often flatter ourselves that their mental processes must be a lot like ours. But Louis Jolyon West and Chester Pierce were baffled by one aspect of elephant behavior: musth. A male elephant in musth becomes violent and temperamental, liable to attack anything, human or elephant, that gets in his way. The animal becomes watery-eyed and produces a foul-smelling secretion, brown and sticky, from a gland just behind his eye. Sexually mature male elephants are like this for several weeks each year. To humans in charge of elephants, whether in zoos, wildlife preserves, or as working animals, musth is a hazardous and unwelcome episode. In the United States, safety concerns have led to several captive animals in musth being killed.

To West and Pierce, two psychiatrists at the University of Oklahoma in Oklahoma City, musth seemed like madness—a pointless state in wild elephants and a dangerous one for captive animals and their handlers. It was the early 1960s, and the hallucinogen LSD, lysergic acid diethylamide, was a recent addition to the psychopharmacologist’s



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In the Beat of a Heart: Life, Energy, and the Unity of Nature 4 SEARCHING FOR SIMILARITY UNFORTUNATELY, NOT EVERYONE kept up with developments in the biology of body size. Elephants are intelligent, long-lived animals with complex social and emotional lives. We often flatter ourselves that their mental processes must be a lot like ours. But Louis Jolyon West and Chester Pierce were baffled by one aspect of elephant behavior: musth. A male elephant in musth becomes violent and temperamental, liable to attack anything, human or elephant, that gets in his way. The animal becomes watery-eyed and produces a foul-smelling secretion, brown and sticky, from a gland just behind his eye. Sexually mature male elephants are like this for several weeks each year. To humans in charge of elephants, whether in zoos, wildlife preserves, or as working animals, musth is a hazardous and unwelcome episode. In the United States, safety concerns have led to several captive animals in musth being killed. To West and Pierce, two psychiatrists at the University of Oklahoma in Oklahoma City, musth seemed like madness—a pointless state in wild elephants and a dangerous one for captive animals and their handlers. It was the early 1960s, and the hallucinogen LSD, lysergic acid diethylamide, was a recent addition to the psychopharmacologist’s

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In the Beat of a Heart: Life, Energy, and the Unity of Nature tool chest. To investigate musth, the researchers decided to manipulate elephant behavior with LSD, which seemed to send people and other animals mad. The psychiatrists teamed up with Warren Thomas, a veterinarian at Lincoln Park Zoo, to test their ideas on Tusko, the zoo’s male Asian elephant. If musth was madness, perhaps a dose of LSD would send Tusko into musth. Perhaps it would point to a “cure” for animals of their affliction. Perhaps Tusko would even produce that brown and smelly gunk. In humans an oral dose of 0.1 milligrams of LSD induces several hours of delirium. Higher doses, of a milligram or more, have powerful physical effects, including raised blood pressure and body temperature, sweating, and dilated pupils, along with psychosis and the drug’s trademark visions. But other animals, such as cats and monkeys, seem to be much less sensitive to LSD. Relative to body weight, the dose needed to produce a similar effect in a macaque monkey or a cat was 10 or more times that needed for a human. The psychiatrists decided to inject Tusko with 0.1 milligrams of LSD for every kilogram of his body weight. Since he weighed a shade under 3 tonnes, this worked out to 297 milligrams. Compared with the doses needed to affect cats or monkeys, this amount seemed conservative. “We considered that we were unlikely to see much reaction with this dosage of LSD,” they wrote. At 8:00 in the morning on August 3, 1962, someone at the zoo shot the 297 milligrams of LSD into Tusko’s rump with a dart from an air rifle. West, Pierce, and Thomas described what happened next in what must be one of the oddest papers ever published by the venerable American journal Science: Tusko began trumpeting and rushing around the pen…. His restlessness appeared to increase for 3 minutes after the injection; then he stopped running and showed signs of marked incoordination. His mate (Judy, a 15-year-old female) approached him and appeared to attempt to support him. He began to sway, his hindquarters buckled, and it became increasingly difficult for him to remain upright. Five minutes after the injection he trumpeted, collapsed, fell heavily onto his right side, defecated and went into status epilepticus. Tusko lay on his right side, shuddering, his eyes askew and pupils dilated, his left legs stretched out and his right legs curled up. His breathing was labored and his tongue, which he had bitten, had turned

