The Equations of Life

by Charles S. Cockell

  The mole is an engineering solution to the compromises needed to effectively shift soil by maximizing the force applied over a small area. It is an organic manifestation of P = F/A.

  As an evolutionary consequence of this law, moles look the same regardless of their provenance. The Talpidae, the true moles, which live in Europe, North America, and Asia, appear similar to Australian moles, or the marsupial moles, which are more closely related to kangaroos and koalas. Moleness is a product of physics, and no matter whether you are a small furry mammal burrowing in the dirt of Edinburgh or you are burrowing in the dusty plains of the Australian outback, P = F/A mandates that you end up looking the same.

  I chose the moles because their very specific lifestyle, which is dominated by the need to burrow, brings one law of physics to the forefront of their lives. Their lifestyle demonstrates that evolutionary possibilities, in this case the options for the form of a small burrowing animal, greatly constrain the potential solutions available in this process of convergent evolution. Many other animals that burrow display the same general design.

  Even where two animals seem to have converged on the same shape not because of some physical feature of their environment but because of the influence of other life forms (maybe they are sleek and slender to escape predators), physical principles ultimately lie at the heart of those similarities. Perhaps the animals have bigger muscles to produce greater acceleration or better eyes that are more effective at collecting light with which to see predators.

  Nothing is magical or strange about convergent evolution any more than there is something uncanny about the fact that when both liquid lithium and liquid water are heated, both substances turn into a gas. This latter observation is not bizarre cosmic coincidence; rather, it is just physical processes, the consequence of adding energy to the molecules or atoms of a liquid to overcome their forces of attraction and cause them to dissipate into gas. This is the same with convergent evolution; very often it results when similar physical principles operating on different organic forms drive those forms toward resemblance.

  Often in convergence, more than one law may be at work, complicating our ability to identify a straightforward equation. Explore enough of the workings of any creature, such as ladybugs or moles, and you will find a multiplicity of laws relevant to their survival. If we do not comprehensively grasp the biology or ecology of an organism, we may well not even be able to identify the relevant laws. There may be more than one overall solution to the laws that operate or are exhibited in the organism. However, the ubiquity of convergent evolution across the biosphere and the vast number of examples show that solutions are not limitless, but usually rather few.

  To some people, there may be something strange about the apparent ability of life to navigate the vast landscape of possibilities to arrive at the solution. Is it not amazing how a mole in England looks like a mole in Australia? Across the vast vista of biological potential, how did they “find” the same solution?

  All life on Earth, all life across the universe, conforms to the law P = F/A. Life has no choice in this matter. Whenever this law plays a role in biology, whether it be our burrowing moles, a worm trying to push its way through some sodden soil, or a sand eel slithering silently through sandy sediment under the sea, all of them must observe the rule. Any biological solution that invokes this law by default observes it; life does not navigate to these solutions. The question, then, is whether the representation of this principle found in the biological solution allows the organism to survive long enough to reproduce. The mole that is born the shape of a sphere or with no front legs is not going to survive underground long. The sand eel that is too fat to burrow its way into sandy sediment will be gobbled by a larger fish and will also fail in the evolutionary struggle.

  With the mole, for which P = F/A happens to play a prominent role, there is only one broadly acceptable biological solution to the optimization of this equation: small animals with stubby, shovellike front feet. The animal did not navigate across a vast landscape of mole possibilities. It had to conform to the law, and the solutions that emerged through mutations from the first mole-like animal that dug underground optimized the application of this equation that was most effective at escaping predators, finding food underground, and crawling out of collapsed tunnels.

  Variant moles with a genetic mutation that makes them more like spherical moles or fat, cuboid moles are less effective than their conspecifics are at burrowing. Ultimately, they are less competitive for territory and resources than those that apply P = F/A more effectively. But those mutations can become effective if the environment changes. If our mutant moles find themselves in a hypothetical new environment where escaping predators by rolling down a hill, pangolin-style, is now the most effective way to escape, then a spherical mole, the circumference of whose rotund body is nicely described by 2πr, may well find that its somewhat porky profile, which disadvantaged it in the burrowing struggle for life, now provides it with an advantage in the hill-rolling escapades demanded of it. Its circumference being described by the relationship 2πr is now the pertinent relationship that operates on the rolling mole. In this hypothetical way, organisms move from one set of conditions to another, across a chessboard of environments, observing a vast variety of rules; those mutants that either use these laws to their advantage or conform to them with greater effectiveness survive to reach breeding age.

  Even with a narrow, straightforward set of physical laws, we can observe wonderful diversity. Increase the importance of P = F/A to a burrowing animal whose ancestors had legs, and we end up with moleness. Do this to a legless invertebrate, and we end up with wormness. Earthworms are long and slender, but as limbless invertebrates, they move not with robust little legs like moles, but by using muscles that contract and expand, forcing their way through the ground. A juvenile worm can push more than five hundred times its body mass as it drives itself through the soil. Variety manifests itself in how laws operate on biological material with prior history, but convergence makes these different branches of life very similar. Even between different groups of life, vertebrates and invertebrates, moles and worms share long, slender, cylindrical shapes with pointy fronts.

