i-/eredity72 (1994) 86—94 Received 1 June 1993 Genetical Society of Great Britain

Mating system of : an autotetraploid

DARLYNE A. MURAWSKI*, THEODORE H. FLEMING, KERMIT RITLAND1 & J. L. HAMRICK Departments of Botany and Genetics, University of Georgia, Athens, GA 30602, tDepartment of Biology, University of Miami, Coral Gables, FL 33124, USA and Department of Botany, University of Toronto, Toronto Canada, M5S3B2

Themating system of the Mexican subdioecious columnar cactus, Pachycereus pringlei (Cardón), was examined by allozyme analysis. Tetrasomic patterns of inheritance were found for all polymorphic loci, indicating that the is an autotetraploid. A model is presented that expands Ritland's (1 990a) mixed mating model for autotetraploids to incorporate an arbitrary number of alleles per locus. This model is applied to the progeny arrays of several female and hermaphroditic individuals of P. pringlei. P. pringlei exhibited a complex breeding system; the multilocus estimate of outcross- ing, tm, for female cacti was 0.949 and for hermaphrodites was 0.30 1. These results are discussed in the context of inbreeding depression and the evolution of the breeding system of this species.

Keywords:allozymeanalysis, autotetraploidy, cactus, mating system, Mexico, Pachycereus.

