For example, for the ant species Messor sancta: Quantified and discussed in Buhl J, Gautrais J, Deneubourg JL, Theraulaz G. (2004) Nest excavation in ants: Group size effects on the size and structure of tunnelling networks. Naturwissenschaften 91, 602–606; and Buhl J, Deneubourg JL, Grimal A, Theraulaz G. (2005) Self-organised digging activity in ant colonies. Behavioral Ecology and Sociobiology 58, 9–17.
Perhaps best known: Willmer P. (2009) Environmental Physiology of Animals. Wiley-Blackwell, Chichester.
The exact physical underpinnings: However, some excellent papers explore the basis of these laws and themselves are usually based on physical models. One example is West GB, Brown JH, Enquist BJ. (1997) A general model of allometric scaling laws in biology. Science 276, 122–126, which proposes that the basis of many of the physiological power laws in life are rooted in the need to transport materials through linear networks that then branch out to supply all parts of the organism. They use this supposition to develop a model that predicts a variety of structural features of living forms, from plants to insects and other animals.
Many fixed relationships: For a good account of these ideas and the past literature on allometric power laws and their physical basis, I very much recommend West GB. (2017) Scale: The Universal Laws of Life and Death in Organisms, Cities and Companies. Weidenfeld & Nicolson, London.
Attempting to reduce: The classic paper that proposed a model of simple particle motion that would make the transition from disordered to ordered behavior using some basic rules was Vicsek T et al. (1995) Novel type of phase transition in a system of self-driven particles. Physical Review Letters 75, 1226–1229, and was applied to biological systems in Toner J, Tu Y. (1995) Long-range order in a two-dimensional dynamical model: How birds fly together. Physical Review Letters 75, 4326–4329. The transitions that give rise to this sort of self-organized behavior were further elaborated on by Grégoire G, Chaté H. (2004) Onset of collective and cohesive motion. Physical Review Letters 92, 025702. There are, of course, many other papers exploring the physics of self-organization applied to both nonliving and biological systems.
This field strives: Self-organization can be observed at many scales, not just in biology, but in all physical systems, including weather systems: Whitesides GM, Grzybowski B. (2002) Self-assembly at all scales. Science 295, 2418–2421. For a nice short summary of how systems far from equilibrium are relevant to biology, see Ornes S. (2017) How nonequilibrium thermodynamics speaks to the mystery of life. Proceedings of the National Academy of Sciences 114, 423–424. His missive also contains some other relevant citations on nonequilibrium systems in biology.
Like other aspects: This formulation has been shown to predict behaviors in, for example, the Argentine ant (Iridomyrmex humilis): Deneubourg JL, Aron S, Goss S, Pasteels JM. (1990) The self-organizing exploratory pattern of the Argentine ant. Journal of Insect Behaviour 3, 159–168.
Like a miniature computer: A discussion of the differences between ants and molecules, as well as principles of interactions between ants is Detrain C, Deneubourg JL. (2006) Self-organized structures in a superorganism: Do ants “behave” like molecules? Physics of Life Reviews 3, 162–187.
The reactions complicate: Models can be made that take into account how memory, for example in bird flocks and schooling fish, affects subsequent group behavior. Random fluctuations that cause large-scale gross changes in animal groups can also be investigated. These attributes add complexity to models, but at their core, the models are still constructed on the basic principles of how the component organisms interact: Couzin ID et al. (2002) Collective memory and spatial sorting in animal groups. Journal of Theoretical Biology 218, 1–11.
Hampering efforts: A paper that reviews this history as well as some of the theories on bird flocking is Bajec IL, Heppner FH. (2009) Organized flight in birds. Animal Behaviour 78, 777–789.
At the core: A detailed paper looking at some of these assumptions is Chazella B. (2014) The convergence of bird flocking. Journal of the ACM 61, article 21. Also see Barberis L, Peruani F. (2016) Large-scale patterns in a minimal cognitive flocking model: Incidental leaders, nematic patterns, and aggregates. Physical Review Letters 117, 248001.
Yet rules applied: A model that examines how vertebrates can organize, find new food sources, or navigate to new places with only a few individuals in the group with access to the necessary information is Couzin ID, Krause J, Franks NR, Levin SA. (2005) Effective leadership and decision-making in animal groups on the move. Nature 433, 513–516.
