Abstract
We study the behavior of Φ_{ft}, a recently introduced estimator of instantaneous pollen flow, which is basically the intraclass correlation of inferred pollen cloud genetic frequencies among a sample of females drawn from a single population. Using standard theories of identity by descent and spatial processes, we show that Φ_{ft} depends on the average distance of pollen dispersal (δ) and on the average distance between sampled mothers (
OST authors (e.g., Slatkin and Barton 1989; Nath and Griffiths 1996; Beerli and Felsenstein 1999) have focused on the estimation of the “historical” migration rate, i.e., the effective longterm average (Hudson 1998). Such estimates are useful in understanding the evolutionary history of a set of populations, but they say nothing about the current level of gene flow, a more relevant predictor of contemporaneous (realtime) genetic exchange among a set of populations.
The need to estimate “realtime” gene flow rates has led to the design of direct estimators of gene flow. The most used method is currently paternity analysis (Schnabel and Hamrick 1995), which is especially efficient when highly polymorphic markers are used (Dow and Ashley 1996; Streiffet al. 1999), but conclusive analysis requires knowledge of the identity and genotypes of all potential males who could have contributed pollen to females within the stand.
Several studies (Dow and Ashley 1996; Streiffet al. 1999) have now shown that a substantial portion of pollen comes from outside the stand, and it has become clear that characterizing all potential fathers that might have contributed pollen to the mothers within the stand is virtually a hopeless task. Lacking the ability to identify and/or evaluate the external males, paternity analysis can provide only a minimum estimate of average pollination distance across a landscape, sometimes a serious underestimate. Clearly, the estimation of realtime pollen dispersal must be addressed by some other means.
Smouse et al. (2001) have proposed a new estimation procedure, which uses only the genotypes of the mothers and of seedlings derived from them, along with the spatial positions of the mothers; the potentially contributing males are ignored. Using the sampled mothers as strata and their seedlings as replicates, one estimates the intraclass correlation of paternal gametes drawn from a single mother, Φ_{ft}, which is then used to estimate average pollination distance. Based on a simulation study, Smouse et al. (2001) have shown that Φ_{ft} is directly related to the decay parameter of the pollen dispersal curve.
The aim of this article is to address several questions that have been raised by that first study:

How is Φ_{ft} affected by the chosen dispersal function? It will be difficult to derive a valid estimate of the average pollen dispersion from an estimate of Φ_{ft} if that parameter is overly sensitive to the shape of the (usually unknown) distribution.

For any particular pollen dispersal distribution, what is the precise relation between Φ_{ft} and the dispersal rate?

How is the estimate of Φ_{ft} affected by the average physical distance from one sampled mother to another?

