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2Department of Geography, University of Georgia, Athens, Georgia 30602 USA; 3Plant Biology Department, University of Georgia, Athens, Georgia 30602 USA
Received for publication May 11, 2006. Accepted for publication November 17, 2006.
ABSTRACT
Understanding gene movement patterns in unidirectional flow environments and their effect on patterns of genetic diversity and genetic structure is necessary to manage these systems. Hypotheses and models to explain genetic patterns in streams are rare, and the results of macrophyte studies are inconsistent. This study addresses Ritland's (Canadian Journal of Botany 67: 2017–2024) unidirectional diversity hypothesis, the one-dimensional stepping stone model, and the metapopulation model within and among populations. Hymenocallis coronaria, an aquatic macrophyte of rocky river shoals of the SE USA, was sampled in four river basins. Within populations and among populations <16.2 km apart had significant isolation by distance. However, the rate of gene flow decay was not consistent with a one-dimensional stepping stone model, nor was evidence strong or consistent for Ritland's hypothesis. Some evidence indicates that localized metapopulation processes may be affecting genetic diversity and structure; however, gene flow patterns inconsistent with the assumptions of the linear and unidirectional models are also a possible influence. We discuss three variants on the one-dimensional stepping stone model. Future research in linear environments should examine the expectations of these models. This study is also one of the first efforts to calculate population genetic parameters using a new program, TETRASAT.
Key Words: Amaryllidaceae • aquatic plant • Cahaba-lily • gene flow • genetic differentiation • shoals spider-lily • southeastern United States
Despite the long-recognized significance of seed dispersal by water, referred to as hydrochory (e.g., Ridley, 1930
; Schneider and Sharitz, 1988
; Nilsson et al., 1991
; Griffith and Forseth, 2002
), there is a particular dearth of research concerning evolutionary processes of macrophytes inhabiting streams. Because the effect of moving water on gene flow is unknown, prediction of the pattern of genetic structure and diversity within a particular stream, river, or drainage basin is difficult (Gornall et al., 1998
). Few investigations to date have used molecular markers to examine how gene flow and genetic structure and diversity are related to unidirectional stream flow in aquatic environments either within or among populations (Ritland, 1989
; Gornall et al., 1998
; Lundqvist and Andersson, 2001
; Tero et al., 2003
), and the findings from those studies are inconsistent. Without a more complete knowledge of the effect of stream flow on gene flow, we cannot predict how macrophyte populations will be affected by flow alteration and damming. Not only is it difficult to assess whether populations are imperiled by potential flow changes, but performing restoration efforts in disturbed streams currently proceeds without an understanding of the microevolutionary processes in these systems.
A number of approaches have been utilized to examine the accumulated effects of stream flow on genetic patterns. Ritland (1989)
hypothesized a downstream increase in effective population size due to the movement of seeds and propagules from upstream to downstream populations, resulting in increasing downstream genetic diversity. Contrary to his initial hypothesis, however, Ritland found no downstream trend in allozyme diversity among populations of Mimulus caespitosus in the Washington Cascades, which he attributed to high statistical variance in heterozygosity estimates. Lundqvist and Andersson (2001)
, on the other hand, did find evidence of increasing diversity downstream among populations of Angelica archangelica, a species with seeds that can float up to a year and disperse over long distances in running water. At the within-population scale, Gornall et al. (1998)
concluded that prevailing water currents and wind direction resulted in unidirectional gene flow, causing increased diversity at two allozyme loci in a population of Potamogeton coloratus in a network of drainage ditches.
