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In the Beat of a Heart: Life, Energy, and the Unity of Nature 5 NETWORKING THERE WAS A SAYING going around the Los Alamos National Laboratory in the early 1990s: Biology would be to science in the twenty-first century what physics had been in the twentieth. The researchers at the lab—a physics powerhouse, where the atomic bomb was invented—realized the gathering momentum of fields such as genetic engineering and neuroscience and wondered whether it was biology’s turn to change the world the way quantum theory and relativity had. If biology was truly about to supersede physics as the cradle of scientific revolutions, Geoffrey West had reason to be glum. West had been working at Los Alamos since 1976 and had become head of the lab’s particle physics group. But there were signs that the tide was turning against his discipline. At the end of 1993, the U.S. Congress pulled the financial plug on the Superconducting Supercollider, a giant particle accelerator intended to be built near Dallas, Texas. The public, it seemed, was unwilling to spend $11 billion to probe the building blocks of matter. There would be no new accelerators—and so probably no major developments in particle physics—for more than adecade.
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In the Beat of a Heart: Life, Energy, and the Unity of Nature The talk of biology’s coming preeminence was probably right, thought West. But the idea that this meant physics would go out of fashion struck him as ridiculous. Like D’Arcy Thompson, he saw mathematical descriptions as the root of any true science, and most areas of biology were still lacking in such theories. To West’s mind this gave physicists an entry into biology. He began thinking about what sort of biology problems he might tackle. West had recently turned 50, and his mind had turned to thoughts of mortality. He decided to look into what determined life span. On the one hand, longevity is predictable. Insurance firms and actuaries can calculate life expectancy based on diet, occupation, income, and so on. Half-an-hour’s research would have given him a good idea of how much longer he had left. On the other hand, although biologists have some ideas about the mechanisms of aging, no one could explain why a mouse should live for a year or two and a man, built from much the same molecules, with much the same genes and biology, should live for a century. Explanations based in genetics or physiology struck him as superficial. “If biology is to be a real science, you ought to have a theory that can predict why we live 100 years,” says West. “That’s real science, not some qualitative nonsense about gene expression. That’s not an explanation of anything.” He began to teach himself biology in the evenings and on weekends, picking up bits and pieces of knowledge wherever he could. It was, he says, “like learning about sex on the street.” The library at Los Alamos is devoted almost entirely to physics, so he resorted to reading his children’s high school biology textbooks. He discovered that larger species live longer and that across species life span increases proportional to the 1/4 power of body mass. And he discovered that many researchers had linked life span to metabolic rate, believing that animals that burn energy relatively quickly live shorter lives than the slow burners. But he also discovered that no one could explain why a species’ average life span was as long as it was. West, too, has a background in scaling. Like living things, the behavior of physical systems depends on their size, and the laws of physics are a matter of scale. Quantum theory is most useful for describing what happens inside the atom. Relativity comes into its own
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In the Beat of a Heart: Life, Energy, and the Unity of Nature when considering interstellar distances and velocities close to light speed. Classical Newtonian physics is good for everything in between, from colliding billiard balls to satellite trajectories. And since the Big Bang, the Universe has been growing. “I viewed the unification of forces, and the origin of the Universe, as a scaling problem,” West says. “Since the Universe defines space and time, and it is continually changing its size, what space and time are becomes a scaling problem.” His fumble with the biology literature convinced West that, to understand life span, he needed to understand metabolic rate. To understand metabolic rate, he needed to understand biological scaling. And to understand scaling, he needed to understand the ubiquitous power laws. There was, he realized, a branch of mathematics devoted to just this task—fractal geometry. Branching into Biology Crudely put, a fractal is a branching shape that divides and divides, becoming more intricate by producing an ever-larger number of ever-smaller branches akin to the first branch. Since their discovery, such shapes have been associated with natural forms, and many plant and animal structures, such as fern leaves, tree branches, and the air passages of the lungs, look a lot like the fractals drawn by computers, and vice versa. The power of fractals—and a property that suggested nature would use them—is that they use a small amount of information to generate a large amount of complexity. All you need do is repeat a few simple rules: Grow, branch, and shrink; grow, branch, and shrink; grow, branch, and shrink. Instead of encoding an entire shape in their genes—every leaf, every branch and tube—organisms would need just the rules for generating the fractal, in the same way that the equation for a spiral describes the shape of shells and horns. This repetitive rule means that every bit of a fractal looks the same, regardless of how far back you stand from it or how closely you zoom in on it. And if you chop it into bits, like the sorcerer’s apprentice, you get new fractals that look like copies of the original. Identical branches and trees appear ad infinitum. This property is called self-similarity, and it is the link between fractals and power laws. Physicists realized
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In the Beat of a Heart: Life, Energy, and the Unity of Nature that, when a system can be described using power laws, it is a clue that it is made up of many self-similar parts interacting with one another. Like a power law, a fractal is a rule that generates a hierarchical structure and describes how each level in that structure relates to those above and below it. The mathematics of fractals opened up new ways of thinking about biological scaling, and of connecting what happens at small scales with what happens at large ones. Before the arrival of fractals, structures such as lungs had been described using vague metaphors such as tree-or cloudlike. But fractal geometry provided a way to replace verbal with mathematical descriptions, just as allometry did for simpler biological forms half a century earlier. West’s chain of reasoning went something like this. Scaling laws are power laws. Fractals describe power laws. So, he asked himself, what fractal might lie behind the scaling of metabolic rate? The answer, he decided, was the system of blood vessels, the network of pipes that carry food and oxygen to the cells, and take carbon dioxide and other waste away from them. This system is obviously like a fractal. It branches from one large tube, the aorta that leaves the heart, and becomes a series of smaller blood vessels. Arteries, such as the carotid, which takes blood to the head, or the femoral, which runs down the leg, lead to narrower arterioles, which lead eventually to capillaries, which reach between the cells and deliver their life-giving cargo to the tissues. West defined metabolic rate as the rate at which resources were supplied to the cells, via the blood system, and he reasoned that the scaling law behind metabolic rate—Kleiber’s rule—was a consequence of how the geometry of this supply network changed as animal size changed. Blood vessels start with one wide tube and branch, like a fractal, to create many small vessels. Credit: Reprinted with permission from Science, vol. 272, p. 122. Copyright 1997 by the American Association for the Advancement of Science.