Introduction and unbalanced heterozygotes at all of its loci, an observation that suggested autopolyploidy. Consistent Naturallyoccurring autopolyploids have long been with this observation was the haploid chromosome considered rare and maladaptive on the theoretical number of 22 reported for P. pringlei (Pinkava & grounds that chromosome pairing difficulties during McLeod, 1971; Pinkava et al., 1973) which is double meiosis would prevent stable and reliable genetic trans- that of the majority of cacti including other members of mission. Studies that used morphological traits to Pachycereus (Katsgiris, 1952; Pinkava et a!., 1977). P. distinguish auto- and allopolyploid modes of speciation pringlei is also an unusual cactus because it is subdioe- were often speculative as experimentally derived auto- cious with male, female, hermaphroditic and neuter polyploids do not always resemble the parental diploid individuals occurring at varying frequencies in its (Grant, 1981). However, a growing number of studies populations (T. H. Fleming, unpublished data). using genetic information in addition to cytological and In this paper allozyme assays of progeny arrays morphological traits confirm that autotetraploids are produced by controlled crosses are used to confirm the more common than was originally thought. Species that nature of polyploidy (auto vs. alloploidy). Additional have been shown to be autopolyploid or to have allo- progeny arrays derived from open-pollinated flowers zyme banding patterns that are consistent with auto- are used to compare the outcrossing rates of female ploidy include alfalfa, Medicago sativa (Quiros, 1982), and hermaphroditic individuals. A multiallelic auto- media (Bayer et al., 1990), Heuchera tetraploid mating system model is developed to utilize micrantha (Ness et al., 1989), Maclura pomifera the highly polymorphic allozyme loci found in Cardón. (Schnabel et al., 1991) and Dipteryx panamensis (Murawski & Hamrick, unpublished data). During an initial allozyme survey of three large, columnar cacti of the Sonoran desert (J. L. Hamrick & Materialsand methods T. H. Fleming, unpublished data), the Cardón cactus, Cardón,Pachycereus pringlei, is a large columnar Pachycereus pringlei, was found to have an unusually cactus (up to 15 m) restricted to the coastal desert por- low proportion of homozygosity and both balanced tions of the Mexican states of Sonora and Baja Califor- nia. Its breeding system, which will be described in *Correspondence: Dr D. A. Murawski, Harvard University detail elsewhere, is cryptically subdioecious with four Herbaria, 22DivinityAvenue, Cambridge, MA 02138, USA. sexual types in the Bahia Kino region of Sonora. These 86 MATING SYSTEM OF CARDON CACTUS 87 types and their relative frequencies for 211 adult indi- for mating system analyses, flower bud tissue was viduals in our main study area include male steriles (42 collected from 14 females and 14 hermaphrodites on per cent), hermaphrodites (28 per cent), female steriles 10 May 1990. Tissue was refrigerated until it was air- (28 per cent) and neuters (male and female steriles, 2 shipped on ice to the laboratory at the University of per cent). The Cardón flowering season lasts from late Georgia. To estimate outcrossing rates, on 17 June March through late May with a flowering peak in late 1990 two fruits from open-pollinated flowers were April. Flowers open shortly after sunset and close by collected from each of these individuals and their seeds noon the next day. At night they are visited by the nec- were removed and air-dried. 17 June was early in the tar and pollen-eating bat Leptonycteris curasoae and fruit ripening period and although care was taken to several species of small moths. Beginning at sunrise, select the largest, ripest fruit from each , not all of flowers are visited by a variety of bird species and bees these fruits contained mature, germinable seeds. (primarily the exotic Apis mellifera). Pollinator exclu- sion experiments indicate that nocturnal and diurnal visitors are equally effective pollinators although the Electrophoresis large, pollen and nectar-rich flowers are most strongly Seedswere germinated on moist filter paper in petri coevolved with bats (T. H. Fleming, Homer & Tuttle, dishes inside growth chambers. Germination occurred unpublished data). within 1 week. Seedlings at the cotyledon stage (2 weeks old) were prepared for electrophoress by crush- ing with the PVP extraction buffer of Mitton et a!. Studyarea and population samples (1979). Adult (bud) material was crushed in liquid Fieldaspects of this study were conducted between 15 nitrogen before adding the extraction buffer. Wicks April and 30 June 1989 and 1 April and 29 June 1990 containing seedling and adult material were sorted in near Bahia Kino, Sonora, Mexico (28° 50' N). This site an ultracold freezer (—70°C)until needed for electro- is part of the Central Gulf Coast region of the Sonoran phoresis. Our goal was to assay 