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A hapless mathematical contribution to biology

Chromosome inversions in Drosophila, 1937–1941

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Abstract

This is the story, told in the light of a new analysis of historical data, of a mathematical biology problem that was explored in the 1930s in Thomas Morgan’s laboratory at the California Institute of Technology. It is one of the early developments of evolutionary genetics and quantitative phylogeny, and deals with the identification and counting of chromosomal inversions in Drosophila species from comparisons of genetic maps. A re-analysis of the data produced in the 1930s using current mathematics and computational technologies reveals how a team of biologists, with the help of a renowned mathematician and against their first intuition, came to an erroneous conclusion regarding the presence of phylogenetic signals in gene arrangements. This example illustrates two different aspects of a same piece: (1) the appearance of a mathematical in biology problem solved with the development of a combinatorial algorithm, which was unusual at the time, and (2) the role of errors in scientific activity. Also underlying is the possible influence of computational complexity in understanding the directions of research in biology.

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Notes

  1. Theodosius Dobzhansky, letter to Milislav Demerec 1936

  2. According to the vocabulary of Zuckerkandl and Pauling (1965), this is the directly transmitted hereditary material, and not one of its products, see also Dietrich (1998).

  3. Despite crucial differences in the biological objects have also been discussed (Darden, 2005).

  4. I.e not explicit.

  5. The term homology, in the sense of “common evolutionary origin”, was not commonly used at the beginning of the 20th century. The terminology was discussed and ranged from “allelomorph” to “corresponding”. I use the current terminology for the sake of consistency and clarity.

  6. Renamed melanogaster shortly after.

  7. Later named the X-chromosome in order to emphasize its peculiarity. The link between chromosomes and linkage groups was already well established, as can be seen by the natural use of “chromosome” in genetic studies from the 1910s onward.

  8. I.e homologous, see footnote 3. Homology was deduced from the similarity of phenotype variations during crossing experiments.

  9. Several types of translocations, i.e. other mutations of the linear organization of genes along chromosomes, were predicted at the same time (Bridges, 1917; Mohr, 1919; Morgan, Bridges, & Sturtevant, 1925) and later demonstrated using cytology (Muller, 1929; Dobzhansky, 1930). They were generally considered to be “deficiencies”, or abnormalities of karyotypes, possibly resulting from mutagenic conditions. By contrast, inversions were immediately seen as evolutionary patterns susceptible to being used in differentiating species, and thus be a character for taxonomy. Translocations were later used in plant taxonomy by Babcock and Stebbins (1938).

  10. Examined by Kohler (1994), who writes that the unpublished part is of wider significance.

  11. It is a coincidence that the maximum number found in 1995 is precisely the one that biologists struggled with in 1937. That we have not been able to greatly improve our handling of the data is indicative of the inherent computational complexity of the problem.

  12. Note that the corrected values given here were obtained only with the published data and the statistical test proposed by the original authors. However this analysis requires computational tools that were not available at the time. There would probably be a lot more to discover if we were to redo this analysis with new data.

  13. A bona fide statistical test in this case would require a p-value rather than standard deviations. This was not considered in the 1937 and 1941 articles but it is possible to compute an empirical p-value from a sample of 1,000 uniformly sampled random permutations. This gives a probability of achieving six or less inversions for 13 genes of 0.06, a probability of achieving two or less inversions for six genes of 0.2, and a probability of achieving three or less inversions for seven genes of 0.35. Considering each chromosome independently is hardly conclusive. When all chromosomes are taken into account, gene inversions can be considered significantly different from what would be expected at random based on the usual significance thresholds.

  14. This story is attributed to Charlie Munger in Belevin (2007).

  15. Note that this contrasts with the history of protein sequence alignment, where it became possible to compare two related sequences without excessive mathematical involvement (see, for example, (Margoliash, 1963)). I am not saying that sequence alignment did not pose an interesting mathematical problem but it was inherently easier to solve with the intuitive ideas of biologists than computing an inversion distance.

  16. Finding the minimum number of inversions to transform a sequence of letters into alphabetical order is provably intractable (Caprara & Lancia, 2000).

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Acknowledgements

Thanks to Istvan Miklos for showing me the 1937 article by Sturtevant, and to Vincent Daubin and Bastien Boussau for giving me the opportunity to present part of this work at the Jacques Monod conference in 2016, “Molecules as documents of evolutionary history: 50 years after”. Thanks also to several anonymous historians who have kindly helped me improve the historical aspects of this article, find the relevant secondary literature and get rid of most teleological and anachronistic arguments.

Funding

The funded was grant by Agence Nationale de la Recherche (ANR-19-CE45-0010 Evoluthon).

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  1. Laboratoire de Biométrie et Biologie Évolutive UMR5558, Université de Lyon, 69622, Villeurbanne, France

    Eric Tannier

  2. Centre de Recherches de Lyon, Inria, Institut National de Recherches en Informatique et Automatique, Villeurbanne, France

    Eric Tannier

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Tannier, E. A hapless mathematical contribution to biology. HPLS 44, 34 (2022). https://doi.org/10.1007/s40656-022-00514-x

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