Abstract
The classical island and onedimensional steppingstone models of population genetic structure developed for animal populations are extended to hermaphrodite plant populations to study the behavior of biparentally inherited nuclear genes and organelle genes with paternal and maternal inheritance. By substituting appropriate values for effective population sizes and migration rates of the genes concerned into the classical models, expressions for genetic differentiation and correlation in gene frequency between populations can be derived. For both models, differentiation for maternally inherited genes at migrationdrift equilibrium is greater than that for paternally inherited genes, which in turn is greater than that for biparentally inherited nuclear genes. In the steppingstone model, the change of genetic correlation with distance is influenced by the mode of inheritance of the gene and the relative values of long and shortdistance migration by seed and pollen. In situations where it is possible to measure simultaneously F_{st} for genes with all three types of inheritance, estimates of the relative rates of pollen to seed flow can be made for both the short and longdistance components of migration in the steppingstone model.
A variety of models have been formulated to analyze the development of population genetic structure under a balance between drift and migration. To date they have tended to concentrate on the problem of differentiation for nuclear genes and are appropriate for situations in animals where diploid individuals migrate between populations. The simplest island model comprises many discrete populations with a certain proportion of migrants interchanging between them irrespective of their spatial proximity (Wright 1969). The steppingstone model (Kimura and Weiss 1964) deals with a more realistic situation in which a certain proportion of migration occurs strictly between neighboring populations, while the remainder takes place by longdistance migration, with migrants being drawn randomly from a migrant pool.
While these classical models are appropriate for investigating differentiation for nuclear markers in animal populations, they are inadequate for fully describing genetic differentiation under drift/migration in plant populations. For these situations models that explicitly incorporate seed and pollen flow as agents of migration are needed. In addition the models must address the cases of differentiation for the uniparentally inherited (both maternal and paternal) chloroplast and mitochondrial markers that can now be detected in natural plant populations through the application of molecular techniques (Neale et al. 1986, 1991; Neale and Sederoff 1989; Dong and Wagner 1993, 1994; Powellet al. 1995).
Recently the classical island models that deal with population differentiation for nuclear genes have been extended to consider differentiation for uniparentally inherited organelle genes in animal and plant populations (Takahata and Palumbi 1985; Birkyet al. 1989; Petitet al. 1993). Petit et al. (1993) showed that the effects of gene flow on G_{st} at equilibrium depend on the relative rates of pollen and seed migration, as well as the mode of inheritance of genes (McCauley 1995). Further insight into this area was given by Ennos (1994), who used an island model to show that a comparison of F_{st} values for markers with different modes of inheritance could provide an estimate of the relative rate of pollen flow to seed pollen flow among populations.
The purpose of this article is to consolidate and develop further theories required for understanding and interpreting population genetic structure of nuclear, chloroplast, and mitochondrial genes in plant populations under drift/migration equilibrium. In the first part we derive expressions for population differentiation for plant populations under the island model using the rigorous approach of Wright (1969) and show that these are completely compatible with previous results found by Petit et al. (1993) and Ennos (1994). We then incorporate migration by seed and pollen flow into the steppingstone model of Kimura and Weiss (1964), look at the implications for the behavior of biparentally, paternally, and maternally inherited genes, and explore the inferences that can be drawn about the relative rates of pollen and seed flow among populations from data on these differently inherited genes.
ISLAND MODEL
The rate of gene migration in the classical expression for F_{st}, derived by Wright (1969) for an island model, refers to the simple migration of diploid individuals between populations before mating takes place. When dealing with hermaphrodite plants, it is necessary to model gene migration as a twostep process, which occurs both by migration of haploid pollen before fertilization and by migration of diploid seeds after fertilization.
Drift/migration balance can be reached by seed flow, by pollen flow, or by both forms of gene migration combined. In the following we rederive an equation for F_{st} under drift/migration balance that applies to hermaphrodite plants following the method used by Wright (1943, 1969). We also demonstrate that complex expressions for F_{st} derived for each genome can be reduced to Wright’s general equation by substitution of appropriate values for effective population size and migration rate specific to the different genomes.
Assumptions: The model deals with a hermaphrodite population of plants showing random mating. Paternally and maternally inherited genes are assumed to be haploid, while biparentally inherited genes are considered to be diploid. Two alleles per selectively neutral gene are considered in each case. The mutation rate for each gene is assumed to be much smaller than the migration rate and is therefore not considered. There is no association among genes differing in mode of inheritance. We consider initially that all the populations have already become established and contain the same effective number of adult plants, N. The effective number of paternal and maternal genes is considered to be N because they are effectively haploid. This assumption can be relaxed if they are not the same by letting N = N_{f}, the effective number of maternally inherited genes, and N = N_{m}, the effective number of paternally inherited genes.
