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

by Geoffrey West

energy minimization principle: namely, that the total volume of the network—that is, the total volume of blood in the body—must be directly proportional to the volume of the body itself, and therefore proportional to its weight, as observed.

the volume of blood is a constant proportion of the volume of the body, regardless of size.

fractality— are, in fact, ubiquitous features of the complex world we

Furthermore, the mathematics that describes self-similarity and its implicit recursive rescaling is identical to the power law scaling

His main thesis was that the dynamics of conflict are primarily governed by the rates at which nations build up their armaments and that their continued accumulation is the major cause of war.

viewed the accumulation of weapons as a proxy for the collective psychosocial forces that reflect, but transcend, history, politics, economics, and culture and whose dynamics inevitably lead to conflict and instability.

he hypothesized that the probability of war between neighboring states was proportional to the length of their common border.

In fact, he discovered that the finer the resolution, and therefore the greater the expected accuracy, the longer the border got, rather than converging to some specific value!

In designing and manufacturing human-engineered artifacts, whether primitive pots and tools or modern sophisticated automobiles, computers, and skyscrapers, we employed and aspired to the simplicity of straight lines, smooth curves, and smooth surfaces.

“Smooth shapes are very rare in the wild but extremely important in the ivory tower and the factory.”

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.

Being overly rigid and constrained means that there isn’t sufficient flexibility for the necessary adjustments needed to withstand the inevitable small shocks and perturbations to which any system is subjected.

Think of the stresses and strains your heart is exposed to every day, many of which are unexpected. Being able to accommodate and naturally adapt to these is critical for your long-term survival. These continuous changes and impingements require all of your organs, including your brain as well as its psyche, to ...

Resilient ecosystems have greater diversity of species.

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.

The Fractal Geometry of Nature.25

general argument to address this can be made by recognizing that, in addition to minimizing energy loss, natural selection has also led to a maximization of metabolic capacity because metabolism produces the energy and materials required to sustain and reproduce life.1 This has been achieved by maximizing surface areas across which resources and energy are transported. These surfaces are in actuality

Terminal units therefore play a critical role not only because they are invariant but also because they are the interface with the resource environment, whether internal as in the case of capillaries or external as in the case of leaves.

Natural selection has taken advantage of the fractal nature of space-filling networks to maximize the total effective surface area of these terminal units and thereby maximize metabolic output.

Even though your lungs are only about the size of a football with a volume of about 5 to 6 liters (about one and a half gallons), the total surface area of the alveoli, which are the terminal units of the respiratory system where oxygen and carbon dioxide are exchanged with the blood, is almost the size of a tennis court and the total length of all the airways is about 2,500 kilometers, almost the distance from Los Angeles to Chicago, or London to Moscow.

Even more striking is that if all the arteries, veins, and capillaries of your circulatory system were laid end to end, their

total length would be about 100,00...

or nearly two and a half times around the Earth or over a third of the distance to the moon . . . and all of this neatly fits ins...

the total effective area of two-dimensional sheets filling three-dimensional washing machines scales like a volume rather than an area, so in this sense, we have turned an area into a volume. The reason for this is that we have taken smooth Euclidean surfaces, the sheets, and crumpled them up to create a huge number of crinkles and wrinkles, thereby turning them into fractals.

the distribution of the sizes of the wrinkles follows a classic power law:

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

The vast majority of organisms exhibit scaling exponents very close to 3⁄4 for metabolic rate and 1⁄4 for internal times and distances.

can break down broccoli only so far before it eventually loses its self-similar characteristics and reveals the underlying structure and geometry of its tissues, cells, and ultimately its molecular constituents.

Unlike most biological networks, mammalian circulatory systems are not single self-similar fractals but an admixture of two different ones, reflecting the change in flow from predominantly pulsatile AC to predominantly nonpulsatile DC as blood flows from the aorta to the capillaries.

all mammals have roughly the same number of branching levels, about fifteen, where the flow is predominantly steady nonpulsatile

The distinction among mammals as their size increases is the increasing number of levels where the flow is pulsatile AC.