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In the Beat of a Heart: Life, Energy, and the Unity of Nature blue. The researchers administered massive doses of anticonvulsant drugs but to no effect. At 10:40 a.m., an hour and 40 minutes after the dart was fired, the elephant died. An autopsy revealed that Tusko’s larynx had gone into spasm, blocking his windpipe and strangling him. Poor Tusko, and poor Judy. Even so, the researchers were unwilling to simply write off Tusko’s death as an experiment gone wrong. Perhaps, they speculated, the natural musth-triggering hormone was chemically similar to LSD. And perhaps some good could come from Tusko’s demise. “It appears that the elephant is highly sensitive to the effects of LSD—a finding which may prove to be valuable in elephant-control work in Africa,” they concluded. No conservation organization has pursued this suggestion. (West, who died in 1999, went on to become one of America’s most eminent psychiatrists. An expert on cults and brainwashing, he examined Jack Ruby after Ruby shot Lee Harvey Oswald and also Patty Hearst, the heiress-turned-terrorist.) Yet an understanding of the effects of body size on biology might have saved the trio their place in scientific infamy. You might need a tenth of a milligram of LSD per kilogram of body weight to give a cat a psychotic episode, but to expand Tusko’s consciousness, the researchers were wrong to simply give him a proportionate dose, the biggest single (recorded) hit of LSD ever administered—enough to trip out about 1,500 people. Because larger animals have proportionately much slower metabolisms, their cells degrade drugs more slowly, and so they need proportionately smaller doses. Using a calculation based on the relative metabolic rates, rather than the body sizes, of elephants and cats, an effective dose of LSD for an elephant would be 80 milligrams. Injected with nearly four times this amount, small wonder Tusko got the bad trip to end them all. If Tusko’s dose had been extrapolated from the amount typically given to humans, it would have been less than 10 milligrams, even without adjusting for metabolic rate. The knowledge that human metabolisms burn more slowly than those of the smaller animals used in experiments has now percolated through the scientific community more widely than it had by 1962, and today body size is routinely considered in drug trials and dosage calculations on humans and animals.

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In the Beat of a Heart: Life, Energy, and the Unity of Nature Tusko’s story is a gaudy and catastrophic footnote in the history of biological scaling. But over a half-century of experiments ranging from the quirky to the grisly, researchers’ efforts to work out what controlled an animal’s metabolic rate, and where Kleiber’s rule came from, were just as inconclusive. Inside or Out? Max Rubner originally thought that temperature sensors in an animal’s skin controlled its metabolic rate, stoking the fires in cold weather. Each sensor controlled a small fraction of total metabolism, he reasoned, so as an animal’s area increased, so would its number of sensors and its metabolic rate. This turned out not to be true, however, and eventually Rubner changed his mind and concluded that an animal’s metabolic rate was instead an intrinsic part of its biology. Much of the earliest thinking about metabolism focused, like Rubner, on the regulation of body temperature. But in the early twentieth century, biologists stopped believing that an animal’s metabolic rate was a reflection of how hard it was working to keep warm. After all, if that were so, the rate would fall as the animal got hotter, and this doesn’t happen. By the time Kleiber, DuBois, and Benedict were working on the problem, the belief that an animal’s basal metabolic rate was something that came from within was widespread. Researchers thought that most of the heat produced by animals was a sort of waste product, a thermal background hum emanating from all the body’s chemical reactions and physical activity, like the low buzzing noise a fridge makes. From measuring the volume of this metabolic hum, the next step was to locate its source and work out what controlled its volume. Biologists were searching for similarity: They wanted to find some feature of animals that varied with body size in the same way as metabolic rate. Rubner had thought that this feature was surface area; as his theory fell from favor, scientists began casting around for something to replace it. Having been let down by animals’ outsides, biologists went looking inside, dismantling their subjects into tissues and organs and measuring each component’s energy consumption. This promised to show