Streams are environments where seeds and propagules are assumed to move in a linear pattern. Thus, the one-dimensional stepping stone model is often applied (Fig. 1A), which assumes an infinite linear arrangement of populations that can only share genes with adjacent populations, and gene flow is bidirectional. These restrictions on gene flow result in isolation by distance with a decrease in gene flow with distance approximating a –1.0 regression slope, as predicted from simulation modeling by Slatkin (1993)
. Macrophyte research has failed to support this model as yet. Kudoh and Whigham (1997)
rejected a one-dimensional stepping stone model because the regression of gene flow on geographic distance did not approximate the slope expectations. However, they concluded that hydrochory was important in determining the spatial genetic structure of Hibiscus moscheutos because populations separated from the channel had lower gene flow rates than those adjacent to the estuary channel. More recently, Tero et al. (2003)
rejected the one-dimensional stepping stone model in their analysis of seven populations of Silene tatarica in the Oulankajoki River. Rejection was based on a lack of correlation between genetic and geographic distance and because genetic diversity was not higher in the middle of the habitat than at the ends as suggested by Wilkins and Wakeley's (2002)
stepping stone model for continuous linear populations. They concluded that metapopulation processes, rather than unidirectional stream flow, influenced the observed genetic structure.
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, most of the research to date has focused on a single stream or network. Ignoring issues of scale and excluding replicates can lead to potentially spurious generalizations of patterns to an entire species or environment.
The objective of this research was to characterize patterns of gene flow, genetic structure, and diversity both within and among populations of the aquatic macrophyte Hymenocallis coronaria (J. LeConte) Kunth (Amaryllidaceae), to determine the influence of unidirectional stream flow on population genetic patterns and processes. Specific questions addressed include (1) Is the distribution of genetic diversity reflective of the pattern expected in Ritland's (1989)
hypothesis? (2) Do patterns of gene flow support a one-dimensional stepping stone model? (3) Is the available evidence consistent with a metapopulation model? (4) Are these various models consistent at both the within- and among-populations scales of analysis? This is one of the first studies to use the program TETRASAT (Markwith et al., 2006
) to calculate genetic diversity and genetic structure parameters for an allotetraploid organism. We also introduce new hypotheses to an area of research gaining increasing attention and provide additional much needed analysis in an inadequately studied area of plant microevolution.
MATERIALS AND METHODS
Species description
Hymenocallis coronaria is an emergent bulbous macrophyte anchored in rocky shoals of the Piedmont, Ridge and Valley, and Cumberland Plateau provinces of Georgia, South Carolina, and Alabama, USA. (Fig. 2). The fragrant flowers of H. coronaria open overnight, when they are visited and apparently pollinated by the plebian sphinx moth (Paratrea plebeja). The pipevine swallowtail butterfly (Battus philenor) visits flowers during the evening and morning (Davenport, 1996
). The opening of each bud on the flower stalk occurs sequentially, so that each day a new flower is open to pollination (Davenport, 1990a
). Self-pollination may occur, because flower disintegration after one night allows the stigma and anthers to come into contact, although experiments have not been performed to test this possibility (L. Davenport, Samford University, Department of Biology, personal communication). Two Mexican species of Hymenocallis have been observed to produce seed in the absence of pollination, a process called apomixis (Bauml, 1979
). However, seeds collected from four different maternal H. coronaria plants possessed alleles not found in the maternal plant, thus indicating that seeds were produced by outcrossing events (S. Markwith, unpublished data). The plant is known to create bulblets, that may detach from the original bulb and if uprooted may reestablish downstream (Aulbach-Smith, 1998
). The large seeds of H. coronaria, which can be greater than 4 x 3 cm, are released from a capsule upon maturity and sink after release from the mother plant (Davenport, 1990b
). There is no recorded biotic seed disperser for this species. Although the literature provides ample evidence of zoochory for many aquatic organisms (e.g., Green et al., 2002
), because the protective membranous capsular walls split when seeds mature (Flint, 1943
) the succulent integument structures are left vulnerable to damage, thereby possibly limiting the effectiveness of animal-mediated dispersal. Higher than average magnitude stream flows may provide an effective dispersal mechanism.
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All samples were transported on dry ice to the Department of Plant Biology, University of Georgia, for DNA extraction and molecular analysis. Genomic DNA extraction, amplification of six polymorphic microsatellite markers, and capillary acrylamide electrophoresis were performed in accordance with the protocols of Markwith and Scanlon (2006)
.
