In Part I we reviewed how the work of Benoit Mandelbrot was developed into complexity theory and crafted the Fractal Lens. We now observe how Scale applies this to Biological Systems and Companies.
Scaling Biological Adaptive Systems
There is a scaling law for the metabolic rate vs the mass of a living thing known as Kleiber’s law, which states that if you double the size of a plant or animal, it’s metabolic rate will not double, but only increase by 75%. This law holds true for all types of plants and animals, from bacteria to trees to fish to crustaceans to mammals. Another way of saying it, is that it follows a power-law distribution with an exponent of ¾. Thus, a cat is 100 times heavier than a mouse, but, as to its metabolic rate, only generates 32 watts to the mouse’s 1 watt. Similarly, a cow that is 100 times the size of the cat generates wattage of 32 x 32, or 1024 watts. For every scaling up of 100 times in mass, the wattage goes up 32 times.
Similar power laws govern the scaling of essentially all physiological quantities and life-history events, including growth rate, heart rate, evolutionary rate, genome length, mitochondrial density, gray matter in the brain, life span, the height of trees and even the number of their leaves. What they all have in common is that the scaling factor is a multiple of ¼.
"[T]he exponent for growth rates is very close to ¾, for lengths of aortas and genomes it’s ¼, for heights of trees ¼, for cross-sectional areas of both aortas and tree trunks ¾, for brain sizes ¾, for cerebral white and gray matter 5⁄4, for heart rates minus ¼, for mitochondrial densities in cells minus ¼, for rates of evolution minus ¼, for diffusion rates across membranes minus ¼, for life spans ¼ . . . and many, many more. The “minus” here simply indicates that the corresponding quantity decreases with size rather than increases, so, for instance, heart rates decrease with increasing body size following the ¼ power law[.]"
"Particularly fascinating is the emergence of the number four in the guise of the ¼ powers that appear in all of these exponents. It occurs ubiquitously across the entire panoply of life and seems to play a special, fundamental role in determining many of the measurable characteristics of organisms regardless of their evolved design. 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."
Indeed, just by knowing the size of a mammal, one can know "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."
But why is four or ¼ the basic scaling law of biology? The answer as it turns out goes back to Benoit Mandelbrot’s geometry – fractals. Kleiber’s law follows from requiring that the energy needed to pump blood through mammalian circulatory systems, including ours, is minimized so that the energy we devote to reproduction is maximized. Examples of other such networks include the respiratory, renal, neural, and plant and tree vascular systems. By organizing these networks into fractal shapes that are self-similar but get smaller and smaller, fractal shapes fill the spaces they serve in the body so efficiently that they effectively behave as if they had four dimensions – not three. They are evolved in this manner to optimize their functions.
"Optimization principles lie at the very heart of all of the fundamental laws of nature, whether Newton’s laws, Maxwell’s electromagnetic theory, quantum mechanics, Einstein’s theory of relativity, or the grand unified theories of the elementary particles. Their modern formulation is a general mathematical framework in which a quantity called the action, which is loosely related to energy, is minimized. All the laws of physics can be derived from the principle of least action which, roughly speaking, states that, of all the possible configurations that a system can have or that it can follow as it evolves in time, the one that is physically realized is the one that minimizes its action. Consequently, the dynamics, structure, and time evolution of the universe since the Big Bang, everything from black holes and the satellites transmitting your cell phone messages to the cell phones and messages themselves, all electrons, photons, Higgs particles, and pretty much everything else that is physical, are determined from such an optimization principle."
Another feature of these networks is the invariance of the terminal units where points of delivery are located. What this means is that the smallest blood vessels, mitochondria and nerve endings are virtually the same in every organism, regardless of its size. It is only the larger units – like hearts and aortas – that vary.
These three characteristics: space filling, optimization and invariance of terminal units, lead to the same sort of structures in every organism.
But how does the number 4 come out of this? "It 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. The resulting magic number four emerges as an effective extension of the usual three dimensions of the volume serviced by the network by an additional dimension resulting from the fractal nature of the network."
Yes, that’s a mouthful – but rest assured the rest of the math is in the book.
"[D]riven 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. Thus, instead of scaling with classic ⅓ exponents, as would be the case if they were smooth nonfractal Euclidean objects, they scale with ¼ exponents. Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional."