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In the Beat of a Heart: Life, Energy, and the Unity of Nature whether an animal’s metabolic rate is a product of its parts or its whole: whether mouse cells are always hungry for energy, even when they are not part of a mouse and whether cow organs are inherently sluggish. In the 1920s, separate French and German teams measured the rate at which slices of tissue from organs including brain, kidney, liver, and heart used up oxygen in a laboratory dish. The tissues came from species ranging from mice to cattle. In their home bodies these tissues from different animals would have burned energy at speeds varying by a factor of 10. In isolation they ran at more or less the same speed. Score one for metabolic rate as a property of whole animals, not their individual organs. But another camp claimed the opposite, that the metabolic rates of an animal’s parts matched its whole. In 1941, Max Kleiber did an experiment which, he claimed, showed that even in a laboratory dish a slice of sheep liver used energy more slowly than a slice of mouse liver. As with surface area measurements, the use of different species and experimental techniques made it difficult to compare the results coming out of different labs. How you treat a slice of tissue—what chemicals you bathe it in, for example—has a big effect on how much oxygen it consumes. Also like the surface law, the debate puttered on for decades. Probably the best experiments were done in 1950 by Hans Krebs at the University of Sheffield in England. Krebs, who won a Nobel Prize in 1953 for unraveling the chemical reactions that turn food into energy inside cells, tested a range of different tissues from different species under different conditions. He concluded that tissues from larger animals had slower metabolic rates but that the deceleration was too small to account for the pattern seen in whole animals. A logical, if gruesome, conclusion to this line of inquiry was to look at all the organs of an individual animal at once, to see if the metabolic rate of its parts summed to that of its whole. In 1955, Arthur Martin and Frederick Fuhrman, colleagues of Kleiber’s at Davis, did just that. They dissected 17 dogs into 23 different organs and measured the metabolic rate of each organ. They did the same for 15 mice, except they were able to extract only 21 organs (dogs were divided into testes and bladder; in mice they left the urogenital system intact). Martin and Fuhrman came down in favor of Kleiber’s earlier experiment, and

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In the Beat of a Heart: Life, Energy, and the Unity of Nature on the opposite side of Krebs, concluding that a disembodied organ had the same metabolic rate as the body it came from. Rather than coming to any conclusions, this line of inquiry fizzled out. The experiments were too labor-intensive and difficult to repeat and interpret. Again, some of the methods were questionable. Today, laboratories use animals whose origins and biology are well understood. Often the beasts are inbred for generations to make them genetically uniform. Martin and Fuhrman, on the other hand, got their dogs from the local pound. Maybe a still more detailed view would solve the problem. Metabolic rate, after all, depends on what happens in cells. And a small animal’s cells do have more metabolic equipment. They have more of the energy generators called mitochondria and higher concentrations of the enzymes that turn food into fuel, so they use energy more quickly. The concentrations of both mitochondria and enzymes fall with total body size in proportion with mass raised to the power of −1/4, identical to the rate at which relative metabolic rate slows with size. This, you could say, explains why relative metabolic rate slows with size, but it really just shifts the problem to explaining why the amount of energy-burning equipment declines. And disembodiment does have an effect. Cells and organs are sensitive to their surroundings. Many of the experimenters who studied isolated organs tried to take their measurements as soon as possible after the tissue came out of the animal. The metabolic rates of these organs would have carried a hangover from their previous surroundings. But if you take some cells from an animal and culture them in a Petri dish, they make more mitochondria and enzymes and their metabolic rate rises. It’s as if they think they have become part of some smaller organism. Over time, the metabolic rates of cells from different-sized animals growing in a Petri dish converge on the same high level. A cell’s energy consumption depends on how many cells surround it. A natural example occurs every time a baby mammal leaves its mother’s womb. In the womb a fetus has the same metabolic rate as its mother—it behaves as if it were one of her organs. Pregnant women are advised to eat 300 extra kilocalories per day, enough to both feed

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In the Beat of a Heart: Life, Energy, and the Unity of Nature their developing baby and increase their own weight by about 14 kilograms (30 pounds) during pregnancy. But when a human baby is born, its metabolic rate rises rapidly. A baby weighing 3.5 kilograms (7.5 pounds) needs about 400 kilocalories a day on its own. It would be foolish to push the comparison too far. Unlike cells in dishes, babies have sophisticated sensory equipment, brains, and a battery of nerves and hormones with which to fine-tune their metabolic responses. But it does suggest that an animal’s metabolic rate is more than the sum of its parts. Around the same time that Kleiber, Krebs, Martin, and Fuhrman were cutting animals up, other biologists were arguing that small animals’ bodies weren’t made of different stuff than those of large ones, but from different amounts of stuff. As bodies got bigger, their composition should change so that the quantity of what was called “active protoplasm” declined. The most metabolically active tissues, such as brain, heart, and liver, would represent a smaller percentage of the whole, and drowsy fat and bone a greater proportion. As we saw, big animals do have proportionately more bone. And big animals also have relatively smaller brains. For mammals, relative brain mass declines with size: A rat’s brain accounts for about 1.5 percent of its body weight, an elephant’s only about a tenth of this. But other organs do not change their relative size. The heart accounts for about 0.6 percent of the body mass in all mammals. Nor do small animals power their more rapid metabolisms with relatively larger lungs. Nor do they have proportionately more blood to carry oxygen. Obesity researchers are investigating the relationship between body composition and metabolism. Studies on humans have shown that large people do indeed have less metabolically active tissue, but also that their energy-hungry tissues—basically the major organs—consume less energy, pound for pound, than the same tissues in smaller people. So simple changes in tissue proportions seem not to account for the relative decline in metabolic rate. Fudge Factors Another idea, proposed by several biologists, was that Kleiber’s rule of body size and metabolism could be a sort of average of all the body’s scaling rates, the end result of several different rules acting at once. If

