Scale: The Universal Laws of Life and Death in Organisms, Cities and Companies

by Geoffrey West

Viewed through the lens of scaling, a remarkably general universal pattern emerges, strongly suggesting that evolution has been constrained by other general physical principles beyond natural selection.

Doubling the mass of a mammal increases all of its timescales such as its life span and time to maturity by about 25 percent on average and, concomitantly, decreases all rates, such as its heart rate, by the same amount.

If you tell me the size of a mammal, I can use the scaling laws to tell you almost everything about the average values of its measurable characteristics: how much food it needs to eat each day, what its heart rate is, how long it will take to mature, the length and radius of its aorta, its life span, how many offspring it will have, and so on. Given the extraordinary complexity and diversity of life, this is pretty amazing.

Julian Huxley, himself a very distinguished biologist, was the grandson of the famous Thomas Huxley, the biologist who championed Charles Darwin and the theory of evolution by natural selection, and the brother of the novelist and futurist Aldous Huxley. In addition to the word allometric, Julian Huxley brought several other new words and concepts into biology, including replacing the much-maligned term race with the phrase ethnic group.

here’s something truly extraordinary that you should know about yourself: every day you typically make about 2 × 1026 ATP molecules—that’s two hundred trillion trillion molecules—corresponding to a mass of about 80 kilograms (about 175 lbs.). In other words, each day you produce and recycle the equivalent of your own body weight of ATP!

As its name suggests, this takes a large-scale, top-down systemic approach to understanding ecosystems, having much in common with the philosophy inherent in complexity science, including an appreciation of using a coarse-grained description of the system. Macroecology has whimsically been referred to as “seeing the forest for the trees.”

Like many physicists, I was horrified to learn that there were serious scientists who put Darwin on a pedestal above Newton and Einstein.

answers. In high energy physics, where we struggle to unravel the basic laws of nature at the most microscopic level, we mostly know what the questions are and most of one’s effort goes into trying to be clever enough to carry out the highly technical calculations. In biology I found it to be mostly the other way around: months were spent trying to figure out what the problem actually was that we were trying to solve, the questions we should be asking, and the various relevant quantities that were needed to be calculated, but once that was accomplished, the actual technical mathematics was relatively straightforward.

Of equal importance was their appreciation that, to varying degrees, all theories and models are approximate. It is often difficult to see that there are boundaries and limitations to theories, no matter how successful they might have been. This does not mean that they are wrong, but simply that there is a finite range of their applicability.

This simplified toy model, which played a seminal role in the development of modern physics, is called the “kinetic theory of gases” and was first proposed independently by two of the greatest physicists of all time: James Clerk Maxwell, who unified electricity and magnetism into electromagnetism, thereby revolutionizing the world with his prediction of electromagnetic waves, and Ludwig Boltzmann, who brought us statistical physics and the microscopic understanding of entropy.

Physicists take for granted the concept of the “theorist” who “only” does theory, whereas by and large biologists do not. A “real” biologist has to have a “lab” or a field site with equipment, assistants, and technicians who observe, measure, and analyze data.

In other words, there must be a common set of network properties that transcends whether they are constructed of tubes as in mammalian circulatory systems, fibers as in plants and trees, or diffusive pathways as in cells.

For example, the capillaries of all mammals, whether children, adults, mice, elephants, or whales, are essentially all the same despite the enormous range and variation of body sizes.

To put it slightly differently: of the infinite number of possibilities for the architecture and dynamics of circulatory systems that could have evolved, and that are space filling with invariant terminal units, the ones that actually did evolve and are shared by all mammals minimize cardiac output.

Fundamental to the conceptual framework of the theory is that, despite these completely different physical designs, both kinds of networks are constrained by the same three postulates: they are space filling, have invariant terminal units, and minimize the energy needed to pump fluid through the system.