64 seeds from each desert and is dominated by Bursera microphylla and plant (32 seeds from each fruit if two fruits were two species of Jatropha (Shreve & Wiggins, 1964). collected) although actual progeny numbers were Other common include Larrea tridentata, Cerci- reduced in some families because of poor germination. dium microphyllum, Olneya tesota and Fouquieria Seven enzyme systems with eight loci were assayed. splendens. In addition to Pachycereus pringlei, three Additional polymorphic loci were detected but not other columnar cactus species, Carnegia gigantea, assayed due to inconsistent resolution. The enzyme Stenocereus thurberi and schotii are systems and locus designations are as follows: phos- common in the area. Annual rainfall at Bahia Kino phoglucomutase (Pgml, Pgm2), phosphoglucoiso- averages about 90 mm; most rain falls in July and merase(Pgi),alcoholdehydrogenase (Adhi), August. Topography in the area includes gravelly flat- isocritrate dehydrogenase (Idh), shikimate dehydro- lands punctuated by rugged hills 200—450 m above sea genase (Skdh), malate dehydrogenase (Mdh), aspartate level. Our main study area, located about 9 km amino transferase (Aat2). Adult genotypes were scored northeast of Nuevo Bahia Kino, encompassed about for PGM, PGI, IDH and SKDH, and inferred from 100 ha and contained 211 marked and sexed adult progeny genotypic proportions for MDH and AA T Cardóns. The buffer systems and staining recipes of Soltis et a!. To determine rates of self-fertilization by herma- (1983) were employed: system 4 for IDH and SKDH, phrodites in the absence of pollinators, four flowers on system 7 for MDH and AA T and a modification of each of 14 individuals were bagged with bridal veil system 6 for PGM, PGI and I4DH. netting shortly before opening; the netting was removed the following afternoon after the flowers had Data analysis closed. Half of the flowers were bagged on 30 April 1990 and the other half on 8 May. Seeds from the five Genotypefrequencies resulting from the selfed crosses surviving fruits (from three individuals) were collected were compared with expected ratios based on tetra- in late June. These were then used to test progeny somic inheritance. Expected ratios for balanced heter- genotypic proportions against those expected given ozygotes (aabb) were 1(aaaa): 8(aaab): 18(aabb): tetrasomic inheritance. 8(abbb): 1(bbbb) and for unbalanced heterozygotes An initial survey of allozyme polymorphism was were 1(aaaa or bbbb): 2(aaab or abbb): 1(aabb). These based on a sample of seeds from 14 open-pollinated ratios were determined by assuming that diploid individuals(eight females,six hermaphrodites) gametes are produced. A given allele of the diploid pair collected in 1989. To determine maternal genotypes is chosen at random without replacement with a proba- 88 D. A. MURAWSKI ETAL. biity of 1/4 and a second allele is chosen at random families because of the low number of families from the remaining genes with probability 1/3 (Wendel analysed. Consequently, standard errrors were not &Weeden, 1989). calculated for ovule allele frequencies. Fifty bootstraps Family genotypic data from open-pollinated fruits of were used for population and individual fruit estimates. females and hermaphrodites were analysed separately Differences between ovule and pollen allele frequen- with a multiallelic extension of Ritland's (1 990a) 'tetrat' cies and between the pollen allele frequencies received program based on the procedures described in the by female and hermaphrodite plants were tested with Appendix. Multilocus and single-locus estimates of the chi-squared statistic, where x2= NFST(a —1); outcrossing rates were obtained for females and d.f. =(a—1).FST is a measure of the genetic diversity hermaphrodites. In addition, multilocus estimates were between the two groups (i.e. pollen vs. ovule or female calculated for individual fruits of hermaphroditic vs. hermophrodites), N is the sum of the gametes in the plants. two groups. The bootstrap method was used to determine the variance of the estimates by creating replicate data sets Results from progeny sampled randomly with replacement from the original data set. The distribution of the Allpolymorphic enzyme systems observed in P. sample estimates determines the variance of estimates. pringlei showed evidence of the tetrasomic inheritance This sampling was within families rather than between patterns indicative of autotetraploids (Fig. 1 and Table 1). Both balanced and unbalanced heterozygotes (corresponding to equal and unequal dosages of differ- ent alleles) were present. Owing to the isozyme diver- I sity found in this plant, we observed as many as four I COM 1' a ii' allozyme alleles per locus in an individual plant (Fig. la). Additional polymorphic loci with similar tetra- 1223 2223 1224 1224 1224 2222 2222 1234 2234 1222 2244 2333 somic patterns were found but not scored because of inconsistent resolution. There was no evidence of fixed