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In the Beat of a Heart: Life, Energy, and the Unity of Nature cells in different-sized animals used energy at the same speed, metabolic rate would scale simply with body mass. Tusko would not have overdosed, but he probably would have starved, as he would have needed to eat at the same rate as a shrew. On the other hand, cells and animals must avoid overheating, and they can only do so through their surface areas, which are proportional to their body mass raised to the power of two-thirds. Kleiber pointed out that if a steer had the metabolic rate of a mouse, its body surface would be hotter than the boiling point, and a mouse with the energy consumption of a steer would need fur 20 centimeters thick to keep warm. Perhaps Kleiber’s rule is the result of a balance between these two forces: 3/4 lies between the 1 predicted by the maximum metabolic rate and cell number and the 2/3 predicted by the surface law. A similar suggestion was that the 3/4 arose from a balance between the surface law and the force of gravity. As we saw, gravity places a greater burden on large animals. Biologists can change gravity: Raising animals in a centrifuge, a device like a merry-go-round for rodents, increases the force on them, and firing them into space eliminates it. The results have been mixed. After two weeks of spinning around and around, rats consume more oxygen, supporting the idea that gravity influences metabolic rate. But another study found something like the opposite: Rats living at G-forces ranging from twice Earth’s gravity to weightlessness on the Space Shuttle seemed to increase their metabolic rates as gravity’s pull weakened. These experiments give no evidence for gravity as a general explanation for Kleiber’s rule—and they were not designed to, being more concerned with finding out what gravity does to the body, often with a view toward predicting the medical hazards of space travel. Another, more damning strike against this argument is that aquatic animals, buoyed by water, are effectively weightless, and so their metabolic rates should follow the surface law. Yet seals and porpoises follow Kleiber’s rule just as closely as their land-living counterparts. Kleiber thought that some average of scaling effects was the strongest contender to explain his rule. But this hypothesis never attracted much support. The reasoning behind the ideas about averaging is hand-waving, working backward from 3/4 by trying out sums

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In the Beat of a Heart: Life, Energy, and the Unity of Nature . until we get the right number. Ideally, we would like to work forward toward scaling laws, without any preconceived pattern in mind—scientists prefer elegant laws over post hoc fudges. In fact, Kleiber put most of his efforts into arguing for his rule’s usefulness and validity, rather than trying to explain it. In his 1961 book on energy in biology, The Fire of Life, he spends most of the chapter on the relationship between metabolic rate and body size criticizing the surface law, which by then had nearly expired, rather than presenting any alternative theory. Kleiber was not mathematically inclined and seems to have seen his law as a useful way to work out how much to feed an animal, rather than anything that demanded theoretical explanation. Biologists tried to pin metabolism down in ever more detail—dissecting bodies, organs, tissues, and cells—hoping an explanation for the trends might pop out of the ever-decreasing circles of precision. Such measurements are doubtless challenging, but there’s something unsatisfying about this path. A better description is not an explanation. Even if changing body composition, or an average of different rates, did account for the trend in metabolic rate, it still wouldn’t tell us why. Why does metabolic rate seem to be proportional to the 3/4 power of body mass? Where does 3/4 come from? In physics, when an experiment produces something unexpected, the theorists’ juices start flowing. In the science of metabolism, strange measurements led mostly to more measurements. The Metabolic Ark And my, how biologists measured. As the decades passed, ever more species lined up to have their metabolisms recorded. We have seen that Kleiber’s rule applies to both mice and elephants, but this range covers only a small number of species and a narrow spectrum of life. All the investigations considered until now have dealt with mammals—moreover, only the placental mammals like us, accounting for a mere 4,000 of Earth’s several million animal species. What metabolic rates did Noah assign to the rest of the animal kingdom? Marsupials—kangaroos, koalas, possums, and so on—have a lower body temperature than placental mammals: 34°C to 36°C, compared

