蜘蛛抱蛋亚族的花粉形态

本文研究并报道了百合科蜘蛛抱蛋亚族的花粉形态,主要结论如下：1．开口箭属含两种花粉类型即：(1)单槽、椭圆形;(2)无萌发孔、球形。具单槽、椭圆形花粉的种类含单型性核型,花形较小,柱头小而花柱短,雄蕊高于柱头;无萌发孔、球形花粉的种类则为单型性核型,花形较大,柱头常高度膨大,雄蕊常着生在花被筒基部。2．万年青属花粉为椭圆形,具单槽;蜘蛛抱蛋属花粉则为球形,无萌发孔;开口箭属花粉一部分与万年青属相似,另一部分则与蜘蛛抱蛋属雷同。3．蜘蛛抱蛋亚族中最原始的花粉类型为单槽椭圆形,具孔状或网状外壁纹饰。球形,无萌发孔,外壁具皱波状、瘤状或芽孢状纹饰的花粉为派生类型。4．蜘蛛抱蛋亚族种下花粉形状、萌发孔类型和外壁纹饰基本稳定,可作为分种特征。     

Pollen Morphology in the Subtribe Aspidistrinae (Liliaceae s. l.)

Pollen morphology in the subtribe Aspidistrinae is reported and the main viewpoints are summarized as follows: (1) Two major types of pollen grains, i.e., monosulcate,ellipsoidal pollen and inaperturate, spheroidal pollen, are identified in the genus Tupistra. The species with monosulcate, ellipsoidal pollen usually have monomodal karyotypes, brightcolored flower, smaller stigma, and stamens inserted at upper or middle part of perianth tube, while the inaperturate and spheroidal pollen is always found in the species with bimodal karyotypes, dingy-colored flower, swollen stigma and stamens inserted at the base of perianth tube. (2) Pollen grains in the genus Rhodea are monosulcate and ellipsoidal, while those in the genus Aspidistra are inaperturate and spheroidal, correspondingly similar to the second type of pollen grains in the genus Tupistra. (3) The most primitive pollen in the subtribe Aspidistinae is regarded as monosulcate and ellipsoidal, having perforate or reticulate exine sculpture. The inaperturate, spheroidal pollen with verrucate, gemmate or rugulate exine sculpture is considered derived; (4) Unlike those reported in other groups of the family Liliaceae, the infraspecific pollen shape, aperture type and exine sculpture in Aspidistrinae are basically stable and may serve as a taxonomic character.     

Early cretaceous fossil evidence for angiosperm evolution

Morphological, stratigraphic, and sedimentological analyses of Early Cretaceous pollen and leaf sequences, especially from the Potomac Group of the eastern United States, support the concept of a Cretaceous adaptive radiation of the angiosperms and suggest pathways of their initial ecological and systematic diversification. The oldest acceptable records of angiosperms are rare monosulcate pollen grains with columellar exine structure from probable Barremian strata of England, equatorial Africa, and the Potomac Group, and small, simple, pinnately veined leaves with several orders of reticulate venation from the Neocomian of Siberia and the basal Potomac Group. The relatively low diversity and generalized character of these fossils and the subsequent coherent pattern of morphological diversification are consistent with a monophyletic origin of the angiosperms not long before the Barremian. PatuxentArundel floras (Barremian-early Albian?) of the Potomac Group include some pollen and leaves with monocotyledonous features as well as dicotyledonous forms. Patuxent angiosperm pollen is strictly monosulcate and has exine sculpture indicative of insect pollination. Rare Patuxent-Arundel angiosperm leaves are generally small, have disorganized venation, and are largely restricted to sandy stream margin lithofacies; the largest are comparable to and may include ancestors of woody Magnoliidae adapted to understory conditions. Patapsco floras (middle to late Albian?) contain rapidly diversifying tricolpate pollen and several new complexes of locally abundant angiosperm leaves. Ovate-cordate and peltate leaves in clayey pond lithofacies may includeancestors of aquatic Nymphaeales and Nelumbonales. Pinnatifid and later pinnately compound leaves with increasingly regular venation which are abundant just above rapid changes in sedimentation are interpreted as early successional “weed trees” transitional to but more primitive than the modern subclass Rosidae. Apparently related palmately lobed, palinactinodromous leaves which develop rigidly percurrent tertiary venation and become abundant in uppermost Potomac stream margin deposits (latest Albian-early Cenomanian?) are interpreted as riparian trees ancestral to the order Hamamelidales. Comparisons of dated pollen floras of other regions indicate that one major subgroup of angiosperms, tricolpate-producing dicots (i.e., excluding Magnoliidae of Takhtajan) originated in the Aptian of Africa-South America at a time of increasing aridity and migrated poleward into Laurasia and Australasia. However, the earlier (Barremian) monosulcate phase of the angiosperm record is represented equally in Africa-South America and Laurasia before marked climatic differentiation between the two areas. These trends are considered consistent with the hypothesis that the angiosperms originated as small-leafed shrubs of seasonally arid environments, and underwent secondary expansion of leaf area and radiated into consecutively later successional stages and aquatic habitats after entering mesic regions as riparian “weeds,” as opposed to the concept that they arose as trees of mesic forest environments.     