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In the Beat of a Heart: Life, Energy, and the Unity of Nature When he started looking at the blood system, West had to leave Euclidean geometric similarity behind. To the trained eye, a cat’s femur and a cow’s femur are recognizably the same bone. The big animal needs proportionately thicker bones, to resist the stronger pull of gravity, but the basic structure of its skeleton does not change. But unlike bones, you cannot look at a blood vessel and tell what size of animal it came from. A wide tube and a narrow one might be the same artery from different-sized beasts or tubes from different bits of the same animal. Such big and small blood vessels are, however, geometrically similar to one another. Euclidean similarity is lost; self-similarity is gained. So West tried to build a fractal that described blood vessels and relate that to metabolic rate. This would, he realized, be an abstraction, a sort of cartoon of how animals worked. It would not account for all the complexity and variability seen in real vascular systems, but West thought that, by ignoring the details of particular systems, he could get to deeper principles underlying them. Physics has often worked like this, by discarding much of the detail in the systems it seeks to study, such as friction in mechanics. For example, when Galileo dropped objects off the Leaning Tower of Pisa, he ignored their differences in air resistance and concluded that they all fell at the same speed, thus paving the way for a theory of gravity. West is fond of saying that, had Galileo been a biologist, paying more attention to the details, he would have ended up writing tomes on how every object falls at its own unique speed. He (West, that is, not Galileo) can talk like this because, as a physicist, he is not too concerned about annoying biologists. So West began building a fractal that described blood vessels, to work out how fluids would flow through such a network and to relate that to metabolic rate. He eventually designed such a network, but it was a sorry creature, even as a cartoon, bearing little resemblance to any real blood vessels. West realized that he didn’t understand the biology of blood vessels very well and furthermore that some of his mathematics and physics were mistaken. As a scientist looking to jump fields, however, he had one big advantage: the Santa Fe Institute. The institute was set up in 1984 by a group of senior researchers at Los Alamos, along with other eminent physicists from across the United States, to address the problems in
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In the Beat of a Heart: Life, Energy, and the Unity of Nature chaos theory, complex systems, and emergence that were beginning to make an impact across the physical, biological, and social sciences—problems that seemed somehow to combine hideous complexity with tantalizing flashes of order. Such complex dynamics and emergent order were a common feature of any system consisting of many interacting parts, be it stock markets, cells, ecosystems, or societies. The institute was intended to be truly multidisciplinary—it has no departments, only researchers. Since then, Santa Fe and complexity theory have become almost synonymous. Twenty years on, the institute, now situated on a hill on the town’s outskirts, must be one of the most fun places to be a scientist. The researchers’ offices, and the communal areas they spill into for lunch and impromptu seminars, have picture windows looking out across the mountains and desert. Hiking trails lead out of the car park. In the institute’s kitchen, you can eavesdrop on a conversation between a paleontologist, an expert on quantum computing, and a physicist who works on financial markets. A cat and a dog amble down the corridors and in and out of offices. The atmosphere is like a cross between the senior common room of a Cambridge college and one of the West Coast temples of geekdom, such as Google or Pixar. At the time he began thinking about metabolism, West was a visiting fellow at the institute, spending one day a week there. Through Santa Fe, he met two biologists, based 60 miles down I-25 at the University of New Mexico in Albuquerque, who had also been thinking about metabolic rate. Enter the Ecologists Brian Enquist began his undergraduate education hungry for big unsolved problems that would need big ideas to explain them. The last place he expected to find them was in biology—Darwin seemed to have sewn up the market more than a century before. Enquist thought he might major in philosophy. But then in his initial biology lectures he saw the graph of body mass versus metabolic rate and learned that there were still big patterns in biology that cried out for big ideas to explain them. He got hooked on scaling.
