Journal of Heredity Advance Access originally published online on April 13, 2005
Journal of Heredity 2005 96(4):430-433; doi:10.1093/jhered/esi063
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Brief Communication |
Single-Locus Gametophytic Incompatibility in Autotetraploids
From Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK (Ridout); and East Malling Research, New Road, East Malling, Kent ME19 6BJ, UK (Xu and Tobutt)
Address correspondence to Xiangming Xu at the address above, or e-mail: xiangming.xu{at}emr.ac.uk.
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Abstract |
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 Top Abstract A Deterministic GSI Model...Results and DiscussionReferences |
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It is known that a single-locus gametophytic self-incompatibility (GSI) system can persist with just two distinct alleles in an autotetraploid population, in contrast to diploid GSI systems, assuming "competitive interaction" in which heteroallelic pollen is universally compatible. The steady-state population structure of a GSI system in autotetraploids was investigated in an undivided population assuming "competitive interaction." A deterministic model was developed to predict the frequencies of genotypes with two, three, or four distinct S alleles, assuming no mutation or population subdivision. The model showed that unlike in diploid GSI systems, the limiting values of the frequencies of genotype classes do not minimize pollen wastage.
Many plant species have evolved mechanisms that prevent self-fertilization (Richards 1986). One such mechanism is gametophytic self-incompatibility (GSI). In the simplest GSI system, which occurs in many angiosperm families and has been well studied in, for example, Onagraceae, Rosaceae, Solanaceae, and Papaveraceae, there is a single locus, termed the S locus, which determines compatibility relationships. The S locus is considered to have two complementary parts, one expressed in the style and the other in the pollen. In diploid plants, pollen carrying a particular allele, S1 say, at the S locus is unable to fertilize ovules from any plant that also carries this allele (East and Mangelsdorf 1925). A single-locus GSI system has several simple, but important properties. First, it is clear that all individuals must be heterozygous at the S-locus. Moreover, an ovule from an S1S2 plant cannot be fertilized by pollen from an S1S2 plant; therefore there must be at least three distinct S alleles in the population.
The allelic frequency dynamics of the GSI system in diploids has been analyzed by many researchers, the first being Wright (1939). A brief review on the deterministic behavior of the allelic frequencies was given by Nagylaki (1975). Under random mating, pollen carrying a rare S allele has a selective advantage, because it is less likely to encounter an incompatible female parent than pollen carrying a frequent allele. This "balancing selection" has two consequences. First, an S allele is less likely to become extinct through random genetic drift than a neutral allele. As soon as an S-allele becomes rare, it acquires a selective advantage, and its frequency therefore tends to increase again. Second, by the same reasoning, new alleles arising through mutation or immigration are more likely to become established within the population than those at a neutral locus. Thus there are likely to be more distinct alleles at the S locus than at a typical neutral locus, and many species do indeed exhibit a very high degree of polymorphism at the S locus. For example, in a pioneering article, Emerson (1939) found 37 distinct alleles in a sample of about 500 individuals of evening primrose (Oenothera organensis). Values for several more species are given by Lawrence (1996). Recently Schierup (1998) used extensive computer simulations to show that the predictions of Wright's theory are quite accurate for undivided populations, but underestimate the total number of distinct S alleles in subdivided populations, sometimes substantially.
Several authors (Boucher 1993; Moran 1962; Nagylaki 1975; Steiner and Gregorius 1994, 1995) have undertaken a deterministic analysis of an incompatibility system with k distinct S alleles, in the absence of mutation. It is clear from symmetry that a random mating population will be in equilibrium if all the possible heterozygous genotypes SiSj (i 
j), of which there are k(k – 1)/2, occur with equal frequency in the population. This has been shown to be the only equilibrium solution (provided that the frequency of each allele exceeds zero), and moreover to be globally asymptotically stable. From an evolutionary viewpoint, Steiner and Gregorius (1994) note that this configuration minimizes the proportion of pollen that is wasted through incompatibility.
Studies of population genetics of the S locus have been largely restricted to diploids. However, it is estimated that 30%–80% of all angiosperm species are polyploids and many of these polyploid species are important grain, vegetable, and fruit crops (e.g., Stace 1993). For example, in the Rosaceae, polyploid series occur in such important genera as Fragaria, Prunus, Rosa, and Rubus. One of the simple forms of polyploidy is autotetraploidy, in which inheritance is tetrasomic, though it is unclear what proportion of tetraploids are autotetraploids. Population genetic theory of autopolyploids has been studied, for example, by Bever and Felber (1992), but we are not aware of any studies of theoretical population genetics of the S locus in autotetraploids. Autotetraploidy typically results in partial self-compatibility (Golz et al. 2000; Lewis 1947). This has been attributed to the heteroallelic pollen grains succeeding on self styles because of "competitive interaction" such that neither pollen S allele is expressed, though other interactions, such as dominance, are possible (Lewis 1947). In crosses, homoallelic diploid pollen is rejected if the female parent carries one or more copies of the same allele; all other pollen types are compatible. Direct experimental evidence for the self-compatibility of heteroallelic pollen grains has recently been presented (e.g., Chawla et al. 1997), although the nature of the competitive interaction has been reformulated as the "heteroallelic pollen effect" (Luu et al. 2001).
In this article, we investigate one aspect of the long-term population structure of an autotetraploid GSI locus in an undivided population, namely the frequency of genotypes with different numbers of S alleles at equilibrium. Autotetraploid individuals can be classified with respect to the number of distinct alleles present at a locus. For an undivided population with no mutation, we develop a deterministic model for the changes from one generation to the next in the relative frequencies of the different classes of S locus genotypes. From this, we calculate the equilibrium genotype structure.
