The Equations of Life

by Charles S. Cockell

  We have not talked: Weihmann T et al. (2015) Fast and powerful: Biomechanics and bite forces of the mandibles in the American Cockroach Periplaneta americana. PLoS One 10, e0141226. The physics of the food that insects eat might itself influence the physics of insect mandibles required to eat it. For a discussion of the physics of grass, this splendidly titled paper is worth reading: Vincent JFV. (1981) The mechanical design of grass. Journal of Materials Science 17, 856–860. Physical principles have been shown to be behind the evolution of the jaws of many organisms, such as the intriguing extinct giant otters: Tseng ZJ et al. (2017) Feeding capability in the extinct giant Siamogale melilutra and comparative mandibular biomechanics of living Lutrinae. Scientific Reports 7, 15225.

  And the physics: Already partly taken on in Gutierrez AP, Baumgaertner JU, Hagen KS. (1981) A conceptual model for growth, development, and reproduction in the ladybird beetle, Hippodamia convergens (Coleoptera: Coccinellidae). Canadian Entomologist 113, 21–33.

  I suspect three: Although the topic is not explored from the point of view of physics, an excellent text that does bring one face-to-face with the incredible complexity of insects and their extraordinary abilities is Chapman RF. (2012) The Insects: Structure and Function. Cambridge University Press, Cambridge.

  Natural selection: Physical principles described in the forms of equations are ultimately mathematical, and so we would not be completely misled in describing life as merely the manifestation of mathematics (see du Sautoy M. [2016] What We Cannot Know. Fourth Estate, London, for a rather eloquent exposition of the idea that the universe is just a manifestation of mathematics). But here, I generally focus on physical principles to draw attention to the physical manifestations of life’s mathematical relationships and equations and their impact on the form and function of organic matter with a code that evolves. At this point in the text, however, mathematical seems apposite to emphasize the interconnected mathematical relationships between the different terms of equations manifest in the living form.

  Such efforts would take us: There are two forms of prediction. One is reductive prediction, for example, the ability to predict cell membrane structure using the knowledge that living things are made from cells. In other words, prediction at lower levels of complexity according to knowledge at higher levels. The other form, predictive synthesis, is the ability to predict complex structures according to knowledge about lower levels of its hierarchy. The latter is generally much more difficult than the former since how simple things assemble into complex structures is less well understood. However, increasingly the capacities for prediction in both directions are being improved. An eloquent exploration of these forms of prediction is made by Wilson EO. (1998) Consilience. Abacus, London, 71–104.

  genes are involved: The modularity of developmental processes and phenotypic characteristics may make this task more tractable than it might first seem. This modularity is reviewed in Müller GB. (2007) Evo-devo: Extending the evolutionary synthesis. Nature Reviews Genetics 8, 943–949.

  Nevertheless, this evolutionary physics: I refer the reader to a fascinating paper in which the authors attempt to capture adaptation using statistical physics, illustrating the sort of approaches that might aid a description of biological processes in the form of equations that can yield predictive power: Perunov N, Marsland R, England J. (2016) Statistical physics of adaptation. Physical Review X 6, 021036. Regarding evolutionary changes in quantitative, physically circumscribed terms, what I describe here can be thought of as a type of optimality model (see, for example, Abrams P. [2001] Adaptationism, optimality models, and tests of adaptive scenarios. In Adaptationism and Optimality, edited by SH Orzack and E Sober. Cambridge University Press, Cambridge, 273–302). In the real world, of course, knowing which traits are important to an organism and what their optimized properties are is extraordinarily difficult, particularly for poorly studied organisms. Thus, these approaches have their limitations. Nevertheless, my suggestion might allow us to understand better certain physical trade-offs and create a more converged area of thought between physicists and biologists. This approach can work particularly well where empirical information is forthcoming, such as the heat balance in ladybirds and its link to wing-case thickness and metabolic rates in given environmental conditions. We may not know optimal conditions, but we can measure quantities in given organisms and use these measurements to define a parameter space of interacting equations that can be used to investigate how different physical constraints in the environment and physical quantities in organisms interact to influence its adaptive traits.

