D'arcy Thompson and the Theory of Transformations

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70. Colot, V., Maloisel, L. & Rossignol, J. L. 73. Bender, J. & Fink, G. R. Epigenetic control of an Competing interests statement Interchromosomal transfer of epigenetic states endogenous gene family is revealed by a novel blue The author declares no competing financial interests. in Ascobolus: transfer of DNA methylation fluorescent mutant of Arabidopsis. Cell 83, 725–734 is mechanistically related to homologous (1995). DATABASES recombination. Cell 86, 855–864 74. Melquist, S. & Bender, J. Transcription from an The following terms in this article are linked online to: (1996). upstream promoter controls methylation signaling Entrez Gene: http://www.ncbi.nlm.nih.gov/entrez/query. 71. Suter, C. M., Martin, D. I. & Ward, R. L. Germline from an inverted repeat of endogenous genes in fcgi?db=gene epimutation of MLH1 in individuals with multiple Arabidopsis. Genes Dev. 17, 2036–2047 (2003). NR3C1 cancers. Nature Genet. 36, 497–501 (2004). Acknowledgments FURTHER INFORMATION 72. Das, O. P. & Messing, J. Variegated phenotype and I am grateful for the helpful comments of G. Allen and the Eric J. Richards’s homepage: developmental methylation changes of a maize allele anonymous reviewers. My laboratory’s experimental work on http://www.biology.wustl.edu/faculty/richards originating from epimutation. Genetics 136, epigenetic variation and inheritance is funded by the US Access to this links box is available online. 1121–1141 (1994). National Science Foundation.

and development can be gauged from the SERIES ON HISTORICAL PROFILES — TIMELINE dedication at the front of Gould’s Ontogeny and Phylogeny: “To the philomorphs of D’Arcy Thompson and the theory Cambridge, the world, and beyond, where D’Arcy Thompson must lie in the bosom of transformations of Abraham.”7 For those unfamiliar with the theory of transformations, here is a brief overview. Wallace Arthur You take either the outline of an entire Abstract | D’Arcy Thompson was a biologist, a mathematician and a classicist. His animal or plant, or the outline of one of its component parts such as a bone or a leaf, writing was great literature as well as great science. He is primarily known for a and draw this against the background of single book — On Growth and Form — and indeed for a single chapter within it, on a Cartesian grid (for example, ordinary his ‘theory of transformations’, which shows how the differences between the graph paper). Then you submit the forms of related species can be represented geometrically. This theory cries out for grid to some systematic mathematical causal explanation, which is something the great man eschewed. Perhaps the time transformation, such as stretching it in one dimension or distorting it so that its is close when comparative developmental genetics will be able to provide such an squares become rhombuses. You inspect explanation. the transformed outline of the animal that you drew faithfully on the original Evolutionary and developmental biology novel theory of transformations has, for grid, and in many cases note that, far from parted company from each other around nearly a century, been an inspiration being just a weird shape, the transformed 1900 and remained largely separate for about to biologists who are interested in how outline corresponds closely to the shape three-quarters of the twentieth century. development and morphology evolve. The of another related animal. Clearly, this They only began to reintegrate in the late esteem in which D’Arcy Thompson is held intriguing finding is telling us something 1970s and early 1980s, eventually producing by those who are interested in furthering about how evolution works — but what? the interdisciplinary endeavour that we the reintegration of theories of evolution This is the key question. now know as evolutionary developmental biology or ‘evo–devo’1–6. The two main catalysts of reintegration were a series of books, most notably Stephen Jay Gould’s Ontogeny and Phylogeny in 1977 (REF. 7), and the advances in developmental genetics that were made possible by the discovery of the homeobox in the early 1980s (REFS 8,9). In the period between 1900 and 1975, only a few lone voices had intermittently reminded biologists that the two great processes of biological creation — evolution and development — were deeply intertwined. D’Arcy Thompson (1860–1948) was one of them10 (FIG. 1). Others included the neo-Darwinians Huxley 11 and de Beer12, the mutationist Goldschmidt13, and the hard-to-classify Waddington14. D’Arcy Thompson was unique; no one before him had attempted the kind Figure 1 | D’Arcy Thompson in the early 1900s of geometrical approach to development and in the 1940s. Reproduced with permission and evolution that he did. His entirely from REF. 15 © (1958) Oxford University Press.