Formation and function of a new pollen aperture pattern in angiosperms: The proximal sulcus of Tillandsia leiboldiana (Bromeliaceae)

Pollen grains are generally surrounded by an extremely resistant wall interrupted in places by apertures that play a key role in reproduction; pollen tube growth is initiated at these sites. The shift from a proximal to distal aperture location is a striking innovation in seed plant reproduction. Reversals to proximal aperture position have only very rarely been described in angiosperms. The genus Tillandsia belongs to the Bromeliaceae family, and its aperture pattern has been described as distal monosulcate, the most widespread aperture patterns recorded in monocots and basal angiosperms. Here we report developmental and functional elements to demonstrate that the sulcate aperture in Tillandsia leiboldiana is not distal as previously described but proximal. Postmeitotic tetrad observation indicates unambiguously the proximal position of the sulcus, and in vitro germination of pollen grains confirms that the aperture is functional. This is the first report of a sulcate proximal aperture with proximal germination. The observation of microsporogenesis reveals specific features in the patterns of callose thickenings in postmeiotic tetrads.     

Aperture variation in the pollen of Nelumbo (Nelumbonaceae)

The evolutionary and developmental origin of tricolpate pollen is of great interest because pollen of this kind defines a major clade of angiosperms (eudicots), a clade that is also well supported by molecular data. We examined evidence that tricolpate and monosulcate pollen types are produced alongside each other in the anthers of Nelumbo flowers, as has previously been reported. Observations of pollen in situ within individual anthers revealed mainly tricolpate pollen produced in tetrahedral tetrads, but also a small percentage of clearly aberrant pollen grains that have a great variety of aperture configurations. Previously published evidence for tetragonal tetrads is not supported, and previously reported monosulcate grains are part of a continuum of variation among the aberrant grains in aperture number, position and form. Other eudicots show similar variability in their pollen apertures. The variation in the pollen of Nelumbo is not exceptional, and may not be more significant than variation seen in the other taxa with regard to the origin of the tricolpate and tricolpate‐derived pollen characteristic of eudicots. Nevertheless further studies of aberrant pollen in Nelumbo and other eudicots, together with comparisons of pollen development in “normal” eudicots and closely related species that show radical, and developmentally fixed, reorganization of apertures and pollen polarity, may be helpful in understanding the processes that controlled the transition from the monosulcate to the tricolpate condition.     

鹅掌楸的传粉环境与性配置

近年来研究表明,动物传粉者对植物花部诱物特征（花冠形状大小,花蜜产量和花序大小）有潜在的选择作用。不同效率的传粉者可能导致植物性配置的变化,不同传粉效率的环境下,两性花植物鹅掌楸的花粉和胚珠的配置不同,居群Ｚ在主要访花者是传粉效率较低的类的传粉环境下,花粉粒小,花粉数量相对较高;另一方面胚珠投资的减少,缓解了低效的传粉（增大授粉的几率）,增加了受精的机会。相对而言,具有高效访花者的两个居群,资源较     

Benefits and costs of floral visitors to Chilopsis linearis: pollen deposition and stigma closure