  Throughout this tantalizing foray: I say “often” because some examples of convergence appear to be driven more by biological interactions than by the physical environment. Mimicry is an example found various places, from butterfly wings to stick insects that look like twigs or leaves. Many of these examples are generally about the cosmetic appearance of organisms rather than their fundamental structure and mechanics. However, we might still think about them from a physical perspective. Although the selection that gives rise to these phenomena is driven by interactions between organisms, at a fundamental level a larva that evolves to look like a twig is an example of one lump of matter containing a code evolving to look like another lump of matter. This similarity means that the lump of twiglike matter called a larva is less likely to be destroyed and so its abundance correspondingly increases. This process is understood just as easily in simple physical terms. Once we think about organisms as lumps of matter with a code, the distinction between biology and physics becomes less distinct.

  CHAPTER 4

  Evolutionary convergence: A comprehensive and excellent account of convergence is found in Conway-Morris S. (2004) Life’s Solution: Inevitable Humans in a Lonely Universe. Cambridge University Press, Cambridge. An erudite review can also be found in McGhee G. (2011) Convergent Evolution: Limited Forms Most Beautiful. Massachusetts Institute of Technology, Cambridge, MA. For a discussion of convergence, its potential confusion with other mechanisms of similarity, and the question of whether some instances of convergence are not truly independent but might emerge from the same developmental possibilities and phylogenetic constraints, particularly in closely related organisms, see Wray GA. (2002) Do convergent developmental mechanisms underlie convergent phenotypes? Brain, Behavior and Evolution 59, 327–336.

  This is the same with convergent evolution: Mayr E. (2004) What Makes Biology Unique? Cambridge University Press, Cambridge, 71, states that “the physicochemical approach is totally sterile in evolutionary biology. The historical aspects of biological organization are entirely out of reach of physicochemical reductionism.” I agree with Mayr that contingency is not easy to reduce to equations, but I disagree that evolution is only historical. Many examples of convergence are underpinned by physical constraints and therefore do allow us to bring physicochemical explanations forcefully into the evolutionary synthesis.

  Yet the question they ask: Another question is why large animals, such as mammals and reptiles, have four legs and insects and other arthropods have six (or many more). The canonical response is that here at last we have an example of contingency at work, the number of limbs reflecting past body designs from which evolution cannot escape. In the case of arthropods, the segmented body plan allows for many pairs of limbs to be added or removed, leading to a range of numbers of legs among arthropods, from six-legged insects to animals with just over seven hundred legs (e.g., the millipede Illacme plenipes). In vertebrates, ancestors with two sets of pectoral and pelvic fins led to a four-legged design. This theory of contingency for legs may well be the case—we could imagine, for instance, that a mammal version of a praying mantis that could run on four legs and have two appendages left free for predation might be very successful. More biomechanical studies may help discover to what extent leg number affects locomotion and whether, aside from contingency, there are any physical reasons for limb-number choices. See, however, Full RJ, Tu MS. (1990) Mechanics of six-legged r
unners. Journal of Experimental Biology 148, 129–146, which suggests that the energy needed to move a center of mass a given distance is unchanged across many animals with different leg numbers.

  So why has nature: LaBarbera M. (1983) Why the wheels won’t go. American Naturalist 121, 395–408.

  Richard Dawkins: Dawkins R. (1996) Why don’t animals have wheels? (London) Sunday Times, November 24.

  There is no selection pressure: I note anecdotally tales of animals such as wolves using roads to get across forests more easily. Once these structures are built by an intelligent life form, other life may find roads useful for locomotion.

  Compare this: I simplify somewhat. The menu of possibilities that fish and other aquatic animals use for locomotion is varied and impressive. Many fish, such as tuna, rely more on tail movements (thunniform swimming) than on body waves, and in this sense, their propulsion is closer to a propeller’s.