Figure 1 illustrates the processes that occur from generation to generation and influence rates of gene migration and genetic drift among hermaphrodite plant populations. The gene frequencies in migrating pollen grains or seeds are equal to the average of gene frequencies over all populations. The male gametes, including those from migrated pollen grains, are assumed to combine randomly with female gametes (ovules) during the formation of seeds. The gene frequency in ovules before mating with pollen grains is assumed to be the same as that in the preceding generation.
Biparentally inherited diploid nuclear genes: The following derivation is based on the method used by Wright (1969, p. 292). Suppose that there are an infinite number of populations. At generation t (t ≥ 1), let p_{i.t} be the gene frequency of population i in adults. Each population contains the same number of adults, N. After pollen flow, the gene frequency in male gametes (pollen) of population i at generation t + 1,
Similarly, after seed flow the gene frequency in seeds of population i, p′_{i.t}_{+1}, is
Thus, the variance of gene frequencies among populations after seed flow can be obtained via Equations 2 and 3,
Therefore, the total variance of gene frequencies among populations after random sampling can be obtained by putting Equation 8d into Equation 7,
Paternally inherited haploid organelle genes: Using the same method as in the case of biparentally inherited diploid genes, we can obtain F_{st(p)} for paternally inherited genes at steady state,
Maternally inherited haploid organelle genes: Similarly for maternally inherited haploid genes, we can obtain
STEPPINGSTONE MODEL
Assumptions: The basic assumptions are similar to those in the classical steppingstone model (Kimura and Weiss 1964; Weiss and Kimura 1965). An infinite array of populations lie on a Cartesian grid. Only the onedimensional case is considered. Both forms of migration have two components: migration between populations one step apart (m_{p1} for pollen and m_{s1} for seeds) and longdistance migration (m_{p∞} for pollen and m_{s∞} for seeds) that draws pollen and seed from all populations. For the onestep migration, half of the pollen and seed comes from each side. The number of seeds produced in each population is assumed to be large enough to allow us to ignore sampling effects of pollen and ovules before seed formation.
Biparentally inherited diploid nuclear genes: Using the same notation as Weiss and Kimura (1965), let p(i) be the gene frequency in population i and p(i + k) be the gene frequency in the population k steps away from population i. Initially we assume that all populations comprise adult plants. Upon reaching the reproductive stage, each adult produces pollen. Let p^{p}(i) be the gene frequency in pollen grains after pollen flow, which can be written according to the steppingstone model (Kimura and Weiss 1964),
As in the case of the island model, after random combination between pollen and ovules the gene frequency in sampled seeds, formed as p^{s}(i), is
Similarly, after seed flow and then sampling, the gene frequency in adults at the next generation, p′(i), which is assumed to be the same as in seeds after seed flow, can also be expressed by
It can be seen from Equation 21 that the gene frequency in adults at the next generation is ultimately affected by populations up to two steps away due to the two processes of gene flow (pollen and seed flow), even though only onestep migration is considered for each process. This is because these two processes of gene flow are connected via the stage of random mating. Obviously, the situation is different from animal populations where only the two neighboring populations exchange genes with the studied population if only onestep migration is considered (Weiss and Kimura 1965).
Because the L in Equation 21 satisfies the relationship
Justification of Equation 26 can be easily obtained by comparing Equation 26 with Equation 4.4 of Weiss and Kimura (1965).
If the rates of shortdistance migration are much larger than those of longdistance migration for both seed and pollen flow, i.e., m_{s1} ⪢ m_{s∞}, m_{p1} ⪢ m_{p∞}, according to the discussion of Weiss and Kimura (1965, p. 136) we can see that A_{1}(k) is much greater than A_{2}(k). Therefore, r(k) can be approximated by
Now, consider population differentiation. Notations the same as those of Weiss and Kimura (1965) are used. Let ρ(0) be the variance of gene frequencies among populations. Using a method similar to that of Weiss and Kimura (1965), according to Equation 18 we can obtain the variance of gene frequencies in seeds among an infinite number of populations after pollen flow and random mating, ρ^{s}(0),
Similarly, according to Equation 20 we can obtain the variance of gene frequencies among populations in adults at the next generation, ρ′(0),
If the rates of shortdistance migration are much larger than those of longdistance migration, i.e., m_{s1} ⪢ m_{s∞}, m_{p1} ⪢ m_{p∞}, and the β_{2} is very small, according to Equation 3.11 and the discussions from Equation 4.3 to Equation 4.6 in Weiss and Kimura (1965, p. 134), we can obtain
Paternally inherited haploid organelle genes: To avoid repeating procedures similar to those in the case of biparentally inherited genes, the main results are listed below. After pollen flow, the gene frequency in pollen of population i, p^{p}(i), is
Maternally inherited haploid organelle genes: The case of maternally inherited genes is exactly the same as the case of paternally inherited haploid genes except that no pollen flow occurs. The correlation of gene frequencies between populations k steps apart, r(k), and the population differentiation, F_{st(m)}, can be immediately obtained by m_{p1} = m_{p∞} = 0 in corresponding equations of paternally inherited haploid genes.
