Chapter XII: Appendix: To Chapter IX

As it is possible that some readers may care, in spite of its complexity, to enter rather more fully into the peculiar phenomenon {96} of the coupling of characters, I have brought together some further data in this Appendix. In the case we have already considered, where the factors for blue colour and long pollen are concerned, we have been led to suppose that the gametes produced by the heterozygous plant are of the nature 7 BL : 1 Bl : 1 bL : 7 bl. Such a series of ovules fertilised by a similar series of pollen grains will give a generation of the following composition:--

49 BBLL + 7 BBLl + 7 BbLL + 49 BbLl
+ 7 BBLl + 7 BbLL + BbLl
+ BbLl
+ 49 BbLl
\---------------------------------/
177 purple, long

+ BBll + 7 Bbll + bbLL + 7 bbLl + 49 bbll
+ 7 Bbll + 7 bbLl
\-------------/ \-----------/ \-----/
15 purple, 15 red, 49 red,
round long round

and as this theoretical result fits closely with the actual figures obtained by experiment we have reason for supposing that the heterozygous plant produces a series of gametes in which the factors are coupled in this way. The intensity of the coupling, however, varies in different cases. Where we are dealing with another, viz. fertility (F) and the dark axil (D), the experimental numbers accord with the view that the gametic series is here 15 FD : 1 Fd : 1 fD : 15 fd. The coupling is in this instance more intense. In the case of the erect standard (E) and blueness (B) the coupling is even more intense, and the experimental evidence available at present points to the gametic series here being 63 Eb : 1 EB : 1 eB : 63 eb. There is evidence also for supposing that the intensity of the coupling may vary in different families for the same pair of factors. The coupling between blue and long pollen is generally on the 7 : 1 : 1 : 7 {97} basis, but in some cases it may be on the 15 : 1 : 1 : 15 basis. But though the intensity of the coupling may vary it varies in an orderly way. If A and B are the two factors concerned, the results obtained in F_2 are explicable on the assumption that the ratio of the four sorts of gametes produced is a term of the series--

3 AB + Ab + aB + 3 ab
7 AB + Ab + aB + 7 ab
15 AB + Ab + aB + 15 ab, etc., etc.

In such a series the number of gametes containing A is equal to the number lacking A, and the same is true for B. Consequently the number of zygotes formed containing A is three times as great as the number of zygotes which do not contain A; and similarly for B. The proportion of dominants to recessives in each case is 3 : 1. It is only in the distribution of the characters with relation to one another that these cases differ from a simple Mendelian case.

As the study of these series presents another feature of some interest, we may consider it in a little more detail. In the accompanying table are set out the results produced by these different series of gametes. The series marked by an asterisk have already been demonstrated experimentally. The first term in the series, {98} in which all the four kinds of gametes are produced in equal numbers is, of course, that of a simple Mendelian case where no coupling occurs.

+-------+------------------+---------+---------------------------------+
|No. of | Distribution of | No. of | |
|Gametes|Factors in Gametic| Zygotes | Form of F_2 Generation. |
| in | Series |produced.| |
|series.| | | |
+-------+------------------+---------+---------------------------------+
| | AB. Ab. aB. ab. | | AB. Ab. aB. ab. |
| 4 | 1: 1: 1: 1 | 16 | 9 3 3 1 |
| 8 | 3: 1: 1: 3 | 64 | 49 7 7 9 |
| 16 | 7: 1: 1: 7 | 256 | 177 15 15 49* |
| 32 | 15: 1: 1: 15 | 1024 | 737 31 31 225* |
| 64 | 31: 1: 1: 31 | 4096 | 3009 63 63 961 |
| 128 | 63: 1: 1: 63 | 16384 | 12161 127 127 3969* |
| 2n |(n-1): 1: 1:(n-1) | 4n^2 |3n^2-(2n-1) 2n-1 2n-1 n^2-(2n-1)|
+-------+------------------+---------+---------------------------------+

Now, as the table shows, it is possible to express the gametic series by a general formula (n + 1) AB + Ab + aB + (n - 1) ab, where 2n is the total number of the gametes in the series. A plant producing such a series of gametes gives rise to a family of zygotes in which 3n^2 - (2n - 1) show both of the dominant characters and n^2 - (2n - 1) show both of the recessive characters, while the number of the two classes which each show one of the two dominants is (2n - 1). When in such a series the coupling becomes closer the value of n increases, but in comparison with n^2 its value becomes less and less. The larger n becomes the more negligible is its value relatively to n^2. If, therefore, the coupling were very close, the series 3n^2 - (2n - 1) : (2n - 1) : (2n - 1) : n^2 - (2n - 1) would approximate more and more to the series 3n^2 : n^2, _i.e._ to a simple 3 : 1 ratio. Though the point is probably of more theoretical than practical interest, it is not impossible that some of the cases which have hitherto been regarded as following a simple 3 : 1 ratio will turn out on further analysis to belong to this more complicated scheme.

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