The Equations of Life

by Charles S. Cockell

  However, about a decade later in Austria, a conservation group was training some other birds to follow a microlight. This time, it was not for filming, but part of an attempt to get the northern bald ibis (Geronticus eremita) to return to Europe by retraining them in their migration route. The conservation group hoped that once the birds had been shown the way, they would remember it and their journeys to Europe would be reestablished.

  Attaching data loggers to the birds that monitored not just heart rate but also the position, speed, heading, and every wing flap, the research team from the Royal Veterinary College in the United Kingdom tested Weimerskirch’s earlier findings. They too found that the birds saved energy by flying in formation, the upwash from the birds helping lift each other in their long migratory journey. More than that, it also seemed that they coordinated wing flapping to ensure that the beats were not merely a jumbled mess of flapping appendages, churning and turning the air around them, but instead synchronized to each other, apparently to minimize the possible effects of downwash, which would achieve the opposite of what the birds wanted.

  In the majestic beauty of geese flying in formation in the dying light of a summer evening, we find physical principles at work: the need to stay aloft, the aerodynamic lift of a wing countering the weight of the bird, and formation flying to reduce their energy needs. Birds can follow these imperatives apparently by observing some arrestingly simple rules of engagement.

  In the same way we thought about the ants, we might wonder where the individuality of birds, their personal idiosyncrasies, fit into all this. When I was eighteen, I traveled to the Canadian north to spend a month in an isolated wooden hut on the banks of the Horn River. Equipped with a marsh boat, I and three colleagues who, like me, found their way to this isolated spot through the US Fish and Wildlife Service, aimed to catch ducks and ring them as part of a survey to figure out their migration routes across North America. To the farmers living below, they were vermin, stopping off en masse to feast on corn. By establishing their routes and times of migration more clearly, we might perhaps help mitigate their damage or devise better conservation efforts. Each morning, we would buzz across the marshland with our hovercraft-like contraption to pull the ducks out of the nets we had erected across the landscape. We would then band their legs and set them free.

  One thing that stuck with me, more as a point of amusement than any notable insight, was the individuality of ducks. Pull them out of the trap, and some would peck and the terrified others would freeze utterly. A few would quack uncontrollably, some would scratch, and others would sit there, humming or murmuring softly. Each animal behaved differently and had its own unique personality. I am not sure why that should surprise me. Cats and dogs have personalities, and perhaps it is rather species-ist to assume blandly that they would all look and behave the same.

  I recall the Canada experience here because despite the variation in duck character, which I’m sure applies to starlings, pelicans, and northern bald ibises too, duck behavior as a collective can be predicted, and their movements and the physics behind those movements explained. The imperative a goose feels to minimize its energy expenditure while flying across continents is little affected by how it feels that morning or what its experiences have been. They are nothing more than cold calculations of energy use and aerodynamic lift. So although there may be a tendency to claim that life is highly unpredictable, that the vicissitudes of individuals make living things as far removed from atoms of a gas as they could be, these idiosyncrasies are mere tinsel on the underlying patterns that organize collections of creatures. This same idea inheres from communities of organisms right down to the biochemistry of the subatomic particles they shift around to make energy. Physics trumps individuality.

  The extraordinary self-organizing patterns seen in ants and birds are to be found at lower levels of life’s hierarchy as well. Although the macroscopic manifestations of these regular arrangements catch our eyes as we see them day to day, their principles unite life at all its organizational levels, an example of physical rules threaded through life.

  Nothing could be further removed from the murmurations of starlings than the behavior of the slender filaments that hold together the cells that make up you, me, and the birds. Although this excursion takes me prematurely to the molecular level, a journey we will embark on as we travel down through the hierarchies of life, this brief diversion here will show the importance of self-organization as one unifying theme in the predictability of living things.

  Running through and along the edges of your cells, the units from which your body is composed, are long, thin filaments. Like microscopic scaffolding, these microfilaments are made up of actin, protein tendrils stuck together in a long, winding spiral. They provide structure to the cell.

  This cellular skeleton, the cytoskeleton, seems a thing of exceptional abilities. How does something made from mere spirals of a protein organize and direct so many functions of the cell? As for ants and birds, our instinctive answer is that the cytoskeleton must be controlled. To a certain degree, it is. The molecules from which the cytoskeleton is constructed are programmed by the cell’s DNA. But lurking among these filaments are rules of self-organization, those same shadowy capacities that emerge in the ants and birds when we put things together at high numbers.

  We can see this behavior in a laboratory. Filaments are a little easier to manipulate than flocks of birds, and in a significant paper, Volker Schaller from the Technical University in Munich and colleagues reported some simple lab experiments. They placed actin filaments on glass surfaces with some myosin, a protein that will bind to actin, and with the addition of some chemical energy in the form of ATP (adenosine triphosphate), the myosin will begin to “walk” along the filaments. In the real world, this march of myosin along actin filaments is what drives muscle contraction when you walk about or flex your arm.