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Timeline | D’Arcy Wentworth Thompson

Begins a medical Becomes Professor of Zoology Born in Edinburgh, course at Edinburgh at University College Dundee, Dies on 21 June at Scotland, 2 May. University. Scotland. Marries Maureen Drury. Receives a knighthood. St Andrews.

1860 1870 1878 1880 1885 1889 1901 1917 1937 1942 1948

(1870–1877) Educated (1880–1883) Transfers to natural Describes ‘taking Publishes On Growth and Form. Publishes a much-enlarged second at Edinburgh Academy, sciences at Cambridge University, to mathematics’ Takes up the Chair of Natural edition of On Growth and Form. and by his father in England. Graduates with a first as an approach to History at St Andrews Galway, Ireland. class honours B.A. in zoology. morphology. University, Scotland.

The above account might give the impres- old, his father was appointed to the Chair relative frequency of papers that deal with sion that D’Arcy Thompson’s approach was of Greek at Queen’s College Galway, now the molecular details of developmental gene more ‘blue sky’ than it actually was. He did the National University of Ireland, Galway. interactions and those that deal with their not try out countless transformations from As a child, D’Arcy spent some of his time quantitative dynamics. the starting point of any particular animal with his father in Galway (his first visit Most of On Growth and Form deals with and look at which of them produced the being in 1867), but most in the home of his the shapes of various parts of organisms: from form of a related animal. Rather, he took grandfather, Joseph Gamgee, who practised cells and tissues to spicules, shells, horns, teeth two (or more) related forms, and tried to as a veterinary surgeon in Edinburgh. He and bones. In each case, D’Arcy Thompson determine whether one could be produced acquired a love of classics from the former, a attempts succinct mathematical descriptions, from the other by some simple transforma- love of science from the latter. with their elegance and efficiency surpassing tion. The forms he compared were typically D’Arcy was educated at Edinburgh those of what he saw as ‘mere words’. For from the same family or order; he did not Academy, and subsequently began to study example, in his chapter on spirals, he notes believe that his approach was appropriate medicine at Edinburgh University. However, that the typical molluscan shell corresponds for higher-level taxonomic comparisons. he later switched to reading science at to the equiangular spiral, in which the breadth His transformations suggest coordinated Cambridge University, graduating with a of a whorl increases as the spiral proceeds, as rather than piecemeal changes to develop- B.A. in zoology in 1883, and spending opposed to the spiral of Archimedes, in which ment in the course of evolution, an issue a further year at Cambridge working as a it does not. This distinction connects with the which almost completely disappeared from demonstrator. In 1884, he was appointed to broader one of isometric versus allometric view in the era of the ‘modern synthesis’ of a professorship at University College (proportionate and disproportionate, respec- evolutionary theory, but which is of central Dundee, and in 1917 he took up the Chair tively) growth11, which has had an important importance again in the era of evo-devo. of Natural History at the University of St role in many subsequent studies of both Here I take a brief look at D’Arcy Andrews. He produced some 300 publica- evolution and development. Thompson’s life, his general philosophy, and tions, including his magnum opus, On This mathematical philosophy was D’Arcy his approach to studying morphological Growth and Form10, first published in 1917, Thompson’s major strength and weakness. evolution. I then examine the limitations and with a second edition in 1942, and with His theory of transformations would have problems of his theory of transformations. many abridged editions since. (His other been impossible without it. But he allowed This examination leads into an investigation publications were diverse; many of them this philosophy to dominate his approach to of how modern comparative develop- derived from his long sea voyages as a mem- the problems of evolution and development mental genetics might be able to tackle ber of various government commissions to the extent that he sometimes seemed to the outstanding problem of the molecular concerning fisheries.) D’Arcy Thompson ‘set little store’ by genetic or biochemical causality of morphological transformations. died at St Andrews in 1948, at the age of 88. approaches. In his introduction he makes the I end with a short discussion of large-scale For further information on his life, see following point: “…in dealing with the facts of evolutionary changes to which the theory of the TIMELINE; for a detailed biography, embryology or the phenomena of inheritance, transformations does not apply. see the book written by his daughter Ruth15. the common language of the books seems to deal too much with the material elements A brief biography D’Arcy Thompson’s philosophy concerned.”10 And he goes on to explain that This is a story of five places — three in The philosophy that pervades On Growth in his view biologists should place less empha- Scotland, one in England and one in Ireland. and Form, and indeed D’Arcy Thompson’s sis on matter (such as a piece of embryonic D’Arcy Thompson was born in Edinburgh publications in general, is the explanation of tissue) and more on the forces that shape it. in 1860. His father was also called D’Arcy natural phenomena in terms of physical, and We have now reached a stage in the elabo- — indeed the names of both men were especially mathematical, laws. His mathe- ration of biological knowledge at which we D’Arcy Wentworth Thompson. His mother, matical approach was unusual among biolo- can try to knit the two approaches together. née Fanny Gamgee, died about a week after gists then; and it is still a minority approach A gene is indeed a material thing. But its his birth. When D’Arcy junior was 3 years in the present day: compare, for example, the pattern of expression during development

