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

by Geoffrey West

Consequently, the amount of energy needed to support an average person living in the United States has risen to an astounding 11,000 watts.

German physicist Rudolf Clausius in 1855.

To maintain order and structure in an evolving system requires the continual supply and use of energy whose by-product is disorder.

Scaling simply refers, in its most elemental form, to how a system responds when its size changes.

This systematic “value-added” bonus as size increases is called increasing returns to scale by economists and social scientists, whereas physicists prefer the more sexy term superlinear scaling.

This kind of scaling is referred to as sublinear scaling.

A typical complex system is composed of myriad individual constituents or agents that once aggregated take on collective characteristics that are usually not manifested in, nor could easily be predicted from, the properties of the individual components themselves.

The economic output, the buzz, the creativity and culture of a city or a company all result from the nonlinear nature of the multiple feedback mechanisms embodied in the interactions between its inhabitants, their infrastructure, and the environment.

These simulations have given credence to the idea that the bewildering dynamics and organization of highly complex systems have their origin in very simple rules governing the interaction between their individual constituents.

In general, then, a universal characteristic of a complex system is that the whole is greater than, and often significantly different from, the simple linear sum of its parts.

This collective outcome, in which a system manifests significantly different characteristics from those resulting from simply adding up all of the contributions of its individual constituent parts, is called an emergent behavior.

The important lesson that we learn from these investigations is that in many such systems there is no central control.

Scaling up from the small to the large is often accompanied by an evolution from simplicity to complexity while maintaining basic elements or building blocks of the system unchanged or conserved.

In a nutshell, the problem is that the theory also predicts that unbounded growth cannot be sustained without having either infinite resources or inducing major paradigm shifts that “reset” the clock before potential collapse occurs.

Unlike the cubic scaling law, the conventional definition of the BMI has no theoretical or conceptual underpinning and is therefore of dubious statistical significance.

This is characteristic of many scaling arguments: general results can be derived, but details of their mechanistic origins sometimes remain hidden.

Understanding more generally the emergence of complexity from simplicity, an essential characteristic of adaptive evolving systems, is one of the founding cornerstones of the new science of complexity.

No one wants to die. Even people who want to go to heaven don’t want to die to get there. And yet death is the destination we all share. No one has ever escaped it, and that is how it should be, because death is very likely the single best invention of life. It’s life’s change agent. It clears out the old to make way for the new.

In his pioneering work, Kleiber surveyed the metabolic rates for a spectrum of animals ranging from a small dove weighing about 150 grams to a large steer weighing almost 1,000 kilograms. Over the ensuing years his analysis has been extended by many researchers to include the entire spectrum of mammals ranging from the smallest, the shrew, to the largest, the blue whale, thereby covering more than eight orders of magnitude in mass. Remarkably, and of equal importance, the same scaling has been shown to be valid across all multicellular taxonomic groups including fish, birds, insects, crustacea, and plants, and even to extend down to bacteria and other unicellular organisms.6 Overall, it encompasses an astonishing twenty-seven orders of magnitude, perhaps the most persistent and systematic scaling law in the universe.

Thus, even though the cat is 100 times heavier than the mouse, its metabolic rate is only about 32 times greater, an explicit example of economy of scale.

We just saw that a cat that is 100 times heavier than a mouse requires only about 32 times as much energy to sustain it even though it has approximately 100 times as many cells—a classic example of an economy of scale resulting from the essential nonlinear nature of Kleiber’s law.

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.

Whales live in the ocean, elephants have trunks, giraffes have long necks, we walk on two legs, and dormice scurry around, yet despite these obvious differences, we are all, to a large degree, nonlinearly scaled versions of one another. 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.

as allometric scaling laws.

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.

At the most fundamental biochemical level metabolic energy is created in semiautonomous molecular units within cells called respiratory complexes. The critical molecule that plays the central role in metabolism goes by the slightly forbidding name of adenosine triphosphate, usually referred to as ATP.

life. At any one time our bodies contain only about half a pound (about 250 g) of ATP, but 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! Taken together, all of these ATPs add up to meet our total metabolic needs at the rate of the approximately 90 watts we require to stay alive and power our bodies.

It is within this context that we should view allometric scaling laws: their systematic regularity and universality provides a window onto these emergent laws and underlying principles. As external environments change, all of these various systems must be scalable in order to meet the continuing challenges of adaptability, evolvability, and growth. The same generic underlying dynamical and organizational principles must operate across multiple spatial and temporal scales. The scalability of living systems underlies their extraordinary resilience and sustainability both at the individual level and for life itself.