- heterozygotes as might be seen in allotetraploids. 0• Further evidence for autotetraploidy in P. pringlei is N) provided by the genotypic frequencies of selfed crosses that did not differ significantly from the expected fre-

1112 1111 1222 1222 1112 1122 1222 1112 2222 1122 1122 1122 1112 quencies of 1:8:18:8:1 for balanced heterozygous mothers and 1:2:1 for unbalanced heterozygotes (Table 1). These genotypic proportions in selfed pro- -L geny are indicative of independent chromosomal segregation (as in Weeden & Wendel, 1989). For most loci the allele frequencies of the pollen and 1111 1111 1112 1112 1111 1112 1111 1112 1122 1111 1112 1111 ovule contributions to the progeny genotypes are similar (Table 2). There are, however, a few loci for & I 4 I which pollen and ovule allele frequencies are signifi- a. ru

Ui cantly different (P < 0.05). These include the Pgi locus for the female individuals and the Pgm2, Pgi and Aat2

2222 2222 2222 2222 2222 2222 2233 2233 2223 2222 2223 2222 loci for hermaphroditic individuals. Pollen allele fre- quencies received by females and hermaphrodites are also similar for the most part. Only the Skdh and Aat2 CD 0 $ ft U $1 94 • $4 loci are significantly different (P <0.05) between ft Ce hermaphroditic and female individuals. The multilocus outcrossing estimate (tm =0.301)for 2223 2233 2233 2233 2233 2233 2223 2223 2233 2233 2333 2333 333 hermaphrodites was much lower than the estimate for Fig. I Photographs of representative starch gels of Pachy- females (tm =0.949)(Table 3). The average single locus cereus pringlei. Numbers on the right side of the photographs estimate did not differ significantly from the multilocus designate alleles. Individual tetraploid genotypes are labelled estimate in females but was significantly lower in beneath each lane. Note the presence of balanced and unbalanced heterozygotes at each locus. (a) Skdh, a mono- hermaphrodites (P <0.05). This difference suggests the possibility of consanguineous matings but may also be mer; (b) Aat2, a dimer; (c) Mdhl, a dimer; (d) Idh, a dimer; (e) Pgi, a dimer. due to correlated or non-random mating. Examining MATING SYSTEM OF CARDON CACTUS 89

Table I Progeny genotypic frequencies of selfed crosses of Pachycereus pringlei. Expected progeny genotypic frequencies for balanced heterozygote mothers (aabb) assuming tetrasomic inheritance is the ratio 1:8:18:8:1. For unbalanced heterozygotes (e.g. abbb and aaab) the expected ratio is 1:2:1

Significance No. ofoffspringby genotype level of deviation MaternalMaternal from expected Locus ID genotypeaaaaaaabaabbabbbbbbbprogeny frequency