Some properties of r(k): The onedimensional steppingstone model can provide some insight into the way that pollen and seed flow affect the genetic structure of natural plant populations. The model predicts how the correlation of gene frequencies between populations k steps apart, r(k), varies with seed and pollen flow. It can be seen from Equation 27b that r(k) decreases monotonically with the ratio of long to shortdistance migration (α_{0}/β_{1}). Thus, the ratio of α_{0}/β_{1} is more important than either of them separately in determining the correlation of gene frequencies. Generally an increase in the rate of longdistance migration (α_{0}) reduces the correlation (r(k)), while an increase in migration from neighboring populations strengthens the genetic correlation. Therefore long and shortdistance migration have completely different effects on r(k).
Another important point is that r(k) values differ for genes with different modes of inheritance and the relative values depend on the extent of long and shortdistance migration by seed and pollen. Denote the correlation of gene frequencies between populations k steps apart by r_{b}(k), r_{p}(k), and r_{m}(k) for biparentally, paternally, and maternally inherited genes, respectively. We can roughly obtain
Separate effects of pollen flow and seed flow on r(k) can also be seen, which can be investigated by holding pollen flow but changing seed flow or by fixing seed flow but changing pollen flow. Figure 3a shows that an increase in onestep seed flow (m_{s1}) may increase the genetic correlation for each of the three inherited types of genes. Similarly, Figure 3b shows that an increase in onestep pollen flow (m_{p1}) can increase r(k) for paternally and biparentally inherited genes, but has no effect on maternally inherited genes.
Ratio of pollen to seed flow: The onedimensional case of the steppingstone model may allow us to estimate the ratio of pollen to seed flow for both short (R_{1}) and longdistance dispersal (R_{∞}). This is an extension of the results obtained by Ennos (1994) for the island model.
Suppose that F_{st} can be estimated for each of the three plant genomes using selectively neutral genetic markers. If m_{s1} ⪢ m_{s∞} and m_{p1} ⪢ m_{p∞}, F_{st} can be approximated by the general formula in Equation 31b for each of the three genomes. In the island model, Wright’s F_{st} value has been used to estimate the average number of migrants (Nm = (1/F_{st} –1)/4) (Slatkin and Barton 1989). We show next the application of Equation 1.12 of Kimura and Weiss (1964), i.e.,
According to Equation 31b and the equivalent equations for paternally and maternally inherited genes, we can obtain
DISCUSSION
The first aim of this article is to consolidate work on the use of the classical island model to predict and contrast population genetic structure for the biparentally, paternally, and maternally inherited genes of plants. In contrast to previous studies by Petit et al. (1993) and Ennos (1994), we employ the basic method used by Wright (1951, 1969) that analyzes in detail the variance of gene frequencies among populations. The theoretical analysis incorporates the basic biological process responsible for gene flow in plants (Figure 1) and provides comprehensive equations (12a, 13a, and 14a) for describing population differentiation of biparentally, paternally, and maternally inherited genes in plants. As in previous studies population differentiation at equilibrium is larger for maternally inherited than that for paternally inherited genes, which in turn is greater than for biparentally inherited genes. If migration rates of both seed and pollen are small enough to be able to ignore their product or secondorder terms in each, Wright’s existing F_{st} equation (Wright 1969, Equation 12c) may be used to predict differentiation at driftmigration equilibrium by substituting in appropriate terms for the effective population size and migration rate of plant genes. These results fully support the conclusions of previous extensions of the island model to plant populations that were based on different approaches to the problem (Petitet al. 1993; Ennos 1994).
The second aim of this article is to adapt the onedimensional steppingstone model (Kimura and Weiss 1964) to make predictions about the genetic structure of plant populations linked simultaneously by two forms of gene flow, short distance (between adjacent populations) and long distance (from a combined pollen pool). This model is arguably more biologically realistic than the island model, in which only longdistance migration is considered. As for the island model it is possible to show that the standard expression for F_{st} derived by Kimura and Weiss (1964) for biparentally inherited genes in animal populations (Equation 31b) can be used to predict F_{st} for genes in plants by substituting relevant expressions for effective population size and migration rates. Again genetic differentiation for maternally inherited genes is larger than that for paternally inherited genes, which in turn is larger than that for biparentally inherited genes.