The infant state: In the case of bird flocking, a forceful paper that examines their collective behavior as a physical process (with a wonderful title that only a physicist can muster) is Cavagna A, Giardina I. (2014) Bird flocks as condensed matter. Annual Reviews of Condensed Matter Physics 5, 183–207.
However, evidence: This idea was first elaborated by Wynne-Edwards VC. (1962) Animal Dispersion in Relation to Social Behaviour. Oliver & Boyd, Edinburgh. One of the idea’s problems is that it suggests a form of bird behavior directed to the good of the group (a theory that was at the forefront of Wynn-Edwards’s writing), a form of self-censorship on breeding behavior. A bird that took part in the census but then cheated by having a few more offspring than other birds would quickly spread in the population, potentially vitiating the whole strategy. Furthermore, clutch (egg number) has not been shown to regulate in response to the numbers of birds in a murmuration, making the idea difficult to test empirically.
Weimerskirch saw: Weimerskirch H et al. (2001) Energy saving in flight formation. Nature 413, 697–698.
time, it was not for filming: Portugal SJ et al. (2014) Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature 505, 399–402.
Filaments are a little easier: Schaller V et al. (2010) Polar patterns of driven filaments. Nature 467, 73–77.
Tim Sanchez: Sanchez T et al. (2012) Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–435.
About four times: That’s 0.000000025 meters.
The rules and principles: A comprehensive text that synthesizes information on self-organization in diverse organisms, including ants, bees, fish, and beetles is Camazine S et al. (2003) Self-Organization in Biological Systems. Princeton University Press, Princeton, NJ. The book also discusses the general reasons and principles behind self-organization, including its ability to enhance the formation of stable structures. The book contains an wide-ranging set of references to various works covering self-organization. A highly comprehensive study of self-organization is to be found in Kauffman S. (1993) The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, Oxford, which is beautifully summarized in his popular science book: Kauffman S. (1996) At Home in the Universe: The Search for Laws of Self-Organization and Complexity. Oxford University Press, Oxford. And see the work by Ao: for example, Ao P. (2005). Laws of Darwinian evolutionary theory, Physics of Life Reviews 2, 117–156.
It is easy to think: Despite our desire to consider ourselves separate from “mere” natural processes, human populations are amenable to modeling as well, such as this fascinating study of city size and shape shows: Bettencourt LMA. (2013) The origins of scaling in cities. Science 340, 1438–1441.
The self-organization of life: Although I have focused on aspects of self-organization to illustrate physical principles at work, many other areas of physics and mathematics may be applied to understanding the operation of groups of organisms. One major contribution has been the biological and evolutionary application of game theory, which seeks to understand the evolutionary benefits of different choices taken by organisms—and for which there is a vast amount of literature. See Maynard Smith J, Price GR. (1973) The logic of animal conflict. Nature 246, 15–18. A book looking at the application of game theory to biology is Reeve HK, Dugatkin LE. (1998) Game Theory and Animal Behaviour. Oxford University Press, Oxford. A thoroughgoing technical text that explores these evolutionary inte
ractions and other aspects of the application of mathematical theory to evolution is Nowak MA. (2006) Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press of Harvard University Press, Cambridge, MA. I discovered his book after the decision on the title of my book was long since committed. However, I feel no proprietary concern. The “equations of life,” I think, is a natural phrase that succinctly captures the manifestation of life in physical principles given expression in mathematical relationships that can be written in equations. Moreover, The Equations of Life is the title of a novel by Simon Morden. Set in a postnuclear apocalypse, the book’s plot involves a link between physics and evolutionary biology—a link that is perhaps best avoided.
CHAPTER 3
In the winter of 2016: And I’d like to thank the members of this group for their work, on which this chapter is based: Julius Schwartz, Hamish Olson, Danielle Hendley, Emma Stam, Rodger Watt, and Laura McLeod. They did a very fine job and wrote a splendid report.
With so many degrees: Cruse H, Durr V, Schmitz J. (2007) Insect walking is based on a decentralized architecture revealing a simple and robust controller. Philosophical Transactions of the Royal Society A 365, 221–250.
Wind speed: The physics and mathematics of insect legs and locomotion is a fertile area of research, driven by an interest in creating legged robots that will more effectively navigate terrain. See, for example, Ritzmann RE, Quinn RD, Fischer MS. (2004) Convergent evolution and locomotion through complex terrain by insects, vertebrates and robots. Arthropod Structure and Development 33, 361–379.