Statistical analysis
Three models were explicitly tested, Ritland's hypothesis, the one-dimensional stepping stone model, and the metapopulation model, with some overlap in analysis methods. Often when the one-dimensional stepping stone is tested the two-dimensional stepping stone model is concurrently tested because it provides an alternative model that can be examined based on a similar regression analysis. The distribution of H. coronaria within drainage basins does not offer enough populations in tributary streams to constitute a spatial arrangement considered two dimensional, and thus the analysis for this study concentrates on models concerning linear environments.
Ritland's hypothesis was tested by examining a number of standard diversity indexes within and among populations. The one-dimensional stepping stone model was tested by regression of gene flow on geographic distance, and a Mantel test for significance of the relationship. The slope of the regression should approximate –1.0, per Slatkin's (1993)
one-dimensional stepping stone model simulation analysis. Evidence for the metapopulation model was obtained with the Mantel test for significant isolation by distance and regression of genetic diversity on population size. Malécot (1955)
indicates that isolation by distance is expected under a stepping stone model. Thus, lack of significant isolation by distance would indicate drift and gene flow non-equilibrium, a characteristic of some metapopulation patterns (Hutchinson and Templeton, 1999
). Genetic diversity is hypothesized to be related to population size (Nei, 1987
) because smaller populations are more susceptible to the effects of genetic drift. The lack of a significant relationship between genetic diversity and population size may indicate that recently large diverse populations are proceeding through a bottleneck or that new habitats have been recently colonized and populations rapidly expanded in size.
We calculated a number of commonly used parameters to analyze gene flow between populations, genetic structure, and genetic diversity. Hymenocallis coronaria is a tetraploid species (Joye and Smith, 1993
); tetraploid genotypes were not reduced to the two corresponding homeologous diploid loci because the inheritance pattern is not clearly understood for this species, and the allele copy number is uncertain (see following paragraph). Percentage polymorphic loci (PP), mean number of alleles per locus (AP), mean number of alleles per polymorphic locus (APP), and multilocus observed heterozygosity (HO) were calculated for each population. Multilocus observed heterozygosity was calculated within each population by averaging across all six loci the proportion of heterozygotes, including both partial and full heterozygotes, at each locus. For the large populations sampled as three subpopulations each, PP, AP, APP, and multilocus HO were calculated for the three subpopulations separately and for the population as a whole by pooling the three sampled subpopulations in each population.
Genetic diversity and population differentiation analysis of tetraploids can be difficult when fluorescently tagged microsatellite markers are used. With these markers, the allele copy number information derived from peak heights is not reliable due to exponential PCR amplification. Thus, absolute allele frequencies cannot be calculated for populations containing partial heterozygotes. To overcome this problem the program TETRASAT (Markwith et al., 2006
) was used to calculate mean multilocus expected heterozygosity (HE), Shannon–Wiener diversity index (H'), and Nei's (1986)
pairwise population differentiation (GST), and standard deviations (see Markwith et al., 2006
for equations and further explanation of calculation methodology). TETRASAT analyzes tetraploid microsatellite marker genotypes by making iterative substitutions of all possible allele configurations based on the alleles found for each partial heterozygote, then calculates every combination of allele frequencies and the corresponding HE, H', and GST values, and provides means and standard deviations from the set of all possible values. The mean multilocus HE and H', and standard deviations, were calculated from a 10 000 value subset of all possible multilocus values for each population. GST values were calculated in TETRASAT by randomly choosing a maximum of 10 000 allele frequency configurations for each locus; from these, values were calculated for each randomly selected configuration, and mean multilocus population values and standard deviations were computed from a 10 000 value subset of all possible multilocus values for each population. For the large populations sampled as three subpopulations each, multilocus HE, H', and GST were calculated for the three subpopulations separately. TETRASAT could not process the pooled subpopulations due to memory constraints, thus, multilocus HE, H', and GST are calculated for the whole population by taking the average of the three subpopulations. This procedure is not the most desirable for the pooled subpopulations. However, substitution of the large population's mean value with any of the subpopulation values (i.e., treating individual subpopulations as representative of the entire population), does not affect the overall conclusions of the study; indicating the method is an acceptable compromise.
Diversity can be measured in a number of different ways. Ritland (1989)
used expected heterozygosity (HE) to measure genetic diversity. We also use the Shannon–Wiener diversity index, which is also based on allele frequencies. Both of these parameters are sensitive to both richness and evenness of alleles, which can complicate interpretation. Because measures of richness, including PP, AP, APP, and multilocus HO are not directly affected by changes in evenness and do not have variation associated with iterative substitution in TETRASAT, they may provide a clearer trend than the more complex diversity parameters.
Ordinary least squares (OLS) regression analysis of the dependent variables PP, AP, APP, multilocus HO, mean multilocus HE, and mean multilocus H' on the log of population size was conducted using STATA Release 8.0 (STATA Corp., 2003
). Population sizes were obtained by directly counting bulb clumps for populations under 50 individuals. For larger populations, estimates were obtained by counting bulb clumps in a representative contiguous group of clumps and multiplying that number by the number of groups.
A geographic information system (GIS) database was created containing population location data for H. coronaria, which was partially provided by the Georgia Natural Heritage Program, and supplemented by digitizing populations based on both USGS 7.5-minute topographic quadrangle digital raster graphics (DRGs) and the demographic study of Larry Davenport (1997)
. Distance along river courses between populations was measured with the population location database and 1:24 000 scale USGS National Hydrography Data coverages.
OLS regression analyses were performed with STATA release 8.0 to test for adherence to the one-dimensional stepping stone model. We calculated Slatkin's 
(Slatkin, 1993
), pairwise estimates of among-population gene flow, from mean multilocus GST between all pairs of populations and between all pairs of subpopulations within the three intensely sampled populations. Regression of pairwise log values of Slatkin's on the log of stream distance was conducted for each of the river systems separately, one each in the Cahaba River, Flint River, and the Savannah River. Data from the three basins were also pooled to create three data sets: one containing all pairwise comparisons within the three basins, another containing only those with geographic distances <16.2 km, and the third containing only those with geographic distances >55.6 km. These two distance classes were selected because they constitute two groups based on a natural break in the distance data. Pairwise comparisons for populations not connected by stream flow were excluded. Regression was also performed at the within-population scale by pooling the subpopulations in each of the three intensively sampled populations into one data set.
Since pairwise comparisons of populations are not independent, the significance of the linear regression analysis of the log of on the log of stream distance is not reliable. Thus, Mantel tests were used to assess whether significant isolation by distance is present. PopTools version 2.6.9 (Hood, 2005
) was used to conduct the Mantel tests on the same data sets analyzed in the regression analysis. The Mantel test uses a two-tailed test for significance, and 999 iterations were conducted to calculate the observed test statistic and the distribution of test statistics.
RESULTS
Genetic diversity
Analyses of gene flow, genetic structure, and diversity were performed on 15 populations of Hymenocallis coronaria within the Cahaba, Catawba, Flint, and Savannah freshwater stream systems in Alabama, Georgia, and South Carolina, U.S. (Fig. 2). In general, the most diverse (sub)populations sampled were the LANDS population on the Catawba River, followed by the farthest upstream sampled population in the Cahaba basin, BCRHE (Table 1). As a whole drainage system, the Savannah basin from the tributary Broad River through the main stem populations had the most consistently high diversity across populations. However, the population, STEV, in the Stevens Creek tributary stream in this basin had an anomalously low diversity across all parameters compared to the other sampled populations. Also, there was not a significant relationship between any of the genetic diversity parameters and population size (Table 2 and Fig. 3).
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None of the diversity parameters had a clear downstream trend among subpopulations within each of the three large intensively sampled populations (Table 1). PP decreased downstream among subpopulations, with no change to the farthest downstream subpopulation in the CRWR P1 and FLINT P3 populations. The opposite is true in the LANDS subpopulations. All other parameters (AP, APP, multilocus HO, mean multilocus HE, and mean multilocus H') failed to have a monotonic decrease or increase in a downstream direction, except APP for the LANDS subpopulations, which decreased downstream.
Genetic structure and gene flow
Pairwise population genetic differentiation (GST) in H. coronaria was relatively low, especially considering the geographic distance separating some of the populations (Table 3). The highest GST value among populations was 0.0960 between the STEV and SAV P4 populations, at a stream distance of 55.6 km. In comparison, the highest 
(theta) value, which is an equivalent genetic differentiation parameter to GST, among populations of the macrophyte Hibiscus moscheutos is 0.2280 at a geographic distance of 0.2 km (Kudoh and Whigham, 1997
). The average GST across all subpopulations within the three intensively sampled H. coronaria populations was 0.0287 (Table 4), compared to the among population average GST of 0.0404. The highest pairwise gene flow estimate, = 17.61, is found between two populations that were the farthest apart in geographic distance, BR and SAV P1. Even this value was considerably smaller than the highest among H. moscheutos populations, = 83.06 (Kudoh and Whigham, 1997
). The mean for H. coronaria within populations was 8.82, and among populations it was 7.38. These values are small in comparison to Kudoh and Whigham (1997)
, where the mean = 41.35 for populations adjacent to the studied stream.
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and stream distance in all three drainage basins, indicating decreased gene flow with distance (Table 5 and Fig. 4). However, the regression models and Mantel tests were only significant in the Cahaba basin. The correlation coefficient was strongly negative in the Flint; however, there were few observations for this basin. In the Savannah basin the relationship of the BR population to the other populations in the basin, all of which were >82.7 km distant, caused the correlation to be weak and nonsignificant (Table 3 and Fig. 4). The analysis was also conducted for a number of pooled data sets. The pooled data set containing all pairwise comparisons within the three basins and the data set containing only those with geographic distances <16.2 km both had significant negative log-log relationships between and stream distance in the regression analysis and Mantel tests. However, the slope was more strongly negative for the data set with geographic distances <16.2 km. The pooled data set containing pairwise comparisons >55.6 km had a strong significant positive log-log relationship. None of the regression slopes approached the –1.0 slope expected from Slatkin's (1993)
one-dimensional stepping stone model simulation study.
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and stream distance (Table 5). Again, the regression slope did not approach the –1.0 slope expected from Slatkin's (1993)
one-dimensional stepping stone model simulation study. However, the Mantel test did indicate significant isolation by distance. DISCUSSION
Genetic diversity
The various tested models predict a number of different genetic diversity, structure, and gene flow patterns under the influence of stream environmental constraints. Ritland's (1989) hypothesis predicts increasing diversity with successive downstream populations. Unlike Gornall et al. (1998)
, whose work supported Ritland's hypothesis, we found no trend in genetic diversity statistics from upstream to downstream subpopulations within any of the three large populations of Hymenocallis coronaria examined in this study. Like Ritland (1989)
and Tero et al. (2003)
, but in opposition to Lundqvist and Andersson (2001)
, we found that at the among-population scale of analysis the diversity trend was highly variable within and among streams. The evidence did not support a strong role for asymmetrical flow, over other processes, in shaping genetic diversity in the manner expected by Ritland at either scale in H. coronaria. Based on the results of assignment analysis, Tero et al. (2003)
concluded that gene flow was bidirectional in the stream they studied; and without evidence supporting a strong influence of unidirectional flow on genetic diversity in this study, a bidirectional gene flow pattern influenced by pollinator behavior or zoochory also cannot be excluded here at either the among or within population scales.
Evidence also indicated that processes other than gene flow may be involved in shaping diversity patterns in streams. As mentioned previously, genetic diversity is hypothesized to be related to population size (Nei, 1987
) because smaller populations are more susceptible to the effects of genetic drift and lose variation in the absence of frequent gene flow. Much of the available research reflects this positive relationship (e.g., a review by Frankham, 1996
; Berge et al., 1998
; Fischer and Matthies, 1998
; Persson et al., 1998
; Prober et al., 1998
; Fischer et al., 2000
); however, some studies have found no relationship or a negative relationship between genetic diversity and population size (e.g., Oostermeijer et al., 1994
; Schmidt and Jensen, 2000
; Müller-Schärer and Fischer, 2001
; Tero et al., 2003
). Schmidt and Jensen (2000)
concluded that historical population processes may have much to do with the current relationship between population size and genetic variation. Small populations with high diversity may currently be proceeding through a population bottleneck with an expected lag period in genetic erosion. Large populations with low diversity may result from recent colonizations that have quickly expanded in size. Based on the nonsignificant regression analyses of genetic diversity on the log of population size, it is possible that metapopulation processes are occurring locally at the sub-basin scale in H. coronaria and affecting genetic diversity of populations.
An example of the effect of recent colonization is found in the Savannah basin, where the most support for Ritland's hypothesis was found. In the Savannah basin, the STEV population has a different cpDNA haplotype than any other population currently found in the basin (unpublished data) and the lowest nDNA diversity of any sampled population. Although this population is relatively large, it is likely a recent colonization event from a different basin, having clear evidence of founder effects. Also, because the STEV haplotype is not found in any of the downstream populations and the geographic distance is almost certainly too far for pollen flow, any gene flow from this population to the downstream populations is unlikely. Consequently, despite the increased diversity from the STEV upstream tributary population through the downstream SAV populations, the STEV population cannot be used as evidence to support Ritland's hypothesis.
One other factor that may influence the results and conclusions found herein involves the manner of analysis and the associated assumptions. The mean multilocus HE and H' values calculated using TETRASAT are the means of a 10 000 value subset of all possible multilocus HE and H' values for each population or subpopulation. The logic behind selecting this approach to analyzing allotetraploid microsatellite data was provided previously. The standard deviation associated with each mean provides an indication of the amount of variability created in the analysis of each population or subpopulation. Although the standard deviations associated with these parameters are small, the possibility of alternative patterns within the set of all possible values of multilocus HE and H' must be noted. In addition, even though the variability in mean multilocus HE and H' values accommodate the possibility of different patterns with stream position, the other parameters used in this research (i.e., PP, AP, APP, and HO) parallel the patterns derived with TETRASAT, therefore confirming rejection of Ritland's hypothesis. The same caveat concerning statistical variation applies to mean multilocus GST and the derived parameters, i.e., Slatkin's , discussed in the following section.
Genetic structure and gene flow
Tero et al. (2003)
accepted the model they referred to as the "classical" metapopulation model (referred to here simply as the metapopulation model) in part because they found no correlation between genetic and geographic distance. As mentioned previously, a lack of significant isolation by distance indicates drift–gene flow non-equilibrium. Given the major Quaternary climate changes resulting in range contractions and expansions (Watts, 1980
), drift–gene flow non-equilibrium is likely to be observed in many temperate and subarctic environments similar to the area studied by Tero et al. (2003)
. The evidence of non-equilibrium found herein is the nonsignificant isolation-by-distance analysis within the Flint and Savannah River basins and among populations >55.6 km distant. For the Flint River basin, the number of pairwise observations is small and probably accounts for the nonsignificant correlation, but there is a trend of decreasing gene flow with geographic distance in this stream. The results in the Savannah River and among populations >55.6 km apart are related and may be explained by (1) an expected increase in statistical variance with increasing distance (Hutchinson and Templeton, 1999
) and (2) inadequate sampling at distances >16.2 km to characterize the increasing variance. The existence of significant isolation by distance at distances <16.2 km provides evidence that the populations of H. coronaria are generally in drift–gene flow equilibrium and the evidence of widespread Holocene range expansion is lacking for this species.
The regression slopes of the log of on the log of geographic distance are equivalent at both the within- and among-population scales, and the slopes do not correspond with expectations of the one-dimensional stepping stone model. Although large-scale range adjustments may not be a factor, localized metapopulation processes may be affecting population differentiation and gene flow values in addition to genetic diversity. The lack of a relationship between population size and genetic diversity, indicating recent population expansions or bottlenecks and the evidence of founder effects in the STEV population support this conclusion. However, a formal direct test for population bottlenecks, such as those performed by the program BOTTLENECK (Cornuet and Luikart, 1997
), could not be performed due to the nature of the allotetraploid microsatellite data. Another possible explanation for rejection of the one-dimensional stepping stone model may be that gene flow is occurring in a non-stepping stone manner. Seeds and propagules may establish in the next adjacent subpopulation or population or be transported farther and establish in other habitats beyond the nearest neighbor subpopulation or population with diminishing frequency. This non-stepping stone flow may occur in an asymmetrical flow pattern or, as mentioned previously, due to the rejection of Ritland's hypothesis, bidirectional flow among non-adjacent populations cannot be excluded. The possible applicability of variation on the one-dimensional stepping stone model to gene flow is discussed in the following section.
Although significant physical barriers that may affect genetic structure or gene flow (including distance) are not expected within most H. coronaria populations, evidence of isolation by distance is found within the three large populations. Outcrossing is apparently a major component of the reproductive strategy of H. coronaria, and the species is known to be visited by a moth and butterfly (Davenport, 1996
). Insect-mediated pollen movement tends to be effective over a few tens of meters and diminishes with distance in a manner approximating a leptokurtic distribution curve (Fenster, 1991a
, b
; Richards et al., 1999
; Fenster et al., 2003
). Large H. coronaria populations that are spread across shoals upwards of 0.75 km to 2.0 km in length consist of clumps of individuals that are 10's to 100's of meters apart. Pollen flow may be high within the clumps; however, a significant decrease in pollen movement is probable among more distantly spaced clumps. In addition, because seeds sink and water flow likely plays an integral role in gene flow, seeds may be consigned to their natal subpopulations in the majority of years without higher than mean stream flows. Thus, even within large populations, distance acts as a graded barrier between upstream and downstream subpopulations.
Despite geographic distances among populations that are generally greater than those within populations, the amount of gene flow among populations <16.2 km distant and the rate at which gene flow diminishes with distance are similar to that found at the within-population scale. At the among-population scale, the sampling units are delineated based on the individual shoal; shoals are separated by pools, which do not support populations of H. coronaria and thus serve as natural breaks. Within individual drainage basins, these inhabited shoals can be slightly less than 1 km apart, and for the analysis at the <16.2 km scale, populations are generally 1–5 km from the next closest population. Even at these distances, rare pollen flow may still be possible, but among-population seed and/or propagule flow equivalent to that within the three large populations may be an important influence on the similar values at the two scales. The higher magnitude and lower frequency stream discharges necessary to move seeds and propagules among subpopulations may be equivalent to those necessary to initiate among-population seed and propagule flow.
Compared to Kudoh and Whigham's (1997)
study, the gene flow rates both within and among H. coronaria populations are relatively low. They found mean gene flow rates among populations along the estuary channel to be about five times greater than those of this study. Some reasons for this discrepancy among aquatic species may be: (1) the research on Hibiscus mosceutos was conducted in an estuary, where flows are bidirectional with tides, whereas in upland streams the water always moves downstream and may result in more restricted gene flow; and/or (2) unlike H. mosceutos, the seeds of Hymenocallis coronaria are not buoyant, restricting seed flow to years with the necessary magnitude flows, or potentially to rare zoochory events.
Alternative linear models
When applied to stream environments, the assumptions of the one-dimensional stepping stone model are often viewed as the model's strengths, but it may lack the necessary attributes to be a useful representation of these systems. The model assumes that gene flow occurs only between adjacent populations; therefore, genetic distance increases monotonically with geographic distance between populations (Kimura, 1953
; Kimura and Weiss, 1964
). Not only is this model attractive for the linear spatial pattern, but established simulation modeling by Slatkin (1993)
makes fitting observations to an expected value relatively simple. However, research has shown that hydrochory can result in long-distance dispersal (e.g., Waser et al., 1982
; Schneider and Sharitz, 1988
; Griffith and Forseth, 2002
); therefore an assumption of gene flow predominantly between adjacent populations may be inaccurate. Additionally, a one-dimensional stepping stone model of gene flow incorporates bidirectional gene flow, which may be uncommon in upland stream populations if pollen dispersal and animal-mediated seed dispersal are limited.