In pollination mutualisms, nectar-robbers are usually considered antagonists; visitors that enter flowers (legitimate visitors) are usually considered mutualists. However, nectar-robbers may provide some benefits to plants, whereas legitimate visitors may inflict some costs. The costs and benefits of floral visitors to Chilopsis linearis (desert willow) were compared by number of pollen grains deposited and their effect on stigmas. Because these plants are self-incompatible and pollen-limited, they depend on visitors for services by pollinators to reproduce. On a per-visit basis, only one legitimate visitor, Bombus sonorus (bumblebees) generally benefited plants in terms of pollen deposition. However, no species of visitor was consistently beneficial; every one was at least sometimes ineffective in terms of pollen deposition. Chilopsis had sensitive stigmas that closed immediately upon touch and sometimes reopened later. Whether stigmas remained permanently closed or reopened depended on number of pollen grains deposited and tended also to be affected by source either, outcross or self. Legitimate visitors sometimes cost plants by causing stigma closure without depositing enough pollen to set a fruit. When abundant, a visitor such as Apis mellifera (honeybees) that was only occasionally beneficial on a per-visit basis may have provided a greater benefit to plants as a population than a more effective, but rare visitor. In contrast, nectar-robbers did not benefit plants by depositing pollen grains, but they also did not inflict costs on plants by causing stigmas to close without adequate pollen. There are two general implications of these results. First, individuals of species that appear to be mutualists can vary greatly in the benefits that they give to their partners. Second, apparent mutualists can inflict costs on plants that apparent exploiters (nectar-robbers) do not.     

Arecaceae potential for production of monofloral bee pollen

Bees often collect pollen from plants that are near their colonies or apiaries. Arecaceae are characterised by having species that flower throughout the year and produce large quantities of pollen. Pollen grains of Arecaceae have two main forms, namely monosulcate and trichotomosulcate. The study area is located in the coastal region of the state of Sergipe, Brazil, where large amounts of bee pollen are produced by Apis mellifera. This study aims to evaluate how Arecaceae contribute to the production of monofloral bee pollen. Samples of bee pollen loads were collected each month for two years (totalling 24 samples), acetolysed, and mounted on permanent slides. At least 500 grains of pollen were counted for each sample, the data was analysed, and figures were made using the Tilia software. The pollen grains of Cocos nucifera were present in 23 samples (96%), and concentrations of this pollen type varied between 1.1 and 46.8% (but were commonly above 20%). The concentration of Arecaceae pollen grains is the result of the large-scale production of coconuts along the coastline. However, the presence of other pollen types, such as Mimosa arenosa, Mimosa pudica, Mikania and Myrcia, compliments the diets of bees in the region. The results show that even when in low frequency, Cocos nucifera pollen grains represent a large proportion of pollen content compared with other pollen types, because of the large size of Arecaceae pollen grains.     

Gametophytic vs. sporophytic control of pollen aperture number: A generational conflict

In flowering plants, the haploid phase is reduced to the pollen grain and embryo sac. These reproductive tissues (gametophytes) are actually distinct individuals that have a different genome from the plant (sporophyte), and are more or less independent. The morphology of pollen grains, particularly the openings permitting pollen tube germination (apertures), is crucial for determining the outcome of pollen competition. Many species of flowering plants simultaneously produce pollen grains with different aperture numbers in a single individual (heteromorphism). In this paper, we show that the heteromorphic pollen aperture pattern depends on the genetic control of pollen morphogenesis. This points out a conflict of interest between genes expressed in the sporophyte and genes expressed in the gametophyte. More generally, such a conflict should exist whenever heteromorphism is an ESS resulting from a bet-hedging strategy. For pollen aperture, heteromorphism has been observed in about 40% of angiosperm species, suggesting that conflicting situations are the rule. In this context, the sporo-gametophytic conflict could be one of the factors that led to the reduction of the haploid phase in plants.     