Oxygen molecules bind to the iron-rich hemoglobin in blood cells, which act as the carriers of oxygen. It is this oxidation process that is responsible for our blood being red in much the same way that iron turns red when it oxidizes to rust in the atmosphere.

After the blood has delivered its oxygen to the cells, it loses its red color and turns bluish, which is why veins, which are the vessels that return blood back to the heart and lungs, look blue.

Thus, hearts beat approximately four times for each breath that is inhaled, regardless of the size of the mammal.

Although a blue whale’s aorta is almost a foot in diameter (30 cm), its capillaries are still pretty much the same size as yours and mine. This is an explicit example of the invariance of the terminal units in these networks.

So to ensure that there is no energy loss via reflections as one progresses down the network, the radii of successive vessels must scale in a regular self-similar fashion, decreasing by a constant factor of the square root of two (√2) with each successive branching.

An interesting consequence of area-preserving branching is that the cross-sectional area of the trunk is the same as the sum of the cross-sectional areas of all the tiny branches at the end of the network (the petioles). Amazingly, this was known to Leonardo da Vinci.

For example, telephone network systems use matched impedances to minimize echoes on long-distance lines; most loudspeaker systems and musical instruments contain impedance matching mechanisms; and the bones in the middle ear provide impedance matching between the eardrum and the inner ear.

Without the gel, the impedance mismatch in ultrasound detection would result in almost all of the energy being reflected back from the skin, leaving very little to go into the body to be reflected back from the organ or fetus under investigation.

However, there were good scientific reasons for favoring AC transmission, especially for long distances, not least of which is that one can take advantage of its pulsatile nature and match impedances at branch nodes in the power grid so as to minimize power loss, just as we do in our circulatory system.

His research and speculations on lightning, death rays, and improving intelligence via electrical impulses, as well as his photographic memory, his apparent lack of the need for sleep or close human relationship, and his Central European accent led him to become the prototype of the “mad scientist.”

Tesla

But what’s really surprising is that blood pressures are also predicted to be the same across all mammals, regardless of their size.

In the nonpulsatile domain where the flow is dominated by viscous forces, minimizing the amount of power being dissipated leads to a self-similarity in which the radii of successive vessels decrease by a constant factor of the cube root of two 3√2 (= 1.26 . . .), rather than the square root √2 (= 1.41 . . .) as in the pulsatile region.

is the mathematical interplay between the cube root scaling law for lengths and the square root scaling law for radii, constrained by the linear scaling of blood volume and the invariance of the terminal units, that leads to quarter-power allometric exponents across organisms.

Mandelbrot did not get his first tenured professorial appointment until he was seventy-five years old, thereby becoming the oldest professor in Yale’s history to receive tenure.

When he plotted the number of deadly quarrels of a given size versus their magnitude on a logarithmic scale, he found an approximately straight line just like the straight lines we saw when physiological quantities like metabolic rate were plotted in this way versus animal size (see Figure 1).

In trying to develop a theory, he hypothesized that the probability of war between neighboring states was proportional to the length of their common border. Driven by his passion to test his theory, he turned his attention to figuring out how the lengths of borders are measured . . . and in so doing inadvertently discovered fractals.

Lewis richardson

The take-home message is clear. In general, it is meaningless to quote the value of a measured length without stating the scale of the resolution used to make it. In principle, it is as meaningless as saying that a length is 543, 27, or 1.289176 without giving the units it’s measured

Mandelbrot introduced the concept of a fractal dimension, defined by adding 1 to the exponent of the power law (the value of the slopes). Thus the fractal dimension of the South African coast is 1.02, Norway 1.52, and so on. The point of adding the 1 was to connect the idea of fractals to the conventional concept of ordinary dimensions discussed in chapter 2. Recall that a smooth line has dimension 1, a smooth surface dimension 2, and a volume dimension

You might have thought that the healthier the heart the smoother and more regular would be the EKG, that is, that a healthy heart would have a low fractal dimension compared with a more diseased one. Quite the contrary. Healthy hearts have relatively high fractal dimensions, reflecting more spiky and ragged EKGs, whereas diseased hearts have low values with relatively smooth EKGs.