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Harpinia plumosa Oithona nana Geryon Corystes Scyramathia

Stegocephalus inﬂatus

Sapphirina Paralomis Lupa Chorinus

Hyperia galba

Figure 2 | Transformations that are used to relate different crusta- complex, as in the case of the three amphipod genera Harpinia, ceans to each other. Some of the transformations required are simple, Stegocephalus and Hyperia. Reproduced with permission from REF. 10 as in the case of the copepods Oithona and Sapphirina; others are more © (1917) Cambridge University Press.

is dynamic. And the forces that cause this Phylogeny. Working as he was in the early Of course, we cannot blame D’Arcy dynamism include such things as transcrip- twentieth century, D’Arcy Thompson was Thompson for a failure to incorporate tion factors and morphogens, which are well within the era of evolutionary trees ideas about transcription factors into his themselves material things. Old barriers (a generalized one of which is, famously, theory, as they were then unknown. Nor are breaking down. There is no need to the only picture in the whole of Darwin’s can we blame him, at least in the first edi- see biochemical and biomathematical Origin of Species16). But he was working well tion of On Growth and Form, for omitting approaches as antagonistic. before the rigorous treatment of such trees the embryological ideas of gradients, fields But the specific question of most interest that began with the advent of phylogenetic and morphogens, as these also awaited is: can our modern developmental genetic systematics in the mid-twentieth century 17. articulation18–20. But there is one thing that approach to understanding how organisms This limitation shows in the fact that he was we can and perhaps should blame him for grow and form themselves, and how evolu- usually content to note that the morphologies — a neglect of juvenile morphologies. tion modifies this process, give us a signifi- of a group of related genera could be derived Notice that all the forms shown in cant causal insight into D’Arcy Thompson’s from each other by appropriate transforma- FIGS 2–4 are those of adults. This is generally transformations? This is the question to tions, and he was not terribly concerned with true of the other transformations pictured in which we will turn, after first examining mapping the genera to a phylogeny so that the relevant chapter of On Growth and Form, some problems in D’Arcy’s theory, which it became apparent which way round the which are not reproduced here. This concen- help to point the way forward. transformations had taken place (FIG. 4). tration on adults is strange, given two facts. First, it must have been as obvious to D’Arcy Limitations and problems of the theory Direct versus indirect development. It is Thompson as it is to biologists today that Some examples of particular transformations interesting that D’Arcy Thompson generally there is no way evolution can turn one sort are given in FIGS 2,3. It can be seen that, in used, as examples, direct rather than indirect of adult form into another except by modify- each case, a particular distortion of the initial developers. He used many crustaceans as ing the course of development. Second, Cartesian grid on which the outline of one examples (FIG. 2), but no insects. Likewise, many of the chapters leading up to the one body form was plotted leads to the appear- in the vertebrates, he used many fish (FIG. 3), on transformations, such as the one on ance of another. Often, the different forms but no amphibians. This is a limitation of the spirals mentioned above, did explicitly deal are related at about the level of the genus or theory (so far anyway), but not a problem with both adult and juvenile morphology. It family — give or take a taxonomic level. That — indeed it was a wise strategy to limit his is as if the developmental and evolutionary is, the method works reasonably well for examples in this way. It is difficult enough parts of the book were disconnected from species, genera, families and orders. It works to understand how a directly developing each other. less well for intraspecific variation, which is system evolves in quantitative terms, without That is probably too harsh a criticism. typically rather minor. It also works less well adding the complexities of metamorphosis. In the pre-computer age, there was a limit for differences at very high taxonomic levels, to the dimensionality of a problem that such as those between classes and phyla, Morphology of sub-adult stages. The main could reasonably be dealt with. A series where there are often qualitative, as well as deficiency of the theory of transforma- of growth stages of a species of gastropod quantitative, differences in form. This in itself tions, from a genetic or developmental constituted a tractable problem. So did the poses an interesting evolutionary question point of view, is that no causal mechanism relationships between the adult forms of — one that I discuss in a later section. was proposed for their occurrence. several kinds of fish. But putting together the