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In the Beat of a Heart: Life, Energy, and the Unity of Nature Every biologist has his or her favorite branch of life—birds, bees, moths, water lilies, coral reef fish, or whatever. For Enquist it was trees. Ever since he began climbing them as a child, he had had a thing about them. They were impressive and important. If you stand in a forest, the world seems to be made of trees. He was also impatient, and botany is a good discipline for a naturalist in a hurry, because you don’t have to sit in a hide all day waiting for trees to come to you, and it doesn’t take a week to get your sample size into double figures. And for a young biologist beginning his career and looking for big questions to answer, botany was fertile ground. The main thrust of the science has been to classify and describe plants, and there is relatively little theory or mathematics in the discipline; new ideas in biology have tended to be developed and explored by researchers working on animals. Enquist decided to investigate scaling in forests, to see how the sizes of individual trees influenced the form of the whole forest. For graduate school he went to work in Jim Brown’s lab at the University of New Mexico. More than a decade before Enquist joined his lab, Brown had decided that energy was the key to understanding biodiversity. He had begun his career in the 1960s as a physiological ecologist, studying the biology of energy in animals, looking at how the challenge of keeping warm affected their energy budgets and how it would affect their behavior and where they could live. One of his early studies was on the metabolic rate of weasels, and the cost of being long and thin, with a relatively large surface area—a weasel burns energy twice as quickly as a round animal of the same weight, he found. Lately, he had come to see energy as an organizing principle for the whole of nature. How organisms got energy and divided it between themselves could, he believed, explain biodiversity: Why different environments contain different numbers of species, why species live where they do, why certain species are found together or not, and why some are common and some are rare. And the foundation of any investigation into energy and biodiversity, Brown believed, should be body size. Body size controls how much energy plants and animals need and so how much is left over for other individuals and species. Body size is also closely related to virtually everything else that ecologists are interested in, such as how much
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In the Beat of a Heart: Life, Energy, and the Unity of Nature land animals need, how quickly their populations grow, and how many young they produce. The ecological importance of body size makes it of practical as well as academic interest. To conserve a species, we need to understand it. As the biologist William Calder wrote: “A conservation biologist trying to prepare a protection plan without data on the species’ biology is like an insurance underwriter issuing a policy in ignorance of the applicant’s age, family status and medical history.” Yet conservationists often work in almost perfect ignorance of what they are trying to save. It is likely that at least 90 percent of species, and possibly a much higher proportion, have not yet been discovered, described, and named. And our knowledge of most of those that have stops with their existence—we have only a pressed leaf or a bug pinned out in a draw. No one knows how widely they are spread, how great their numbers are, what they eat, what parasites assail them, how many young they produce and how often, what foods or habitats they prefer, or how they behave. Of the species we do know something about, the majority are birds, mammals, and flowering plants. Of the most diverse groups, such as insects, fungi, and bacteria, we know practically nothing. Most species live in tropical forests, through which it is difficult to travel and in which it is difficult to spot things. The number of biologists and the resources they have to fund their discovery and description of species are both limited, particularly in tropical countries, which tend to be among the poorer nations. Our destruction of biodiversity outpaces our knowledge of it. We need rules of thumb that, in the absence of detailed knowledge, can help us predict which species are most vulnerable to threats such as habitat destruction or hunting and so which are most in need of conservation. Rules based on body size have two great advantages. Size is informative, and it is easy to measure. Even if we know nothing else about a species, we almost always know how big it is. It’s been said that ecologists report body size in the same way that journalists report their subjects’ ages—practically as a reflex. It takes only a moment to weigh an animal; taking such a measurement requires little equipment or expertise, its accuracy can usually be trusted, the animal doesn’t have to be alive to be weighed, and if it is, you don’t need to interfere
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In the Beat of a Heart: Life, Energy, and the Unity of Nature with it or detain it for long. Size is the only universal piece of biological data. One of the most reliable rules of conservation is that it is bad to be big. Across the animal kingdom, big species are more at risk than their smaller brethren. Orangutans and elephants are disappearing more quickly than rats and rabbits. Rhinos are more endangered than antelopes. Large birds, such as albatrosses and eagles, are in more trouble than warblers and finches. Whales are more endangered than porpoises. This rule holds even for reptiles and insects: The planet’s largest known earwig, a 3-inch-long giant from the Atlantic island of St. Helena (being an island species is another almost guaranteed recipe for trouble), has not been seen for 20 years. It’s easy to see what makes large species more prone to extinction. They need more food and so more land. They are often found higher up the food chain and so depend on everything below them staying in good shape. All of this means that there are fewer of them to start with. Big animals take longer to reach breeding age, breed less often, and produce smaller numbers of young when they do. The fossil record shows