Aat2 634 aabb 2 15 37 8 1 n.s. Aat2 666 aabb 1 17 33 12 1 n.s. Aat2 623 aabb 4 1 12 4 0 n.s. Pgi 634 aabb 1 16 34 10 2 n.s. Pgi 666 aabb 1 18 32 11 1 n.s. Pgml 666 aabb 2 14 22 12 7 n.s. Skdh 634 abbb 21 27 14 n.s. Adhi 634 aaab 10 15 8 n.s. Adhi 623 aaab 7 9 5 n.s.

individual fruits of hermaphroditic mothers, the multi- analytical advance for allele-rich tetraploid species locus outcrossing estimates varied extensively from such as P. pringlei. complete selfing to 72 per cent outcrossing. Hetero- The most striking result of our analysis is the signifi- geneity in outcrossing rates was also found among cant difference in the estimates of outcrossing for fruits of the same plant, most notably 3 per cent and 52 female and hermaphroditic individuals. Females, as per cent in plant no. 614. expected, had estimated rates of outcrossing that were The two-locus fixation index in the pollen pool was close to 1.0 (0.949 0.0 19). In sharp contrast, individ- positive for both females (F =0.1180.016) and uals with bisexual flowers had much lower multilocus hermaphrodites (F =0.1840.040), indicating a estimates of outcrossing (0.301 0.023) indicating that deficiency of heterozygous gametes compared with the majority of seeds from bisexual flowers result from random expectations. self-pollination. In the apparent absence of a self- incompatibility system there is nothing in the morpho- logy of the bisexual flower to prevent selfing. The Discussion stigma is placed at the same height as the abundant Mostof the quantitative data on plant mating systems anthers. It is thus inevitable that large 'messy' pollina- is based on diploid species or on ancient polyploid tors such as L. curasoae would scatter considerable species with diploid gene expression (see Schemske & autogamous pollen onto the stigma. Lande, 1985 for a review of this literature). The It is also possible that additional inbreeding may absence of mating system analyses of polyploids is result from the mating of related individuals. A lower largely because the variety and complexity of segrega- single locus estimate of outcrossing (0.238 for tion patterns in polyploids make quantitative esti- hermaphrodites) relative to the multiocus estimate mates of the mating system difficult. It is only recently (0.30 1 for hermaphrodites) is usually indicative of that algorithms have been developed to analyse the biparental inbreeding (Brown et al., 1990). However, mating system of species with tetrasomic inheritance as we have independent estimates of outcrossing for (Barrett & Shore, 1987; Ritland, 1990a). As far as we females and bisexual individuals, we can calculate the are aware this is the first application of procedures that level of biparental inbreeding more directly based on jointly infer selfing rate and parameters of maternal deviation of the mating system estimates of the females and paternal parentage under tetrasomic inheritance. from unity (Sun & Ganders, 1988). Because the mean Inparticular, tetrasomy provides an unique single locus (0.979) and the multi-locus estimate opportunity to infer correlations between paternal (0.949) are only marginally significantly different from alleles, as paternal gametes possess two alleles per 1.0 we conclude that biparental inbreeding plays a rela- locus. The methods developed in the Appendix also tively small role in the breeding system of this species. represent a modification of the original algorithms to This result is consistent with what is known of the include more than two alleles per locus, a significant natural history of P. pringlei. Dense populations of this 90 D. A. MURAWSKI ETAL.

Table 2Allele frequency estimates for pollen (with SE.) and ovules of females and hermaphroditic Pachycereus pringlei. Standard errors are not given for ovule allele frequencies as bootstrap resampling was done within families due to the small number of families

