Impacts of Seed and Pollen Flow on Population Genetic Structure for Plant Genomes With Three Contrasting Modes of Inheritance

Corresponding author: Xin-Sheng Hu, The Research Institute of Forestry, Chinese Academy of Forestry, Wan Shou Shan, Beijing 100091, China., xinsheng{at}rif.forestry.ac.cn (E-mail)

Communicating editor: R. G. SHAW

	ABSTRACT

TOP ABSTRACT ISLAND MODEL STEPPING-STONE MODEL DISCUSSION LITERATURE CITED

The classical island and one-dimensional stepping-stone 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 migration-drift equilibrium is greater than that for paternally inherited genes, which in turn is greater than that for biparentally inherited nuclear genes. In the stepping-stone 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 short-distance 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 long-distance components of migration in the stepping-stone model.

Avariety 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 stepping-stone 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 long-distance 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 , NEALE et al. 1991 ; NEALE and SEDEROFF 1989 ; DONG and WAGNER 1993 , DONG and WAGNER 1994 ; POWELL et 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 ; BIRKY et al. 1989 ; PETIT et 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 stepping-stone 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

TOP ABSTRACT ISLAND MODEL STEPPING-STONE MODEL DISCUSSION LITERATURE CITED

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 two-step 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 , WRIGHT 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.

View larger version (11K): In this window In a new window Download PPT slide   Figure 1. The basic processes of pollen flow and seed flow occurring among populations within the life cycle of a hermaphrodite plant population.

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, p^p_i.t+1, is

(1)

where m_p is the rate of pollen flow and is the mean gene frequency over all populations. It can be shown that the gene frequency in seeds formed by random combination between pollen and ovules at generation t + 1, p^s_i.t+1, is the arithmetic average of the gene frequencies in male and female gametes, i.e.,

(2)

Similarly, after seed flow the gene frequency in seeds of population i, p^'_i.t+1, is

(3)

where m_s is the rate of seed flow. The variance of gene frequencies in seeds over an infinite number of populations after seed flow, ²_p^'t+1, is

(4a)

(4b)

(4c)

where E stands for expectation with respect to the gene frequency distribution among populations, (p^'_t+1) is the probability density of the gene frequency at generation t + 1, and n is the number of populations. Equation 4a is the expression of the variance of gene frequencies in the case of the island model. The method for the use of Equation 4b to approximate Equation 4a can be found in KIMURA and WEISS 1964 (pp. 562–563), and also in NAGYLAKI 1979 (p. 168) in the derivation of his Equation 22, although KIMURA and WEISS 1964 and NAGYLAKI 1979 did not indicate this approximation clearly.

Thus, the variance of gene frequencies among populations after seed flow can be obtained via Equation 2 and Equation 3,

(5)

It can be seen from Equation 5 that the variance of gene frequencies among populations is reduced due to different contributions of seed flow (1 - m_s) and pollen flow (1 - m_p/2). However, after randomly sampling N seeds in each population, the variance of gene frequencies among populations will increase because of genetic drift. Thus the gene frequency of population i adults, p_i.t+1, which can be regarded as the same as that in a corresponding sample of seeds for selectively neutral genes, is

(6)

where _p^'t+1 is the change due to sampling. Similarly, according to Equation 6 we can obtain the variance of gene frequencies among populations in adults,

(7)

The second item on the left-hand side of Equation 7 is equal to zero because of the independence of (p^'_i.t+1 - ) and _pi.t+1. The third item on the left-hand side of Equation 7 is

(8a)

(8b)

(8c)

(8d)

where E stands for expectation with respect to -distribution within population. Use of Equation 8aEquation 8bEquation 8cEquation 8d to approach the variance of 's over an infinite number of populations can also be found in KIMURA and WEISS 1964 (p. 563) and NAGYLAKI 1979 (p. 168).