that, throughout history, carnivorous mammals, such as cats and dogs, have experienced a high degree of evolutionary churn. Species come along, dominate for a few million years, and then disappear, at which point a new group comes along to do the same job. There are obvious benefits to a predator in being big and fierce, but this might also paint a species into an evolutionary corner. Populations become smaller and more spread out, and anything that reduces their food supply or splits a population into isolated fragments, as deforestation and urbanization are doing for large carnivores these days, will hit them harder than smaller species with less grandiose diets. There’s no such thing as small-game hunting, and human hunters’ lust for large prey has exacerbated big species’ vulnerability. Since humans appeared on the scene, a swath of large mammals, such as mammoths, mastodons, and giant ground sloths, have gone extinct. The same goes for the largest birds—the moas of New Zealand, the elephant birds of Madagascar, and perhaps the largest of them all, Genyornis, an Australian species that weighed 100 kilograms. When humans arrived in Australia the continent was also home to a lizard,
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In the Beat of a Heart: Life, Energy, and the Unity of Nature Megalania, which was 5 meters long and weighed 600 kilograms. This too is no more. The trend continues to this day: Fish stocks of large-bodied, slow-growing species are slower to recover from fishing and so more vulnerable to overfishing. Big charismatic animals are the poster children of conservation. We study them more intensively, bias our conservation efforts in their direction, and value them aesthetically and spiritually, but it hasn’t done them much good. So when we plan where to spend scarce conservation money, working out which species are most likely to be at risk of extinction, there is a good case for biasing our efforts toward large species. And when we find a new species or population, we should be more concerned for its future if it is an ape or a deer than if it is a rodent. The Insurance Man When Jim Brown and Brian Enquist began thinking about metabolism, they were more interested in explaining nature than saving it. The search for general principles that apply across the living world, the pair believed, should focus on a combination of energy and allometry. This call to make energy the center of ecology was not unprecedented. Ecology is the study of nature’s economy, and energy has long been one of the currencies tracked by ecologists, through food webs, for example, to explain how much life a habitat can support or as a way to understand why animals prefer to eat certain foods. Ecology has also been one of the areas of biology most receptive to ideas from physics. Around the turn of the twentieth century, ecologists seized on concepts then influential in chemistry, such as equilibrium and thermodynamic descriptions of energy flow, and applied them as metaphors for explaining ecological phenomena such as the stability of populations and the coexistence of species. The person who did the most to get ecologists thinking like physicists was Alfred Lotka, born in Austria in 1880. Lotka did nearly all his scientific work in his spare time. He trained as a physical chemist and emigrated to the United States in 1902. There he worked at the General Chemical Company, in a patent office, at the U.S. Bureau of Standards, and as a science journalist. He ended his working life, following a brief
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In the Beat of a Heart: Life, Energy, and the Unity of Nature stint in the 1920s as a full-time researcher at Johns Hopkins University in Baltimore, at the Metropolitan Life Insurance Company, where he made seminal contributions in applying mathematics to the study of human populations, such as calculating how mortality changes with age. Beginning in the early 1900s, Lotka began thinking about how physics might be applied to biology. Like D’Arcy Thompson, he made no distinction between biological and physical systems. But Lotka saw life in terms of the exchange of energy, rather than being governed by physical forces and geometry. Lotka was not the first person to have this idea, but he pursued it harder and farther, and with greater mathematical rigor, than anyone else. Also like Thompson, he worked outside the academic mainstream and had little contact with other scientists. A quarter of a century of such work culminated in his 1925 book, Elements of Physical Biology. In the book Lotka imagines physical biology, which he defines as “the application of physical principles in the study of life-bearing systems,” as analogous to the then-voguish physical chemistry. Scarcely any life-bearing system escaped his gaze. He tackled growth and population dynamics, the cycling of chemical elements from the environment into life and back again, behavior, the senses, communication, travel, and consciousness. Often he took examples from economics and sociology. Lotka showed that the sheep population of the United States matches the sheep consumption of its citizens and argued that predators similarly control the numbers of their wild prey. And he speculated on the immense impact that the newfound ability to take nitrogen from the air and turn it into fertilizer would have on humanity and the planet. Lotka wanted to build an intellectual framework that would unify physics, biology, and the study of human society. As well as trying to understand biology in terms of physical principles, Lotka sought to solve biological problems using the tools of physics. He drew analogies between evolution and thermodynamics, but he also thought that organisms were far too complicated to be understood simply by the application of thermodynamics. It would be “like attempting to study the habits of an elephant by means of a microscope.” But the mathematical techniques of physics could still be