The ladybug, like spiders: Some insects have smooth pads.
With it, we can predict: The development of these models can be found in a number of papers, such as, Zhou Y, Robinson A, Steiner U, Federle W. (2014) Insect adhesion on rough surfaces: Analysis of adhesive contact of smooth and hairy pads on transparent microstructured substrates. Journal of the Royal Society Interface 11, 20140499. The equation shown in this chapter can be found in Dirks JH. (2014) Physical principles of fluid-mediated insect attachment—shouldn’t insects slip? Beilstein Journal of Nanotechnology 5, 1160–1166.
The first term is the surface tension: The Laplace pressure is the pressure difference between the inside and the outside of a curved surface that forms a boundary between a gas and a liquid region. This pressure difference is caused by the surface tension of the interface between the two regions.
To achieve this, the leg: All biological structures, particularly appendages, are evolved to have factors of safety (the ratio of the stress that causes failure to the maximum stresses experienced). This is not to say that evolution has engineering foresight, but these factors are likely to minimize the probability of failure sufficiently not to significantly impinge on survival. For a comprehensive and interesting discussion, see Alexander RMN. (1981) Factors of safety in the structure of animals. Science Progress 67, 109–130, which touches on the field of biomechanics, yet another field that brings together physics and biology, especially at the level of the whole organism, although Alexander also considers seeds and other biological structures.
From the top: Peisker H, Michels J, Gorb SN. (2013) Evidence for a material gradient in the adhesive tarsal setae of the ladybird beetle Coccinella septempunctata. Nature Communications 4, 1661.
equations: Federle W. (2006) Why are so many adhesive pads hairy? Journal of Experimental Biology 209, 2611–2621.
Yet at the scale: I do not exaggerate when I say that one of my favorite scientific papers, which explores this topic exactly, is Went FW. (1968) The size of man. American Scientist 56, 400–413. Went draws our attention to the different physics principles operating at the small and large scales and their biological implications, discussing the forces of gravity at the large scale and molecular forces that dominate at the small scale. Particularly entertaining is his thought experiment on the ant preparing to go to work. If you want his explanation on why the ant can’t kiss his wife good-bye or have a sneaky cigarette on the way to work, you’ll have to read the paper yourself. Another earlier paper in the same vein is Haldane JBS. (1926) On being the right size. Harper’s Magazine 152, 424–427. Here Haldane pays particular attention to insects and argues that the size of an organism mandates what sorts of systems it must have to exist. Implicitly, he is recognizing that physical size pulls into play physical principles that ultimately decide how a living thing is constructed, not mere contingency.
However, we can unravel: When I use the term contingency throughout this book, I mean an evolutionary development that was a quirk of history, a chance path that could have been very different. Stephen Jay Gould and other scientists who believe that contingency is an important driver in evolution theorize that if the tape of evolution were rerun, a completely different set of paths might be followed. Note some subtlety here. Contingency could refer to two similar or identical evolutionary experiments changed by chance mutations on their course, or it could refer to small, different historical conditions, such as at the start of an evolutionary experiment, radically changing the outcome of evolution. Usually in this book, I am referring generally to both possibilities.
If the insect is distracted: Jeffries DL et al. (2013) Characteristics and drivers of high-altitude ladybird flight: Insights from vertical-looking entomological radar. PLoS One 8, e82278.
Rapid advances: I have deliberately not written equations for insect flight here since the equation of lift, which I use later, is too simple to capture the complexity of insect aerodynamics. To list one equation would also force me to list many more to even do the subject cursory justice. However, for details on the phenomenon, I refer the reader to the following papers, although there are many more: Dickinson MH, Lehmann F-O Sane SP. (1999) Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954–1960; Sane SP. (2003) The aerodynamics of insect flight. Journal of Experimental Biology 206, 4191–4208; Lehmann F-O. (2004) The mechanisms of lift enhancement in insect flight. Naturwissenschaften 91, 101–122; Lehmann F-O, Sane SP, Dickinson M. (2005) The aerodynamic effects of wing–wing interaction in flapping insect wings. Journal of Experimental Biology 208, 3075–3092.
Its solution, chitin: Mir VC et al. (2008) Direct compression properties of chitin and chitosan. European Journal of Pharmaceutics and Biopharmaceutics 69, 964–968.
The severity of collisions: Henn H-W. (1998) Crash tests and the Head Injury Criterion. Teaching Mathematics and Its Applications 17, 162–170.