Almost all of your cardiac output goes into pumping blood through the much smaller vessels of the nonpulsatile regime, whose number of levels is approximately the same for all mammals.

Relatively speaking, then, the proportion of the network where the heart expends most of its energy systematically decreases as the size of a mammal increases, illustrating again that larger mammals are more efficient than smaller ones: the whale needs only one hundredth the amount of energy needed by a shrew to supply blood to

This argument shows that only mammals that are large enough for their circulatory systems to support pulsatile waves through at the very least the first couple of branching levels would have evolved, thereby providing a fundamental reason why there is a minimum size.

weight of an animal increases faster than the ability of its limbs to support it so that if the design, shape, and materials remain unchanged, it will collapse under its own weight as its size increases.

there are additional constraints on maximum body size that transcend ecological biomechanics and that arise from the fundamental need to ensure that all cells are supplied with sufficient oxygen.

One of the more arcane results of the network theory is that the average distance between terminal units such as capillaries scales with body mass as a power law with an exponent of 1⁄12 (= 0.0833

although a blue whale is a hundred million times (108) heavier than a shrew, the average distance between its capillaries is only about (108)1⁄12 = 4.6 times larger.

Capillaries service cells, so the opening up of the network means that there is increasingly more tissue that needs to be serviced situated between adjacent capillaries as size increases. So on average, each capillary systematically has to service more cells, another reflection of the increasing economy of scale

there is a limit to how far this can be pushed. Each capillary, being an invariant unit, can deliver only...

the group of cells needing to be supplied by a single capillary becomes too large, some of them will inevitably become oxygen deprived, a situa...

The physics of how oxygen diffuses across capillary walls and through tissue to supply cells was first quantitatively addressed more than a hundred years ago by the 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 imaginar...

capillary is about half a millimeter long and about five times longer than its diameter). Based on this, one can calculate how large an animal could be before the separation distance between its capillaries gets so large that significant hypoxia develops. This leads to an estimate of about 100 kilograms for the maximum size, roughly equivalent to the larg...

describe how such characteristics change during growth within species. Biologists use the term ontogenesis to describe the developmental process that occurs within an individual during growth, beginning with the fertilization of an egg through birth and up to maturity. The onto part of this word is derived from the Greek word for “being,” while genesis means “origins,” so ontogenesis, or ontogeny, connotes the idea of the study of how we came to be.

You eat, you metabolize, you transport metabolic energy through networks to your cells, where some is allocated to the repair and maintenance of existing cells, some to replace those that have died, and some to create new ones that add to your overall biomass. This sequence of events is the template for how all growth occurs, whether for an organism, a city, a company, or even an economy, as symbolized in the figure below.

Because the supply of metabolic energy is apportioned between the maintenance of existing cells and the creation of new ones, the rate at which energy is used to create new tissue is just the difference between metabolic rate and what is needed for maintenance of existing cells. This latter term is directly proportional to the number of existing cells and therefore increases linearly with the mass of the organism, whereas metabolic rate increases sublinearly with a 3⁄4 power exponent. This difference in the way these two contributions scale with increasing size plays a central role in growth, so to ensure that you understand what it implies here’s a simple example to illustrate the point. Suppose the size of the organism doubles; then the number of cells also doubles, so the amount of energy needed for their maintenance increases by a factor of 2. However, metabolic rate (the supply

increases by only a factor of 23⁄4 = 1.682 . . . which is less than 2. So the rate at which energy is needed for maintenance increases faster than the rate at which metabolic energy can be supplied, forcing the amount of energy available for growth to systematically decrease and eventually go to zero, resulting in the cessation of growth. In other words, you stop growing because of the mismatch between the way maintenance and supply scale as size increases.

So as the organism grows and size increases, each capillary systematically has to service more cells following 1⁄4 power scaling.

sublinear scaling and the associated economies of scale arising from optimizing network performance lead to bounded growth and the systematic slowing of the pace of life.

metabolic rate scales exponentially with temperature rather than as a power law as it does with mass. Because metabolic rate—the rate at which energy is supplied to cells—is the fundamental driver of all biological rates and times, all of the central features of life from gestation and growth to mortality are exponentially sensitive to temperature.