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In the Beat of a Heart: Life, Energy, and the Unity of Nature described, the way to make more capillaries is to build a bigger network with more branches. But this is not the network’s only job. It must also fill three-dimensional space and have tubes that narrow and shrink so as to make the flow of fluid as smooth as possible and the journey to the capillary as short as possible. West, Brown, and Enquist found that this combination of criteria meant that a large body’s network produced proportionately fewer capillaries than a small one. In a large animal each capillary must supply more cells. So each cell gets a smaller proportion of that capillary’s supply and must slow down its energy consumption accordingly. An animal’s metabolic rate is proportional to its number of capillaries. If the number of capillaries was a simple linear function of body size every animal would have the same relative metabolic rate, directly proportional to body mass. But bigger animals have relatively fewer capillaries. In fact, the rules for the optimal network produce a number of capillaries that is proportional to body mass raised to the power of 3/4. This, West, Brown, and Enquist believe, is why metabolic rate is proportional to the 3/4 power of body mass. It is the number, and therefore surface area, of capillaries that is similar to body size in the same way as metabolic rate, not the relationship between their external surface and volume, as Rubner thought, or the amount of muscle needed to stop them from buckling under their own weight, as McMahon suggested. In bigger animals, resources have a longer journey to the cells and each capillary must supply more cells, so the cells must slow their energy consumption. The network theory predicts that, as body size grows, cells must slow their metabolic rates proportional to body mass raised to the power of −1/4. A cell’s metabolism goes as fast as it can, given the rate at which its body can provide it with resources. West compares cells to cars. A car might have a top speed of more than 120 mph, but it will not go nearly that fast in city traffic. Cells are similarly limited by all the other stuff around them. But if you take cells out of bodies and grow them in a dish, like a sports car on a country road, their metabolic rate speeds up until it hits maximum. Other scaling relationships, such as the breadth and length of the aorta, also fall out of the mathematics used to describe the ideal tubes. The
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In the Beat of a Heart: Life, Energy, and the Unity of Nature theory predicts that they should be proportional to the 3/8 power and the 1/4 power of body mass, respectively. Measurements of different species show that this is indeed so. West, Brown, and Enquist’s model is an abstraction. It is an engineering solution derived from the laws of physics, rather than from any foreknowledge of biological systems. If you were going to design an animal, the model says, this is how you should do it. It is impressive that such a design reproduces key properties of living things, such as the way that blood vessels and metabolic rates change with body size. Real animals are bound to fall short of the theoretical ideal—their networks will not reproduce exactly the geometry of the model, nor will they change with size in the way it predicts. And in nature, metabolic rate is seldom if ever exactly proportional to the 3/4 power of body mass. But this shortcoming does not sink the whole theory. It would be very surprising to find a moose that ate the exact proportion of pondweed and land plants that, to the nearest gram, gave it the best diet. To see if moose eat optimally, we need to look at lots of them and see if their average diet is close to the theoretical prediction. Likewise, no animal is going to have a blood system built exactly according to the fractal rules. Network design is flexible—long-term exercise leads to more capillaries in a muscle, for example. And cells are flexible in the amount of energy they demand. But if there are underlying rules of metabolism, if we look at enough animals across a broad enough range of sizes, we should expect to see this theme underlying the variations that different species and individuals can play on it. And we do. After doing some gnarly mathematics to explain how properties of the blood system, such as the jerky flow produced by a beating heart, would affect nutrient supply to the cells, West, Brown, and Enquist sent their model off to Science. Their paper was accepted, but only after several rounds of review involving eight reviewers (three or four is more usual). Three of the eight thought it was outstanding, revolutionary even. The Nobel Prize may have been mentioned. Three thought it was good. Two thought it was the dumbest thing they had ever seen. Lots of things about the fractal theory rub some biologists the wrong way. It is based on mathematics and physics, it is an abstraction, and it is a generalization. It throws away most of the nitty-gritty
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In the Beat of a Heart: Life, Energy, and the Unity of Nature of biology and most of what most biologists spend their time doing. None of these characteristics are likely to recommend it to biologists who favor staring over squinting—the physiologist who has spent a career describing the quirks of the cardiovascular system or a biochemist who spends his or her time describing the reactions of cellular respiration. And not everyone believes in Kleiber’s rule. Some biologists still think there is no one scaling law for metabolism and so nothing to explain. To them, West, Brown, and Enquist are squinting so hard that their eyes have closed. It’s fair to say, however, that D’Arcy Thompson would have been delighted with the network model. Among his papers is the following note, similar although not identical to the passage in On Growth and Form that discusses blood vessels: To keep up a circulation sufficient for the part and no more, Nature has not only varied the angle of the origin of the arteries to suit her purpose; she has regulated the dimensions of every branch and stem, and twig or capillary. The normal operation of the heart is perfection itself; we are told that even the amount of oxygen which enters and which leaves the capillaries is such that the work involved in its exchange and transport is a minimum. This perfect fitness, this maximal efficiency, we accept as a cardinal hypothesis; and we come to understand the form and dimensions of this structure or that by solving the problem of the work which they do. That’s what I was trying to say. Sometimes I wonder why I bother. Networks in Plants A tree looks like a naked fractal. In its branches can be seen the form of its distribution networks. But unlike blood vessels, the network of xylem tubes that carry water around the tree does not start with one big tube in the trunk and split into lots of narrow tubes as it approaches the leaves. Xylem is more like a bundle of electrical cables than plumbing; it is a group of tubes of the same size that split from each other as they near their destinations. Trees prompted Enquist to start thinking about networks, but could his theory apply to them? In fact, plant networks reproduce many of the same features as animal networks, although for different reasons. As Leonardo saw, trees have area-preserving branching. And xylem vessels also taper, getting