  Dangling from the sides: Berg HC, Anderson RA (1973) Bacteria swim by rotating their flagellar filaments. Nature 245, 380–382 and Berg HC. (1974) Dynamic properties of bacterial flagellar motors. Nature 249, 77–79.

  Despite the superficial similarities: A rather famous paper that describes these two very different worlds of low and high Reynolds numbers is Purcell EM. (1977) Life at low Reynolds Number. American Journal of Physics 45, 3–11.

  However, we find: We could imagine wheels being useful on a planet dominated by smooth surfaces. However, even a planet without plate tectonics to create mountain ranges, such as Mars, has an irregular surface that would suit legged animals better than wheeled ones. A flat surface at the macroscopic scale does not change the irregular structure of a planet at the millimeter scale. As for propellers, there are other physical reasons for doubting that bacterial-like flagella motors can simply be scaled up. In a delightful and elegant essay on the topic of wheels, Gould S. (1983) Kingdoms without wheels. In Hen’s Teeth and Horse’s Toes, 158–165, suggests that flagella evolved and larger rotating structures did not because the bacterial apparatus depends on diffusion, which is too slow at large scales to enable the evolution of an analogous, but larger structure. Gould’s argument suggests a physical barrier to explain something evolutionary, which in itself is interesting since he was fond of relegating physical processes to being an irrelevance against contingency.

  In 1917, a brilliant: Thompson DW. (1992) On Growth and Form. Cambridge University Press, Cambridge.

  He explores the shapes: The mathematical relationships in the growth of plants and their physical and biological bases have been a subject of fascination for centuries. Many plants exhibit growth that has a spiral pattern in leaves. You can even see it in pine cones. The arrangement of leaves in a plant is known as phyllotaxis. Often, two spirals—one clockwise and one anticlockwise—can be discerned in a plant leaf pattern if you look down toward its apex. Extraordinarily, the number of spirals running clockwise and anticlockwise is found to be two adjacent terms in the Fibonacci series. Each number in this special sequence is the sum of the preceding two numbers, so, for example, 1, 1, 2, 3, 5, 8, 13, 21, and so on, is a Fibonacci series. A plant might have 8 and 13 spirals. This arrangement is (8, 13) phyllotaxis. Why this should be the case is thought to be related to the packing of leaves to acquire maximum sunlight, to maximize efficiency of stacking, or both. This is reflected in the angle through which each leaf is rotated relative to the previous leaf. It is determined by physical principles, not a contingent product of natural selection. This intriguing mathematical relationship has long attracted the attention of physicists and mathematicians. Two examples to whet the appetite further: Newell AC, Shipman PD. (2005) Plants and Fibonacci. Journal of Statistical Physics 121, 927–968; and Mitchison GJ. (1977) Phyllotaxis and the Fibonacci series. Science 196, 270–275. A classic paper offering a physical model is Douady S, Couder Y. (1991) Phyllotaxis as a physical self-organised growth response. Physical Review Letters 68, 2098–2101. The link between Fibonacci series and biology is another beautiful example of the connection between biological form and predictable patterns in which the action of genes is to impose shape, structure, color, and other characteristics on an otherwise physically determined process.

  In his wonderful book: Carroll SB. (2005) Endless Forms Most Beautiful. Quercus, London. He also explores the fascinating way in which wings in vertebrates developed in different ways from their limbs and digits. The transition between gills and wings in insects also shows the quite astonishing adaptability of the modules of animals to shape-shift into entirely new structures (Averof M, Cohen SM. [1997] Evolutionary origin of insect wings from ancestral gills. Nature 385, 627–630).

  Some of this variation: I do not discuss it here, but recent research on the development of flight in birds shows how specific gene regulatory elements are associated with the development of wings and feathers, another extraordinary example of how modest changes in genetic regulation and units can drive the acquisition of characteristics that allow organisms to tap into physical laws, in this case the rules of aerodynamics. For example, see Seki R et al. (2016) Functional roles of Aves class-specific cis-regulatory elements on macroevolution of bird-specific features. Nature Communications 8, 14229.

