A population of autotetraploid maize may consist of a bare majority of euploids and many different aneuploids. Aneuploidy is the result of the numerical non-disjunction (a 3 to 1 division) of the chromosomes of a quadrivalent. The frequency of numerical non-disjunction has been estimated (Doyle, 1973 Theor. Appl. Genet. 43:139-146) to be about 3% per quadrivalent.

A theoretical study of aneuploidy in autotetraploid populations even using the simplest assumptions requires a great amount of calculations. If we limit our consideration to genotypes that have four or less chromosomes in excess or in deficiency or a combination thereof, we must deal with 15 different gametic types and 30 zygotic types (these genotypes are shown along the left margins of the first two tables).

If we assume that sets of homologous chromosomes disjoin independently of each other and that they all have the same frequency of numerical nondisjunction, then the gametic output of a 4n plant may be calculated by expanding the trinomial, (m + t + d)n, where m, t, and d are the frequencies of functioning gametes with 1, 3, or 2 chromosomes, respectively, from a set of four homologous chromosomes, and n is the number of chromosomes in the genome. This is shown in Table 1.

The gametic output of aneuploid plants may be calculated by separating sets with different chromosome numbers and then combining them (Table 2).

Thus the genotype I (4n-1-1 + 1) has the formula
(m + t + d)^{7}(M + D1)^{2}(T
+ D_{2}). The seven 4-chromosome sets are
represented by (m + t + d)^{7}, the two 3-chromosome
sets are expressed (M + D_{1})^{2}
andthe 5-chromosome set is (T + D_{2}). A
3-chromosome set will give gametes with 1 and 2 chromosomes from the set
with frequencies symbolized by M and D1 respectively.
Likewise a 5-chromosome set will give gametes with 3 (T) and 2 (D_{2})
chromosomes. In this case the expected frequency of 2n (a) gametes would
be d^{7}D_{1}^{2}D_{2},
for example. Space does not permit an expansion of these formulae here.
Genotypes that have 2-chromosome sets as in P (4n-2) would form a bivalent
for that set and only one chromosome would go to all gametes. Genotypes
with 6-chromosome sets Q (4n + 2) are assumed to always contribute 3 chromosomes
from the set to all gametes. This is probably a poor assumption, but is
necessary for the simplicity of the model.

After computing the expected frequencies of these 15 kinds of gametes, they must be united in 120 different combinations (See Table 3). Depending on whether the excess and deficient chromosomes are homologous or not, different zygotic genotypes are possible in most cases. A union of b (2n-1) and c (2n + 1) gametes will yield A (4n) and F (4n-1 + 1) zygotes with frequencies of 1/n and n-1/n, respectively. This situation is quite complex and the relative frequencies of different zygotes that result from the union of gametic types require elaborate formulas. Along with the zygotes A to D', there are a great number of aneuploids with more than 4 plus or minus chromosomes, symbolized in Table 3 by #.

To run through one generation of random mating or self-fertilization of an autotetraploid population using a desk calculator required two days. Consequently, my son, Ted Doyle, devised computer programs that can do a generation in 20 seconds. The program has nine inputs, m, t, d, D1, D2, M, T, n, and g (the number of generations desired-essentially unlimited). The genome chromosome number may vary from 8 to 32. Under 8 chromosomes the aneuploidy component is too limited and over 32 chromosomes overloads the combination-figuring component of the program.

A number of insights into the behavior of aneuploidy in autotetraploid populations have been made. Some of these insights are obvious after they have been demonstrated in the model. For example, the greater the value of n the more aneuploids there will be in the population, but the relative frequency of genotypes P through D' will be less. Random mating populations achieve an equilibrium state after which there is no change in the relative frequencies of the various genotypes. Self-fertilizing populations continue to become more and more aneuploid until they all become B' (4n-2-2), C' (4n + 2 + 2), or D' (4n-2 + 2). The approach to this state is very slow.

Copies of these programs will be sent on request.

Table 1. Formulae for estimating the gamete production of a euploid (4n) autotetraploid.

Table 2. Formulae for gamete production for 30 possible chromosome constitutions.

Table 3. Zygotes formed from combining gametes

G. G. Doyle

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