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PolyprionScorpaena Argyropelecus olfersi Diodon

Pseudopriacanthus altus Antigonia capros Sternoptyx diaphana Orthagoriscus

Figure 3 | Transformations that are used to relate different fish to produce the form of Sternoptyx diaphana from that of Argyropelecus each other. Again, some of the transformations are more complex olfersi. Reproduced with permission from REF. 10 © (1917) Cambridge than others. The simplest transformation is the ‘shear’ required to University Press. evolutionary and developmental problems, modified to produce morphological trans- humans, mice, chicks and zebrafish, but especially in relation to the two-dimensional formations. However, achieving such an preferably not frogs or fruitflies (indirect and three-dimensional morphologies that understanding is not straightforward — if developers), or nematodes (minimalist D’Arcy Thompson focused on (in contrast to it were, given the array of techniques now morphology). univariate measures like body length), was, at our disposal, the relevant research would There are already two problems. First, at the time, a step too far. already have been carried out. in most cases there is not a species that is The main problem lies in the fact that an clearly related to our starting-point species Partial transformations. One final dif- approach to analysing the mechanistic basis by a transformation and is equally well ficulty with the theory of transformations of transformations is a multi-step process, known genomically. A system that comes deserves a brief mention. Not all intergenus with the results of step 1 determining what particularly close is the human–chimpanzee morphological differences within a family should be done at step 2, and so on. The sug- one (FIG. 4). However, ethical considera- are readily explicable in terms of a single, gestions that follow must be considered in tions preclude much of the desired work simple transformation. Often some part that light. At each step, I assume a particular on embryos in this case. For this reason, or other does not ‘fit’, unless, as well as the outcome and proceed accordingly. This is and also because of the greater visibility of whole-body transformation, some other, a necessary restriction, for otherwise my embryos, a system that is centred on the more spatially limited transformation is account would have to deal with all of a large zebrafish would be a better choice. Second, applied — for example to the head. The series of bifurcating possibilities and would the relevant molecular methods can only need for such applications is reminiscent, therefore become far too long for a paper of be used on living species (notwithstanding perhaps, of attempts to save Ptolemy’s earth- this kind. our ability to extract DNA from some centred solar-system model by introducing fossil material), but transformations ‘epi cycles’ into originally smooth orbits. Choosing a system. The first task is to select occur between ancestor and descendant, Although the comparable conclusion — that an appropriate system, and, in doing so, to not between extant ‘cousins’. So we should the model was fundamentally flawed — maximize the range of molecular techniques use a system in which information on would be inappropriate in relation to trans- that could be used, while also putting the appropriate outgroups indicates that one of formations, this line of thinking does lead to whole endeavour in the strongest possible our two species has changed little since their another interesting, and so-far unanswered, phylogenetic context. These points argue lineages diverged, and the other much more question: what proportion of interspecies for using two (or more) species in a well- so. In such cases, the comparison of the two comparisons (at the level of the genus, family established clade for which molecular and extant forms might act as a surrogate for the or order) can be treated in terms of transfor- morphological phylogenies agree, and for comparison of ancestor with descendant mations, whether whole and/or partial, and which the species concerned preferably have that we would really like to make. what proportion cannot? well-characterized genomes. Already, these requirements severely limit our options. A When does development diverge? Assuming Investigating causality genus or family level comparison where one that we can find an appropriate pair of spe- Ideally, we would like to be able to under- of the species being compared is a ‘model cies, the next step is to identify the stage in stand, in molecular terms, the ways in which organism’ (from a genetic perspective) development during which a transformation developmental processes are evolutionarily would be best. Possibilities therefore include can first be seen. This involves plotting