Therefore, the total variance of gene frequencies among populations after random sampling can be obtained by putting Equation 8d into Equation 7,

(9)

Substituting Equation 5 into Equation 9, we may obtain

(10)

According to Equation 10, the variance of gene frequencies among populations at steady state can be obtained by letting ²_pt+1 = ²_pt = ²,

(11)

Using Wright's notation (WRIGHT 1969 , pp. 295 and 299) according to Equation 11 we can obtain F_st(b) (= ), the population differentiation for biparentally inherited diploid genes, i.e.,

(12a)

(12b)

if migration rates of seed and pollen flow are small. To obtain a comparable expression between genes differing in mode of inheritance, denote the effective number of genes by Ñ = 2N, which is total effective number of genes in each population in adults. Denote the effective migration rate by = m_s + , which is due to diploid seed flow and haploid pollen flow. Thus Equation 12b can be rewritten as

(12c)

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,

(13a)

(13b)

if m_s and m_p are small. It is necessary to note that in deriving Equation 13a the gene frequency in seeds after random combination between male and female gametes, p^s_i.t+1, is equal to the gene frequency after pollen flow, p^p_i.t+1, because of haploid genes. This is different from the case of biparentally inherited genes (Equation 2). Similarly, let Ñ = N be the effective number of haploid genes. Let = m_s + m_p be the effective migration rate. This is because both seed and pollen flow contribute to migration of paternally inherited haploid genes. Thus, using these equalities, F_st(p) can be written in the same form as Equation 12c.

Maternally inherited haploid organelle genes:
Similarly for maternally inherited haploid genes, we can obtain

(14a)

(14b)

if m_s is small. It is necessary to note that in deriving Equation 14a pollen flow and random mating can be ignored because only seed flow contributes to migration of maternally inherited haploid genes. Let Ñ = N be the effective number of haploid genes and = m_s be the effective migration rate. Then F_st(m) can be written in the same form as Equation 12c. The above results show that the 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.

	STEPPING-STONE MODEL

TOP ABSTRACT ISLAND MODEL STEPPING-STONE MODEL DISCUSSION LITERATURE CITED

Assumptions:
The basic assumptions are similar to those in the classical stepping-stone model (KIMURA and WEISS 1964 ; WEISS and KIMURA 1965 ). An infinite array of populations lie on a Cartesian grid. Only the one-dimensional 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 long-distance migration (m_p for pollen and m_s for seeds) that draws pollen and seed from all populations. For the one-step 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 stepping-stone model (KIMURA and WEISS 1964 ),

(15)

where m_p1 stands for the rate of pollen migration per generation one step away from the source population, and m_p stands for the rate of long-distance pollen dispersal per generation. Here the long- and short-distance pollen migrations are assumed to occur at each generation. For simplicity in mathematical treatment, the shift operator S used by WEISS and KIMURA 1965 is also employed here. Equation 15 can be rewritten as

(16)

where ^p(i) = p^p(i) - , (i) = p(i) - and L_p = (1 - m_p1 - m_p) + m_p1(S^-1 + S). The shift operator S is defined by the following properties: Sp(i) = p(i + 1), S^-1p(i) = p(i - 1) (WEISS and KIMURA 1965 , p. 132).

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

(17)

which is the arithmetic average of the gene frequencies in male and female gametes. Substituting Equation 15 into Equation 17, we then obtain

(18)

where

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

(19)

where m_s1 stands for the rate of seed migration per generation one step away from the source population, m_s stands for the rate of long-distance seed migration per generation, and the _s(i) is the change of gene frequency due to sampling, with mean E[_s(i)] = 0 and variance V[_s(i)] Again, the long- and short-distance seed migrations are assumed to occur at each generation. Putting Equation 17 into Equation 19, we can obtain

(20)

where

(21)

in which

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 one-step 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 one-step migration is considered (WEISS and KIMURA 1965 ).

Because the L in Equation 21 satisfies the relationship

(22)

where (k) is the unnormalized correlation function (WEISS and KIMURA 1965 , p. 133), we can directly obtain the solution of the correlation of gene frequencies between populations k steps apart at steady state, r(k), by substituting

(23)

into Equation 3.10 of WEISS and KIMURA 1965 (p. 134). Equation 23 was developed by WEISS and KIMURA 1965 (p. 134) to obtain the exact solution of r(k). According to WEISS and KIMURA 1965 , the general solution to the r(k) can be obtained,

(24)

where

(25a)

(25b)

Equation 24 provides an exact solution for r(k). However, by ignoring the very small part ß₂, we can obtain

(26)

Justification of Equation 26a can be easily obtained by comparing Equation 26 with Equation 4.4 of WEISS and KIMURA 1965 .

If the rates of short-distance migration are much larger than those of long-distance 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 A₁(k) is much greater than A₂(k). Therefore, r(k) can be approximated by

(27a)

(27b)

Equation 27b is equivalent to Equation 1.13 developed by KIMURA and WEISS 1964 for animal populations.