Allele frequency Locus Allele Group (n seeds) designation Pollen pool Ovule pool

Females PgmJ(663) 1 0.988 (0.003) 0.994 2 0.012(0.003) 0.006 Pgm2(684) 1 0.014(0.004) 0.021 2 0.577 (0.015) 0.550 3 0.289(0.014) 0.355 4 0.096 (0.007) 0.086 5 0.011 (0.003) 0.008 Pgi(698) 1 0.001 (0.001) 0.000 2 0.538 (0.016) 0.488 3 0.458(0.016) 0.511 4 0.001 (0.001) 0.000 AdhJ(691) 1 0.955(0.006) 0.968 2 0.042 (0.006) 0.032 Idh(700) 1 0.87 1 (0.009) 0.917 2 0.137(0.009) 0.083 Skdh(691) 1 0.083(0.009) 0.086 2 0.590(0.017) 0.594 3 0.167 (0.013) 0.185 4 0.130(0.010) 0.134 5 0.002 (0.002) 0.001 Mdhl(702) 1 0.932 (0.008) 0.945 2 0.068 (0.008) 0.055 Aat2(679) 1 0.518(0.019) 0.511 2 0.482(0.019) 0.489 Hermaphrodites Pgml(5 12) 2 0.000 (0.000) 0.000 3 1.000(0.000) 1.000 Pgm2(508) 1 0.000 (0.000) 0.004 2 0.608 (0.034) 0.480 3 0.366 (0.034) 0.38 8 4 0.043(0.019) 0.128 5 0.000 (0.000) 0.000 Pgi(512) 2 0.498(0.037) 0.430 3 0.502 (0.037) 0.559 4 0.000(0.000) 0.011 AdhI(496) 2 0.916 (0.022) 0.909 3 0.093 (0.022) 0.091 Jdh(512) 2 0.884(0.021) 0.904 3 0.138(0.021) 0.096 Skdh(495) 1 0.203 (0.036) 0.227 2 0.520 (0.037) 0.476 3 0.237 (0.026) 0.216 4 0.054(0.016) 0.080 6 0.004(0.003) 0.00 1 Mdhl(512) 1 0.983(0.007) 0.998 2 0.017(0.007) 0.002 Aa12(492) 1 0.547 (0.037) 0.469 2 0.453 (0.037) 0.531 MATING SYSTEM OF CARDON CACTUS 91

Table 3 Multilocus and average single-locus outcrossing heterozygous diploids. Progeny of selfed unbalanced estimates for females and hermaphroditic individuals of heterozygotes (aaab or abbb) would have 25 per cent Pachycereuspringlei. Multilocus estimates are also given for homozygous progeny, still significantly less than that individual hermaphroditic cacti by fruit for selfed diploids. Individuals heterozygous for three or more alleles at a locus would have even lower pro- Population or portions of homozygous progeny. Thus, as most of the n individual tm (SE.) t (S.E.) adults in the population are heterozygous, we conclude Females 6270.949 0.979 that the potential cost of self-fertilization for species (0.019) (0.023) with tetrasomic inheritance is much lower than that Hermaphrodites 5650.301(0.023)0.238(0.021) Individual (fruits ID) experienced by most diploid species. 614 (1) 320.03 (0.04) The correlation between the two alleles within 614 (2) 320.52 (0.09) parental gametes was observed to be about F =0.12.In 621 (1) 320.19 (0.07) a mixed-mating allotetraploid population with selfing 621 (2) 320.38 (0.09) rate s and random outcros sing rate t =1—s,this corre- 623 270.15 (0.07) lation changes among generations according to the 630 640.72 (0.07) recursion in the absence of selection (Ritland, 1990b). 634 250.43 (0.11) In this relation, F' denotes the correlation in the 638 640.18 (0.05) 646 (1) 160.08 (0.07) F 1+3F 646 (2) 480.17 (0.08) 663 520.26 (0.10) F'=t+s 666(1) 320.18 (0.09) gametes produced by the offspring (second genera- 666 (2) 320.00 (0.00) tion). Note that in contrast to diploid inheritance, random outcrossing does not completely eliminate this correlation. If selfing rates are consistent among species consist of reproductive individuals separated generations, at equilibrium by distances of 10—30 m. Leptonycteris curasoae is also 3s F= an important agent for seed dispersal. These strong fly- 8—5s ing animals have an average seed passage time of about 30 mm (V. Sosa, unpublished data), making it unlikely In our study, the mean selfing rate was about 30 per that seeds would be deposited in the immediate vicinity cent (60 per cent male sterile with s 0 and 40 per cent of the fruiting adult from which they originated. Seeds hermaphrodite with s =0.7).If this adequately repre- defaecated together could be genetically related (Levin, sents the selfing practised the last few generations, we 1981) but the relatively infrequent recruitment success expect F =0.14.Values of observed parental F which of this cactus would most likely preclude the establish- are much less than that expected based on the selfing ment of related individuals adjacent to one another. rate are indicative of selection against selfed progeny One question that arises from our results is how the (which reduces parental F). In our study, the observed largely selfed progeny of the hermaphroditic individ- F(0.1 2) was only slightly less than the expected uals could compete with outcrossed progeny derived F(0.14), supporting the above arguments for reduced from females. In other words, why does inbreeding inbreeding depression in autotetraploids. depression acting on the progeny of the hermaphro- The subdioecious breeding system of P. pringlei, to dites not reduce the fitness of bisexual individuals and our knowledge, is unique among the columnar cacti. lead to their exclusion from the population? Yet, in this Most cactus species have perfect flowers and have well population and in others that have been surveyed (T. H. developed self-incompatibility systems. Controlled Fleming, unpublished data) hermaphroditic individuals selfed crosses of one species of Pachycereus, P. pectin- makes up a substantial proportion (up to 67 per cent) aboriginum and two of the columnar cactus species of the reproductive individuals. The key to this that co-occur with P. pringlei, Carnegia gigantea question probably lies in the ploidy level of P. pringlei. (sagauro) and Stenocereus thurberi (organ pipe), failed With tetrasomic inheritance selfing leads to lower to produce seed or fruit whereas similar crosses of P. levels of individual homozygosity than with diploids pringlei produced abundant seeds (T. H. Fleming, and, as a result, there should be less inbreeding depres- unpublished data). Although the breeding system of all sion. With balanced heterozygotes (aabb) only 1 /18of but one diploid species of Pachycereus is unknown, we the progeny of a self-fertilization will be homozygous can speculate that the diploid ancestor of P. pringlei, (aaaa or bbbb) rather than the 1/2 expected for selfed like other columnar cacti, was most probably a self- 92 D. A. MURAWSK} ETAL.

