Severely Reduced Sexual Reproduction In Northern Populations of a Clonal Plant, Decodon verticillatus (Lythraceae)

Marcel E. Dorken and Christopher G. Eckert

Appendix

Here we calculate the probability of obtaining only heterozygotes in a small sample of ramets in a population practicing a mixture of clonal and sexual reproduction. If an isolated population is founded by a single genotype heterozygous at one marker locus, the genotype frequencies at the marker locus will eventually return to Hardy-Weinberg proportions if there is any sexual recruitment at all. The speed at which equilibrium is attained depends on both the proportion of mature ramets surviving from year to year (R) and the frequency of sexually-produced individuals among the new recruits (S). The recursion equation for the proportion of genotypes heterozygous at the marker locus (H) from year to year is:

 

,

Where t is the number of years since the population was founded, the starting point is H0 = 1, and He is the frequency of heterozygotes at sexual equilibrium. When sex involves only random outcrossing with respect to the marker loci He is the H at Hardy-Weinberg equilibrium (i.e. He = 0.5 with a single heterozygous founding genotype).

At any given time after the population was founded, the probability of obtaining only heterozygotes in a sample of n ramets is:

.

If the founding genotype was heterozygous at k marker loci, the probability of obtaining only one genotype heterozygous at the same k loci among a sample of n ramets is:

.

This probability (b) can be viewed as the likelihood of making a type II error: failing to reject the null hypothesis of no sex when in fact sexual recruitment is occurring at some rate S. If there is some inbreeding in the population, He is reduced and b will be lower than in the case of random outcrossing for any level of S.

We can now roughly calculate b for different values of S and k from our data from New England populations of D. verticillatus. Because D. verticillatus is a long-lived species, it is likely that R ~ 0.9 (Eckert & Barrett 1995). Populations practice a mixture of selfing (30%) and outcrossing (70%), however inbreeding depression is so strong that the genotype frequencies of mature plants conform to Hardy-Weinberg expectations (i.e., He = 0.5, Eckert & Barrett 1993b, 1994b, 1994c). If we assume that the populations in New England have existed for at least 200 years (t = 200, an arbitrary but conservative value), we can use the recursion equation given above to calculate H200 for various levels of S. For each value of S, we then calculate b from H200 for a sample of n ramets (n = 12 in this study). These calculations indicate that it is very unlikely that a sample of 12 ramets would only include heterozygous genotypes if sexual recruitment were occurring even at very low levels. In eight monomorphic populations we detected only a single genotype which was heterozygous for one marker locus (k = 1). This result would be highly unlikely (b = 0.046) even if the level of sexual recruitment (S) was as low as 0.03. In eight populations, the single genotype detected was heterozygous at two loci (k = 2); an unlikely outcome (i.e. b < 0.05) even when S = 0.014. In another population the single genotype detected was heterozygous at all three loci (k = 3). This is unlikely to happen even when S = 0.009. If populations are older than 200 years, as is probably the case since most are found in stable lakes and streams with relatively little human disturbance, then sexual recruitment probably occurs even less frequently.