Now, consider population differentiation. Notations the same as those of WEISS and KIMURA 1965 are used. Let (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, ^s(0),

(28)

Similarly, according to Equation 20 we can obtain the variance of gene frequencies among populations in adults at the next generation, '(0),

(29)

By putting Equation 28 into Equation 29 and by letting '(0) = (0), we can obtain the general solution of the F_st(b) (= ) at steady state:

(30)

If m_s1 = m_p1 = 0, but m_s 0 and m_p 0, according to Equation 25aEquation 25b we can obtain

and

Thus Equation 30 reduces to Equation 12b in the infinite island model.

If the rates of short-distance migration are much larger than those of long-distance migration, i.e., m_s1 >> m_s, m_p1 >> m_p, and the ß₂ 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

If we ignore the small part of the ()(1 - L²_s)r(0) (= O(m_p1m_p, m²_p1, m²_p)), which is introduced by pollen flow, Equation 30 becomes

(31a)

(31b)

where ₁ = m_s1 + , = m_s + , and Ñ = 2N. Equation 31b is equivalent to Equation 1.12 of KIMURA and WEISS 1964 for animal populations.

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

(32)

where ^p(i) = p^p(i) - , (i) = p(i) - , and the L_p is the same as that in Equation 16. The gene frequency in resident seeds formed by random combination between pollen and ovules is the same as that in male parents (pollen), i.e., p^s(i) = p^p(i). After seed flow the gene frequency in adults at the next generation can be written as

(33)

where

in which

and

The _s(i) is the change of gene frequency by sampling, with mean E[_s(i)] = 0 and variance V[_s(i)] The correlation of gene frequencies between populations k steps apart at steady state can be obtained by substituting the ₀ and ß₁ in Equation 33 into Equation 27aEquation 27b. However, F_st(p) at steady state is different from the case of biparentally inherited genes because of paternally inherited haploid genes and is shown to be

(34)

If m_s1 = m_p1 = 0, but m_s 0 and m_p 0, then F_st(p) is the same as that in the island model (Equation 13aEquation 13b). If m_s1 >> m_s and m_p1 >> m_p and we let ₁ = m_s1 + m_p1, = m_s + m_p, Ñ = N, then F_st(p) can be written in the same form as that of Equation 31b.

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 one-dimensional stepping-stone 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 short-distance migration (₀/ß₁). Thus, the ratio of ₀/ß₁ is more important than either of them separately in determining the correlation of gene frequencies. Generally an increase in the rate of long-distance migration (₀) reduces the correlation (r(k)), while an increase in migration from neighboring populations strengthens the genetic correlation. Therefore long- and short-distance 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 short-distance 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

(35)

according to Equation 27aEquation 27b for each of the three genomes. Let R₁ = and R = . If R₁ > R, it can be shown from Equation 35 that the correlation of gene frequencies for maternally inherited haploid genes is smaller than that for biparentally inherited diploid genes, which in turn is smaller than that for paternally inherited haploid genes, i.e., r_m(k) < r_b(k) < r_p(k). However, if R₁ < R, then r_m(k) > r_b(k) > r_p(k). To confirm these inferences about the properties of r(k) outlined above, we directly calculated the genetic correlation according to Equation 24. Let m_p1 = 10^-2, m_s1 = 10^-4, m_p = 10^-5, and m_s = 10^-6, i.e., R₁ > R. The results indicate that r_p(k) > r_b(k) > r_m(k) (Figure 2A). Letting m_p1 = 10^-2, m_s1 = 10^-3, m_p = 10^-5, and m_s = 10^-7, i.e., R₁ < R, the results indicate that r_m(k) > r_b(k) > r_p(k) (Figure 2B). These are consistent with the inferences above. The correlation for biparentally inherited diploid genes, however, is very close to that for paternally inherited haploid genes.

View larger version (25K): In this window In a new window Download PPT slide   Figure 2. Comparison of the genetic correlation with distance between populations r(k) for biparentally, paternally, and maternally inherited genes at migration/drift equilibrium. Levels of short-distance pollen migration, short-distance seed migration, long-distance pollen migration, and long-distance seed migration are as follows: (a) mp1 = 10-2, ms1 = 10-4, mp = 10-5, and ms = 10-6; (b) mp1 = 10-2, ms1 = 10-3, mp = 10-5, and ms = 10-7.