  Keep the actin filaments at a low concentration, and little happens. They move around haphazardly across the surface, randomly shunting this way and that. Now increase the concentration so that the actin filaments are bundled together, and an impressive transformation comes over them. They show coordinated movements. Giant waves and vortices of filaments begin to self-organize spontaneously as the movement of every filament influences those around it. Swirling and gyrating, these structures are an elaborate mixture of both short- and long-distance interactions between the filaments. Even the computer models made by the researchers to simulate the filaments could not completely replicate the mingled mosaic of the bundles.

  Other parts of the cytoskeleton show these same extraordinary powers. Tim Sanchez from the University of Brandeis, Massachusetts, and his colleagues played around with filaments of the protein tubulin. About four times the diameter of actin, or about twenty-five nanometers across, tubulin filaments also form part of the cell’s skeletal structure, providing minute motorways along which molecules, essential to the cell, are transported. They organize and direct the movements of chromosomes, that is, packages of DNA, when the cell divides. When the scientists added the tubulin filaments to a dish with kinesin, a molecule that, like myosin, can crawl its way along the filaments, they too observed patterns of self-organization. Long, streaming tapestries of filaments buckled and folded, driven by the active behavior of their marching molecules.

  In these astounding experiments, we see how even at the molecular level, biological entities that affect one another can spontaneously form ordered structures. Gone is the view that anything organized in life must be overseen and that this supervision, when stopped, leads to a cessation of living processes.

  The rules and principles that come to the fore in a cellular filament, an ant, a bird, or even shoals of fish or migrations of wildebeests show the power of physical processes to direct and shape life at the scale of whole populations. There are common themes: the role of a critical number in transforming one type of behavior to another and the place of small, random fluctuations in dramatically driving a c
ollection of living things into a new state. Of course, we find variety in the other rules that impose themselves on these collectives. In the air, aerodynamics, irrelevant to an ant, takes center stage in imprinting itself on a family of geese. But barring the obvious influences of other physical laws that life must also observe in different environments, is it so mad to think of the murmurations of molecules in a cell and starlings in the sky as similar?

  It is easy to think that much of the complex behavior in the natural world—the murmurations of starlings, the underground dominions of ants, and the swarming of cellular filaments—is the product of something far from physics. Step out one evening, and watch the swooping and gentle swaying of thousands of starlings making an aurora-like pattern through the sky. The sight is mesmerizing, a show of such unpredictability and beauty that anyone would be forgiven for thinking this was some gift of life on Earth, something that stands above physics, something rooted in a higher order of organization. Yet within these mass organizations, there is simple order, predictability, and physical principles. Yes, there is room for chaotic behavior. The chance movement of a starling here or the collision of several there might well send the flock flowing in a new direction. It is the propagation of these small chance alterations to the whole flock that gives the whole edifice that lure of something more than physics.

  It we were to travel to an alien world and observe ant-like creatures—simple things that communicate using chemicals they pass to one another—building a nest, we would see the same feedback processes giving rise to ordered communities. So too with larger creatures that may have taken to the skies of that distant planet. We might see variations between species in how they fly, for sure, but at their heart, there are physical principles that guide them in their flight, equations that order and shape societies and collections of living things. The self-organization of life shows stunning diversity entrenched in fundamental rules we might reasonably conclude are likely to be universal.

  CHAPTER 3

  THE PHYSICS OF THE LADYBUG

  IN THE DEPARTMENT IN which I work, we offer undergraduates an option called Team Projects. In essence, you find them something interesting to work on, and they spend a semester digging around and, hopefully, in the process, learn something new.

  Moving down one level of the hierarchy from populations of creatures to individuals, I thought that this general question of which laws and equations fashion life at the level of a single organism had some merit as a project. In the winter of 2016, I set my group a simple objective: spend a few months investigating the physics of a ladybug. Metaphorically take the ladybug apart bit by bit, consider every facet of its life: its airborne life in the Meadows, its perambulations on leaves, its breathing, the strength of its protective wing cases. Write down the relevant laws and equations with which that little insect is constructed and that play a dominant role in defining its day-to-day life. Tell me everything I need to know about physics to apprehend how a ladybug works. This was no small undertaking, and I was aware before they arrived in my office for the first meeting that the task I had set them probably covered an enormity of knowledge. Aerodynamics, diffusion, locomotion, thermal inertia—it was easy to casually list a mélange of physics that had at least some part to play. As I expected, the project took them on a fascinating journey into the many ways in which physical principles shape life.

  A good place to start is its legs. Simple though they may look, packed into those appendages is a fascinating array of physics. Because each little leg of the insect has three joints, it can move them around in all sorts of interesting contortions. With so many degrees of freedom, the ladybug has multiple options for how it might just put its foot down. In its head, a computer is at work deciding among those possibilities. Wind speed, surface irregularities, bits of leaves, and no end of other fine details probably feed into the decision on how to move each of those six very adaptable legs.

