Excerpt

How does structure arise in an embryo? How do the chemicals present in a domain form patterns? In this paper, Alan Turing proposes an answer. This mechanism is now called diffusion-driven instability. Turing’s paper contains other ideas that have triggered large areas of scientific and mathematical research. Here I try to explain Turing’s theory and to discuss very briefly the research it inspired. I will also mention a setback in the search for Turing patterns.

Alan Turing was born in 1912 in Maida Vale, London. He studied logic at King’s College, Cambridge, and proposed the concept of the Turing machine. During the war he worked for the Foreign Office and, in 1951, he was elected a Fellow of the Royal Society.

“The Chemical Basis of Morphogenesis” (published in August 1952) was the penultimate journal article that Turing published before his death in 1954 (Turing 1953; Newman 1955). It pre-dates Watson and Crick’s (1953) paper on the structure of DNA and Crick and others’ (1961) discovery of the genetic code.

The key ideas in the paper are described in sections 1, 4, and 11. The key mathematics is in sections 6, 7, and 8.1. Section 8.1 describes the stationary patterns now referred to as Turing patterns. Section 11 gives a biological interpretation of his results. Finally, section 12 aims to explain gastrulation, a process that initiates the formation of the major axis in amniote embryos (e.g., those of birds, reptiles, and mammals). Turing focuses on the breakdown of symmetry on the surface of a sphere, which is of approximate relevance to gastrulation during the development of mouse embryos. However, human and chick embryos are more disc-like initially (e.g., Page et al. 2001).

Bibliography

Balaskas, N., A. Ribeiro, J. Panovska, E. Dessaud, N. Sasai, K. M. Page, J. Briscoe, and V. Ribes. 2012. “Gene Regulatory Logic for Reading the Sonic Hedgehog Signaling Gradient in the Vertebrate Neural Tube.” Cell 148 (1–2): 273–284. https://doi.org/10.1016/j.cell.2011.10.047.

Biancalani, T., D. Fanelli, and F. Di Patti. 2010. “Stochastic Turing Patterns in the Brusselator Model.” Physical Review E 81 (4): 046215. https://doi.org/10.1103/PhysRevE.81.046215.

Castets, V., E. Dulos, J. Boissonade, and P. De Kepper. 1990. “Experimental Evidence of a Sustained Standing Turing-Type Nonequilibrium Chemical Pattern.” Physical Review Letters 64 (24): 2953–2957. https://doi.org/10.1103/PhysRevLett.64.2953.

Collier, J. R., N. A. M. Monk, P. K. Maini, and J. H. Lewis. 1996. “Pattern Formation by Lateral Inhibition with Feedback: A Mathematical Model of Delta-Notch Intercellular Signalling.” Journal of Theoretical Biology 183 (4): 429–446. https://doi.org/10.1006/jtbi.1996.0233.

Crampin, E. J., E. A. Gaffney, and P. K. Maini. 1999. “Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation.” Bulletin of Mathematical Biology 61 (6): 1093–1120. https://doi.org/10.1006/bulm.1999.0131.

Crick, F., L. Barnett, S. Brenner, and R. J. Watts-Tobin. 1961. “General Nature of the Genetic Code for Proteins.” Nature 192 (4809): 1227–1232. https://doi.org/10.1038/1921227a0.

Dawes, J. H. P. 2016. “After 1952: The Later Development of Alan Turing’s Ideas on the Mathematics of Pattern Formation.” Historia Mathematica 43 (1): 49–64. https://doi.org/10.1016/j.hm.2015.03.003.

Dessaud, E., A. P. McMahon, and J. Briscoe. 2008. “Pattern Formation in the Vertebrate Neural Tube: A Sonic Hedgehog Morphogen-Regulated Transcriptional Network.” Development 135 (15): 2489–2503. https://doi.org/10.1242/dev.009324.

Glover, J. D., K. L. Wells, F. Matthäus, K. J. Painter, W. Ho, J. Riddell, J. A. Johansson, et al. 2017. “Hierarchical Patterning Modes Orchestrate Hair Follicle Morphogenesis.” PLoS Biology 15 (7): e2002117. https://doi.org/10.1371/journal.pbio.2002117.

Karig, D., K. M. Martini, T. Lu, N. A. DeLateur, N. Goldenfeld, and R. Weiss. 2018. “Stochastic Turing Patterns in a Synthetic Bacterial Population.” Proceedings of the National Academy of Sciences 115 (26): 6572–6577. https://doi.org/10.1073/pnas.1720770115.

Lengyel, I., and I. R. Epstein. 1992. “A Chemical Approach to Designing Turing Patterns in Reaction–Diffusion Systems.” Proceedings of the National Academy of Sciences 89 (9): 3977–3979. https://doi.org/10.1073/pnas.89.9.3977.

Müller, P., K. W. Rogers, B. M. Jordan, J. S. Lee, D. Robson, S. Ramanathan, and A. F. Schier. 2012. “Differential Diffusivity of Nodal and Lefty Underlies a Reaction–Diffusion Patterning System.” Science 336 (6082): 721–724. https://doi.org/10.1126/science.1221920.

Murray, J. D. 1993. Mathematical Biology. 2nd ed. Berlin, Germany: Springer-Verlag.

Newman, M. H. A. 1955. “Alan Mathison Turing, 1912–1954.” Biographical Memoirs of Fellows of the Royal Society 1:253–263. https://doi.org/10.1098/rsbm.1955.0019.

Nüsslein-Volhard, C., and E. Wieschaus. 1980. “Mutations Affecting Segment Number and Polarity in Drosophila.” Nature 287 (5785): 795–801. https://doi.org/10.1038/287795a0.

Page, K. M., P. K. Maini, N. A. M. Monk, and C. D. Stern. 2001. “A Model of Primitive Streak Initiation in the Chick Embryo.” Journal of Theoretical Biology 208 (4): 419–438. https://doi.org/10.1006/jtbi.2000.2229.

Painter, K. J., J. M. Bloomfield, J. A. Sherratt, and A. Gerisch. 2015. “A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations.” Bulletin of Mathematical Biology 77 (6): 1132–1165. https://doi.org/10.1007/s11538-015-0080-x.

Raspopovic, J., L. Marcon, L. Russo, and J. Sharpe. 2014. “Digit Patterning is Controlled by a Bmp-Sox9-Wnt Turing Network Modulated by Morphogen Gradiants.” Science 345 (6196): 566–570. https://doi.org/10.1126/science.1252960.

Turing, A. M. 1952. “The Chemical Basis of Morphogenesis.” Philosophical Transactions of the Royal Society of London B 237 (641): 37–72. https://doi.org/10.1098/rstb.1952.0012.

—. 1953. “Some Calculations of the Riemann ζ-Function.” Proceedings of the London Mathematical Society 3 (1): 99–117. https://doi.org/10.1112/plms/s3-3.1.99.

Watanabe, M., and S. Kondo. 2015. “Is Pigment Patterning in Fish Skin Determined by the Turing Mechanism?” Trends in Genetics 31 (2): 88–96. https://doi.org/10.1016/j.tig.2014.11.005.

Watson, J. D., and F. H. C. Crick. 1953. “Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid.” Nature 171 (4356): 737–738. https://doi.org/10.1038/171737a0.

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