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outline morphologies for a series of stages, we can rise to, in the knowledge that studies embryos that are the cause of the transfor- not just the adult, as was D’Arcy Thompson’s of these intermediate stages might ultimately mations D’Arcy Thompson observed at the usual practice. Such a multiple-plot compar- help to unite these two currently rather morphological level in adults. But, as ever ison identifies the point at or before which disparate fields of genetics. in development, there is the problem of one or both ontogenies have been modified potentially infinite regress (to the egg), in in a way that causes them to be relatable by a Cells and genes: a four-way comparison. the sense that an altered spatial distribu- transformation. As well as simple graphical The best strategy now would be to make a tion of a hormone receptor at stage Y (for plots, it is now possible to look at shapes in four-way comparison. The four embryos/ example) is itself caused by something terms of landmark features21, a method that juveniles to be compared would be ‘stage X’ in else upstream of it. And so, our search has been profitably used in relation to the each of our two species (pre-transformation) would continue. mouse mandible22. However, a drawback and ‘stage Y’ in each species (post- with this method for use in tracing transfor- transformation). Confirmation of the The origin of new body plans mations back in developmental time is that causality residing in patterns of cell prolif- What of comparisons between distantly at early stages some landmarks will not yet eration could be achieved using a standard related taxa, such as chordates and echi- have formed. technique for examining such patterns (for noderms, where transformations do not Again, we must assume an outcome in example, labelling with 5-bromo-uracil). ‘work’ at all? Here we come face to face order to proceed to the next step. Looking Assuming a positive result here, the next with that age-old problem in evolutionary at the transformations of FIGS 2–4, and all stage in our causal analysis of the transfor- biology of whether those few phylogeneti- the others that D’Arcy Thompson revealed, mation would be to investigate what causes cally deep lineage splits that resulted in it seems likely that the crucial evolutionary the altered cell-proliferation patterns. One radically different body plans were in some changes occur at a stage in development obvious candidate, given the spatial and way different to most ‘routine’ evolutionary that is after the key early formative stages temporal scale involved, is the interaction changes26. In other words: is mega- of gastrulation and neurulation, after the between hormones or growth factors and evolution27 fundamentally different from highly conserved phylotypic stage23 (or their receptors. Because these agents are microevolution? Or, conversely, is evolution phylotypic period24) and probably also after relatively well known, in a wide range really ‘scale-independent’28? organogenesis, but early in the subsequent of species, they can be investigated with D’Arcy Thompson seemed to take the growth phase. If this is true, then in cellular several established molecular techniques, former view — that there is a fundamental terms transformations are probably caused both observational (for example, in situ difference — when he said: “Our geometri- by changes in the rates, durations and hybridizations to look at the expression cal analogies weigh heavily against Darwin’s directions of cell proliferation, and not by patterns of the genes involved, such as those conception of endless small continuous changes in patterns of cell movement, or, for that make receptors) and experimental (for variations; they help to show that discon- that matter, by changes in cell size or shape, example, alteration of hormone concentra- tinuous variations are a natural thing, that although neither of these can be ruled out as tion patterns or signalling mechanisms). ‘mutations’ — or sudden changes, greater or contributory factors. A particularly interesting example of the less — are bound to have taken place, and So, the developmental origins of latter involved the manipulation of insulin new ‘types’ to have arisen now and then.”10 transformations probably lie in the period signalling in the Drosophila wing, resulting I think he was right, although we should all in between the early stages that are typi- in an enlarged anterior, but not posterior, admit that this is still an unresolved issue. cally studied by molecular developmental compartment — in effect a transformation, But in any event, we should be careful to geneticists and the much later stages that are although a rather abrupt one25. distinguish between D’Arcy Thompson’s typically studied, with different methodolo- Future studies of this and other kinds view, which put the discontinuity between gies, by quantitative geneticists. This in itself will hopefully reveal, in many systems, the two types of evolutionary change poses something of a challenge, but one that spatial patterns at the molecular level in at a high taxonomic level, and Richard