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In the Beat of a Heart: Life, Energy, and the Unity of Nature with 36°C to 38°C. They also, it turns out, have relatively slower metabolisms. A marsupial burns energy at two-thirds the speed of a placental mammal of the same size. But for species ranging from the 9-gram marsupial mouse to a kangaroo weighing more than 50 kilograms, the relationship between metabolic rate and size fits nicely on Kleiber’s line. Birds, the other group of animals besides mammals that use metabolism to maintain a constant body temperature, have higher body temperatures than us: 39°C to 41°C. They have faster relative metabolic rates, too, but again, there is good evidence that metabolic rate changes with body size according to Kleiber’s rule. So Kleiber’s rule seems to hold for warm-blooded vertebrates. Some mammal species, however, lie away from Klieber’s line. This doesn’t mean that we should ditch the whole idea. Allometry’s power lies partly in its ability to highlight exceptional cases—the individuals or species that deviate from a general pattern—and challenge us to explain them. There are plausible arguments as to why the metabolic rates of many of the unusual species might buck the trend. Marine mammals such as seals and whales use a relatively large amount of energy for their size, perhaps to keep warm in the sea. Arctic species also have higher metabolic rates than tropical animals of the same size. Desert animals, in contrast, burn calories relatively slowly, which could point to an adaptation to the shortage of food in their environment. The same goes for animals living in deep or cold water and in caves. Sloths, too, have slow metabolisms, probably because they are so, well, slothful. Kleiber’s is not an iron rule—animals can bend it. What about reptiles, fish, amphibians, insects, and other invertebrates, whose body temperatures fluctuate with the ambient temperature? The evidence is not as good as for mammals. The resting metabolic rate of cold-blooded species (technically known as ectotherms, or poikilotherms) is harder to define than it is for birds and mammals—it changes much more with temperature, for example—and studies have estimated wildly different equations relating mass to metabolic rate. There have also been fewer studies, as these animals are not raised on farms, so there is less economic incentive to study them, and they are not such good models for human biology, so there is less

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In the Beat of a Heart: Life, Energy, and the Unity of Nature By the 1960s, Kleiber’s rule had been extended to cover everything from microbes to monsters. Credit: Novo Nordisk, Inc. medical incentive. But one conclusive finding is that a reptile or a fish has a much slower metabolism than a bird or a mammal of the same size, and so needs less food. This is hardly surprising—the ectotherm doesn’t need fuel to maintain its body temperature. Big ectotherms also have slower relative metabolic rates than small ones, and over a large size range and a large number of measurements the metabolic rates again converge on the line of body mass raised to the power of 3/4. Still other studies have showed that Kleiber’s rule even applied to single-celled microscopic organisms such as amoebas. This body of evidence contains a lot of variation. Energy has been found to scale with powers of body weight varying from about 1/2 to about 1 in groups from mammals to microbes. Some biologists believe that no unifying trend in metabolic rate exists and that each species’

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In the Beat of a Heart: Life, Energy, and the Unity of Nature energy needs are the result of its particular physical proportions, chemical makeup, evolutionary history, and environment. Nevertheless, most measurements are close to Klieber’s 3/4, particularly if you look at a large number of species across a large range of body sizes—allowing the big picture to emerge from the fog. By 1960, species ranging in size from elephants weighing several thousand kilograms down to microbes weighing less than a millionth of a millionth of a gram, or about 20 orders of magnitude, had all found a place on Kleiber’s line. Quarters, Quarters, Everywhere It wasn’t just metabolic rate that caught the attention of allometry seekers. Biologists got to measuring and comparing just about everything they could. As the power law equations piled up, a pattern began to emerge. Other biological processes, such as mammals’ dietary requirements for nitrogen and vitamins, and the rate at which they produce urine, also scale to body mass to the power of 3/4. An animal’s heart rate is proportional to its body mass to the power of −1/4, the same as the amount of fuel burned per cell. For every 1,000-fold increase in body mass there is a nearly sixfold decrease in heart rate. Many biological times, such as life span, time spent in the womb, and time between birth and maturity were found to be proportional to body mass raised to the power of 1/4, so a 1,000-fold increase in body mass leads to a sixfold increase in life span. Walking speed also increases with the 1/4 power of body mass. Similar rules apply to biological structures. The cross-sectional area of the aorta, the largest blood vessel, scales as the 3/4 power of body mass. Plants joined in the fun: The area of tree trunks scales in the same way as that of aortas. Other features abandon 1/4 but stick with multiples of it; for example, hemoglobin’s ability to seize hold of oxygen declines slowly as animals get larger, with an exponent of −1/12. The number 4—in 3/4, 1/4, −1/4—or multiples of 4, such as in the case of hemoglobin, came up again and again. It is not surprising that scaling laws should be related, because the different bits of an animal