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In the Beat of a Heart: Life, Energy, and the Unity of Nature The water-carrying vessels of plants begin as a bundle and then branch to go their separate ways. Credit: Reprinted with permission from Science, vol. 272, p. 122. Copyright 1997 by the American Association for the Advancement of Science. narrower as they reach the treetops. The problem with a tube of a constant diameter is that the longer it gets, the more work it takes to force fluids down it. Resistance to flow increases down the tube. If xylem vessels kept the same breadth all along their length, the top leaves of a tree would never get any water. But if a tube tapers at just the right rate, the resistance to flow remains the same along its length. The degree of tapering that keeps the pressure in a tube uniform regardless of how long it is—and so keeps all parts of a plant equally supplied—plugs into the fractal model to give a metabolic rate predicted to be the 3/4 power of the plant’s mass. The mysterious number 4, the reason that scaling laws are built around multiples of one-quarter, and not the one-third powers that regular geometry predicts, is a consequence, as J. B. S. Haldane saw, of the pressure that natural selection exerts to fill an organism’s volume with as much surface area as possible. In combination the networks for transporting resources and the surfaces for processing them become a four-dimensional entity. Three of these dimensions come from the two-dimensional area of surfaces, the capillary walls across which resources travel. They are folded so much, and fill space so well, that they take on the geometrical properties of a three-dimensional solid. It’s as if you saw a ball, but realized on closer inspection that it was a crumpled sheet of paper. The fourth dimension is the distance resources must travel—the length of the tubes leading from point of uptake to point of delivery. Together, internal surface area and transport distance determine the rate at which an animal can burn energy. The number 3 in 3/4 arises because this four-dimensional network
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In the Beat of a Heart: Life, Energy, and the Unity of Nature must fit itself inside a compact body existing in a three-dimensional world. Alternative Routes West, Brown, and Enquist do not have the field to themselves. While some biologists were accusing them of oversimplifying biology in trying to explain metabolic rate, some physicists thought their model wasn’t simple enough. “We read the paper, and it seemed somewhat complicated,” says Jayanth Banavar, a physicist at Pennsylvania State University. “We thought there must be some other explanation.” Banavar teamed up with two other physicists, Amos Maritan and Andrea Rinaldo, and a biologist, John Damuth, to design a network using different criteria. Rather than looking for a network that maximized energy supply, they sought one that minimized the rate of flow in the system and so the volume of blood. They considered a body as a set of delivery stations, cells, serviced by a network of pipes stemming from a single source. Unlike the fractal model, the network can flow through its destination and go on to somewhere else. Cells in this model are not like twigs at the end of a branch. They are more like stations on a railway line, where some people (resources) get off. As the network expands, the amount of fluid in the system must rise. The team showed that, in the most efficient three-dimensional network, where the distance from source to destination is minimized, the amount of fluid must rise proportional to the volume of the network raised to the fourth power. This model got the desired number 4—and, incidentally, also predicted the geometry and flow rates of river networks—but it created a problem. As a network, or an animal, grows, the amount of fluid that is in transit at any time, and so not available to be used, must increase. So for the rate of supply to keep pace with network size, more and more fluid is needed. If animals were like this, big ones would need a volume of blood larger than the volume of their body. So Banavar’s team thought again and forced their model network to stay contained inside the body. In this case, as the network grows, its capacity to supply resources declines, and so the cells receiving the resources must adjust
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In the Beat of a Heart: Life, Energy, and the Unity of Nature their demands accordingly. The network that minimizes the amount of blood in transit delivers resources proportional to the 3/4 power of the body volume it supplies—as predicted by Kleiber’s rule. For Banavar it is the fact that networks are directed—the flow only goes one way—rather than that the network is a fractal that is the key point. “The question,” he says, “is always ‘What are the essential features and what are less important?’”