Using these expressions for genetic differentiation of biparentally, paternally, and maternally inherited genes, it is technically possible to derive expressions for the relative rates of pollen to seed flow for both the short and longdistance components of migration in the onedimensional steppingstone model. To apply these expressions in estimating such parameters, a plant species that displays all three modes of gene inheritance would have to be chosen. Conifers with paternally inherited chloroplast genomes and maternally inherited mitochondrial genes are possible candidates. Independent information is also needed in deciding whether the ratio of pollen to seed flow is greater for short or for longdistance migration before an estimate can be made.
If all these prerequisites are in place, simultaneous estimation of pollen to seed flow ratios for shortand longdistance migration can be conducted, but is likely to be problematic. This is because the estimation expressions include squared terms in F_{st} for all three genomes. Given the enormous limitations on accurate estimation of F_{st}, especially for the organelle genomes, it is unlikely that meaningful measurement of pollen to seed flow ratios for both shortand longdistance migration will be possible. These problems are illustrated by reference to appropriate data from Pinus flexilis (Latta and Mitton 1997). Here the estimated F_{st} value for paternally inherited genes is lower, rather than higher, than the estimated F_{st} value for biparentally inherited genes. Under either the island or steppingstone models this should not occur at equilibrium and indicates either a violation of the model assumptions or inaccurate estimation of the F_{st} parameters. In these circumstances application of the expressions for calculating R_{1} and R_{∞} is invalid.
The development of the onedimensional steppingstone model for plant populations allows us to make predictions not only about genetic differentiation for genes with different modes of inheritance, but also about patterns of correlation in gene frequency between populations. An important point to emerge is that the relative size of the correlation between populations for the three types of genes is dependent upon the relative pollen to seed flow ratios over long R_{∞} and short R_{1} distances. If R_{∞} > R_{1} then the correlation for maternally inherited genes will be greater than that for the other genomes, but when R_{1} > R_{∞} the correlation for maternally inherited genes will be less than that for the other genomes (Figures 2 and 3).
If suitable methods can be found for estimating and comparing the patterns of correlation with distance between populations for the three types of genes (Epperson 1993; Epperson and Li 1996), then it will be possible to determine whether the pollen to seed flow ratio is greater for short R_{1} or for long R_{∞} distance migration. Such a result may help to provide clues about the biology and population dynamics of plant species. R_{∞} is likely to be greater than R_{1} when longdistance gene migration occurs principally by pollen flow and founding of populations is by shortdistance seed migration. On the other hand, R_{∞} is likely to be lower than R_{1} when longdistance pollen flow is limited and new populations are regularly founded by longdistance seed transfer. Evidence that the ratio of pollen to seed flow varies between long and shortdistance migration has already been given by McCauley (1997), who analyzed F_{st} at different spatial scales in Silene alba. He demonstrated that in this species the relative importance of pollen flow declined as distance between populations increased.
One final conclusion from the analysis of the steppingstone model is that, for biparentally inherited genes, the effects of seed dispersal on population structure are more important than the effects of pollen dispersal. However, for paternally inherited markers the effects of seed and pollen flow are equivalent. This can be deduced, for instance, in the case of migration between adjacent populations. Here we roughly obtain ∂F_{st}/∂m_{s1} ζ 2∂F_{st}/∂m_{p1} for biparentally inherited nuclear genes and ∂F_{st}/∂m_{s1} ζ ∂F_{st}/∂m_{p1} for paternally inherited genes according to Equation 31b. Similarly, we can also roughly obtain ∂r(k)/∂m_{s1} ζ 2∂r(k)/∂m_{p1} for the biparentally inherited nuclear genes and ∂r(k)/∂m_{s1} ζ ∂r(k)/∂m_{p1} for paternally inherited genes according to Equation 35. These differences in the influence of seed and pollen flow on both genetic differentiation and patterns of genetic correlation between populations are caused by differences in ploidy level between seed and pollen for biparentally inherited genes, but equivalence of ploidy level for paternally inherited genes. The greater influence of seed compared with pollen dispersal on population structure for biparentally inherited genes has also been emphasized by Doligez et al. (1998).
Acknowledgments
We deeply appreciate Dr. R. G. Shaw and three anonymous referees for valuable comments that substantially improved the earlier versions of this article. Thanks are given to Professor N. H. Barton for helpful comments on this article and to the Overseas Development Administration (ODA), United Kingdom, for financial support.
Footnotes

Communicating editor: R. G. Shaw
 Received April 15, 1998.
 Accepted February 8, 1999.
 Copyright © 1999 by the Genetics Society of America