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In the Beat of a Heart: Life, Energy, and the Unity of Nature all interact with one another. The demand for nutrients, and the rate at which the body processes and gets rid of them, depends on metabolic activity. The heart rate needs to match the rate at which the cells burn oxygen. In combination the scaling laws display a kind of unity. For example, because metabolic rate for each cell decreases as the −1/4 power of body size, but life span increases as the 1/4 power, each cell burns approximately the same amount of energy in its life, regardless of the animal it lives in. What was harder to explain was why scaling laws should all revolve around the number 4. By the mid-1960s, most biologists believed that Kleiber’s rule was an accurate reflection of the relationship between mass and metabolism. We know this because they voted on the question. At an international symposium on energy metabolism held in the Scottish town of Troon in May 1964, Kleiber argued that 0.75 should be adopted as the standard for calculating metabolic rate from body weight. Despite the objections of those who pointed out that metabolic rate varied greatly within species, between species, and at different times in an animal’s life, the conference adopted Kleiber’s motion by 29 votes to nil. Kleiber’s victory reversed a defeat 30 years earlier, when the same conference had voted in favor of Brody’s suggested 0.73. This seems like an odd way for scientists to settle their issues—you would hope that evidence, rather than majority opinion, would be the arbiter. Some of the arguments about Kleiber’s rule have a strong whiff of numerology—Kleiber and Brody had fierce arguments about what the third decimal place of the scaling factor should be, even though there was no way of measuring this number precisely. Agricultural researchers such as Kleiber and Brody needed to know how metabolic rate changed with size, but they weren’t much interested in explaining why. Kleiber’s rule, and other allometries, gave good predictions, and highlighted exceptions, without any theoretical underpinning, so most biologists probably saw little use for one. A serious mathematical attack on the problem would require someone with a different background, and a fresh angle, to the nutritionists and physiologists who until then had dominated the study of metabolism. This person arrived in the early 1970s.

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In the Beat of a Heart: Life, Energy, and the Unity of Nature Intellect and Muscle Scaling is a large part of biomechanics, the discipline that uses physics to explain how animals move and support themselves. Just as the structure of animals needs to change with size, with bigger animals and plants needing more support, the way they move changes too. Small birds hop along the ground; for any bird larger than a robin, it is more efficient to walk. The largest birds are flightless, because the power of muscles increases more slowly than their weight, so a big bird can never have enough flight muscle to lift itself. A recent study argued that Tyrannosaurus rex was too big to run, because it would have needed unfeasibly large muscles to power its massive legs. This might be bad news for the makers of Jurassic Park, but it wasn’t necessarily good news for the herbivorous dinosaurs of the Cretaceous, as a brisk walk would have carried T. rex along at more than 20 kilometers per hour, the speed of a charging rhino. For very small animals, gravity becomes irrelevant, and friction takes over as the dominant physical force in their lives. This is why insects and spiders can run up walls and across ceilings and why the smallest insects have wings like brushes rather than paddles: For them the air is like treacle. Thomas McMahon trained as a physicist. His doctorate, taken at the Massachusetts Institute of Technology, was in fluid mechanics. His move into biology came via rowing. McMahon, a keen rower, came up with a physical model to explain why rowing shells powered by eight oarsmen or -women go faster than those carrying a single sculler and how much faster they go. This sparked a general interest in how animals moved, and McMahon ended up holding chairs in both biology and applied mechanics at Harvard. There he made great strides in working out the physics of walking, showing how the legs act like two pendulums—something D’Arcy Thompson also believed. The standing leg pivots around the ground, and the swinging one pivots around the hip. He put his discoveries to practical use, designing a running track for Harvard, which, by matching the stiffness of the track to the stiffness of the human leg, reduced both times and injuries. McMahon also had a broad playful streak. He went everywhere, even to official Harvard functions, with his golden retriever. And he published four