—which details you need, and which you can junk. The fractal model may be accurate, says Banavar, but he sees it as a special case of a more general set of networks described by his team’s more economical model. Unlike West, Brown, and Enquist’s, this model does not need the complications of fractals, area-preserving branching, or fluid dynamics. Nor does it assume that the networks get resources to the cells as quickly and efficiently as possible. Instead, Kleiber’s rule emerges as a general property of distribution networks—not just the best way to deliver resources but the only way. West, Brown, and Enquist point out that their model is better at predicting specific details about living things—mirroring some of the criticisms aimed at them by biologists who believe that the fractal model is too general and abstract. The original West, Brown, and Enquist model reactivated interest in metabolic rate, and since they launched their theory still others have joined in with ideas and theories quite different from the two network models. One recent model based on the chemical structure of cell membranes argues that small animals’ membranes are softer and leakier than those of large animals, and so they use energy and process chemicals more quickly. Another argues that metabolic rate arises from the way that the amount of DNA in a cell affects its size (the more DNA an organism has, the bigger its cells). Still another looks at metabolic rate from the viewpoint of how body size affects an animal’s ability to obtain food and store it in the body. And another argues that no one factor depends on metabolic rate but rather that Kleiber’s rule is a consequence of the “allometric cascade” of the many different processes, involving the heart, lungs, and cells, that make up metabolism. And there are others I have not mentioned. But in this crowded field the West-Brown-Enquist model is currently the front-runner. Some reasons for its lead are scientific. The
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In the Beat of a Heart: Life, Energy, and the Unity of Nature network model’s mathematics are complex, but its foundations are intuitive and its applications universal—all organisms must get stuff from somewhere to somewhere else. It’s difficult to see how else organisms could be constructed. The fractal model also makes predictions that match other aspects of biology, particularly its calculations of the width of tree trunks and aortas. The weakness of the network models is that they are difficult to test—it’s hard to see a killer measurement that might settle things one way or the other, particularly as these models are generalized abstractions of how life works. Other reasons for its preeminence are political: The team got there first, they have a long list of publications in the most prestigious scientific journals, and, as we shall see, they have been vigorous in using their model to explain other aspects of biology, which has raised their profile. Nine years, at the time of writing, after the fractal model’s debut, researchers are divided along similar lines, in similar proportions, as the original reviewers that Science got to comment on the theory. Some are keen advocates, some are virulent opponents, and the majority are interested but undecided, waiting to see which way things go. The debate is still fierce, but no one has found a fatal flaw in the fractal theory to convince the scientific community that it is invalid. “If it’s wrong, it’s wrong in some really subtle way,” says West. Single Cells, Virtual Networks Another branch of the tree of life challenges the network theory. Many, probably most, species don’t bother sticking their cells together into organs, networks, and bodies. They live as single cells. But they are not beyond the reach of biologists looking for scaling laws. Single-celled organisms certainly come in lots of different sizes. The alga Ostreococcus tauri, discovered in the Mediterranean in 1994, has cells a millionth of a meter across. Another alga, Acetabularia, also known as the mermaid’s wineglass, has cells that are 5 centimeters long and 1 centimeter across. By my reckoning this is almost 10 billion times the volume of Ostreococcus; a whale is a mere 10 million times more massive than a mouse. Other unicellular organisms are even larger, although they pull some fancy tricks in the process. The alga Caulerpa taxifolia grows thin fronds up to a meter long, each of which is basically a single cell,
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In the Beat of a Heart: Life, Energy, and the Unity of Nature although one that is subdivided and reinforced. And some slime molds come together to form blobs that are tens of centimeters across and have no cell walls, although they may contain millions of cell nuclei (the bit of the cell containing the chromosomes). None of them have any blood vessels, xylem, or any other sort of plumbing. Yet there is evidence that the metabolic rates of unicellular organisms also scale as the 3/4 power of their mass. How can a theory based around distribution networks say anything about them? In fact, unicellular organisms are not just blobs. Their resource distribution problems are similar to those of larger organisms, and they have come up with similar solutions. The cells belonging to the kingdom called eukaryotes, which includes plants, animals, fungi, and single-celled organisms such as algae and amoebas, have complex structures that are like miniature organs (the technical term is organelles), such as the fuel-burning mitochondria and plants’ carbohydrate-making chloroplast. Like capillaries in the blood system, these structures provide a surface for the transactions of biochemistry to take place. And like capillaries, these miniature organs keep the same size in different-sized cells—they are the terminal units in the cellular network. Evolution should drive a cell to speed its internal transport and maximize the surfaces where molecules are used—in the same way that larger bodies increase the surface area of their lungs and intestines—so that a cell can get on with its life as quickly as possible. Even in unicellular organisms, the evolutionary drive to use resources as quickly and efficiently as possible should produce some sort of structured distribution network. Such a network could be physical—cells have microscopic cables and packages that shunt molecules around—or it could be virtual, with resources flowing by diffusion from where they are taken up at the cell membrane to the terminal organelles inside the cell. The distance from where cells take up chemicals to where they use them is analogous to the length of the tubes in the blood or xylem system, and as in larger organisms, evolution should seek to minimize it. This means that cells’ metabolic rates should scale as the 3/4 power of their body masses, regardless of the considerations that apply to tubular networks, such as how they branch or how fluids flow through them.