What is the impact of adult density within the reference population?
Answers to these questions should allow us to design a proper estimate of the average pollination distance from Φ_{ft}. We develop the theoretical framework necessary to address these issues.
THE MODEL
General context: Assume that we have an infinite population, with adult individuals randomly distributed across the landscape, at density (d) per squared unit of distance. All individuals are monoecious and selffertile, but they practice no more selfreproduction than would be expected at random. Allele frequencies are uniform across the landscape, all individuals are noninbred, and all have the same male fecundity. We assume that male gametes disperse independently and according to a probability distribution, to which we return below. We consider a sample of mothers drawn from among the adults, spaced an average distance of
The genetic diversity within and among the pollen clouds impinging upon the various mothers depends upon the pollen dispersal distribution. We focus here on two isotropic twodimensional distributions of pollen dispersal: the normal distribution and the exponential distribution. In Cartesian coordinates measured from a single mother, assumed to be at coordinates (0, 0), the normal distribution with parameter σ will be
Figure 1 gives an example of the pollen distributions for both normal and exponential distributions, with the same average pollen dispersal distance, δ = 10. These curves have very different shapes; the exponential distribution has a sharp peak at zero, but it also has a greater probability of reaching large dispersal distances.
The different shapes of the two distributions affect the dispersion of pollen distances for a given value of δ. One can gauge this dispersion via the variance of pollen dispersion, defined as v^{2} = E(z^{2}) — E^{2}(z), where z = (x^{2} + y^{2})^{1/2} is the linear distance from the position of the pollinating male (x, y) to the index female (0, 0). Alternatively, one could use the mean squared distance η^{2} = E(z^{2}) from that same index female. In any case, all three measures (δ^{2}, η^{2}, v^{2}) are simple functions of the squares of the decay rate parameters of both the normal (σ^{2}) and exponential (γ^{2}) distributions. For the normal, the expected squared distance is
The probability that two seedlings, derived from the same mother, have the same father: The “genetic structure” of the pollen clouds of different maternal individuals is a function of the likelihood of drawing two pollen grains with alleles that are identical by descent. The first task is to compute the probability that two pollen grains from a single female are drawn from the same male, using the same reasoning as in Wright (1946, 1969). The father of the first seed has coordinates (x, y) with probability p(x, y). Since we assume a density of d across the landscape, the father of the second seed will be the same as the father of the first seed if his coordinates (x_{2}, y_{2}) fall in the square interval
Note that for both distributions, Q_{0} and N_{ep} are simple functions of σ^{2} or γ^{2}. Whether these results are converted into units of δ, η, or v, the essential information is to be found in the relevant decay rate parameter, σ or γ. The choice of parameterization is largely a matter of convenience, and for both normal and exponential cases, we find it useful to express Q_{0} in terms of the average distance (δ) of pollen flow. For the normal distribution, by substituting (4) into (11), we obtain
The probability that two seedlings, derived from two different mothers, have the same father: Assume that these two females are at a distance x_{1} apart. Without loss of generality, we set our system of Cartesian coordinates so that the first female is at position (0, 0) and the second is at position (x_{1}, 0). For a male at position (x, y), the probability of fertilizing the first female is, as above, p(x, y). For the same male, the probability of fertilizing the second female is p(x — x_{1}, y). The probability Q(x_{1}) that these two females are fertilized by the same male is
Derivation of Φ_{ft}: We define f_{i}, f_{f}, f_{t}, and f_{p} as the probabilities of IBD for different pairs of genes. For two alleles within an individual, pr(IBD) = f_{t}; for two pollen grains drawn at random from the pollen cloud of a single female, pr(IBD) = f_{f}; for two pollen grains drawn at random from the pollen cloud of all females, pr(IBD) = f_{t}; and for two genes sampled at random from the whole population, pr(IBD) = f_{p}. For any of these quantities, h_{x} = 1 — f_{x} will denote the diversity within that same compartment. We can relate Φ_{ft} to these coefficients in the same way that F_{st} (Nei 1973; see also Slatkin 1991) is related to the pr(IBD):
To calculate h_{f}, the diversity within the pollen cloud of a single female, we must consider two genes sampled within this pollen cloud. These two genes come, with probability Q_{0}, from the same father. In this case, they are derived from the same paternal chromosome with probability 1/2 and cannot be different. Also, with probability 1/2, they are derived from the different homologous chromosomes of the father and will then be different with probability h_{i}. Thus, two genes derived from the same father will, on average, be different with probability h_{i}/2.
On the other hand, these two genes will be drawn from different fathers, with probability 1 — Q_{0}, in which case they will be different with probability h_{p}. Averaging over all cases,
To calculate h_{t}, the diversity within the pollen clouds of all sampled mothers, we must consider two genes sampled within the pollen clouds of different mothers. Mothers are, on average, at a distance
We assume here no inbreeding among the adults themselves; i.e., we set the inbreeding coefficient F = 0. This coefficient is defined as F = 1 — h_{i}/h_{p}, so for F = 0, we have h_{i} = h_{p}; i.e., two genes sampled at random from within the same individual are neither more nor less likely to be identical than two genes sampled at random from the whole population, in which case, (21) and (22) become
If the sampled mothers are far enough apart (
DISCUSSION
This study advances our understanding of the various parameters that might affect the estimation of pollen dispersal distance, using Φ_{ft}. One of the most important results is that the Φ_{ft} parameter will not depend upon distance between mothers (
This might sound useless from a practical point of view since in experimental situations δ is the unknown parameter to be estimated. If a rough estimate of this distance is already available, however, based either on previous genetic studies or on the observation of physical dispersal distance of pollen, mothers can be chosen so that all are at least five times more distant from each other than this rough estimate. If it later develops that mothers have been placed too close to one another, the estimate can be adjusted easily. An algorithm is described in the appendix, for which both a C implementation and a DOSexecutable file are available from F.A.
We also showed that the global diversity (h_{p}) of the marker loci employed has no effect on the relation between Φ_{ft} and δ, and thus on the estimation of δ. It is noteworthy that the relation between F_{st} and Nm also does not depend on populationwide genetic diversity (Wright 1951). Nevertheless, estimation variance for Φ_{ft} increases when less polymorphic markers are used (Smouseet al. 2001), as expected, and it will therefore always be more useful to employ highly polymorphic markers when available, or to increase the number of markers employed, when minimally polymorphic markers are used.
An important result is that the basic form of the relation between Φ_{ft} and δ is not affected by the choice of dispersal function, and while there are numeric consequences of the choice, they are not profound. This is of importance, since little is known about the shape of this dispersal function a priori, and to infer it (in detail) from genetic data would be prohibitively labor and cost intensive. The distance, beyond which two mother trees are unlikely to be fertilized by the same male, is only slightly larger for the exponential distribution than for the normal.
It is also interesting to note that, although increasing density (d) decreases Φ_{ft}, it has no impact on the minimum distance at which mothers (
We have assumed here that there is no undue degree of selfing, although an elevated/reduced selfing rate can be incorporated into the model. We have also ignored the possibility of past inbreeding for the adult generation, as well as the possibility that nearby males are more related to the female (and each other) than randomly placed males; but in real populations with restricted propagule flow, the probability of IBD among pollinating adults is expected to decrease with distance (Malécot 1973). If there were “local structure” among the adults, even the pollen drawn from different males would yield increase in Q_{0} and Q, which would affect Φ_{ft}. Variance in male fecundity and correlated dispersal of male gametes (for example for vectorpollinated species) are also likely to occur in some cases, either of which would inflate our estimate of Φ_{ft}. We leave these extensions for later communication.
Acknowledgments
We thank the other members of the TwoGener team, V. Sork, R. Westfall, and R. Dyer, as well as two anonymous reviewers, for helpful comments and suggestions on this manuscript. F.A. received a Formation Complémentaire par la Recherche grant from the French Ministère de l'Agriculture and a complementary grant from the North Atlantic Treaty Organization. P.E.S. is supported by McIntireStennis grant United States Department of Agriculture/NJAES17309.
APPENDIX: ALGORITHM FOR THE ESTIMATION OF THE AVERAGE DISTANCE OF POLLEN DISPERSAL FROM Φ_{ft}
We assume here that an estimate of Φ_{ft} is available from the study and that the average distance between mothers (
This recursive algorithm converges in no more than 100 iterations for the values we have tested and can be executed within a few minutes on an average PC computer. It yields estimated values for λ or σ that can be directly transformed into an estimate of the average distance of pollen dispersal (δ), using (4) or (5), or into η and v, using (7)(9).
Footnotes

Communicating editor: M. A. Asmussen
 Received May 10, 2000.
 Accepted November 10, 2000.
 Copyright © 2001 by the Genetics Society of America