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In the Beat of a Heart: Life, Energy, and the Unity of Nature novels, including one that invented an alternative life for Gordon McKay, the nineteenth-century engineer who endowed his Harvard chair in mechanics. The real-life McKay invented shoe-making machinery; McMahon sent his fictional McKay to Kansas to found a city powered by bees. McMahon explained Kleiber’s rule using a combination of mechanical and metabolic reasoning. Like Karl Meeh, he considered animals as an assortment of interconnected cylinders—limbs, torso, and neck. But whereas Meeh was interested in these cylinders’ surface area, McMahon was interested in their mass and strength. His theory revolved around the need for animal bodies to guard against the threat of buckling. Slender limbs make sense: They use less material and are lighter and more mobile. But although a column, such as a leg or torso, might be able to stand firm if it remains vertical, a small bend can have a catastrophic effect, causing the column to fracture and collapse. The elasticity of bone and muscle can guard against this danger to an extent, making the column spring back to the vertical if it bends, but the risk of buckling still prevents bodies from being too skinny. It’s obvious that a long thick column is less likely to collapse than a long thin one. McMahon calculated the rate at which a limb would need to get wider, to protect it against buckling as it got longer. He found that, to avoid such buckling, the width of a limb or torso should be proportional to body weight raised to the power of 3/8. This looks promising—it’s an allometry equation and, even better, it’s a 1/4 power law. To test his idea, McMahon reanalyzed the data on the dimensions of cattle collected decades earlier by Samuel Brody’s team. The girth of a cow’s chest—treating the torso as the body’s central pillar—was indeed very close to the body mass raised to the power of 3/8. To get from an animal’s geometry to its metabolism, McMahon calculated the power that the muscles around such a column could generate. A muscle’s power is proportional to its cross-sectional area, which is proportional to the square of its diameter. So to avoid buckling, a body would need a quantity of muscle proportional to its mass, raised to the power of 3/8, and squared. This equals mass to the power of 3/4. Bingo! Where Rubner thought that animals were similar in the amount of heat their surfaces produced, McMahon thought each

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In the Beat of a Heart: Life, Energy, and the Unity of Nature animal was similar in the way its limbs and muscles resisted buckling. He called his theory elastic similarity. McMahon’s theory renewed interest in Kleiber’s law. But although his ideas worked well for four-legged land dwellers, it’s harder to see why the same rules regarding buckling bodies and bending limbs would apply to whales, or birds, or amoebas. None of these animals face the same mechanical problems, but all show 3/4 power scaling. Other researchers argued that the relationship between an animal’s size and its shape did not meet the predictions of McMahon’s theory and that big animals could get much longer and still not collapse under their own weight than his model allowed for. Dissenting Voices Meanwhile, of course, the 1964 vote in Troon hadn’t really changed anyone’s opinion or settled the question of whether Kleiber’s rule was valid. A leading skeptic was Alfred Heusner, who, while working toward his Ph.D. at the University of Strasbourg in France, had discovered that rats show a daily rhythm in metabolic rate. A rat’s metabolism runs fastest around the times when it is normally most active, which is at night. Many other species show a similar match between activity cycles and metabolic rate. In 1967, Heusner became a colleague of Kleiber’s at what was by then the University of California, Davis. The two were already close friends, and their families holidayed together. To recap the allometry equation: y = axb. So far we have fretted about the term describing the power in the metabolic power law, b (also called the exponent), which for mass and metabolism seems to be 3/4 and describes the gradient of the line. Heusner turned his attention to the constant at the front, a, where the line crosses the vertical y axis. He looked at studies of mice, rats, cats, dogs, sheep, and cattle. But instead of lumping all the animals in one species into one group, plotting an average value for each species, and drawing a line through the dots, he considered the species separately. Within a species, he argued, metabolic rate was best described by the old value of mass to the power of 2/3. It was only between species that 3/4 emerged, but Heusner argued that this was biologically meaningless. You weren’t comparing

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In the Beat of a Heart: Life, Energy, and the Unity of Nature like with like. Heusner saw the regularity of the mouse-to-elephant line as a mirage. The one line, with a gradient of 3/4, he said, is really a host of parallel lines, each with the gradient 2/3. It was the differences between these lines, and species, represented by the differing values of the constant a in the allometry equation, that truly demanded an explanation. Heusner’s criticisms were themselves soon under fire, as biologists rode to defend Kleiber’s rule. Some said Heusner’s mathematical reasoning didn’t hold up. McMahon entered the fray, along with his Harvard colleague Henry Feldman. They recrunched the same numbers Heusner had used and concluded that he was wrong to dismiss 3/4 as just an artifact of the data analysis. Two-thirds, they said, might work within a species, where individuals maintained the same proportions as they grew, but the 3/4 seen between species was still meaningful. It showed that the bodies’ design did need to change over large size ranges. The trend between species still needed explaining. The argument between Heusner and Feldman and McMahon set the tone for a debate around Kleiber’s rule that continues to this day. Every few months someone unveils a new data set or analysis that, they say, settles the matter in favor of 3/4, 2/3, or neither. Or they claim to be able to demonstrate once and for all that the other guy is talking through his hat. To an outsider everyone seems convincing, or at least convinced. The debate rumbles on, however, and scientists’ ability to draw opposite conclusions from the same data seems baffling. In the three-quarters of a century since Kleiber, the literature on this topic has become rather like the Bible: You can find something to support every possible stance. There are wearisome echoes of the arguments we encountered about the surface law and organ metabolism. But, as we’ve seen, those earlier debates about the surface law and organ metabolism did eventually result in consensus—even if it was a battle of attrition, rather than a coup d’état—and we might expect researchers to eventually come to a broad agreement. So what will history decide? Employing the Troon principle of majority rule, most biologists believe that Kleiber’s rule is a good description of nature. But is it just a handy rule of thumb for farmers and physicians, or is it pointing to an underlying unity of life? Well, for

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In the Beat of a Heart: Life, Energy, and the Unity of Nature a start, living organisms do have an underlying unity. Whales are 1021 times bigger than bacteria, but they are made from much the same stuff and sustain themselves in much the same ways, so we might expect to find principles that apply to both creatures. And because the diversity of organisms’ sizes is one of the most striking aspects of the diversity of life, we also shouldn’t be surprised to find that the patterns in biology are based around changes in size. On the other hand, organisms’ metabolisms depend on a lot of things besides their size. Two animals could be the same size, but their differences in diet, ways of moving about, or breeding behavior, to name a few, could result in very different energy requirements. The high variability between organisms, and each species’ idiosyncratic ecological circumstances and evolutionary history, make it hard to spot regularity. Indeed, it means that looked at over small size ranges we should not expect regularity. But if underlying trends are present, we would expect them to emerge over a wide size range, which gives large-scale order a chance to drown out small-scale noise—and this is indeed what we see. This sounds like special pleading, but the same is true in many physical systems. Some quantum effects make themselves apparent only at unimaginably high energies, and some relativistic effects become apparent only over vast distances. Other physical principles are statistical phenomena, applicable to large groups but meaningless when applied to small groups or individuals. In warm-blooded vertebrates, the organisms whose metabolism is both best studied and easiest to get a handle on, a large body of evidence points to metabolic rate being proportional to mass raised to the power of 3/4. In the 1930s three rival researchers arrived at this conclusion independently. Cold-blooded species are less well studied and their metabolisms are different, but the evidence again points to 3/4 power scaling. Historically, measuring metabolic rate has been difficult, and it is always hard to know how much faith to put into any one data set. But, perhaps most tellingly, even if you ignore metabolic rate and Kleiber’s rule, a wide range of quarter power scaling laws are found throughout biological systems. These scaling laws have been found in many situations that can be measured more easily than metabolic rate, such as the width of a blood vessel or the rate of a heartbeat. It is

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In the Beat of a Heart: Life, Energy, and the Unity of Nature unlikely that everyone who uncovered one of about 200 quarter-power scalings set out to prop up Kleiber’s rule. Animal bodies exist in three dimensions—this is the logic of the surface law. So where does nature’s fixation with the number 4 come from? There’s one simple way to produce scaling laws that revolve around the number 4. Although we live in a world with three spatial dimensions, it’s perfectly acceptable to do geometry on a four-dimensional shape. The proportions of such a shape scale in the same way as those of a three-dimensional one. Just as in a three-dimensional solid, dimension two (surface area) is proportional to dimension three (volume or mass) raised to the power of 2/3, in a four-dimensional shape dimension three is proportional to dimension four raised to the power of 3/4—the very number we are looking for. In 1977, Jacob Blum pointed out that, if you considered living things as having four dimensions, a four-dimensional version of the surface law led simply to a scaling exponent of 3/4. Find an extra dimension from somewhere and all your problems are solved. Blum didn’t have any definite ideas about what this fourth dimension might be. Twenty years later, another team of researchers did.

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