  At each step, the ladybug must ensure that it can hold on, such as when climbing vertically up the hand of a tourist, because otherwise, the insect would fall off. The ladybug, like spiders, lizards, and other beetles, has on each foot a pad covered in tiny hairs, or setae. These hairs play a role in attaching the creature to a surface. The problem the ladybug must solve is to ensure a good connection between the pads on its feet and the surface. To make the connection hold fast, it uses a very thin layer of liquid. Secrete a film of fluid under your foot, and at the small scale of insects, the fluid produces a huge adhesion force by capillary action and the viscosity of the fluid. By producing a liquid in this way, the ladybug fills in irregularities on the surface, essentially making the surface behave as if it were flat. The layer of fluid is thin and keeps the friction high enough to prevent the insect from slipping down the vertical face.

  By combining all this newfound knowledge about the ladybug’s foot and the behavior of fluids, we can even write an equation for the total adhesive force that the ladybug leg can produce. With it, we can predict its ability to master and own the terrains that constitute its little world. That memorable equation is:

  F(adhesion) = 2πγR + πγ(2cosθ / h − 1 / R)R2 + dh / dt 3πη R4 / 2h3

  where F(adhesion) is the adhesive force. γ is the surface tension of the fluid under the foot. R is the radius of the foot, considered a simple disk, although this can be made more complicated and realistic in shape. The value η is the viscosity of the fluid under the foot, h is the distance between the foot and the surface, that is, the thickness of the liquid layer, and t is time. The first term is the surface tension, the second one the Laplace pressure, and the last term the viscous forces.

  But in walking across its varied and unpredictable terrain, the ladybug has a little problem. The legs must be rigid or the insect will collapse, but we want malleable feet that can move flexibly across those surface irregularities. To achieve this, the leg contains a protein called resilin, an elastic biological polymer that enables fleas to jump and other insects to do contortions when they need to. From the top of the ladybug leg to its feet, a gradient of resilin has been discovered, with more of the substance nearer the feet, where elasticity is needed, and less toward the top of the leg, where rigidity is preferred. The top of the leg contains more of the tough chitin that makes up the rest of the insect’s exoskeleton, increasing the Young’s modulus, a measure of the stiffness of a material. Here we find that the physics of material properties have evolved to suit the needs of the perambulatory part of the insect’s lifestyle.

  The forces that attach the foot to the ground are impressive, but we also must be able to pull the legs back off the surface again. Otherwise, our insect is rigidly transfixed to one location, pulling and tugging in vain against the laws of physics. Simple equations can again define the energy needed (W) to pull the feet away from these adhesive forces:

  W = F2NAlg(θ) / 2πr2E

  where NA is the density of setae on the feet, l is the length of the setae, E is the Young’s modulus of the setae, a measure of their tendency to deform, and r is the radius of the seta. The term g(θ) describes the angle of the setae to the surface and is given by g(θ) = sinθ[4/3(l/r)2cos2θ + sin2θ].

  Now the story is not over, because the insect, to really hold on, wants to have lots of those hairs, but not too many. If it has too many, they will all get stuck together. They must be far enough away from one another that the forces of attraction between them are not too great, but close enough to maximize the number on the feet. That theoretical maximum density of the setae is given by:

  maximum NA ≤ 9π2r8E2/64F2l6

  where F is the adhesive force of one seta.

  To really pack these setae in, the insect can evolve some modifications. If the setae are sticky on only one side, the chances that they will get stuck together can be minimized. Protrusions and nodules on the foot further keep them separated.

  The insect must evolve to ensure that all the solutions to these equations are optimized in a way so that the feet can hold on tight but
be pulled away as the ladybug moves across the ground. The setae, those hairs on its feet, do exactly this, a superb and exquisite example of the honing of biological form by simple physical principles that can be expressed in equations. It is no surprise, then, that hairy pads on insect feet have evolved time and time again completely independently. Those physical limitations are unyielding in their grip, and yet the solutions to them in the living form are limited, bounded by some straightforward outcomes in the insect. Convergent evolution is here exposed in all its glory as the mere channeling of biological form into similar outcomes determined by physical principles. All insects converge on some relationships, complex and fascinating in their totality and yet reducible to understandable simplicity.

  It is not immediately obvious to you and me that a small insect would use these cunning means to walk across surfaces, up walls and even upside down on leaves and ceilings. Try covering your feet in a thin layer of water and then walking vertically up the outside wall of your house, and you will soon be sorely disappointed. At human scale, the forces of gravity dominate, yanking you remorselessly off the side of the building before you have had time to take a single step forward. For the ladybug, a thing about seventy-five thousand times less massive than you and me, molecular forces dominate. Surface tension, capillary action, van der Waals forces, and others. All these forces that yank and pull molecules together shape a ladybug’s universe and lead inevitably to the little tricks of using the strength of these molecular forces to stick to walls. Not that gravity is an irrelevance to ladybugs. Fall off a leaf or a wall, and they will fall as surely as you and I, although slower and with less damage. Gravity is inescapable. Yet at the scale of most insects, it takes back stage as molecular forces come to the fore.