The one-dimensional stepping stone model's two main components, other than a linear spatial arrangement, are (1) direction of movement and (2) pattern of movement among populations. For this model, these characters are bidirectional or symmetrical movement and movement only among adjacent populations. However, three other linear models can be derived from a combination of these components and their opposite characteristics. The alternative models are (1) asymmetrical and adjacent gene flow (linear asymmetrical adjacent flow model [LAAF]); (2) symmetrical and non-adjacent gene flow (linear symmetrical non-adjacent flow model [LSNF]); and (3) asymmetrical and non-adjacent gene flow (linear asymmetrical non-adjacent flow model [LANF]) (Fig. 1B, C, and D, respectively). Each of these models may be applicable under different linear environmental conditions and dispersal traits of the studied organisms. Variation in spatial distribution of populations and dispersal mechanisms between pollen and seed movement may also create hybrid models. Species found in rivers with some closely spaced populations and some at a greater distance may have a pattern with symmetrical flow among close populations due to insect-mediated pollen flow or zoochory and asymmetrical flow among distantly spaced populations due to the dominance of stream-mediated seed flow.
Although they did not recognize that asymmetrical flow may be dominant in linear environments, Kimura and Weiss (1964)
did recognize that more than one step along the stepping stone of populations may be taken per generation. They also suggested modifications to their equations so that expectations for rates of isolation by distance could be examined. However, at this point, little can be said about the expectations of these three alternative models concerning estimates of genetic differentiation and gene flow without computational and simulation analysis concerning the effects of skipping steps altogether in the stepping stone and the constraints created by asymmetrical flow. It is assumed that in the case of the two models with asymmetrical flow, LAAF and LANF, Ritland's hypothesis will apply in the absence of other evolutionary processes. Thus, due to the rejection of Ritland's hypothesis, the findings of the current analysis are not consistent with either the LAAF or LANF models. Because flow occurs only between adjacent populations in the LAAF model, but is restricted further by asymmetrical flow, the rate at which gene flow decreases with distance may be greater than the rate of the one-dimensional stepping stone model. For the LSNF model, the pattern of gene movement is very similar to that of the island model, with populations potentially sharing genes directly with all other populations. In keeping with the analysis of Hymenocallis coronaria, the rate at which gene flow decreases with distance for the LSNF model should be less than the expected rate for the one-dimensional stepping stone model. The major departure of the LSNF model from island model type gene movement occurs when tributary streams are added onto the simplistic linear arrangement and a barrier to gene flow exists between populations of different tributaries. Inclusion of tributaries in one-dimensional type models is a fundamental challenge to realistically modeling flow in stream systems, and hopefully will gain greater attention in the future.
Conclusion
The one-dimensional stepping stone model has not been accepted in any studies on aquatic plants inhabiting streams to date (e.g., Kudoh and Whigham, 1997
; Tero et al., 2003
; and the present study). Although the metapopulation model is a population dynamics model and is poor in predicting gene flow patterns, its introduction by Tero et al. (2003)
and the introduction of the alternative linear models found herein are necessary for further development in this field of research. While indirect methods will continue to provide valuable insight into the accumulated effects of combined gene flow and metapopulation processes, by incorporating direct gene flow analysis, we will have a clearer understanding of yearly gene flow patterns of macrophytes inhabiting streams.
With the current state of knowledge, predictions concerning genetic diversity, structure, and gene flow in streams cannot be made confidently, which hinders decision-making about anthropogenic impacts. Questions about impacts of an upstream population extinction or the interruption of downstream dispersal by dam construction on the genetic diversity of downstream aquatic plant populations cannot be convincingly addressed by resource managers without a greater understanding of gene flow patterns. The importance of unidirectional flow in structuring genetic patterns in aquatic plants remains an essential question for further population research.
FOOTNOTES
1 This work was supported by NSF dissertation improvement grant no. 0401799, Sigma Xi Grants-In-Aid of Research, and Georgia Museum of Natural History's Joshua Laerm Academic Support Award. The authors thank K. C. Parker for review of an earlier draft and the anonymous reviewers for their assistance. 
4 Current address: Department of Plant Biology, Cornell University, Ithaca, New York 14853 USA
5 Author for correspondence (markwith{at}uga.edu
)
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= each of the 15 sampled populations