The morphology and ultrastructure of Jurassic in situ ginkgoalean pollen

A fragmentary pollen organ with four to six microsporangia is discovered from the Middle Jurassic of the Irkutsk coal basin, Siberia. The in situ pollen grains are boat-shaped, monosulcate, and with a nearly psilate surface. The non-aperture ectexine is composed of a thick solid tectum, a thin infratectum, and a thin foot layer. The infratectum includes one row of small rare alveolae. The supposedly poorly preserved endexine is thin and grainy. The ectexine reduces greatly in the aperture area, where only homogeneous ectexinal patches are present over the endexine. The pollen grains under study resemble in their exine ultrastructure pollen grains of the modern Ginkgo biloba and pollen grains from dispersed seeds of a presumably ginkgoalean affinity from the Middle Jurassic of Uzbekistan. This suggests that the ginkgoalean exine ultrastructure of the modern type existed as early as the Middle Jurassic. The exine ultrastructure under study is also similar, though to lesser degree, to that of dispersed pollen grains of a presumed ginkgoalean affinity from the Cretaceous of the Russian Far East. The diversity of such a long-living group as ginkgoaleans is apparently reflected in the diversity of their exine ultrastructure. To the present knowledge, ginkgoalean pollen grains can be differentiated from similar boat-shaped monosulcate pollens by the following co-occurring characters: a thick homogeneous tectum, a thin infratectum with one row of structural elements, a thin foot layer, and an ectexine that is reduced in the aperture region to patches.     

Studies on the Pollen Morphology of the Chinese Palmae

This paper deals with morphology of the recent pollen grains of 26 species representing 16 genera and some fossil pollen of the Chinese Palmae. They are examined with the light microscope and scanning electron microscope. On the basis of the types of aperture and the distribution of fossil pollen and lithophyll in the strata, the trends of the evolution of the family Palmae are discussed. This paper points out that the monosulcate pollen belongs to the most predominent and more primitive aperture type in this family and that there is a trend in which the trichotomosulcate, annulocolpate, disulcate etc. are derived from the monosulcate pollen.     

花粉形态与风媒植物母体地理分布间的关系

本文对部分风媒植物花粉形态特征,及其母体植物地理分布情况进行了对照研究。在此基础上发现,同一分类系统内,小个体花粉者,其母体植物地理分布趋向于较优越环境。对于具气囊花粉而言,具较大气囊者,或者花粉本体较小者,其母体植物趋向于分布在低纬度或较低海拔环境。并且,产大个体或球状花粉者,往往具有较强的单种成林倾向。本文首次指出了花粉形态与母体植物地理分布间存在着一定的统计学关系,并首次将动态概念引入植物生态学领域,认为花粉形态特征作为一项重要的参变量,通过种间生存竞争过程对母体植物适应倾向施加影响,进而影响物种的生活习性。那些在花粉的传播及授粉过程中稍占优势的物种,其自然分布域趋向于比较优越的环境。本文仅作为一种尝试,试图为动态植物生态学研究,乃至植物生态演化学研究提供一个新的研究思路。     

Pollen Morphology and Epidermal Characters of Leaves in Convallarieae (s. l.)

The tribe Convallarieae (sensu Krause 1930) consists of 7 genera, i.e. Convallaria, Speirantha, Reineckia, Theropogon, Tupistra, Rohdea and Aspidistra, but now generally recognized as two tribes, Convallarieae (the former 4 genera) and Aspidistreae (the rest). Observed in this work were pollen morphology of 17 species and epidermal characters of leaves of 12 species. All the 7 genera are covered in observations. Pollen grains in Convallarieae (s. str.) are all monosulcate and boat-shaped (Plate 1: A-F). The exine is rather uniformly microperforate (Plate 1: A-F); only Theropogon is exceptional in this respect: it has rugulate exine (Plate 1: O, P). Tang and Zhang (1985) have pointed out the heterogeneity of Theropogon in this tribe. Pollen morphology in the tribe Aspidistreae is widely variable. The genera Tupistra and Rohdea were shown to have monosulcate and boat-shaped pollen grains. Their exine is perforate or reticulate (Plate 1: G-N). Pollen grains in the genus Aspidistra, however, are nonaperturate and spheroidal. The exine in the genus varies from crass-rugulate, variously gemmate to tuberculate-baculate (Plate 2; A-H). The pollen morphology of Aspidistra is therefore distinctly different from that of Tupistra and Rohdea, which supports the Nakai's (1936) establishment of the tribe Rohdeae for Tupistra and Rohdea. Therefore, Krause's Convallarieae is reasonably divided into at least three tribes, Convallarieae (Speirantha, Convallaria, Reineckia and Theropogon), Aspidistreae (Aspidistra) and Rohdeae (Rohdea and Tupistra). The pollen characters of all the 7 genera are shown in Table 1. The evolutionary trends of pollen morphology (aperture and exine) in the three tribes are discussed and our major view-points are shown in Fig. 1. Observations on epidermal characters of leaves show that in the Convallarieae (s. 1.) stomatal apparatuses are all anomocytic; cuticular layer on the upper epidermis is mainly striatethickened or rather uniformly thickened (Plate 2: J--P; Plate 3: A-C, F-N), whereas in the genus Convallaria the cuticular layer is squamosely thickened (Plate 2: I; Plate 3: D, E).The epidermal characters of leaves in the 7 genera are summarized in Table 2.     