  One transition is impressive: Denny MW. (1993) Air and Water: The Biology and Physics of Life’s Media. Princeton University Press, Princeton, NJ, doesn’t explicitly address the move from water to land, but his treatise examining the physics of biology in water and in air, and often comparing both media and the implications for the structure of biological systems, is an impressive work. In many ways, it is one of the most detailed and ambitious paeans to the link between biology and physics to be written. Denny also explored aspects of life at the air-water interface. See, for example, Denny MW. (1999) Are there mechanical limits to size in wave-swept organisms? Journal of Experimental Biology 202, 3463–3467.

  But the complete transition: A very cogent account of this transition in the context of locomotion is given in Wilkinson M. (2016) Restless Creatures: The Story of Life in Ten Movements. Icon Books, London.

  The scientists found: A series of papers document these insights. Just some worth reading are Freitas R, Zhang G, Cohn MJ. (2007) Biphasic Hoxd gene expression in shark paired fins reveals an ancient origin of the distal limb domain. PLoS One 8, e754; Davis MC, Dahn RD, Shubin NH. (2007) An autopodial-like pattern of Hox expression in the fins of a basal actinopterygian fish. Nature 447, 473–477; Schneider I et al. (2011) Appendage expression driven by the Hoxd Global Control Region is an ancient gnathosome feature. Proceedings of the National Academy of Sciences 108, 12782–12786; Freitas R et al. (2012) Hoxd13 contribution to the evolution of vertebrate appendages. Developmental Cell 23, 1219–1229; Davis MC. (2013) The deep homology of the autopod: Insights from Hox gene regulation. Integrative and Comparative Biology 53, 224–232.

  The slightly more graceful: A review of these different forms of movement is found in Gibb AC, Ashley-Ross MA, Hsieh ST. (2013) Thrash, flip, or jump: The behavioural and functional continuum of terrestrial locomotion in teleost fishes. Integrative and Comparative Biology 53, 295–306. This paper was part of a wider symposium, and the review paper on the whole meeting is worth reading for an insight into the general problem that faced life in the transition from water to land: Ashley-Ross MA, Hsieh ST, Gibb AC, Blob RW. (2013) Vertebrate land invasions—past, present, and future: An introduction to the symposium. Integrative and Comparative Biology 53, 1–5.

  Nonetheless, in the history of animal life: Of course, arthropods also made this transition, giving rise to insects.

  Not surprisingly: Cockell CS, Knowland J. (1999) Ultraviolet radiation screening compounds. Biological Reviews 74, 311–345.

  In experiments that tracked: A fascinating paper that describes these findings is Leal F, Cohn MJ. (2016) Loss and re-emergence of legs in snakes by modular evolution of Sonic hedgehog and HOXD enhancers. Current Biology 26, 1–8.

  Almost certainly, the secrets: A very readable book describing the histo
ry of research investigating the invasion of land and the return to water is Zimmer C. (1998) At the Water’s Edge. Touchstone, New York.

  Charles Darwin concluded: Darwin C. (1859) On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. John Murray, London.

  CHAPTER 5

  However, if I tell you: Bianconi E et al. (2013) An estimation of the number of cells in the human body. Annals of Human Biology 40, 463–471.

  Hooke had no inkling: Hooke R. (1665) Micrographia. J Martyn and J Allestry, printers to the Royal Society, London.

  What he found: Van Leeuwenhoek published a prodigious number of letters on many things he observed under his microscopes, not merely his animalcules. One seminal paper on observations about microbes is Leeuwenhoek A. (1677) Observation, communicated to the publisher by Mr. Antony van Leuwenhoek, in a Dutch letter of the 9 Octob. 1676 here English’d: concerning little animals by him observed in rain-well-sea and snow water; as also in water wherein pepper had lain infused. Philosophical Transactions 12, 821–831.