incompatible hermaphrodite. Immediately after its Inuleae) detected by the segregation of genetic markers. development as an autotetraploid, P. pringlei was Am.]. Bot., 77, 1078—1083. probably also a self-incompatible hermaphroditic BROWN, A. H. D., BURDON, J. J. AND .JAROSz, A. M. 1989. Isozyme species. However, as selection pressures against analysis of plant mating systems. In: Soltis, D. Soltis, P. inbreeding and, in particular, selfing should be much (eds), Isozymes in Plant Biology, Dioscorides Press, Port- land, Oregon, pp. 73—86. lower in autotetraploids, selection to maintain the self- GRANT, V. 1981. Plant Speciation. Columbia University Press, incompatibility system may have been relaxed. Further- New York. more, for a species whose seeds may be dispersed long KATAGIRIS, s. 1952.Studieson the chromosome number in distances by bats there may have been positive selec- some Cactaceae species. Jap. J. Breeding, 1,233—236. tion for isolated individuals to produce seeds via self- LEVIN, D. A. 1981. Dispersal versus gene flow in plants. Ann. fertilization. This conclusion is supported by the Mo. Bot. Gard., 68, 233—25 3. observation of Levin (1983) that breakdown of self- LEWN, D.A. 1983.Polyploidy and novelty in flowering plants. incompatibility is the most common alteration of the Am. Nature, 122, 1—25. reproductive system following chromosome doubling MIlTON, J. B., LINHART, Y. B., STURGEON, B. K. AND HAMRICK, J. L. in plants. 1979. Allozyme polymorphism detected in mature needle Later mutations could have been introduced in tissue of ponderosa pine, Pinus ponderosa Laws. J. Heredity, 70, 86—89. established populations that produced male sterility NESS, B. D., SOLTIS, D. E. AND SOLTIS, P. A. 1989. Autopolyploidy in (i.e. females). In established populations females may Heuchera micrantha (Saxifragaceae). Am. J. Bot., 76, have somewhat higher fitnesses relative to hermaphro- 6 14—626. dites due to marginally more fit progeny and because PINKAVA, D. J., MCGILL, L. A. AND REEVES, T. 1977. Chromosome less energy is expanded to produce pollen. At Bahia numbers in some cacti of western ifi. Bull. Kino, for example, females produce 1.6 times more TorreyBot. Club, 104, 105—110. seeds annually than hermaphrodites (T. H. Fleming, PINKAVA, D. J. AND MCLEOD, M. G. 1971. Chromosome numbers Buchmann and Tuttle, unpublished data). Genes in some cacti of Western North America. Brittonia, 23, influencing female sterility (i.e. males) probably arose 17 1—176. at a later time as males are not as geographically wide- PINKAVA, D. J., MCLEOD, M. G., MCGILL, L. A.AND BROWN,R. c. 1973. spread as are hermaphrodites or females (T. H. Chromosome numbers in some cacti of Western North Fleming, unpublished data). We have no information America II. Brittonia, 25, 2—9. QUIROS, C. F. 1982. Tetrasomic inheritance for multiple alleles concerning the genetic control of male or female steri- in alfalfa. Genetics, 101,117—127. lity but the ratios seen in this population and in others RITLAND, K. 1990a. A series of FORTRAN computer programs are consistent with a two nuclear gene, dominant— for estimating plant mating systems. J. Heredity, Si, recessive system. The presence in this population of 235—237. functionally neuter individuals is consistent with this RITLAND, K. 1 990b. Inferences about inbreeding depression hypothesis. based on changes of the inbreeding coefficient. Evolution, 44,1230—1241. SCHEMSKE, D. W. AND LANDE, R. L. 1985. The evolution of self- Acknowledgements fertilization and inbreeding depression in plants. II. Fieldassistance was provided by P. Homer, T. May, J. Empirical observations. Evolution, 39, 41—52. Debelak, C. Sahley, T. Finn, and L. Sechback. G. SCI-INABEL, A., LAUSHMAN, R. H. AND HAMRICK, J. L. 1991. Com- Nabhan kindly transported plant tissues back to the parative genetic structure of two co-occurring species Maclura pomifera (Moraceae) and Gleditsia triacanthos U.S.A. We also thank a reviewer for suggesting a clarifi- (Leguminosae). Heredity, 67, 3 57—364. cation in the notation used in the Appendix. Financial SHREVE, F. AND WIGGINS, I. L. 1964. Vegetation and Flora of the support (to TF) was provided by the National Geo- Sonoran Desert, vols 1 and 2. Stanford University Press, graphic Society (grants 4024—89, 4263-90), National Stanford, . Fish and Wildlife Foundation (grant 90-015) and a SOLTIS, D.E., HAUFLER,C.H.,DARROW,D. C. AND GASTONY, 0.J. University of Miami General Research Support Award. 1983. Starch gel electrophoresis of ferns: a compilation of grinding buffers, gel and electrode buffers and staining schedules. Am. Fern. J., 73, 9—15. References SUN, M. AND GANDERS, F. R. 1988. Mixed mating systems of Hawaiian Bidens(). Evolution, 42, 5 16—527. BARRETr,S. AND SHORE, j.s.1987.Variationand evolution of WEEDEN, N. F. AND WENDEL, J. F. 1989. Genetics of plant iso- breeding systems in the Turnera ulmifolia L. complex zymes. In: Soltis, D. and Soltis, P. (eds), Isozymes in Plant (Tumneraceae). Evolution, 41, 340—354. Biology, Dioscorides Press, Portland, pp. 46—72. BAYER, R. J., RITLAND, K.AND PURDY,B. G. 1990. Evidence of partial apomixis in Antennaria media (Asteraceae: MATING SYSTEM OF CARDON CACTUS 93