Pollen morphology inParis and its related genera

Pollen grains ofParis, Medeola, andScoliopus have been examined with light, scanning electron, and transmission electron microscopies.P. quadrifolia L.,P. verticillata Bieb.,P. delavayi Franch. andP. polyphylla Smith have monosulcate pollen characterized by psilate, foveolate or reticulate exine sculpture. In contrast to the former species,P. japonica (Fr. et Sav.) Franch. andP. tetraphylla A. Gray have monosulcate pollen with gemmate (rarely rugulate) exine.Medeola has monosulcate pollen with reticulate exine that is distinct from that ofParis. Scoliopus has monosulcate pollen characterized by a peculiar reticulate exine pattern. The palynological evidence suggests thatParis andTrillium are closely related to each other, andMedeola andScoliopus should be separated fromParis andTrillium.     

Stable isotope ratios of carbon and nitrogen in pollen grains in order to characterize plant functional groups and photosynthetic pathway types

Measurements of delta(13)C, delta(15)N and C : N ratios on modern pollen grains from temperate plants, including whole grains as well as extracted sporopollenin, were analysed in order to characterize physiological plant types at the pollen level and to determine the variation of these parameters in modern pollen grains of the same climatic area. Measurements are presented for 95 batches of whole modern pollen from 58 temperate species and on the stable fraction of modern pollen grains, chemically extracted sporopollenin, for two modern species. Fourier transform infrared (FTIR) and cross-polarization and magic-angle spinning (CP/MAS) sporopollenin spectra were conducted in parallel. C(3) and C(4) plants can be separated by delta(13)C measurements based on pollen. Probabilistic assignments to plant functional groups (herbaceous, deciduous woody, evergreen woody) of C(3) plants by the means of a discriminant analysis can be made for C : N ratios and for delta(13)C. The results are related to other studies on sporopollenin in order to use this method in future work on fossil samples. Stable isotope measurements on pollen allow improved pollen diagrams, including forms that cannot be differentiated at species level, increasing the accuracy and resolution of plant physiological type distribution in quaternary and older fossil sediments.     

Pollen Organs and Seeds with Decussosporites Brenner from Lower Cretaceous Potomac Group Sediments of Eastern USA

Microsporangiate structures, Brenneria potomacensis gen. et sp. nov., containing pollen grains similar to dispersed Decussosporites are described from the Lower Cretaceous (Barremian or Early Aptian?) Potomac Group localities at Drewry's Bluff and Dutch Gap on the James River southeast of Richmond, Virginia. These fossils provide the first megafossil evidence of plants producing Decussosporites-type pollen and contribute important new information on the structure and possible systematic affinities of this unique Mesozoic gymnosperm. The microsporangiate structure is composed of an axis with helically arranged synangiate microsporangiate units, each unit consisting of two laterally fused sporangia borne on a short stalk. The pollen grains are very small, bisaccate, distinctly striate (taeniate) and TEM shows that they have partly infilled sacci (quasisaccate). These grains represent the youngest occurrence of saccate, striate pollen, which has not been recorded previously from sediments younger than the earliest Jurassic. Fossil seeds (Brennerispermum potomacensis gen. et sp. nov.) from the same localities as Brenneria contain Decussosporites pollen in the micropyle, and are believed to have been produced by the same plant species. The seeds are small and unitegmic with a distinct megaspore membrane. The occurrence of seeds adhering together in groups indicates that they were borne in aggregations. The microsporangiate structures of Brenneria show some similarity to those of the ginkgophytes (Ginkgoales and Czekanowskiales), but there are substantial differences in the structure of seeds and pollen. The Decussosporites pollen grains together with morphology of the reproductive organs support a closer relationship of the Brenneria-plant to previously described “Mesozoic pteridosperms”.     

POLLEN MORPHOLOGY IN THE GENUS ERYTHRONIUM (LILIACEAE) AND ITS SYSTEMATIC IMPLICATIONS

Pollen morphology was examined in 40 samples from 19 species in the genus Erythronium (Liliaceae) using scanning (SEM) and transmission electron microscopy (TEM). Pollen grains are monosulcate monads, except for a plant of E. oregonum which has dyad grains. North American and European species are characterized by reticulate exine sculpture composed of various degrees of wavy muri and variably-sized lumina. The most distinctive feature of exine sculpture in the genus is exhibited by the Asian species E. japonicum, which has a unique reticulate pattern composed of striate muri and a distinctive exine structure without columellae. The distinctness of E. japonicum pollen suggests that it has specialized in isolation from species in North America and Europe.     

Overexpression of a human α3/4-fucosyltransferase alters pollen development in transgenic tobacco plants

. In plants, the function of Ŏ-fucosylation remains largely unknown. To gain insight into the role of Ŏ-fucosylation during plant development, we generated transgenic tobacco plants overexpressing the human ō/4-fucosyltransferase (hFuc-TIII). Overexpressors clearly contained high amounts of hFuc-TIII and revealed a strong increase in !(1,4)fucosyltransferase activity in plant sexual organs. As a consequence, a more significant staining of Lewis^a motifs, the product of !(1,4)fucosyltransferase activity, was observed in transgenic pollen grains compared to those of controls. Here, we show that pollen grain development was altered in transgenic plants. The average size (polar and equatorial diameters) of mature pollen grains overexpressing hFuc-TIII was smaller than control pollen grains. Furthermore, whereas a reticulate cell wall surface was always observed on control pollen grains, a punctate and disorganized cell wall surface was observed on hFuc-TIII overexpressor pollen grains. In addition, transgenic pollen tube elongation was delayed compared to control pollen tube growth. This latter phenotype could at least explain the 35% reduction of seed production determined for the hFuc-TIII-overexpressing plants.     

The ultrastructure of fossil dispersed monosulcate pollen from the Early Cretaceous of Transbaikalia,Russia

The general morphology, surface sculpturing, and exine ultrastructure have been studied in dispersed monosulcate pollen from the Early Cretaceous of Transbaikalia, Russia. The pollen grains dominate the palynological assemblage extracted from coal deposits of the Khilok Formation in the Buryat Republic, which also contain ginkgoalean leaves of Baierella averianovii as the only constituent of the assemblage of plant megafossils. The relationship between the pollen grains and ginkgoalean leaves from this autochthonous burial is hypothesised on the basis of taphonomical analysis and palaeobiogeographical data. It is shown that the ectexine of the pollen grains includes a thick solid tectum, a thin granular infratectum and a thin foot layer; the endexine is fine-grained, slightly more electron-dense than the ectexine, and is preserved only in places. The distal aperture is formed by a thinning of the exine. No analogous ultrastructure has been described so far in fossil pollen grains of this morphotype studied ultrastructurally from in situ material. For comparison, we also studied the exine ultrastructure of pollen grains Ginkgo biloba. The fossil pollen is not identical to pollen of extant G. biloba, but shows several significant similarities in the exine ultrastructure, which does not contradict the presumable ginkgoalean affinity of the fossil pollen.     

Does differential seed siring success change over time or with pollination history in wild radish, Raphanus sativus (Brassicaceae)?

Previous work with wild radish has shown that pollen donors sire different numbers of seeds and that the condition of the maternal tissue affects seed paternity, suggesting that both pollen donor characteristics and maternal tissue affect mating. However, because these results are from the greenhouse, it is difficult to know whether they would hold true in the field. Here, we performed hundreds of crosses on several maternal plants to simulate changes during the flowering season of field plants. During the experiment, maternal resource availability changed due to the costs of producing fruits, and we determined the pollination history of a plant by performing crosses in specific orders. Examination of seed paternity showed that there were small differences in pollen donor success at the beginning of the experiment when maternal resources were abundant. Differential pollen donor success was greatest slightly later in the flowering period, but declined toward the end of the experiment. Thus, maternal plants may distinguish most among pollen donors when they have both abundant resources and experience with the differences in quality of available pollen donors. In contrast, there were few significant effects of the recent pollination history of plants on pollen donor success. Finally, despite the changes in mating performance over time, there were strong overall differences in pollen donor success, suggesting that seasonal changes in the field will not eliminate the potential for nonrandom mating.