The n model

It is clear that the gene-for-gene relationship evolved to control allo-infection, and that it does so as a system of locking. Mathematically, the most efficient system of locking occurs when:

  • Every individual in both the host and the parasite populations has n/2 genes, where n is the number of pairs of genes in the gene-for-gene relationship.
  • All the n/2 combinations of genes occur with an equal frequency in both the host and the parasite populations.
  • All the n/2 combinations of genes occur with a random distribution in both populations.

Such a system of locking is both efficient and economical. With only twelve pairs of genes, the frequency of matching allo-infections is reduced to 1/924. With twenty pairs of genes, it is reduced to 1/184,756. These frequencies are revealed (Robinson, 1987) by the binomial coefficients of Pascal's triangle (Fig. 4.6).

This postulation of a system of locking is supported by a reductio ad absurdum. Assume that every possible combination of vertical subsystem genes occurs (i.e., 2n combinations, where n is the number of pairs of genes), ranging from an individual with no genes, to an individual with all of the genes. Some combinations of genes would obviously have a survival advantage over others. In particular, the host that possessed every available resistance gene would be matched with the lowest frequency, and would have the highest survival advantage. The host with no resistance genes would be matched with the highest frequency, and would have the lowest survival advantage. Equally, the parasite that possessed every available parasitism gene would be able to match with the highest frequency, and it too would have the highest survival advantage. And the parasite with no genes would have the lowest survival advantage.

These survival advantages would lead both the host and the parasite populations to uniformity. Every individual in each population would possess all of the available vertical genes. But, when this uniformity was reached in both populations, every allo-infection would be a matching infection, and the gene-for-gene relationship would cease to function. "Which is absurd", as Euclid would have said.

n

Binomial Coefficients

0

1

1

1

1

2

1 2

1

3

1

3

3 1

4

1

4

4 1

5

1

5

10

10 5 1

6

1

6

15 20

15 6 1

7

1

7

21

35

35 21 7 1

8

1

8

28

56 70

56 28 8

1

9

1 9

36

84

126 126 84 36 9

1

10

1 10

45

120

210 252

210 120 45

10 1

11

1 11 55

165 330 462 462 330 165 55

11 1

12

1 12 66

220

495

792 924

792 495 220

66 12 1

Figure 4.6 Pascal's Triangle.

Pascal's triangle reveals the binomial coefficients for small samples, such as the number of locks and keys obtained from a given number of vertical resistance or parasitism genes, or the possibilities of boy or girl in single-birth children. For example, when n = 2, in a gene-for-gene relationship, there is one possibility of no vertical genes, two possiiblities of one gene, and one possibility of two genes. Alernatively, with single child births, there is one possibility of two boys, two possiiblities of both sexes (i.e., boy then girl, or girl then boy), and one possibility of two girls. These possibilities are the binomial coefficients which are easily ascertained by adding the two numbers above each, to the right and left. The figures in red show the number of n/2 locks and keys for a given number of pairs of genes in a gene-for-gene relationship.

Consider three possible situations (Table 4.1). In the first situation, every individual in both populations has no vertical genes at all. There is then no vertical subsystem. Every allo-infection is a matching infection.

Table 4.1

Number Of Vertical Genes

Frequency of Matching Allo-infection

Host Population

Parasite Population

None

None

Maximum

n/2

n/2

Minimum

All

All

Maximum

The second situation has already been mentioned. Every individual in both populations has every available vertical gene. There is complete uniformity of vertical genes within the vertical subsystem. Every allo-infection is again a matching infection and the vertical subsystem does not function.

The third situation has already been postulated, and is the middle position. Every individual in both populations has half of the available vertical genes (i.e., n/2, where n is the number of pairs of gene in the gene-for-gene relationship). If we assume also that every combination of n/2 genes occurs with equal frequency, and with a random distribution, in both populations, there will be the maximum possible heterogeneity for that vertical subsystem. The vertical subsystem will then function with its maximum possible efficiency.

It follows that maximum efficiency (i.e., the lowest frequency of matching allo-infection, for a given number of pairs of genes) is obtained with the maximum heterogeneity in both populations. The maximum possible heterogeneity in both populations is achieved with the n/2 model.

Note that, as the number of pairs of genes increases arithmetically, the number of n/2 locks and keys increases geometrically. We can also conclude that the number of pairs of genes in the gene-for-gene relationship is probably related to host population density. Clearly, if the host is a grass growing in a large pure stand, more locks and keys will be necessary than if the host is a tree growing in a mixed forest with a host population density of, say, one tree in ten acres. The number of propagules produced by the parasite will also be related to the number of pairs of genes in the gene-for-gene relationship. These delicate pathosystem balances are maintained by genetic homeostasis and, it should be added, they are easily destroyed in the crop pathosystem.

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