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 one-step 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 one-step pollen flow (m_p1) can increase r(k) for paternally and biparentally inherited genes, but has no effect on maternally inherited genes.

View larger version (19K): In this window In a new window Download PPT slide   Figure 3. Effects of (a) short-distance seed flow and (b) short-distance pollen flow on genetic correlation with distance r(10) for biparentally, paternally, and maternally inherited genes at migration/drift equilibrium. Fixed values for short- and long-distance pollen and seed migration are as follows: (a) mp1 = 10-2, mp = 10-5, and ms = 10-6; (b) ms1 = 10-2, mp = 10-5, and ms = 10-6.

Ratio of pollen to seed flow:
The one-dimensional case of the stepping-stone model may allow us to estimate the ratio of pollen to seed flow for both short- (R₁) and long-distance 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 = ) (SLATKIN and BARTON 1989 ). We show next the application of Equation 1.12 of KIMURA and WEISS 1964 , i.e., F_st = in plant populations for estimating R₁ and R.

According to Equation 31b and the equivalent equations for paternally and maternally inherited genes, we can obtain

(36a)

(36b)

(36c)

Let A = and B = . According to Equation 36aEquation 36bEquation 36c, we can obtain the equations that describe the relationship between R₁ and R,

(37a)

(37b)

where c₁ = 2(B - 2A + 1) and c₂ = 4A - B - 3. Thus, if = c²₂ - 4c₁ 0, there are two solutions for either R₁ or R, i.e.,

(38)

To use Equation 38 to estimate the ratio of pollen to seed flow from both short- and long-distance migration, we must first estimate the F_st value for each of the three inherited types of selectively neutral genetic markers. It is suggested that the method introduced by WEIR and COCKERHAM 1984 is appropriate. Second, we need to decide whether the ratio of pollen to seed flow is greater for long- or short-distance migration. Theoretically, this could be resolved by comparing the correlations of gene frequencies k steps apart among the three plant genomes, as explained in the previous section. Alternatively a judgement on the relative values of R₁ and R could be made based on knowledge of the biological dispersal characteristics of the plant species being studied. The standard errors of the estimates of R₁ and R could be calculated using a jackknife procedure over genetic markers or by bootstrapping methods (WEIR and COCKERHAM 1984 ).

	DISCUSSION

TOP ABSTRACT ISLAND MODEL STEPPING-STONE MODEL DISCUSSION LITERATURE CITED

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 , WRIGHT 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 Equation 12a, Equation 13a, and Equation 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 second-order terms in each, Wright's existing F_st equation (WRIGHT 1969 , Equation 12c) may be used to predict differentiation at drift-migration 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 (PETIT et al. 1993 ; ENNOS 1994 ).

The second aim of this article is to adapt the one-dimensional stepping-stone 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 popu-lations) and long distance (from a combined pollen pool). This model is arguably more biologically realistic than the island model, in which only long-distance 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 long-distance components of migration in the one-dimensional stepping-stone 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 long-distance migration before an estimate can be made.

If all these prerequisites are in place, simultaneous estimation of pollen to seed flow ratios for short- and long-distance 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 short- and long-distance 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 stepping-stone 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 R₁ and R is invalid.

The development of the one-dimensional stepping-stone 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 R₁ distances. If R > R₁ then the correlation for maternally inherited genes will be greater than that for the other genomes, but when R₁ > R the correlation for maternally inherited genes will be less than that for the other genomes (Figure 2 and Figure 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 R₁ 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 R₁ when long-distance gene migration occurs principally by pollen flow and founding of populations is by short-distance seed migration. On the other hand, R is likely to be lower than R₁ when long-distance pollen flow is limited and new populations are regularly founded by long-distance seed transfer. Evidence that the ratio of pollen to seed flow varies between long- and short-distance 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 stepping-stone 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 2F_st/m_p1 for biparentally inherited nuclear genes and F_st/m_s1 F_st/m_p1 for paternally inherited genes according to Equation 31b. Similarly, we can also roughly obtain r(k)/m_s1 2r(k)/m_p1 for the biparentally inherited nuclear genes and r(k)/m_s1 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.

Manuscript received April 15, 1998; Accepted for publication February 8, 1999.

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