As I began to ponder what the origins of these surprising scaling laws might be, it became clear that whatever was at play had to be independent of the evolved design of any specific type of organism, because the same laws are manifested by mammals, birds, plants, fish, crustacea, cells, and so on. All of these organisms ranging from the smallest, simplest bacterium to the largest plants and animals depend for their maintenance and reproduction on the close integration of numerous subunits—molecules, organelles, and cells— and these microscopic components need to be serviced in a relatively “democratic” and efficient fashion in order to supply metabolic substrates, remove waste products, and regulate activity.

I particularly enjoyed being reminded of the primal excitement of why I loved being a scientist: the challenge of learning and developing concepts, figuring out what the important questions were, and occasionally being able to suggest insights and answers.

The challenge at every level of observation is to abstract the important variables that determine the dominant behavior of the system.

Physicists have coined a concept to help formalize a first step in this approach, which they call a “toy model.” The strategy is to simplify a complicated system by abstracting its essential components, represented by a small number of dominant variables, from which its leading behavior can be determined.

A concept related to the idea of a toy model is that of a “zeroth order” approximation of a theory, in which simplifying assumptions are similarly made in order to give a rough approximation of the exact result.

The time seems right for revisiting D’Arcy Thompson’s challenge: “How far even then mathematics will suffice to describe, and physics to explain, the fabric of the body, no man can foresee. It may be that all the laws of energy, and all the properties of matter, all . . . chemistry . . . are as powerless to explain the body as they are impotent to comprehend the soul. For my part, I think it is not so.”

The idea behind the concept of space filling is simple and intuitive. Roughly speaking, it means that the tentacles of the network have to extend everywhere throughout the system that it is serving, as is illustrated in the networks here. More specifically: whatever the geometry and topology of the network is, it must service all local biologically active subunits of the organism or subsystem.

Our circulatory system is a classic hierarchical branching network in which the heart pumps blood through the many levels of the network beginning with the main arteries, passing through vessels of regularly decreasing size, ending with the capillaries, the smallest ones, before looping back to the heart through the venal network system. Space filling is simply the statement that the capillaries, which are the terminal units or last branch of the network, have to service every cell in our body so as to efficiently supply each of them with sufficient blood and oxygen.

Quite analogously, many of the infrastructural networks in cities are also space filling: for example, the terminal units or end points of the utility networks—gas, water, and electricity—have to end up supplying all of the various buildings that constitute a city. The pipe that connects your house to the water line in the street and the electrical line that connects it to the main cable are analogs of capillaries, while your house can be thought of as an analog to cells. Similarly, all employees of a company, viewed as terminal units, have to be supplied by resources (wages, for example) and information through multiple networks connecting them with the CEO and the management.

This simply means that the terminal units of a given network design, such as the capillaries of the circulatory system that we just discussed, all have approximately the same size and characteristics regardless of the size of the organism.

And as in biology, basic terminal units, such as faucets and electrical outlets, are not reinvented every time we design a new building regardless of where or how big it is.

The final postulate states that the continuous multiple feedback and fine-tuning mechanisms implicit in the ongoing processes of natural selection and which have been playing out over enormous periods of time have led to the network performance being “optimized.”

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.

So why not life?

The rate at which oxygen is delivered to cells and likewise the rate at which blood is pumped through our circulatory system are therefore measures of our metabolic rate. Similarly, the rate at which oxygen is inhaled through our mouths and into the respiratory system is also a measure of metabolic rate.

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

The term impedance matching can be a very useful metaphor for connoting important aspects of social interactions. For example, the smooth and efficient functioning of social networks, whether in a society, a company, a group activity, and especially in relationships such as marriages and friendships, requires good communication in which information is faithfully transmitted between groups and individuals. When information is dissipated or “reflected,” such as when one side is not listening, it cannot be faithfully or efficiently processed, inevitably leading to misinterpretation, a process analogous to the loss of energy when impedances are not matched.

So, naturalists observe, a flea Hath smaller fleas that on him prey; And these have smaller still to bite ’em; And so proceed ad infinitum.

Now, the volume of the network is just the sum of the volumes of all of its vessels or branches, and these can be straightforwardly calculated from knowing how their lengths and radii scale, thereby connecting the self-similarity of the internal network to body size. 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.

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 1⁄4 power ones.

Furthermore, the mathematics that describes self-similarity and its implicit recursive rescaling is identical to the power law scaling discussed in previous chapters. In other words, power law scaling is the mathematical expression of self-similarity and fractality. Consequently, because animals obey power law scaling both within individuals, in terms of the geometry and dynamics of their internal network structures, as well as across species, they, and therefore all of us, are living manifestations of self-similar fractals.

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.