Human skull Chimpanzee skull Baboon skull

Figure 4 | Transformations that are used to relate human, chimpan- are two reasons for this non-correspondence: imperfect knowledge of the zee and baboon skulls. Note that the starting point chosen, the mod- relevant ancestor; and D’Arcy Thompson’s generally non-phylogenetic ern human skull, does not represent the evolutionary starting point, approach. Reproduced with permission from REF. 10 © (1917) which would be the last common ancestor of the three species. There Cambridge University Press.

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Goldschmidt’s view13, which put it at a much We owe it to the great man to put these 15. Thompson, R. D’A. D’Arcy Wentworth Thompson, The Scholar–Naturalist, 1860–1948 (Oxford Univ. Press, lower level — between intraspecific and three things together to investigate the London, 1958). interspecific evolutionary changes. Using mechanisms that produce the morphological 16. Darwin, C. On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Simpson’s terminology of micro, macro and changes that he captured so elegantly with Races in the Struggle for Life (J. Murray, London, mega-evolution27, with brackets to denote little more than sheets of graph paper and, of 1859). 17. Hennig, W. Phylogenetic Systematics (Univ. Illinois similarity of mechanism, we can see that course, a brilliant mind. Press, Urbana, 1966). D’Arcy Thompson’s view can be represented 18. Child, C. M. Patterns and Problems of Development Wallace Arthur is at the Department of Zoology, (Chicago Univ. Press, Chicago, 1941). as (micro, macro) versus mega; whereas National University of Ireland, Galway, 19. Wolpert, L. Positional information and pattern Goldschmidt’s can be represented as micro University Road, Galway, Ireland. formation. Curr. Top. Dev. Biol. 6, 183–224 (1971). 20. Meinhardt, H. Models of Biological Pattern Formation e-mail: [email protected] versus (macro, mega). (Academic Press, London, 1982). doi:10.1038/nrg1835 21. Bookstein, F. L. Morphometric Tools for Landmark Published online 11 April 2006 Data: Geometry and Biology (Cambridge Univ. Press, Prospect Cambridge, 1991). Despite the inspiration that D’Arcy 1. Raff, R. A., Arthur, W., Carroll, S. B., Coates, M. I. & 22. Klingenberg, C. P., Leamy, L. J., Routman, E. J. & Wray, G. Chronicling the birth of a discipline. Evol. Dev. Cheverud, J. M. Genetic architecture of mandible Thompson’s theory of transformations has 1, 1–2 (1999). shape in mice: effects of quantitative trait loci analyzed provided, it has given rise to little in the 2. Raff, R. A. Evo–devo: the evolution of a new discipline. by geometric morphometrics. Genetics 157, Nature Rev. Genet. 1, 74–79 (2000). 785–802 (2001). way of experimentation to reveal the causal 3. Arthur, W. The emerging conceptual framework of 23. Sander, K. in Development and Evolution nature of the transformations. Why? Until evolutionary developmental biology. Nature 415, (eds Goodwin, B. C., Holder, H. & Wyllie, C. C.) 757–764 (2002). 137–159 (Cambridge Univ. Press, Cambridge, 1983). recently, say the past 20 years, it could be 4. Carroll, S. Endless Forms Most Beautiful: the New 24. Richardson, M. K. Heterochrony and the phylotypic argued that our knowledge of comparative Science of Evo Devo and the Making of the Animal period. Dev. Biol. 172, 412–421 (1995). Kingdom (W. W. Norton, New York, 2005). 