**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
__H___{0} = 1, and __H___{e} is the frequency
of heterozygotes at sexual equilibrium. When sex involves only random
outcrossing with respect to the marker loci __H___{e} is
the __H__ at Hardy-Weinberg equilibrium (i.e. __H___{e}
= 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, __H___{e} 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.,
__H___{e} = 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
__H___{200} for various levels of __S__. For each value
of __S__, we then calculate b
from __H___{200} 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.