The reason that being healthy and robust equates with greater variance and larger fluctuations, and therefore a larger fractal dimension as in an EKG, is closely related to the resilience of such systems.

It is no accident that successful cities are those that offer a greater spectrum of job opportunities and businesses, and that successful companies have a diversity of products and people with the flexibility to change, adapt, and reinvent in response to changing markets.

However, driven by the forces of natural selection to maximize exchange surfaces, biological networks do achieve maximal space filling and consequently scale like three-dimensional volumes rather than two-dimensional Euclidean surfaces. This additional dimension, which arises from optimizing network performance, leads to organisms’ functioning as if they are operating in four dimensions. This is the geometric origin of the quarter power.

Unlike the genetic code, which has evolved only once in the history of life, fractal-like distribution networks that confer an additional effective fourth dimension have originated many times.

The distinction among mammals as their size increases is the increasing number of levels where the flow is pulsatile AC. For example, we have about seven to eight, the whale has about sixteen to seventeen, and the shrew just one or two.

The very largest of these is the magnificent blue whale, which is a mammal that can be up to 30 meters long (about 100 ft.) and weigh almost 200 tons, more than twenty times heavier than the infamous Tyrannosaurus rex.

Whales need to swim sufficiently fast over long distances to be able to supply themselves with the enormous amount of food they need to support their huge metabolic rate, which is equivalent to almost a million food calories a day or about four hundred times more than you eat. Putting these constraints into mathematics

Danish physiologist August Krogh, who received a Nobel Prize for his work. He recognized that there is a limit to how far oxygen can diffuse before there isn’t sufficient left to sustain the cells that are too far away. This distance is known as the maximal Krogh radius, which is the radius of an imaginary cylinder surrounding the length of a capillary, like a sheath, and which contains all of the cells that can be sustained

All mammals and many other animals share the same kind of growth trajectory that we follow, called determinate growth by biologists to distinguish it from indeterminate growth, typically observed in fish, plants, and trees, where growth continues indefinitely until death.

According to the ancient Romans, the Hour of the Wolf means the time between night and dawn, just before the light comes, and people believed it to be the time when demons had a heightened power and vitality, the hour when most people died and most children were born, and when nightmares came to one.

It may not come as such a big surprise to learn that larger cities require fewer gas stations per capita than smaller ones, but what is surprising is that this economy of scale is so systematic: it is approximately the same across all of these countries, obeying the same mathematical scaling law with a similar exponent of around 0.85.

What is even more surprising is that other infrastructural quantities associated with transport and supply networks, such as the total length of electrical lines, roads, water and gas lines, all scale in much the same way with approximately the same value of the exponent, namely about 0.85.

This savings leads to a significant decrease in the production of emissions and pollution. Consequently, the greater efficiency that comes with size has the nonintuitive but very important consequence that on average the bigger the city, the greener it is and the smaller its per capita carbon footprint.

So in marked contrast to infrastructure, which scales sublinearly with population size, socioeconomic quantities—the very essence of a city—scale superlinearly, thereby manifesting systematic increasing returns to scale. The larger the city, the higher the wages, the greater the GDP, the more crime, the more cases of AIDS and flu, the more restaurants, the more patents produced, and so on, all following the “15 percent rule” on a per capita basis in urban systems across the globe.

Yet despite appearances, the interstate system is in fact a quintessential fractal when viewed through the lens of the actual traffic flowing on it, rather than when viewed simply as a physical road network. The traffic flow is the very essence of the interstate and is the fundamental reason for its existence.

Milgram grew up in modest circumstances in New York City, the son of immigrant Jewish bakers whom I would have enjoyed meeting given my chronic addiction to good bread. He was a high school friend of another eminent social psychologist, Philip Zimbardo, who became famous for his “prison experiments” at Stanford in the early 1970s.