The Equations of Life

by Charles S. Cockell

  J = −D dC/dx

  where J is the flux, D is the diffusion coefficient (again), and dC/dx is the rate of change of concentration of the gas over a given distance. This equation allows us to work out how much oxygen will pass into the insect’s body.

  These equations, applied to insects, tell us a very simple thing: insects are limited in size. Get too big, and you cannot get enough oxygen into your middle in a sensible time without a very cumbersome network of trachea. The distance of diffusion is not the only problem. As oxygen diffuses in, it gets used up before it can reach the deep interior of the animal. This is one explanation for why there are no ants or beetles the size of elephants.

  Working out the largest possible size of insects is difficult since complicating our so-far elementary analysis is a snag: some insects can actively pump air in and out of their bodies by moving their abdomens. In this way, convection, which is the movement of air driven by pressure gradients, can more forcefully be brought into play, allowing many insects, particularly largish ones like cockroaches, to actively pump in oxygen. But even with this aid, insects still have a limited mass they can achieve through the simple networks of trachea.

  Insects can get large, but not nearly as impressive as the most sizable mammals or extinct reptiles, the dinosaurs. The largest recorded insect is a specimen of weta, Deinacrida heteracantha, a cricket-like insect that lives in New Zealand. The 71-gram giant comes in alongside the goliath beetles that routinely weigh in at over 50 grams. But compare these Godzillas of the insect world to the 140,000-kilogram mass of a blue whale.

  Peer back through time, and you see something very odd. About three hundred million years ago, bigger insects, much bigger, inhabited the world. Buzzing through the skies of the rich Carboniferous forests—enormous swampy expanses of trees that would ultimately provide us with coal—were immense dragonflies. The now extinct Meganeura had a wingspan of well over half a meter, and creeping its way through the undergrowth was the terrifying Arthropleura, a millipede that grew to over two and a half meters. What happened at this time? Was evolution on a meander through a random experiment in giant insects? Coincidently, at the same time, the oxygen levels in the atmosphere rose to about 35 percent, compared with today’s 21 percent. That elevated oxygen may well have played a role in allowing for giant insects. As the oxygen levels rose, more oxygen could effectively diffuse into larger insects, allowing the body size to increase.

  Like all ideas, sometimes a beautiful story can be ruined by hard facts. Oxygen has a more complicated role in life than merely influencing how big something gets because of diffusion. Higher oxygen levels can be toxic, producing free radicals that must be quenched lest they attack key biological molecules. Insects might have evolved to grow bigger to minimize these effects of oxygen in the insect body by reducing the amount of oxygen that diffuses into the interior of the animal. Larger insects have other problems, like their need for more food and the possibility of breakage in their exoskeleton. These factors also play into the overall constraints on insect size.

  Nevertheless, however incomplete our knowledge of the ancient landscape for insects and their atmospheric environment, we must conclude that physical principles strongly shape the forms of insects and their maximum size. Therefore, ultimately, in comparing them with reptiles and mammals, we can see how the architecture of an animal is severely limited. We can argue for contingency and chance in molding the fine details of insects, but when confronted with the challenge of their ultimate limits, we must return to basic physics.

  To find its way to its protected group of companions under the moss pile or the autumn leaves blown up against rows of the Meadows’ trees, the ladybug must be able to sense its world, and on its head are a veritable complexity of sensors to do just this. The ladybug has two eyes. Unlike in you and me, these eyes are not one large lens capturing light and transferring it to many receptors beneath, but, like all insects, the ladybug has a compound eye. In a giant cluster of minuscule individual lenses, called ommatidia, light is captured by each one from a different part of the sky. The size of these little lenses is limited. Naturally, the ladybug would like to have as many as possible. The more there are, the higher the resolution that can be achieved. In the other words, the more detail can be collected from the world around it. The angular size of each lens (θ) is given by the simple equation:

  θ = ad/r

  where a is the angular field of view of a row of lenses, d is the diameter of the individual lens, and r is the length of a row of lenses.

  If we pack in more lenses, we can certainly collect more information about the world. But a new problem emerges, since small lenses become subject to diffraction, the process that causes light to be slightly bent and distorted, causing interference. The eye becomes useless. The angle (θd) below which these effects begin to disrupt the ladybug’s vision can be calculated using the wavelength of light (λ):

  θd = 1.2λ/d

  Here again we find a trade-off in physical principles up against one another, doing battle in the evolutionary struggle. Make the individual lenses smaller, and you can fit more of them into the eye and you can see more of the world. But make them too small, and the physics of light behavior renders them ineffective. The evolutionary process is restricted by intersecting principles that forge and hammer its products into predictable and narrow forms.

  The reception of light and colors by insect eyes is a field unto itself. Different ommatidia have receptors for blue, green, and, in many insects, UV light, giving them access to a region of the electromagnetic spectrum that you and I cannot see. In some flying insects, receptors adapted for UV and blue preferentially face toward the sky, perhaps to aid navigation. In the visual capacities of insects, and indeed all animals, the requirements for the physical detection of different regions of the electromagnetic spectrum meet biology.

  In this diminutive ladybug, we have toured and explored just one example of the irrefragable and deep links between evolution and physical processes. After several months of study, the group of students whom I had tasked with exploring the physics of a ladybug had probed just a few principles in a report more than forty pages long. We have not even scrutinized the antennae, packed full of sensilla that can detect chemicals, physically feel their surroundings, sense air speed while they are flying, and, in some insects, pick up sound. Each capability has a set of equations we could list and explore in depth. We have not talked about the mandibles and the mechanics of the mouth used for snipping leaves and crushing food, the processes themselves a concatenation of equations and forces all of which must converge to provide the ladybug with sustenance. The digestion and absorption of food carries us down another path, where diffusion, osmosis, and friction all interplay with other forces to define how well our insect can gain the energy and nutrients it must acquire to grow and reproduce. And what about the insect blood, the hemolymph, that circulates through its teeny vessels to bring vital nutrients to the cells and remove waste? What of the physics of muscle function, the storage of energy, or the details of the tegument, the outer layer of the insect? And the physics of ladybug reproduction? The eggs, their development, and the larval stages of insects? I suspect three or more years of research would be needed to do this task true justice. All these questions lie beyond the scope of this book, but the few efforts we have embarked on in this short exploration illustrate the conclusions we must draw.

  The ladybug is a remarkably complex thing, and packed into this machine, whose mass is a million billion billion billionth of the Sun’s, are many more physical principles than those that define the structure and evolution of a star.

  Those physical principles are not discrete, doing their own thing, but are all interwoven. In the evolutionary process, natural selection operates on each living thing to remove the variants in which the mosaic of principles given expression in them are not sufficiently optimized to allow them to achieve reproduction.

  In surviving collisions that would tear and
otherwise damage its micron-thin wings, the ladybug must grow thick wing cases that can withstand the unpredictability and knocks of the outdoor life. Yet if they are too thick, the greater weight of the insect diminishes its ability to fly and to flick its wings into action quickly to escape a predator. In this conundrum, the Young’s modulus of its material of choice, chitin, must come face-to-face with the equations that define its aerodynamic life. The relationships that describe the strength of chitin will themselves change how that material absorbs heat, linking the equation for the temperature of the ladybug to the effectiveness in surviving collisions.

  We can imagine a giant sheet of paper with many hundreds of equations written down, curved arrows running hither and thither showing how the terms or solutions of one equation influence another. Feedback processes abound as this enormous network of equations shifts and changes, each minuscule alteration in one equation, like “the wave” in a stadium, rippling through the rest. This is life. Mutations alter the solutions to some equations, add new ones, remove others. Natural selection takes the interwoven whole of this mathematical olio and subjects it to the environment. Those tapestries of physics that are manifested in ladybugs that successfully reproduce move on to new experiments. Those that do not are removed.

  A fascinating challenge would be to create a ladybug in a computer with as many physical principles that can be described in equations as possible. Beyond the cursory effort here, we might delve deep down into the genetic code to add mutations and errors in the code. At the higher levels, we might produce populations of ladybugs, simulating their gathering multitudes in the cold. Aside from its scientific use in investigating in more detail the possibility of reducing an entire multicellular animal to physical principles manifested in equations, it would be a profoundly good way to deepen our efforts to understand the various forces and possibilities that shape the living form. Such efforts would take us further into the realms of predictive capability, a fundamental characteristic of science.

  Reducing life to a set of equations may provide an effective means to link genetics with physics. Consider the temperature of our ladybug. Each term in the thermal equation that gives us its temperature can be considered to be controlled by a gene, a set of genes, or the emergent properties that result from the products of many genes. The solar radiation lost on the surface of the insect depends on how much is reflected away, and the amount of reflection depends on the surface’s shininess, itself a product of genes and developmental pathways that decide the surface characteristics of the wing cases. Some radiation may be scattered away if the surface is rough, again controlled by genes that influence how the wing cases are fabricated. The quantity of radiation lost from the body of the ladybug will depend on the thickness of the wing cases, which is determined by the genes that control their development. And so on.

  We must be careful not to reify equations. They do not exist as physical entities; they merely express relationships between different variables. However, an equation that defines a characteristic that we know helps an organism reach reproductive age, such as its thermal balance, can be thought of as a way of coalescing various physical features of an organism. And each of these features makes up the term of an equation and might be assigned to a specific gene or set of genes or the interactions of their products.

  By linking the change in an equation’s term to genes and their ultimate pathways (simplistically, say, a gene or genes determining ladybug wing case thickness that could replace the thickness term in the thermal balance equation), we may even express variables in an equation in terms of the activity of genes. In this way, we truly integrate physical relationships and properties in the macroscopic world with the genome and the pathways that result from it.

  The influence of different environments on these genetic pathways in the whole organism might be a further variable we could add to specify how the environment influences the solution to given equations. Thus we would be incorporating processes that operate from the top down as well as the role of the genes in determining the structure of an organism from the bottom up. In essence, an equation provides a useful means to establish which characteristics of an organism should be considered as a whole system, the terms of which come together to influence an important property that bears on its ability to reach reproductive age. Many genes are involved in more than one process, and the complexity of developmental processes assures us that linking genetics with physics in this way would be a heady ambition, made difficult by the fact that, for many processes, there is no simple link between a single gene and a phenotypic characteristic. Nevertheless, this evolutionary physics or physical genetics approach, however one likes to think of it, may provide just one useful way to encapsulate adaptations and evolutionary changes in quantitative, physically circumscribed terms.

  Throughout this tantalizing foray, we have also seen glimpses into common reasons for evolutionary convergence, the reasons why organisms have analogous structures. Evolutionary convergence is often just another way of saying, albeit more efficiently, similarity caused by the laws of physics. In their sticky, hairy feet, the shape of their wings, and the thickness and the color of their wing cases, the ladybugs show us that the simple equations and mathematical relationships that sculpt their curves are imposed on all insects. The network of equations will always lead to modifications. A larger wing here has a ripple effect on the wing case or the size of the legs there. The color of a beetle here affects thermoregulation or hibernation habits there. Through these small alterations forced on insects by their predators, food, or homes, the vast medley of insect life on Earth is produced. Yet through all this detail, the enduring equations of life channel evolution in narrow ways, bountiful, beautiful, and dominant in the phenomenon of life.

  CHAPTER 4

  ALL CREATURES GREAT AND SMALL

  OUR EXCURSION INTO THE physics of ladybugs has revealed much about why creatures look like they do. But ladybugs are only one type of insect, and we might wonder about the rest of life on Earth. Since Darwin’s landmark insights, evolutionary biology has taken a particular interest in whole creatures, from finches to fish. Like ladybugs, do all these creatures great and small show inklings of physics, and can our ladybug provide us with a foundation to learn anything more general about the link between evolutionary biology and physics? Continuing to look at this level of life—how whole organisms are shaped—we might explore how physics informs our understanding of the rest of the planetary zoo.

  The fields of evolutionary biology and physics do not look, at first glance, like natural twins. However, there seems to be no contradiction between the idea that physical rules drive life into narrowly circumscribed forms and the modern view of the role of evolution and biological development in shaping form, just as ladybugs illustrate. Physics explains much about why living things look like they do; evolutionary biology provides much of the explanation about how they become like they are. Together they constitute a complete picture. A beautiful way to reveal the general harmony between physics and evolution at the scale of whole creatures is to continue to explore evolutionary convergence.

  Evolutionary convergence is rampant in the biosphere, far beyond the construction of ladybugs and other insects. Let’s visit one of my favorite animals (I know not why), the mole.

  Wherever they live, moles have a fairly simple objective in life—to burrow underground, build a nest, and have offspring. Their subterranean lifestyle demands some basic biological features, many of which are underpinned by a straightforward equation in physics—pressure is equal to force divided by area:

  P = F/A

  The mole must burrow with enough pressure to push the soil out of the way and make progress in its desire to build a tunnel or a nest. The equation is self-explanatory—the more force you exert per given area of surface, the more pressure you apply. For the mole, the outcome is very simple: either the pressure is greater than the cohesion of the dirt it seeks to burrow through, in which case the material
in front of it can be displaced, or it is not. If, day after day, the pressure persistently is not high enough, then the mole cannot dig or it will get accidentally buried. Either way, those moley genes will not be passed on to offspring. P = F/A matters to a mole.

  There is therefore a powerful and uncompromising selection pressure to push that muck apart to make an underground mole house, and that selection pressure results in certain predictable features. Moles have short but wide stubby front feet that minimize their cross-sectional area, meaning that ultimately the pressure that can be applied in front of the animal is maximized. Those paddle-like paws simultaneously enable the mole to displace large amounts of soil. It is a compromise between large appendages that get in the way and increase your cross-sectional area and ones that are sizable enough to scrape lots of dirt out of the way.

  At the end of the short and powerful appendages are tough nails that enhance the mole’s ability to dislodge and shift the world around it. It even has an extra thumb, an example of a polydactyl paw, which helps it burrow. To improve the effectiveness of its forward movements, the mole is a slender shape—a large, fat, cube-shaped animal will, all things being equal, exert less pressure on the soil, because the force driving it forward is spread over a larger area.

  I have simplified somewhat. The mole’s underground existence begets other adaptations besides. It has a high tolerance to the carbon dioxide that builds up in the subterranean environment as it breathes. In its blood is a type of hemoglobin, the protein that binds oxygen and transports it around the body, with a very high affinity for oxygen, allowing it to subsist at concentrations of carbon dioxide that would become asphyxiating to us. So P = F/A is not everything, but even these other adaptations turn out to be driven by other laws that for the time being I will bypass.