25. Hafen, E. & Stocker, H. How are the sizes of cells, developmental biology, and especially of 5. Holland, P. W. H. The future of evolutionary organs, and bodies controlled? PLoS Biol. 1, e86–e97 comparative developmental genetics, was developmental biology. Nature 402, C41–C44 (2003). (1999). 26. Arthur, W. The Origin of Animal Body Plans: A Study not sufficiently advanced to be up to the 6. Hall, B. K. Evolutionary Developmental Biology in Evolutionary Developmental Biology (Cambridge job. But that is hardly still the case now, (Kluwer Academic, Dordrecht, 1999). Univ. Press, Cambridge, 1997). 7. Gould, S. J. Ontogeny and Phylogeny (Harvard Univ. 27. Simpson, G. G. Tempo and Mode in Evolution some 20 years after the discovery of the Press, Cambridge, Massachusetts, 1977). (Columbia Univ. Press, New York, 1944). homeobox, and with evo–devo being well 8. McGinnis, W., Garber, R. L., Wirz, J., Kuroiwa, A. & 28. Leroi, A. M. The scale independence of evolution. Evol. Gehring, W. J. A homologous protein-coding sequence Dev. 2, 67–77 (2000). established. in Drosophila homeotic genes and its conservation in All the tools are now in place to examine other metazoans. Cell 37, 403–408 (1984). Acknowledgements 9. Scott, M. P. & Weiner, A. J. Structural relationships I thank H. Arthur, U. Frank, G. McCormack and A. Panchen the mechanistic basis of transformations. among genes that control development: sequence for scientific discussion and comments on the draft manu- Not only do we have phylogenetic systemat- homology between the Antennapedia, script, and the library of Trinity College Dublin for the loan of Ultrabithorax and fushi tarazu loci of Drosophila. copies of the first and second editions (1917 and 1942, ics and evo–devo, but, so obvious that it Proc. Natl Acad. Sci. USA 81, 4115–4119 (1984). respectively) of On Growth and Form. is easy to forget, we have computers, and 10. Thompson, D’A. W. On Growth and Form (Cambridge Univ. Press, Cambridge, 1917). Competing interests statement especially, in this context, advanced com- 11. Huxley, J. S. Problems of Relative Growth (Methuen, The author declares no competing financial interests. puter graphics. (It seems almost incredible London, 1932). that D’Arcy Thompson achieved what he 12. De Beer, G. R. Embryos and Ancestors (Clarendon, Oxford, 1940). FURTHER INFORMATION did without this modern aid to morphology, 13. Goldschmidt, R. The Material Basis of Evolution (Yale Wallace Arthur’s homepage: http://www.nuigalway.ie/ working in an era in which the forms of Univ. Press, New Haven, 1940). zoology/w_arthur.html 14. Waddington, C. H. The Strategy of the Genes (Allen & Access to this links box is available online. animals were all individually hand-drawn.) Unwin, London, 1957).

ERRATUM D’Arcy Thompson and the theory of transformations Wallace Arthur Nature Reviews Genetics 7, 401–406 (2006), doi:10.1038/nrg1835 The two photos published in Figure 1 of this article were reproduced with permission from Oxford University Press, which published the images. The original prints are in fact held with the Papers of D’Arcy Wentworth Thompson and have been reproduced courtesy of the University of St Andrews Library, to which we are grateful.

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