  Physics explains much about what we find in life, but can it also explain what we do not see? There is a conversation that some biologists have over coffee, or sometimes pints of beer. At first blush, the conversation may seem a little esoteric, maybe even somewhat bizarre. Yet the question they ask lies at the heart of everything that concerns this book: why don’t animals like moles have wheels?

  Take a look at everything around you that has to do with transport, and you can see that this question is not so strange. Cars, trains, bicycles, and wagons have them. Even planes, although they fly, land on wheels. These modes of transport, and many more, use these simple circular devices. So why has nature thoroughly rejected the wheel?

  Wheels work tremendously well on roads, railway tracks, and flat surfaces, but most of the world is a haphazard mixture of hills, ditches, and other irregular obstacles. Among all these, the wheel comes up short. A wheel cannot get over any vertical obstacle higher than its radius; it just slams into the obstacle and stops—unless the wheel can be lifted over, much like how someone pushing a shopping cart will pull back on the handle to launch the front wheels over a curb. Without the complexity of shopping cart and curb shenanigans, this simple problem can be described in an equation in which the force F to push a wheel over an obstacle is given by:

  F = √(2rh − h2) mg/(r − h)

  where h is the height of the obstacle, r is the radius of the wheel, m the mass of the wheel, and g the gravitational acceleration (on our planet, 9.8 meters per second per second [m/s2]).

  Make the height of the obstacle the same as the radius of the wheel, and this equation tells us that the force becomes infinite. You’re stuck.

  The landscape of our planet is full of irregularities, and there are more obstacles the smaller you are. A wheeled ant
or ladybug would have a quite terrible time trying to get over grains of sand and soil particles, even if it were a four-wheel-drive creature.

  Other problems confound our wheeled critter. Muddy soils, sand, and anything that offers resistance to the rolling of wheels will slow it down. A hungry, bounding four-legged fox would delight in a wheeled rabbit stuck in an English field, flinging mud all over the place as the rabbit wheels go round and round.

  Legs offer that all-important possibility of essentially prodding your way across the landscape, maneuvering left and right in zigzags and reversing easily whether you are being chased by a predator or avoiding a muddy patch. In extreme situations, like the mountain goat perched high on a ledge a few centimeters wide, its legs deftly navigating the footholds irregularly distributed across the craggy surface, the advantages of legs over wheels are all too clear.

  It is tempting to think that the lack of wheels in life may simply reflect the ancestry of all land animals. Perhaps land animals are merely constrained by the legacy of their ancestors, a limb joint that is simply unsuitable for making wheels in the first place. However, there does not seem to be a fundamental barrier. We could imagine an early fin that flips around in a rotating manner similar to a person swinging the arms in a circle as the first fish began sojourns onto land. Increasing efficiencies would be achieved by better performance in rotation and stronger limbs. Eventually, the structure would evolve into some sort of wheel that would evade the problem of nerves and blood vessels getting entangled into an awful mess as it rotated.

  Evolution does experiment with wheel-like contraptions. Across all continents of the planet, excluding Antarctica, dung beetles gather up balls of animal dung into spheres and diligently push them across the landscape back to their burrows either to eat the dung or to use it as a brooding ball for their young. Navigating using the Milky Way, the navigation itself a stunning feat of the evolution of visual cues, the beetles can push an impressive ball, between about ten and a thousand times their body weight. These balls of dung show us that on flat, dry terrain, evolution does test rolling motions. Rolling works where the ground is predictably flat.

  Similarly, wheeling across the deserts of the Earth are tumbleweeds, near-spherical parts of plants that break away from a mature plant and head off, driven by the wind, across the landscape to colonize new terrain. Most of this plant material is dead, and this arrangement has an advantage, since when the weed has stopped moving, the seeds can become detached from the dead material and drop to the soil to germinate. More than ten families of plants produce tumbleweeds, and all these plants live in arid and steppe-like regions of the world, where vistas of flat horizons provide the ideal environments for these vegetative emissaries to roll unimpeded across the land.

  These examples, although not wheels in the strict sense, show that evolution, for some inexplicable reason, did not overlook the circle as a solution to moving something across a landscape. But evolution just did not go far with this form.

  It would be tempting to follow up this discussion with another question: with all that irregular terrain, why don’t animals just build roads or at least road-like flat thoroughfares? Richard Dawkins suggested in an interesting rejoinder that road building is just not selfish enough. If you build a road, someone else may appropriate your hard-won efforts, meaning the energy put into constructing it was worthless. Roads in our own society are very much a product of government action on behalf of everyone—you pay for the road even if you do not use it. With private roads, you must pay a fee to the builder, a transaction difficult for animals that have not invented economics. The argument is persuasive, but probably a much simpler answer is that for road building to evolve in animals, they must either have wheels to begin with or be evolving wheels to provide a selection pressure for road building. However, as the preceding paragraphs have shown, organisms do not even start to evolve wheels as appendages in the first place. There is no selection pressure for road building.