The upshot is that "organisms operate as if they were in four dimensions, rather than the canonical three. In this sense the ubiquitous number four is actually 3 + 1. More generally, it is the dimension of the space being serviced plus one. So had we lived in a universe of eleven dimensions, as some of my string theory friends believe, the magic number would have been 11 + 1 = 12, and we would have been talking about the universality of 1⁄12 power scaling laws rather than ¼ power ones."
"In marked contrast to this, almost none of our man-made engineered artifacts and systems, whether automobiles, houses, washing machines, or television sets, invoke the power of fractals to optimize performance."
Not everything is built on 1/4, as not everything involves fractal structures. In particular, the strength of animals increases only two orders of magnitude for every three orders of magnitude increase in body weight. Thus, the relevant exponent here is 2/3. An animal that is twice as large will have only about 1.6 times the relative strength of the smaller one. This is the result of the volume of an animal growing at 3 times the rate as its size doubles.
The 2/3rd scaling law also applies to dosages of drugs. Thus to convert a dosage for a 40 pound child to one for an 80 pound child, the increase would not be double, but only 1.6 times.
These observations also call into question the use of BMI as a measurement of obesity. BMI is supposed to be invariant over any range of height, but it actually is not. Thus, it is “of dubious statistical significance,” especially for the very tall or very short. A better measure is weight divided by (height squared), which is known as the Ponderal index. Normal adult values are between 11 and 14 on the Ponderal scale.
Finally, the number of heartbeats in a lifetime is approximately the same for all mammals. . . . Thus, shrews have heart rates of roughly 1,500 beats a minute and live for about two years, whereas heart rates of elephants are only about 30 beats a minute but they live for about seventy-five years. Despite their vast difference in size, both of their hearts beat approximately one and a half billion times during an average lifetime. This invariance is approximately true for all mammals, even though there are large fluctuations for the reasons I outlined above. The greatest outlier from this intriguing invariance is us: for modern human beings, on average our hearts beat approximately two and a half billion times, which is about twice the number for a typical mammal. However, . . .it is only in the past one hundred years that we have been living this long.
Scaling Companies -- As Biological Systems
Companies and businesses scale up remarkably like biological organisms.
"[Companies and businesses] scale following simple power laws and as anticipated they do so with a much greater spread around their average behavior than for either cities or organisms. So in this statistical sense, companies are approximately scaled, self-similar versions of one another: Walmart is an approximately scaled-up version of a much smaller, modest-size company. Even after taking this greater variance into account, this scaling result reveals remarkable regularities in the size and dynamics of companies and is quite surprising given the tremendous variety of different business sectors, locations, and age.
"[L]ike organisms, . . . [companies] scale sublinearly as functions of their size, rather than superlinearly like socioeconomic metrics in cities. In this sense, companies are much more like organisms than cities. The scaling exponent for companies is around 0.9, to be compared with 0.85 for the infrastructure of cities and 0.75 for organisms.
Growth rates of companies also behave like biological organisms. After a period of sharp or exponential growth a company levels off at a certain size. This is the familiar “S-Curve” that appears in almost every growth scenario where the upper end is ultimately limited, as discussed in this blog post.
Also like biological organisms, companies die off at predictable rates: "half of all the companies in any given cohort of U.S. publicly traded companies disappear within ten years, and a scant few make it to fifty, let alone a hundred years." More specifically, "[t]he half-life of U.S. publicly traded companies was found to be close to 10.5 years, meaning that half of all companies that began trading in any given year have disappeared in 10.5 years."
Thus, "[o]f the 28,853 companies that have traded on U.S. markets since 1950, 22,469 (78 percent) had died by 2009. Of these 45 percent were acquired by or merged with other companies, while only about 9 percent went bankrupt or were liquidated; 3 percent privatized, 0.5 percent underwent leveraged buyouts, 0.5 percent went through reverse acquisitions, and the remainder disappeared for “other reasons.”
Indeed, [e]xponential survival curves similar to those exhibited by companies are manifested in many other collective systems such as bacterial colonies, animals and plants, and even the decay of radioactive materials. It is also believed that the mortality of prehistoric humans followed these curves before they became sedentary social creatures reaping the benefits derived from community structures and social organization. Our modern survival curve has evolved from being a classic exponential to developing a long plateau stretching over fifty years . . ." as we saw in Powerlaws and Magic in the Country of Old Men.
That's probably enough for now. But stay tuned for Part III, where we visit Cities and bring it all home . . .
I have always been curious about the way the world works and the most elegant ideas for describing and explaining it. I think I have found three of them.
I was very fond of James Burke's Connections series that explored interesting intersections between ideas, and hope to create some of that magic here.