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In the Beat of a Heart: Life, Energy, and the Unity of Nature The metabolic rate of an isolated mitochondrion, sitting in a lab dish, also falls on Kleiber’s line. Even organelles have their own distribution networks. At the molecular level, their respiration is run by large, complex molecules. And even in these isolated molecules, the reactions of respiration run at the speed predicted by Kleiber’s rule, suggesting that within these molecules’ structure there are nanoscale distribution networks that move individual oxygen and ATP molecules around. From monsters to molecules, this line of reasoning extends the network theory’s reach over a size range of 27 orders of magnitude, or a thousand trillion trillion times. The same logic extends the theory to the bacteria, the vast group of single-celled organisms lacking mitochondria, chloroplasts, or nuclei, but that do have these large and complex molecules. West, Brown, and Enquist’s model describes these as “virtual fractals”: Resources are distributed in a branching pattern through the cell, even if they are not in tubes. For the model devised by Banavar and his colleagues, this isn’t such an issue—resources spread out from a central point, like food being shared from the head of a table to the guests seated around it. It seems that natural selection has such a strong preference for efficient distribution that the same fractal network solution has evolved many times, at scales from molecules, to cells, to plants and animals, taking on many different forms and for many different substances but always converging on the same fundamental properties. These networks are such a versatile solution to the problem of supplying a body with resources that they have allowed life to evolve into a remarkable range of sizes. It’s as if a human engineer had invented a single mechanism that could power everything from silicon chips to supertankers. So model networks can explain why the slope of the line linking mass and metabolism has a gradient of 3/4. But that leaves lots of variation. Animals of the same size in different groups of organisms have very different metabolic rates. Reptiles are slow, birds fast, and mammals somewhere in between. Plants are slowest of all. Even accounting for size, the speediest metabolic rates seen in nature run 200 times more quickly than the slowest. Body size controls the rate at which cells can be supplied with resources, but this is not the only
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In the Beat of a Heart: Life, Energy, and the Unity of Nature form of energy that influences metabolic rate. There is also the rate of the chemical reactions within cells. And this depends on temperature. Of Reptiles and Bananas I met Jamie Gillooly in an Albuquerque coffee shop. He picked up a banana skin. “If you correct for size and temperature,” he said, “this banana on the table, at least before I ate it, was respiring at a rate similar to everyone else in this room.” In autumn 1999, Gillooly was completing his Ph.D. at the University of Wisconsin on the effects of size and temperature on the plankton in lakes. Biologists who study aquatic systems do not pay much attention to the work of those concentrating on land life, so he had not come across the scaling theory until Jim Brown came to Wisconsin to give a talk on it. The two met over breakfast to discuss their ideas; by the time they finished, they were “both foaming at the mouth with excitement,” says Gillooly. A few months later Gillooly was on his way to New Mexico to do a postdoc. Life runs faster in the warm—this is why fridges slow the growth of mold. Temperature has an exponential effect on metabolic rate. A 5°C rise in body temperature will raise metabolic rate by 150 percent. It was West who suggested how to incorporate this effect into the equation for metabolic rate. The answer, he said, was the Boltzmann factor—named after the German physicist who believed that life was a struggle for energy and who also laid the foundations for statistical mechanics, the physical theory that explains how particles behave en masse. The Boltzmann factor is the probability that two molecules bumping into each other will spark a chemical reaction. The higher the temperature, the faster the molecules move, the harder they collide, the greater the probability of a reaction, and so the faster the chemical process. Like metabolism, the effect is exponential. West, Gillooly, and the rest of the team added the Boltzmann factor to Kleiber’s allometry equation relating metabolic rate to body mass raised to the power of 3/4. The results were dramatic. Accounting for temperature in this way
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In the Beat of a Heart: Life, Energy, and the Unity of Nature mopped up much of the variation in metabolic rate that size alone could not explain. Correcting for temperature also brought different groups onto the same line: Reptiles’ relatively slower metabolic rates are a consequence of their lower body temperatures, showing that cold-and warm-blooded animals share fundamental metabolic processes. The same goes for plants and animals. Like the network model, this is another simplification. The chemistry of metabolism involves many reactions, each of which will require a certain amount of energy to get going. Applied to metabolism, the Boltzmann factor is a black box. It could be a kind of average for all the hundreds of chemical reactions in metabolism, or it might be the energy needed to get over one crucial hump in the path. But including it reduces the size-adjusted variation in metabolic rates from a factor of 200 to a factor of 20. Perhaps this remaining 20-fold variability is some indication of the wiggle room that organisms have that enables them to tinker with their metabolic rate to match their circumstances. Animals living in cold environments might crank their metabolic rates above the grand average, whereas plants in impoverished soils might depress theirs below it. Or perhaps it is a measure of how far organisms can deviate from the optimal network before they become so inefficient that natural selection weeds them out. This leftover variation, if you like, is what really needs to be explained—the place where biology will come into its own. “I’d like for biology to have a sense of average idealized organisms that share similar principles, which can be understood mathematically—and that what you should be studying are the deviations from that,” says West. “At the end of this century there’ll be a fantastic theory that’ll predict lots of things—everything from ecology to how genes are turned on and off. And I’d like it if there were a footnote in the textbooks saying: ‘At the end of the twentieth century people started taking the problem seriously, and there were these guys who pointed the way to getting rid of the uninteresting part of the problem, the bit that depends on mass and temperature, that determines 90 percent of what we see. Now this real theory of biology is devoted to the other 10 percent.’”
Representative terms from entire chapter: