The Equations of Life

by Charles S. Cockell

  As some doors open, others close. The ladybug may well use thin layers of a liquid to climb a wall, but those same molecular forces, those identical laws, bring disadvantages too. When you and I and other large animals need to wash, we enjoy a shower, a bath, or the nearest watering hole or pond. Climb out, and most of that water drains away under gravity, with just a thin layer and a few droplets of water remaining. A dog will shake them off and you will use a towel, and if neither of these possibilities are available to you, well, you can just wait for the water to evaporate.

  Ladybugs need be more careful, though. Even a single drop of water will relentlessly cling on, those surface-tension forces too much of a match even for its strong little legs as they attempt to push it away. For something smaller, like an ant, the insect may be completely consumed by such a droplet, trapped inside, the molecular attraction between the water molecules on the surface of the droplet imprisoning it in a water cage, a surface-tension prison. For these reasons, many insects, particularly small ones, dry-clean themselves, scraping dust and dirt from their bodies with their tough legs while avoiding the ensnaring promise of water.

  Some of these observations—that humans cannot walk vertically up walls but flies and ladybugs can—may seem so obvious and taken for granted in our everyday experience that they are hardly worth remarking on. But they define two worlds: the physics operating in the ladybug’s domain and the physics you and I must observe. The same physics, for sure, but different forces dominating in each realm. Yet this physics explains so much about the form and shape of creatures that inhabit these different scales. Nothing is trivial about the design of a ladybug’s legs and the way it must wash itself or the limits of locomotion in a human or a gazelle. However, we can unravel these limits and possibilities not by exploring contingency, in other words, the historical quirks of life’s evolution that could be very different if evolution were rerun, but only by investigating the fundamental physics that rules how these organisms operate.

  Ladybugs can do much more than walk across obstacles.

  Conspicuous among the ladybug’s collage of skills are minuscule wings, which are packed under their wing cases, the elytra. Gossamer thin, a mere half micron thick, they are cleverly folded in thirds, protected by the shiny, hard wing cases. If the insect is distracted by a predator or an Edinburgh tourist, in under half a second, those wings unfold into their articulated state and can carry our little insect high into the sky, if it so wishes, over a kilometer up, traveling at speeds of up to sixty kilometers an hour.

  The ladybug wing is no mere simple fixed structure, like an airplane wing. This is a flapping contraption that, fully expanded, is four times the length of its body. Attached to the body by hinges, the wing swings backward and forward in the horizontal plane, the muscles in the body and the veins in the wing acting as levers to push the wing through the air, generating lift from this airfoil as it launches itself skyward. Running along the front of the wing is a reinforced vein that provides strength and stability during collisions with raindrops and other unexpected objects.

  Although the ladybug wing has great flexibility, the muscles at its hinges and the veins within the wing allow the wings to change shape and twist in response to changing air patterns and wind. However, on the face of it, even these impressive structures seem unlikely to hold such a rotund little ball aloft.

  Trying to work out how insects fly has consumed the efforts of many entomologists. Rapid advances in computer modeling, helped along by high-speed photography of insects in motion, have allowed researchers to explore the exquisite refinements that allow insects to exploit every subtlety and possibility that emerges from the physics of aerodynamics.

  Those wings can do all sorts of clever maneuvers to gain lift. Within their apparently chaotic and mad swishing are intricate coordinated changes, one with the mysterious name of clap and fling. As the wings are pushed into their backstroke, they are clapped together, forcing air out from between them, thrusting the ladybug forward. The wings are flung apart as they begin their front stroke, and the air that rushes in to fill the gap enhances circulation over the wing surface, improving lift. Clap and fling has its problems, not the least the tendency to damage the wing with such violent and abrupt adjustments. Instead, increasing the wingspan or the frequency of wing beats can augment the lift the insect can achieve.

  Not surprisingly, with this knowledge in hand, we can reduce the wings of the ladybugs to equations, and once we do that, we can calculate the lift and power generated in these appendages. By considering the forces around the wing, the angular velocity, and the moments of inertia, we can reduce the flight of the ladybug to a number as simple as about thirty watts per kilogram of power produced by those appendages.

  As the creature touches down, it must now cover up and protect those wings lest they are damaged. Inward they flip, beneath the tough wing cases, safe from harm. The delicate wings are now hidden beneath the two shells, which are cleverly evolved to slot together along the middle in a tongue-and-groove system, similar to the slating in floorboards.

  Nature needs a material to build the insect from, including the wing cases. Its solution, chitin, is a robust sugar material about ten times less strong than steel, but about ten times stronger than the material from which hair is made: keratin (a material made of protein). Unlike the legendary strength of spider’s silk, we do not need super durability here; we just need something that will do the job of protecting the wings and the rest of the ladybug’s vulnerable parts, including its head.

  Throughout the ladybug’s armor, chitin is woven. And sometimes, such as in the antennae and legs, we find resilin mixed in for some flexibility.

  As our ladybug bashes and crashes its way through life, the successful ones that get through to reproductive age without having their wings written off will be those that can withstand all those collisions. The severity of collisions can be calculated with equations such as the Head Injury Criterion (HIC), a practical equation used to work out how effective your bicycle helmet is at protecting your own head:

  where t2 and t1 are times, and a(t) is the acceleration in the collision.

  Albeit more complex in a shifting, dodging ladybug, the strength of the chitin and the thickness of the wing cases must withstand the accelerations of the collision to keep the wings intact. Ultimately, those forces are fashioned by physical properties.

  Now, chitin is a translucent material, and the wing cases, as well as covering the wings, must perform some other functions. Well, yes, attracting a mate is important, as is warding off predators. Impregnated onto the wing cases’ otherwise lackluster colors are reds, blacks, and yellows, themselves a mere part of the palette of colors from iridescent reds, greens, and golds found across the insect world.

  Somehow, we must color those cases in a way that makes sense. Random sheens and splashes might make a nice scene for the artistic observer, but on our ladybug, as with other insects, we want patterns, dots, for example, in particular places. Patterns in animals, including dots on insects, perform numerous roles in camouflage, discouraging predators and attracting mates.

  It was Alan Turing, a physicist and a computer genius, who first proposed how this coloring might be done. Imagine a cell making a colorful pigment in our newly begat ladybug. Slightly further away, another cell might be producing an inhibitor that prevents that pigment from being made. As these pigments and their inhibitors diffuse through the cells of the developing insect, we end up with gradients. At just the right places, the pigments can be produced and maybe a little black spot will appear. In other places, the pigments are suppressed and maybe the insect goes red. By varying the range over which these different activators and inhibitors can work, all manner of patterns can be produced in the final product. Through these simple rules, we can make ladybug dots, leopard spots, fish stripes, and Dalmatian dogs. And these gradients can be written in equations, for example:

  δa/δt = F(a,h) − daa + DaΔa

  δh/δt
= G(a,h) − dhh + DhΔh

  where t is time and a and h are the concentration of activator and inhibitor, respectively. The first term describes the production of the chemical, the second term the loss due to degradation, and the last term the diffusion of the substance. There are many variations on these equations.

  Turing patterns are common in nature, and of course, in reality they are much more complex than mere activators and inhibitors spreading out from some cells. Multiple chemicals play roles, and other metabolic interactions add complexity to the final design. In the developing embryo of many animals, genetic control and regulation—rather than simple gradients of chemicals—take center stage. But the essential idea that Turing put forth about how the interacting gradients of chemicals produce patterns provides a clever basis for understanding how the ladybug got its spots. Although Turing’s simple model has frequently been superseded by more-complex knowledge of genetics, his work was a gallant effort to explain complex biological phenomena with simple physical principles and was one of the early attempts to unite physicists and biologists.

  Having warded off a predator or maybe spotted a mate with its dotty body, the ladybug must take to the air. To launch itself from the awakening tourist in the Meadows, the insect must be warm enough to flap its wings. The insect is an ectotherm, gaining most of its warmth from the environment and not from its own metabolism, unlike you and me—we are members of the endotherm club, warm-blooded animals that regulate their own temperature. The ladybug must rely on its surroundings to keep warm. That is a problem for our friend because below a temperature of 13°C, it will seize up from the cold. Thus, an essential part of a ladybug’s life is to bask in the sun, an activity I’m sure many of us would like to justify during our working hours as a physiological requirement.

  By sitting with its wing cases facing the sun, the ladybug can absorb solar radiation through its chitinous covering and therefore raise its temperature. Some of that radiation will be reflected from the shell’s surface. Those bright, attractive colors come with a cost—they are much more likely to reflect the very radiation needed to keep warm. Indeed, dark ladybugs are known to be more efficient at collecting valuable solar energy than are light-colored ones.

  In you and me, some of our heat is carried away in evaporation. The water in our sweat carries away the energy as it is vaporized from the surface of our skin. The very purpose of sweating is to prevent overheating. Luckily, for the ladybug, little evaporation can occur under its thick wing cases, so it is spared an unnecessary loss of heat. However, some heat will be conducted away from its body through the air.

  Carefully and systematically, we can add, as in a ledger, all the sources and losses of heat in a ladybug. We can think about the sun’s energy coming in and how much is reflected away. We can consider the small amount of warmth coming from the insect’s metabolism and making its way from the inside out. We might factor in that thin layer of air between the body and the wing case—a layer that acts as an insulator, slowing heat loss. We could think about the air blowing over the creature, which will carry off some of the heat.

  When we add, subtract, divide, and multiply the various parts of this tapestry of heat fluxing back and forth, we can arrive at an equation for the temperature of a ladybug (Tb). The little equation captures the difference between life and death for the tiny beast:

  Tb = Tr + tQsRb(Rr − Rb)/kRr

  where Tr is the temperature of the wing cases, t is the transmittance of the wing case, Qs is the incident energy on the ladybug, Rb is the radius of the ladybug body, Rr is the radius of its shell, and k is the thermal conductivity of the air between the body and the wing case.

  As the sun on the Meadows sets in the evening twilight or the cold chill of winter descends on this beautiful part of the city, sending townsfolk and tourists into homes and cafés, the little ladybug has no such respite. Tb drops into the red. But it has other tricks. Almost imperceptible to you and me, ladybugs can shiver. In so doing, they burn up more energy and produce heat, offsetting the loss of heat through their shells and keeping their temperature high enough to move around.

  However, soon, the shivers of a ladybug are no match for the winter and it must go into hibernation. For up to nine months, the temperatures of our Scottish city threaten to force this minuscule critter into torpor. Off it heads, joining forces with friends across the city to find some old leaves, soil, moss, or other insulating material, and together they will huddle, surrendering to the cold, allowing their bodies to cease activity. Walking and flying are now difficult, and within their snuggled-up family of friends, they are protected from predators that would otherwise take advantage of their sluggish form.

  In the dead of winter, at the apex of this long sleep, temperatures may drop below freezing and now the ladybugs have another problem. The cold temperatures that made them listless are a mere picnic against the possibility of temperatures that threaten to freeze them solid, raising the specter of ice crystals, sharp and nasty, crashing through cell membranes and causing irreparable damage.

  The beetles must now call into action another equation:

  ΔTf = Kfm

  Add salts or other antifreeze substances to the body, and the freezing point of water can be lowered. The drop in freezing point (ΔTf) is established simply by the product of the freezing-point-depression constant of a chemical (Kf) and m, the molality of a solution.

  Add a gram of the chemical glycerol to two grams of water, and the solution now freezes at about −10°C. By synthesizing these sorts of compounds in their blood, our ladybugs can withstand the precipitous drop in temperatures, preventing ice crystals from forming, and come out on the other side of winter little worse for wear.

  In these equations, we have a marvelous illustration of how life must deal with the unrelenting and ineluctable principles of physics, but also how, through evolutionary inventions, natural phenomena described by different equations are incorporated into living things to their advantage. The temperature of the ladybug is a concoction of interacting terms settled in final mathematical form by the heat terms that come and go from its body. In the darkening sky, the outcome for the ladybug is a foregone conclusion. The creature will begin to chill. In some part, it can take action against this heat loss by shivering. The simple process of burning more food to make heat will offset some of that cooling. The innovative bug can do more. Legs and wings, the equations of locomotion and aerodynamics, allow it to take action that is more drastic. It may crawl into leaves or take to the air in search of a place to hide. Buried in leaf litter, it has succumbed and accepted the laws of physics, capitulated to the inevitable lethargy that must overcome it. But evolution is no passive onlooker.

  Experiments, driven by mutation, explore available physical principles for possible solutions that will produce organisms capable of more successfully reproducing. They define and fashion future populations that are more robust and successful. In the history of insect biology, the production of compounds such as glycerol to push the freezing point down was one such innovation that employed the freezing-point-depression equation to expand the temperature range of ladybugs, to push their lower limits downward and into the realms of the otherwise-frozen world.

  In this equation for freezing-point depression, we have an exquisite example of how variant offspring in a population will explore new physical relationships hitherto unrepresented in the living forms of their ancestors. Physical principles stumbled on by new variants that enhance their chances of reproducing will be selected for and passed to new generations. In this way, life explores the universe of these principles, which are given expression in equations, and incorporates them within the living form.

  Simple physical relationships have a role to play not merely in movement and temperature regulation, but also in the gases that animals need to survive. The ladybug’s ability to move, heat itself, and reproduce depends crucially on its capacity to get that most vital of gases needed by most animals: oxygen. In you and me,
the lungs pumping air in and out provide us with a continuous source of the gas. Even the fish, in their watery world, flow water through their gills, extracting the oxygen essential for gathering energy; like us, they use this gas to burn organic food.

  Like all insects, the ladybug does not have lungs. Instead, cylinders run through its body like a network of life-giving tubes. These trachea, which themselves connect to tinier tracheoles, traverse its insides. Air travels through these hollow lengths, delivering oxygen to the insect’s innards. The network is so comprehensive that the oxygen can be delivered to within micrometers of where it is needed. However, this transport depends on the passive process of diffusion, the movement of atoms or molecules from a place of high concentration to one of lower concentration.

  The relative simplicity of insect breathing lends itself to uncomplicated equations that deal with this process of diffusion. The time it takes (t) for a molecule to travel a given distance (x) can be worked out using the equation:

  t = x2 / 2D

  where D is the diffusion coefficient, a measure of how quickly the molecule can move in a given medium.

  This simple equation tells us that the time it takes for a gas to travel a given distance is related to the distance it travels squared. Double the distance you want your oxygen to go, and you must wait not just twice as long, but four times the time. And it just gets worse as the numbers are increased. That is why insects are ultimately limited by diffusion.

  The gases going into the ladybug diffuse at a rate that depends on the concentrations inside the body. Adolf Fick, a physiologist, was a pioneer in getting to grips with diffusion. In the nineteenth century, he worked out a great deal of what we understand, including his well-known Fick’s first law, which tells us the amount of a gas, like oxygen, that will flow at any given time. His equation is as follows: