Using clines to estimate dispersal

In a similar manner, evolutionary ecologists have begun to utilize clinal theory to estimate dispersal distances of organisms. This approach may become particularly useful to marine researchers, because most marine invertebrates produce hundreds to thousands of microscopic offspring per parent and as a consequence, there are few empirical estimates of larval dispersal of marine organisms to within several orders of magnitude. However, inferring mean dispersal distance requires a precise estimate of the selection coefficient (se), which is itself a logistically difficult task. Field-based experiments can sometimes detect rather strong selection coefficients, but weaker levels of selection are more difficult to measure. Laboratory-based experiments can be more sensitive, but in many cases, their results may not be generalizable to more natural conditions.

In response to the difficulty of estimating selection coefficients directly, researchers have instead focused on estimating linkage disequilibrium (LD) because LD correlates with the magnitude of selection. Positive values of LD along a cline generally reflect an excess ofparental gametic haplotypes and a reduction of hybrid gametic haplotypes within the cline (see Figure 2 for example). LD is generated when either endogenous or exogenous selection acts within the clines and is weakened by recombination. The net effect is that hybrids are less readily generated or maintained, parental alleles do not readily recombine, and a greater than expected number of parental gametes or haplotypes are encountered within a hybrid zone. Consequently, the higher the rate of migration across the clines of a given width, the larger the number of parental genotypes found within the clines, and the higher the degree of LD (i.e., selection). Because LD is generated by selection after migration in each generation, it is largely equivalent to 'effective' selection when se<0.10. When greater levels of selection maintain the clines, LD is not strictly equivalent to se, but rather approximates selection within an order ofmagnitude. It is not strictly generated by epistatic interactions between loci.

For clines at equilibrium, the balance between selection and dispersal can be represented at the center of a cline by

DABr where r is the rate of recombination and D is the maximum level of LD between loci (A and B). The basic formulation of D is the two-locus deviation from random expectation, or

where pA is the frequency of allele A at locus 1, pB is the frequency of allele B at locus 2, and pAB is the frequency of

AB. If one assumes the loci are unlinked, then the rate of recombination is r = 0.5. Because changes in allele frequencies (p) affect the maximum potential genetic disequilibrium, eqn [2] can be replaced with one that uses the correlation coefficient between loci, RAB:

\/pApB (1 - pA )(1 - pB ) so that at the center of a perfect cline when pA = pB = 0.5

4a2 RABr

The clinal theory outlined here is explained in much greater detail by its architects (see the section titled 'Further reading' for more details), and includes a host of factors that complicate its applicability to particular data sets. For example, the equations largely assume an evolutionarily static balance between selection and dispersal. If the cline is new because a new population is invading a region after human introduction, the relationships between selection, width, and dispersal will be very different. For most clines, however, stabilization of clines occurs very rapidly. It has been estimated, for example, that if s = 0.1, then clines stabilize within on the order of 10 generations.

A second complication is that clinal theory is directly relevant for loci under direct selection, such as some allozymes or morphological and physiological traits. However, many hybrid zones are detected using molecular markers (e.g., microsatellites) that are less likely to be under direct selection. If these neutral genes are not physically linked with loci under direct selection, then recombination quickly breaks up linkage disequilibria between loci, and the introgression of the neutral alleles across the cline will occur unimpeded. Thus, genetic differences among populations may be extremely strong immediately after secondary contact (i.e., after historically separated populations reconnect) and begin to weaken after hundreds or thousands of generations of gene flow. As a consequence, such neutral clines often look like a 'staircase' of several steps of allele frequency across the hybrid zone (neutral markers are rarely fixed on both sides of the hybrid zone) and the introgression will eventually homogenize allele frequencies. The net effect of the flattening of the cline is a rather weak but artificial increase in estimated clinal width, and any given empirical estimate of selection in a snapshot of time will infer a rate of dispersal that is somewhat lower than the actual dispersal. In fact, a neutral cline will flatten at a rate proportional to a predictable product of dispersal and time; the width of a cline of neutral genes t generations after two differentiated populations come together is expected to be about 2.51a (t)_1/2, assuming equal population sizes. On the other hand, clines at neutral genes can be stable if these genes are physically linked to genes under selection. Linkage of neutral markers to many genes under selection results in an effective selection (se) that helps sculpt neutral gene clines in a manner analogous to the action of non-neutral clines.

Third, these relationships are largely based on an underlying diffusion approximation and may be violated by rare long distance dispersal. Fourth, shifts in population densities, physical barriers to dispersal, and asymmetric gene flow can slightly alter the relationships between cline width, dispersal, and selection values.

The consequence of these and other complications is that estimates of dispersal based on selection or LD will be robust and of the right order, but may not be precise. Further, the dispersal estimates reflect the distances travelled within the clinal region only.

Still, in the absence of detailed information on linkage or selection, it is possible to produce a first approximation of larval dispersal distance using clinal theory. Even under high levels of selection (e.g., s ~ 0.25) and at equilibrium, the average geographic distance dispersed by offspring - as measured by the variance in distance between parent and offspring or neighborhood size - is less than about a third of the cline width (s <0.35 w; Figure 4). Although this is a crude approximation; it suggests that in general, populations on the endpoints of clines do not typically disperse offspring across the entire cline width in one generation. Instead, typical propagules may requires 3-5 generations to traverse the cline. Only if selection were very large (e.g., hybrids were infertile) relative to rates of recombination would a cline be maintained by a dispersal distance that was as long as the cline was wide. In cases where selection was measurable but ecologically moderate (s ~ 0.1), then s ~ 0.11 w. In other

CD O

CD C

Linkage disequilibrium (LD ) or selection (s)

Figure 4 The correlation between linkage disequilibrium (LD), selection (s), and dispersal distance within a stable cline. The curve is based on formulas [1] and [2]. Dispersal is given as the proportion of the clinal width (e.g., 20% of a 100 km clinal width is 20 km). Under reasonable levels of selection, the dispersal distance is a fraction of the clinal width. From Sotka EE and Palumbi SR (2006) The use of genetic clines to estimate dispersal distances of marine larvae. Ecology 87: 1094-1103.

Linkage disequilibrium (LD ) or selection (s)

Figure 4 The correlation between linkage disequilibrium (LD), selection (s), and dispersal distance within a stable cline. The curve is based on formulas [1] and [2]. Dispersal is given as the proportion of the clinal width (e.g., 20% of a 100 km clinal width is 20 km). Under reasonable levels of selection, the dispersal distance is a fraction of the clinal width. From Sotka EE and Palumbi SR (2006) The use of genetic clines to estimate dispersal distances of marine larvae. Ecology 87: 1094-1103.

w words, for selection that ranges from moderate to strong, clines are generally several times wider than average dispersal distance. For clines subject to weak selection, the subsequent clines can be an order of magnitude wider than dispersal, or more.

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