where p,, is the frequency of allele n in the outcrossing Appendix pollen pool (or, the prior probability of obtaining A,, from an outcross). The 1/2 enters because there are Progeny probabilities for multiallefic loci in tetraploids two possible origins of the progeny genotype, each with Becauseof the large number of possible genotypes at prior probability 1/2: A,,, derived from selfing and A,, multiallelic tetraploid loci, we develop a general nota- derived from outcrossing, vs. Am derived from out- tion for calculating the conditional probabilities of crossing and A,, derived from selfing. observing progeny genotypes, given parental genotypes For the case of tetraploids, we denote the maternal and mating events. These probabilities are the basic parent as A ,A 1A kA and her offspring as A mA ,,A 0A . elements in mating system estimation. Previous Intetraploids, pairs of alleles are transmitted to workers have given probabilities for the diallelic case gametes and there are three possible ways the four only (c.f. Barrett and Shore, 1987; Ritland, 1990a). alleles AmAnA0Ap were obtained from parents: (1) Our approach is based on defining variables that indi- AmA,, in one gamete and A,,A in the other, (2) AmA,, cate the similarity of alleles observed among parents in one gamete and A,,A in the other, and (3) AmAp in and their progeny. Once these variables are defined for one gamete and A,,A,, in the other. any specific data item, a single formula gives the The first step in finding progeny probabilities is to required probabilities as functions of these indicator specify the probability of observing (AmA,,) gamete variables. This formula reflects the mechanics of gene from an A A1A kA !parent: transmission and is easily encoded in a computer program despite its apparent complexity. + Ojm + 6km + óim)(6in + + &,, + 6/fl) Firstly, for ease of understanding and for compari- [(Oim son, we develop these probabilities for diploids. (óimôin + o ,,,6 ,, + 6km6 kn + óimã in)] -- 12 Consider a maternal parent with alleles A 4andher offspring with alleles AmA,,. The subscripts denote A,,,,, alleles, and take on numerical values from one to some 12 arbitrary number of alleles. For example, if i =1= 2,the mother is homozygous for the second allele in the The subtracted term above arises because in the for- population. Define the indicator variables 6.,,,,6in mation of gametes, the second allele is derived from the and 6.,,, where, for example, 5,,,, =1if i =m;otherwise same tetrad as the first, e.g. alleles are sampled without ôim =0.These 6s indicate the identity-by-state of replacement from parent genotypes. This quantity is alleles. The probability of observing progeny genotype divided by 12 because the first allele is drawn from a AmA,,, given parent A.A1 has selfed, is pool of four gametes (thus 1/4 prior probability) while the second allele is drawn from a pooi of three remain- prob(AA,,IA1A1, self) ing gametes (thus 1/3 prior probability), giving 1/12 total prior probability. Considering all three pairs of diploid gametes, each with a prior probability of 1/3, =(26mn)(6im+6im2 2 the probability of observing AmAnAOAP, given its j• parentage and that it was a self, is In this expression, 6im/2 is the probability, conditioned on the data and mating event, that allele i is the parent prob (A,,,AAOA A ,A1A kAl,self) gamete; it is divided by two because there are two possible parent gametes, each with prior probability 1/2. The probabilities involving the second progeny 3 12 12 12 12 12 12 allele n enter as a product because its transmission is an independent event. The 2— 6mn term enters where because the progeny genotypes AmAn and AnAm are classified together. The probability of observing progeny genotype c=24— 12(6mn+ 6,,,+ 6,,,+ 6+ 6+ 6,,) AmA,,, given parent A1A1 has outcrossed, is + 6(ómnà,jp+ ómOOflp+ ómpOno)+ 16 prob (A mAn IA,A1,outcross) (6,,p,,,+ómnp+ 6mop+ 6,,,,,)— 33&,,,,P (thisc term is due to classifying together equivalent 1 Ojm+ôjm 1 (àin+6jn) progeny genotypes, such as A 1A 1A 1A2 with pm (26mn) 2 2 A 1A 1A 2A1, etc.). In this expression, the indicator nota- 94 D. A. MURAWSKI ETAL. tion is extended to triplets and quadruplets of alleles: males can be estimated from progeny arrays and com- =1 if m =n=o;ó,0 =0otherwise, and pared with two-gene coefficients of females also esti- 1 if m =n=op; otherwise. mated from progeny arrays. (However, note that there To obtain the probability of a progeny derived from are additional inbreeding coefficients in tetraploids, outcrossing, note that we condition upon six possible specifically the three- and four-gene coefficients (see pairs of alleles derived from the maternal parent: Bayer eta!., 1990) that cannot be estimated for males.) A,nA,7, A,A0, A,A0, A,A, and A,A0, each By contrast, in diploids, inbreeding coefficients of with prior probability 1/6. Given a certain pair of males can only be estimated by paternity analysis. maternal alleles, the probability of observing a progeny The selfing and outcrossing probabilities determine is determined by the outcrossing pollen pool frequency the total probability of observing an offspring as for the remaining pair of putative paternal alleles. Thus, the probability of observing A,A,AOAP given its prob(AflA,A)APIAAJAkAl)== parentage and that it was outcrossed is s prob(AmA ,/l OAPIAAIA kA ,self) + t prob(A,A,AOAPIAAJAkA/, outcross), prob (A ,A ,A OAP I kA I, outcross) = AAIA where s is the prior selfing rate and ttheprior out- A A A A crossing rate. The multilocus extension of this formula (c/6)—i- p°,+- p,+ — p,0+ p is of the form

A + p0+ i? prob ( G0IGpu,)= sF1 prob( Go1,1 IGpar,i, self)

+ tFJ whereps denote pollen gene frequencies (e.g. p, is the prob( G0 Girnrj, outcross), frequency of alleles m and n in the paternal gamete pollen pool). where G0ff1 and Gparjdenoteoffspring and parent Interestingly, the inbreeding coefficient of parents genotypes, respectively, at locus i; the prob's are the enters into this frequency as p,=p,p,(l—F01)+ single locus probabilities derived above. At this point, p (the standard formula for genotypic propor- the standard procedure and assumptions for outcross- tions under inbreeding described by Wright's F but for ing estimation is applied (Ritland, 1990a). A program F defined as the 'two-gene' inbreeding coefficient that implements this procedure is available from K. R. among paternal, diploid gametes, Fy01)). This means on receipt of a floppy disk or via e-mail (ritland© that in tetraploids, two-gene inbreeding coefficients of botany.utoronto.ca).