Detection of barriers to dispersal is masked by long lifespans and large population sizes

Abstract Population genetic analyses of species inhabiting fragmented landscapes are essential tools for conservation. Occasionally, analyses of fragmented populations find no evidence of isolation, even though a barrier to dispersal is apparent. In some cases, not enough time may have passed to observe divergence due to genetic drift, a problem particularly relevant for long‐lived species with overlapping generations. Failing to consider this quality during population structure analyses could result in incorrect conclusions about the impact of fragmentation on the species. We designed a model to explore how lifespan and population size influence perceived population structure of isolated populations over time. This iterative model tracked how simulated populations of variable lifespan and population size were affected by drift alone, using a freshwater mussel, Quadrula quadrula (mapleleaf), as a model system. In addition to exhibiting dramatic lifespan variability among species, mussels are also highly imperiled and exhibit fragmentation by dams throughout the range of many species. Results indicated that, unless population size was small (<50 individuals) or lifespan short (<22 years), observing genetic divergence among populations was unlikely. Even if wild populations are isolated, observing population structure in long‐lived mussels from modern damming practices is unlikely because it takes longer for population structure to develop in these species than most North American dams have existed. Larger population sizes and longer lifespans increase the time needed for significant divergence to occur. This study helps illuminate the factors that influence genetic responses by populations to isolation and provides a useful model for conservation‐oriented research.

differentiation and structure between fragments (Slatkin, 1987). This population genetic differentiation is largely due to the evolutionary force known as genetic drift.
The random effects of genetic drift reduce variation within a population and, while not necessarily deleterious, often result in negative impacts (reviewed in Keller & Waller, 2002). Small or isolated populations will experience genetic drift to some degree, typically resulting in an increase in homozygosity. While the resulting increase in inbreeding may lead to inbreeding depression (e.g., O'Grady et al., 2006;Pekkala, Knott, Kotiaho, Nissinen, & Puurtinen, 2014), this is not always the case (e.g., Hedrick et al. 2016;Quaglietti et al., 2017;Smith, 1995). Additionally, given enough time, other forces can ameliorate the impact of increased homozygosity including genetic purging (Boakes, Wang, & Amos, 2007;Glémin, 2003;Lopez-Cortegano, Vilas, Caballero, & Garcia-Dorado, 2016), increased gene flow (Garcia-Navas et al., 2015) or selection that can overwhelm the force of genetic drift (Bouzat, 2010). However, when inbreeding depression is not alleviated, effects such as decreased reproduction can lead to population crashes (Liberg et al., 2005;Räikkönen, Vucetich, Peterson, & Nelson, 2009). Thus, the potential for inbreeding depression in small and isolated populations resulting from genetic drift prompts analyses of genetic structure among populations to study population fragmentation and the erection of barriers to gene flow.
In some instances, analyses of fragmented populations find no evidence of fragmentation or isolation, even though a barrier to dispersal seems apparent. Occasionally, the dispersal ability of the species in question may have been underestimated, or patches may not be as isolated as expected. Reddy et al. (2012), for example, identified little population structure in tigers (Panthera tigris) between two increasingly fragmented preserves. However, low levels of structure between nearby fragments were explained by the high dispersal tendencies of subadult tigers seeking to establish new territories (Reddy et al., 2012). In other cases, failure to observe genetic structure may be due to molecular marker choice that provides little power for observing the effects of isolation.
In some situations, low or no observed structure between clearly isolated populations may be explained by other factors. It is often possible that insufficient time has elapsed to successfully identify divergence of isolated populations due to genetic drift. This may be a result of long generation time increasing the time it takes for a population to respond genetically to isolation. For instance, Wozney, Haxton, Kjartanson, and Wilson (2011) found no structure in lake sturgeon (Acipenser fulvescens) across waterways fragmented by dams for nearly 100 years. Similarly, Marsack and Swanson (2009) observed little genetic differentiation resulting from habitat fragmentation in eastern box turtles (Terrapene carolina carolina). In both cases, the species were long-lived and had strongly overlapping generations, which results in ecological processes, such as fragmentation, occurring at a faster rate than evolutionary processes.
In addition to generation time, genetic response time to isolation is influenced by a population's effective size (N e ). Introduced by Wright (1931), N e is the size of an idealized population size that loses genetic diversity at the same rate as the focal population, with respect to drift. In natural populations, a variety of factors such as unequal sex ratios and temporally variable population size reduces N e relative to the census size (N c ; Frankham, 2007). Furthermore, populations with small N e will lose genetic variation faster than those with larger N e (Allendorf, 1986). Because population fragmentation breaks populations into subsets and generally decreases N e , fragmentation typically results in populations that are more susceptible to the influence of drift. Populations, particularly those with smaller N e , require immigration or gene flow from outside sources in order to negate the influence of drift. However, populations with a large N e , even after fragmentation, are slow to change due to drift, increasing the lag in detectable changes to population genetics. As such, the N e of fragmented populations will also influence the results in structure analyses in conjunction with the long generation time of the organism. Failure to account for long generation times or large population sizes during a genetic structure analysis could cause an inaccurate interpretation of results; sampling a population too early after an isolation event could reveal little or no structure, leading to the inappropriate conclusion that gene flow still occurs between fragments. Thus, the ability to understand and predict the genetic response lag of a population, in years, to connectivity changes based on its lifespan and population size would be a valuable resource for researchers and conservation managers.
Manipulative models and simulations can be used to explore the relationships between multiple variables and generate predictions. Landguth et al. (2010), for example, used a simulative approach to detect gene flow barriers of variable intensities using the response lag of populations to fragmentation. Other models have been used to examine how landscape conditions and gene flow influence genetic structure in eastern Massasauga rattlesnakes (Sistrurus c. catenatus; DiLeo, Rouse, Dávila, & Lougheed, 2013) and American pikas (Ochotona princeps; Castillo, Epps, Davis, & Cushman, 2014). However, models to date have focused on fragmentation in more short-lived species, and operate on the order of generations, rather than years. Models that track a population over years, rather than generations, can show slow change occurring within generations, as opposed to just across them.
Particularly in long-lived species, understanding how fragmentation impacts populations on a yearly scale may be more applicable than generations for conservation and management.
The purpose of this study is to use a modeling approach to evaluate the influence of lifespan and population size on the ability to detect population genetic structure. Herein, we detail an iterative population model to examine the influence of lifespan and population size on response lag to fragmentation and isolation as manifested in population structure. To explore response lag with this model, we use a model system of mapleleaf mussels (Quadrula quadrula), a particularly long-lived species of freshwater mussel, sampled from populations potentially isolated by a low-head (run-of-river) dam. In this case study, our model determined if observable genetic divergence should be expected at this time, and if not, when divergence would be observed with 95% probability. Freshwater mussels vary among species in lifespan by an order of magnitude, making them particularly useful model organisms to assess the influence of lifespan on the lag in response to fragmentation.

| Case study
Quadrula quadrula is a freshwater mussel native to the Mississippi-Ohio river drainage and the central Great Lakes (Figure 1 Wilson, Richardson, Lavender, & Ryan, 2007). Like other unionid mussels, Q. quadrula parasitize the gills of fish during their larval stage of development, using the host as a source of nutrition and mode of dispersal (Haag, 2012 While the channel catfish is capable of long-distance dispersal (Steward & Watkinson, 2004), the species is incapable of dispersing upstream across low-head dams (Butler & Wahl, 2011;Welker, 1967).
Thus, the Dunnville Dam should act as a barrier to upstream dispersal and gene flow for Q. quadrula, isolating upstream mussels from the larger, downstream population.
Samples used in this study were collected as part of a population genetic study of Lake Erie and Lake Ontario populations of Q. quadrula (Hoffman, 2016). Mantle tissue biopsies were collected nonlethally (Berg, Haag, Guttman, & Sickel, 1995) from Q. quadrula at two sites above the Dunnville Dam and one site below in 2008 and 2014, respectively ( Figure 2). For each sample, DNA was extracted using a modified alcohol extraction method similar to Sambrook, Fritsch, and Maniatas (1989). Six microsatellite loci were then amplified for Q. quadrula samples (C4, C114, A112, A130, R9, and D102) using polymerase chain reaction (PCR) and primers designed by Hemmingsen, Roe, and Serb grouped by collection locality were tested for linkage disequilibrium using GENEPOP 4.2 (Raymond & Rousset, 1995), and for deviations from Hardy-Weinberg Equilibrium using GENALEX v.6.5 (Peakall & Smouse, 2012). To determine if the Dunnville Dam acted as a barrier, we compared populations of Q. quadrula above and below the dam using the program STRUCTURE v.2.3.3 (Pritchard, Stephens, & Donnelly, 2000). STRUCTURE analysis was conducted using 100,000 MCMC, 100,000 burn-in, and five iterations for each possible number of populations tested (K), assuming admixture and correlated allele frequencies. Maximum K was equal to the number of sites plus one. Mean log likelihood (mean LnPK) values derived using methods described by Evanno, Regnaut, and Goudet (2005) and calculated in STRUCTURE HARVESTER v.0.6.93 (Earl & vonHoldt, 2012) were used to determine the most likely number of populations across the dam.
The F ST value between the upstream and downstream populations was calculated using GENALEX with 9,999 permutations. A discriminant analysis of principal components (DAPC) was conducted using the R package "Adegenet" (Jombart et al., 2016), and the proportion of individuals in the upstream population assigned to the downstream population was reported.