  People who wonder why rabbits do not have wheels are usually the same type of people who wonder why fish don’t have propellers. This is a slightly surreal, but no less interesting question. After all, propellers are ubiquitous in our ships, boats, and even the smallest little paddleboat on a Greek summer holiday. Why doesn’t fish evolution converge on propellers to get through water as moles converge on small, cylindrical shapes and paddle-like front feet to get through dirt?

  Propellers are not very efficient. Rotate a propeller too fast, and the flow around it is broken up. When the water cavitates, or forms bubbles, they form around the tips of the propeller, reducing the thrust to drive a ship forward. On ships, propellers typically reach about 60 to 70 percent efficiency at best. Compare this with the way many fish swim: by flexing their body and inducing an undulatory wave to pass along it or part of it. This locomotion can be over 95 percent efficient. Contorting your body and writhing through the ocean turns out to be an efficient way to escape predators or reach food before your competitors.

  Before we discard propellers as a seemingly unintuitive possibility—an image that even elicits ridicule when we imagine propellers stuck to a fish—there are examples of rotating structures that drive living things though liquids. Dangling from the sides or ends of some microbes, such as the much-studied species that lives in your gut, Escherichia coli, are flagella (in the singular, flagellum), whiplike appendages that spin at amazing speeds of about a hundred turns per second. They propel the microbe forward through fluid at up to six hundred microns per second or about two meters an hour. But don’t be too unimpressed—seen in another way, that’s about six hundred body lengths per second, which for someone of your size doing the same relative body-length-scale rate of movement equals roughly thirty kilometers an hour, which is a fast human running speed.

  The flagella are an extraordinary product of evolution. Tiny motors embedded in proteins in the microbial cell, itself about a micron long, rotate the long protein units that make up the flagella. Some microbes have just one flagellum, some have many flagella, and some can grow them at will, depending on whether they want to settle down or propel themselves somewhere new, perhaps to escape a toxin or find food. By briefly reversing the rotation of the flagella, a microbe can cause itself to tumble, changing its direction in its quest for optimum conditions for growth.

  Despite the superficial similarities between a ship’s propeller and a microbial flagellum, which might give us pause to wonder why fish didn’t use the same trick to get around, there are vast differences between the environment that a microbe inhabits and that of a fish.

  Imagine you were told to swim the length of an Olympic-sized pool, but not any pool—one filled with molasses. As you climb into the gooey pit, you allow yourself to sink in. Then as your whole body is submerged, with a mighty effort, you push your arms behind you in the first stroke. You move forward, but just a few paltry centimeters. As you pull your arms to the front through the syrup to begin the second stroke, you find that your arms’ forward motion pushes you back again, by the same distance. As you continue your efforts, you remain fixed, oscillating back and forth in insignificant movements as your arms torturously push forward and backward against the gloopy substance.

  At the scale of a microbe swimming in water, this is what everyday life is like. Water behaves like a viscous liquid. A conventional ship propeller at this scale would not work, because it relies on pushing water backward, virtually impossible in a syrup-like liquid. So the comparison between the propeller and the flagellum is misleading. The flagellum should instead be thought of as more of a way of corkscrewing a microbe through the water rather than as behaving like a propeller to push something forward by imparting momentum to the water behind.

  The efficiency of the flagellum is poor, about 1 percent, much less than a typical propeller on a ship. In the microbes, we do not see nature using propellers; instead, we see a device for making something move at small scales, at the syrupy scales of what physicists call low Reyno
lds numbers. A flagellum is a particular solution for little things that want to move, but it nevertheless shows that low-efficiency rotating structures for propelling things through fluids are not unknown in nature. At larger scales, at high Reynolds numbers, when water viscosity becomes less important and the fluid behaves more like the water in a swimming pool, propellers work, but they still remain less efficient than slithering through the water like a fish.

  Wheels and propellers get us into some trouble though. Although we can imagine how a wheeled rabbit or propeller-driven fish might evolve, we still cannot conclusively tell whether the absence of these creatures is caused by a development barrier, an insufficient versatility in the tool kit of life to make the leap to these contraptions. It is much easier to experiment with, and trace the evolutionary path of, something that does exist. However, we find compelling physical reasons to suspect that even if the developmental path to wheels and propellers were available, life would eschew these devices for legs on land and body waves in water.

  The recognition that simple mathematical or physical principles might determine what organisms do and don’t look like has occurred to scientists before. In 1917, a brilliant Scottish mathematician, D’Arcy Wentworth Thompson, published a controversial book titled On Growth and Form. In its prolific and fascinating pages, he demonstrates the myriad ways mathematical relationships and scaling can be found in life. He considers the regular (“equiangular”) spirals that define the shapes of shells, from snails to the extinct ammonites. He explores the shapes of horns and teeth and the mathematical relationships in the growth of plants. He even draws fish on grids and, by shearing the grid in different directions, he shows that the now obliquely shaped novelty resembles many species found in the natural world. The point of his book is simple—that biology can be described by mathematics and that all living things conform to simple patterns of scaling and interrelationships between their different dimensions. When his book was published, it was rather radical in an age transfixed by Darwinism. Even today, people are somewhat unsure what to make of it.