Abstract
The seventh section of Gregor Mendel’s famous 1866 paper contained a peculiar mathematical model, which predicted the expected ratios between the number of constant and hybrid types, assuming self-pollination continued throughout further generations. This model was significant for Mendel’s argumentation and was perceived as inseparable from his entire theory at the time. A close examination of this model reveals that it has several perplexing aspects which have not yet been systematically scrutinized. The paper analyzes those aspects, dispels some common misconceptions regarding the interpretation of the model, and re-evaluates the role of this model for Mendel himself. In light of the resulting analysis, Mendel’s position between nineteenth-century hybridist tradition and twentieth-century population genetics is reassessed, and his sophisticated use of mathematics to legitimize his innovative theory is uncovered.
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Notes
Throughout the paper, the English translations of Mendel’s text are based on Eva R. Sherwood’s translation which appeared in Stern and Sherwood (1966). Nevertheless, several modifications in the wording have been deemed necessary, for reasons that will become apparent further below. Translations from other German works are the author’s, unless stated otherwise.
Premises (ii), (iii) and (iv) are also necessary to justify the starting point of the model, where precisely 1 A, 2 Aa and 1 a are presumed to be result of self-pollinating a single Aa plant.
Gliboff’s term ‘Austro-Ungerian’ is an allusion to Franz Unger, a professor of plant physiology at the Vienna University whose influence on Mendel’s intellectual surrounding is brought to the fore in Gliboff’s paper. Margaret Campbell also noted the importance of Unger, whose “warm endorsement” of theories centering on ratios was probably “influential in the formation of Mendel’s views concerning the nature of a satisfactory theory;” see Campbell (1982, p. 48).
There are various algebraic ways through which one may reach the \( 2^{n} - 1:2:2^{n} - 1 \) ratio, and which one Mendel used will probably remain a matter of speculation. In principle, Mendel could have just written down the numbers of individuals obtained as a result of his assumptions and noted the emerging regularity. But given Mendel’s mathematical training and skills, it seems improbable that he would have presented such regularity without attempting to substantiate it by some kind of proof. The one I present here is what I consider to be the simplest and most intuitive proof for the resulting ratio under the given assumptions.
As was already apparent in the previous passages of this paper, for any given trait Mendel denoted the constant forms by using a single letter (A or a), and the hybrids by using two letters (Aa); see in this respect (Olby 1979, pp. 57–59). After the “rediscovery” of Mendel's work and as a result of an improved understanding of the mechanisms of sexual reproduction it became customary to denote also the constant forms using two letters (AA, aa). In the present paragraph I use the modern accepted symbolism.
More recently, a similar mistake was made by the historian Margaret Campbell. In a footnote in her 1982 article, Campbell wrote that Mendel’s model “starts with 4 plants: by the tenth generation there are 2,048 plants” (Campbell 1982, p. 61, fn. 25). Campbell seems to have misread Mendel’s own example, where Mendel wrote that “In the tenth generation, for example, 2n − 1 = 1,023. Therefore, of each 2,048 plants arising in this generation, there are 1,023 with the constant dominating trait, 1,023 with the recessive one, and only two hybrids” (Stern and Sherwood 1966, p. 17; emphasis added). This was an example of ratio, not of quantity, as the word “each” (in German, je) clearly testifies. The total number of plants in the tenth generation, according to the model, would actually amount to 1,048,576 (= 410), and this, too, only on the condition that we accept the assumption of four progenies per plant, which was never meant to be anything more than an auxiliary assumption. See also the following description by Alain F. Corcos and Floyd V. Monaghan: “In addition, the generalization expressed in the last line of the right half of the table enabled Mendel to calculate how many true-breeding dominating forms, how many true-breeding recessive forms, and how many hybrid forms there would be in any generation” (Corcos and Monaghan 1993, p. 98; emphasis added). The number of forms actually stays the same – there are three only – whereas the formula allows for no prediction of the absolute numbers of individuals falling under each of these forms. On the same page, Corcos and Monaghan also supplied an evidently untenable description for the way in which Mendel arrived at his 2n − 1:2:2n − 1 ratio, one that reinforces the impression that these authors failed to properly distinguish numbers from ratios in the MMM.
With regard to Bateson, there is reason to believe that he indeed did not get to the bottom of the MMM’s meanings. This can be inferred from his conspicuous abstinence to attack Weldon’s misuse of the MMM while reprimanding all other parts of Weldon’s analysis of Mendel’s work. Compare Bateson (1902), Weldon (1902) and see Froggat and Nevin (1971, p. 18) for relevant background. This issue, however, deserves separate analysis, and I intend to address it more thoroughly in a different paper.