Well, yes, attracting a mate: The formation of colors in the natural world, such as in the wings of butterflies, is an exquisitely developed area of physics covering photonics and other fields. Just one such paper is Kinoshita S, Yoshioka S, Miyazaki J. (2008) Physics of structural colors. Reports on Progress in Physics 71, 076401.
It was Alan Turing: Turing AM. (1952) The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society Series B 237, 37–72.
By varying the range: A description of the use of the Turing model for explaining and predicting patterns has even been applied to ladybugs themselves: Liaw SS, Yang CC, Liu RT, Hong JT. (2001) Turing model for the patterns of lady beetles. Physical Review E 64, 041909.
But the essential idea: Rudyard Kipling’s writing preceded Turing’s paper, but if Kipling had been born later, he might have collaborated with Turing in writing his Just So story “How the Leopard Got His Spots.”
Indeed, dark ladybugs: Two papers investigating this effect are Brakefield PM, Willmer PG. (1985) The basis of thermal melanism in the ladybird Adalia bipunctata: Differences in reflectance and thermal properties between the morphs. Heredity 54, 9–14; and De Jong PW, Gussekloo SWS, Brakefield PM. (1996) Differences in thermal balance, body temperature and activity between non-melanic and melanic two-spot ladybird beetles (Adalia bipunctata) under controlled conditions. Journal of Experimental Biology 199, 2655–2666. For observations of the same effects in dark- and light-colored beetles in the Namib Desert, see also Edney EB. (1971) The body temperature of ten
ebrionid beetles in the Namib Desert of southern Africa. Journal of Experimental Biology 55, 69–102.
The little equation: See De Jon PW et al. (1996), above.
Keeping their temperature: For a general paper on insect thermoregulation that also considers the role of shivering, see Heinrich B. Keeping their temperature high enough to move around: (1974) Thermoregulation in endothermic insects. Science 185, 747–756; and his later book Heinrich B. (1996) The Thermal Warriors: Strategies of Insect Survival. Harvard University Press, Cambridge, MA.
The drop in freezing point: The molality is the moles of a chemical divided by its mass.
Instead, cylinders run: An early paper discussing some of the principles for flying insects is Weis-Fogh T. (1967) Respiration and tracheal ventilation in locusts and other flying insects. Journal of Experimental Biology 47, 561–587.
Oxygen has: See, for example Verbeck W, Bilton DT. (2011) Can oxygen set thermal limits in an insect and drive gigantism? PLoS One 6, e22610. A very good paper that explores all the complex factors that may influence the role of oxygen in insect size is Harrison JF, Kaiser A, VandenBrooks JM. (2010) Atmospheric oxygen level and the evolution of insect body size. Proceedings of the Royal Society B, doi:10.1098/rspb.2010.0001.
Larger insects: For a thorough discussion of the problems confronting theoretical one-kilogram grasshoppers, see Greenlee KJ et al. (2009) Synchrotron imaging of the grasshopper tracheal system: Morphological and physiological components of tracheal hypermetry. American Journal of Physiology. Regulatory, Integrative and Comparative Physiology 297, R1343–1350.
The angular size of each lens: Barlow HB. (1952) The size of ommatidia in apposition eyes. Journal of Experimental Biology 29, 667–674.
The reception of light: The evolution of the protein receptors that gather light, the opsins, can occupy an entire tract unto themselves. They demonstrate convergence at the molecular level. So too the evolution of compound eyes and camera eyes. From the scale of the eye to its molecular components, the evolution of eyes is riven with convergence. Because the purpose of the apparatus is to capture electromagnetic radiation, physical principles have very strongly channeled convergence. See, for example, Shichida Y, Maysuyama T. (2009) Evolution of opsins and phototransduction. Philosophical Transactions of the Royal Society 364, 2881–2895; Yishida M, Yura K, Ogura A. (2014) Cephalopod eye evolution was modulated by the acquisition of Pax-6 splicing variants. Scientific Reports 4, 4256; and Halder G, Callaerts P, Gehring WJ. (1995) New perspectives on eye evolution. Current Opinions in Genetics and Development 5, 602–609. There is a plethora of other papers that investigate the details of eye evolution. All of them are a journey into a rich link between biology and physics. A substantive book on the equations of eyes would be entirely merited.
The Equations of Life Page 31