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A Deterministic GSI Model for Autotetraploids |
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TopAbstractA Deterministic GSI Model...Results and DiscussionReferences |
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We consider an autotetraploid species in which gamete formation follows the random chromosome model, with no double reduction (Jackson and Jackson 1996). Thus, given the parental genotype, with four S alleles (not necessarily all distinct), a gamete is obtained by selecting two of the alleles at random, without replacement, giving six possible gametes (not necessarily all distinct). In the simplest model proposed by Lewis (1947), and adopted here, homoallelic pollen (e.g., S1S1) is rejected if the female parent carries one or more copies of the same (S1) allele, but heteroallelic pollen is always fully compatible, irrespective of the female genotype.
It is easy to see that for tetraploids, as for diploids, fully homozygous individuals cannot exist. However, there are up to four distinct heterozygous classes of genotype that can occur for an autotetraploid. Table 1 gives an example of each class and indicates the total number of distinct genotypes within each class for a system with k distinct S alleles. We use the symbols 
, ß, 
, and 
to denote the relative frequency of each class within the population ( + ß + + = 1).
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We shall assume that, for any equilibrium solution, all possible genotypes within a class occur with equal frequency, and consider the limiting values of
, ß, , and under this assumption. The assumption is suggested by symmetry considerations and is supported by numerical studies of systems with small values of k, but we have not obtained a formal proof.
Let n, ßn, n, and n denote the values of , ß, , and after n generations of random mating. To derive the updating equations for n, ßn, n, and n, we begin by listing the types of gamete that can occur, and their relative frequency. Then, by combining all possible types of gamete and eliminating those combinations that are incompatible, we can derive the proportions of each genotype class in the next generation. This procedure, which is straightforward in principle, though rather tedious in practice, leads eventually to the following results.
The proportion of pollen produced by plants of generation n that is homoallelic is
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n
n.
The updating equations for , ß, , and are as follows:
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Results and Discussion |
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TopAbstractA Deterministic GSI Model...Results and DiscussionReferences |
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Table 1 shows the equilibrium proportions of the different classes of genotype for neutral alleles, assuming random mating, based on the results of Haldane (1930). The purely homozygous genotype (class 0), which cannot occur for the S locus, is also rare for a neutral locus, unless there are very few distinct alleles (frequency less than 1% when k
5). For k 
5, the frequency of class 1 genotypes at the S locus (e.g., S1S1S1S2), 
, is lower than at a neutral locus, but the frequency of other heterozygous classes of genotype (ß, , ) is similar to or greater than for a neutral locus (Table 2). As k increases, the frequency of heterozygous genotype classes for a neutral locus becomes gradually closer to that for the S locus.
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In contrast to diploid GSI systems, an autotetraploid GSI system can persist with just two distinct alleles. This is because the selection imposed by the GSI system in autotetraploids is on diploid pollen and not directly on single S alleles. Under the assumption of "competitive interaction," a heteroallelic pollen grain is always compatible and thus a GSI system with only two distinct alleles can operate in autotetraploids. In this case, the limiting values of
= 2/5 and ß = 3/5 are easily found analytically. The proportion of pollen that is lost through incompatibility (given in the final column of Table 2) is much less than the proportion lost for diploids, which is 2/k. However, unlike in the diploid case discussed by Steiner and Gregorius (1994), the limiting values of n, ßn, n, and n do not minimize pollen wastage. For example, with two S alleles, pollen wastage would be minimized when all individuals have genotype S1S1S2S2, that is, with = 0, ß = 1, = 0, rather than at the equilibrium situation where = 0.4, ß = 0.6, = 0. Various strands of evidence (Igic and Kohn 2001) suggest that S loci are of extremely ancient origin, and hence they already existed in the diploid ancestors of the autotetraploids; certainly there is no evidence of GSI arising in a polyploid species when it is absent in diploid progenitors. Thus there is no evolutionary basis for expecting pollen wastage to be optimized in polyploid populations. Currently the extent of the relationship between polyploidy and self-compatibility is receiving renewed attention, though no consensus has yet been reached (e.g., Mable 2004; Miller and Venable 2000). In this work, we focused on autotetraploid GSI systems with a single S locus and assuming "competitive interaction." However, other mechanisms such as dominance (Lewis 1947) may operate. There are other types of GSI systems with different genetic mechanisms, such as the two-locus S-Z system in many Poaceae (Baumann et al. 2000). Furthermore, GSI systems in autopolyploids in which any two of four alleles can pair may differ from those in the more common allopolyploids showing disomic inheritance. Our approach can be extended to consider these alternatives. The model developed here assumes an infinite random mating population without mutation. Further studies are needed to investigate the effects of random genetic drift and the introduction of mutation and population subdivision on the equilibrium frequency of different GSI classes of genotypes. From these studies, together with the current deterministic results, we may be able to make some inferences about the underlying evolutionary forces driving the dynamics of these alleles in polyploid plants. Furthermore, by comparing field data with these theoretical results under various scenarios, we may be able to infer evolutionary forces driving the dynamics of GSI alleles in particular systems.
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Footnotes |
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Corresponding Editor: Reid G. Palmer
Received July 13, 2004
Accepted February 7, 2005
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References |
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TopAbstractA Deterministic GSI Model...Results and DiscussionReferences |
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