Mendel was well aware of this. When discussing his Hieracium experiments, Mendel recalled “the knowledge, obtained by experiment, that in hybrids self-fertilization is always prevented if pollen of one of the parent-forms reaches the stigma.” See Mendel’s “On Hieracium-Hybrids obtained by Artificial Fertilisation” in Stern and Sherwood (1966, p. 51). In a letter written in 1873 Mendel mentioned that he found no reason to refute Gärtner who considered “the pollen of the hybrid to be ineffectual in competition with the parental pollen.” See Mendel’s Letter to Carl Nägeli in Stern and Sherwood (1966, p. 100). The danger which the parental types posed was immaterial if self-fertilization could be strictly maintained, yet this, too, was a non-trivial challenge. Mendel discussed the difficulty of preventing cross-fertilization with regard to Hieracium: “…many forms of insects, notably the industrious hymenoptera, visit the flowers of Hieracia with great zeal and are responsible for the pollen, which easily sticks to their hairy bodies, reaching the stigmas of neighbouring plants” (Stern and Sherwood 1966, p. 54). See also Wiegmann’s claims: “It appears further, from the behavior of the Leguminosae and of cabbage, that agronomists and gardeners cannot be careful enough in the arrangement of their fields in order not to suffer from the great damage through hybrid fertilization occurring even in the first year” (quoted in Roberts 1965, p. 164).
This was as true for Gärtner and Koelreuter as it was for other naturalists of the time. In 1929 the Canadian botanist H. F. Roberts published a survey of “Plant Hybridization before Mendel.” Roberts looked into the works of a wide range of European scholars interested in hybridization; judging by his report, none of them conducted experiments which could fit into the Mendelian scheme in the sense of continuing reproduction of parental (constant) types. See Roberts (1965, pp. 1–240). See also Olby (1985, pp. 1–88, 142–185).
This is especially striking since in his concluding remarks Mendel did discuss the future of hybrids, did refer to his predecessors’ results in this field and did make some mathematical predictions on the numbers of hybrid forms expected as a result of one polyhybrid cross involving seven character pairs in the first generation of hybrids. Yet all along, and although it could have been of clear relevance, Mendel did not refer to the respective results of an obviously undeveloped generalized form of the MMM. See Mendel (1866, pp. 38–47).
Hoffmann’s quotation is taken from Keynes (2001, p. 7).
Other early references to Mendel’s work could also be interpreted as alluding to the MMM, but they are much less conclusive in this regard. For example, Wilhelm Olbers Focke’s 1881 remark that “Mendel believed he had found constant numerical relationships between the types of the crosses” (quoted in Olby 1985, p. 227) could allude to the MMM, but Focke could just as well have had other parts of Mendel’s paper in mind, such as the A + 2Aa + a law. The same holds true for most of the other early allusions to Mendel’s paper, which can be interpreted in both ways, and see Olby (1985, pp. 219–234), Orel (1996, 277ff).
The difference between the two quotations (Hybrids have a tendency/Hybrids are inclined) stems from the fact the Callender relied on another translation; it has no effect on our current discussion.
Aside from in the introduction and conclusion, the only time Koelreuter is mentioned in Mendel’s paper is the one noted here, with relation to the MMM; Gärtner is mentioned once more, just following the definition of the terms “dominant” and “recessive” (Mendel 1866, p. 11).
Mendel explicitly stated that the assumption of four descendants per plant is only supplied “for brevity’s sake” (Der Kürze wegen). See Mendel (1866, p. 18).
Especially conspicuous in this regard is also Jendrassik (1911a), as well as the works mentioned in footnote 7, above.
See Brannigan (1979), Olby (1979), Callender (1988), Corcos and Monaghan (1990). For a concise summary of these authors’ point of view, see Sapp (1990) (Sapp refers to Callender, Brannigan and Olby; Monaghan and Corcos’s article was not yet published at the time). For a critical discussion of these works, see Falk and Sarkar (1991), Orel and Hartl (1994). An attempt of reconciliation between the contrasting views is offered in Müller-Wille and Orel (2007, pp. 211–215).
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Acknowledgments
I would like to thank Raphael Falk, Shulamit Volkov, Eva Jablonka, Snait Gissis, Staffan Müller-Wille, Michael Dietrich and the participants of the 2012 Geneva Second European Advanced Seminar in the Philosophy of the Life Sciences, as well as two anonymous reviewers for their helpful comments and advice on earlier drafts of this paper.
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Teicher, A. Mendel’s use of mathematical modelling: ratios, predictions and the appeal to tradition. HPLS 36, 187–208 (2014). https://doi.org/10.1007/s40656-014-0019-9
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DOI: https://doi.org/10.1007/s40656-014-0019-9