| Population genetic model design
We designed an agent-based, forward time model to simulate the effects of isolation at multiple population sizes on our ability to detect population structuring (available at https://github.com/jwillou/ Isolation-Drift). We began each iteration using allele frequencies from Q. quadrula from sites downstream of the Dunnville Dam, observed in Hoffman (2016), to randomly generate the genotypes of individuals in a large population, representing the contiguous Lake Erie population prior to any dam construction and potential fragmentation (N c = 1,000). This population, encompassing Lake Erie and its tributaries, is likely so large that it has collectively changed little due to drift over the last 200 years, and as such, likely reflects allele frequencies of Grand River Q. quadrula prior to the construction of the dam. The simulated source (Lake Erie) population was allowed to persist for 75 years, slightly more than the longest recorded lifespan for Q. quadrula (65 years, Committee on the Status of Endangered Wildlife in Canada (COSEWIC), 2006), in order to ensure all populations were in Hardy-Weinberg Equilibrium at the start of the fragmentation simulation. After 75 years, we simulated an isolation event by randomly sampling a subset of individuals (parameter was varied within simulations to 50, 100, 200, 350, or 500) to create an isolated, upriver population. In both the source and isolated populations, we assumed random mating and Mendelian inheritance of alleles, and removed individuals from the population with a probability equal to [1-the annual survival rate] (estimated from the closely related species from tribe Quadrulini, Amphinaias pustulosa, 96% annually, Haag, 2012) or when the maximum age was reached (varied from 2 to 102 years using 20-year intervals). Maximum age was determined to be the maximum recorded lifespan-the estimated age of reproductive maturity (3 years for Q. quadrula, Haag, 2012), and individuals entering the population were assumed to have reached reproductive maturity.
We ran each of the 100 replicates of each parameter set out to 400 years (after the 75-year stabilization of the source population).
At the end of the simulation, we estimated average heterozygosity at each year and used STRUCTURE, discriminant analysis of principal components (DAPC), and F ST estimates to determine if the imposed isolation was detectable. For STRUCTURE analyses within each iteration, we compared the starting population (year = 0) with the population at 25-year intervals (between time 25 and 400). DAPC and F ST analyses were performed on a yearly basis, rather than on 25-year intervals. A randomly selected subset of 50 individuals per population (source and isolated) for each analysis was used to maintain consistent sample sizes, regardless of the simulated population size. As in the case-study analysis, we used a standardized set of parameters (K = 1-3; MCMC = 100,000; burn-in = 100,000; iterations = 5) for each analysis. Structure analyses were facilitated within R using the package "strataG" (Archer, 2014). We determined the most likely K by comparing mean LnPK within each replicate. For each tested N c , the frequency of identification of two populations, or the frequency of identifying genetic divergence upstream and downstream of the dam, was calculated. DAPC and F ST estimations were performed in the model using the packages "Adegenet" (Jombart et al., 2016) and "DiveRsity" (Keenan, 2017). We

| Case study
In total, 63 Q. quadrula samples from the two sites above the Dunnville Dam and 27 samples from the site below the dam were successfully amplified for at least four of six microsatellite loci.
Eighty percent of samples were successfully amplified at all loci, while only 5% were amplified at just four loci. One locus, R9, was fixed at all three sampling sites, although greater diversity was observed at other Lake Erie sites (Hoffman, 2016). Potential null alleles were predicted at low frequencies for all six loci (0.09%-7.45%).
However, simulations suggest that null alleles at a frequency <20% have little effect on population-based analyses (Carlsson, 2008;Dakin & Avise, 2004). Following Bonferroni correction (α = 0.0006), no significant deviations from Hardy-Weinberg Equilibrium (HWE) were detected at any loci from any of the three sampling locations (Table 1)

| Population genetic model
In all iterations and simulated population sizes, alleles were steadily lost from the population through time. Smaller populations tended to lose alleles and heterozygosity faster and were more prone to fixation at one or more loci compared to larger populations. Fixation of loci was more common in loci with fewer alleles or a greater disparity among initial allele frequencies. Similarly, heterozygosity of the upstream population fluctuated, but generally trended downward over time (Figure 3). This decrease in heterozygosity followed the loss of alleles or fixation.
When maximum lifespan was 62, as is expected for Q. quadrula, the likelihood of identifying two distinct Q. quadrula populations using STRUCTURE approximated a logistic curve (Figure 4). At a small population size (N c = 50), upstream and downstream mussels could be discerned as individual, divergent populations with 58% accuracy after 175 years of complete isolation (nearly the age of the Dunnville Dam), and with 98% accuracy after 375 years ( Figure 4c). However, with N c = 100, detection of genetic differentiation does not occur at that high a frequency, even after 400 years of isolation (Figure 4f). Over the 400 years of the simulation tested with analyses of genetic structure, structure was not detectable at frequencies higher than 20% for   (Butler & Wahl, 2011;Welker, 1967), and so a similar effect from the Dunnville Dam was expected. Although a denil-style fishway was installed in 1994 to alleviate the barrier on walleye, the fishway was ineffective at facilitating dispersal of the target species (Bunt, Cooke, & McKinley, 2000). As such, it is unlikely that channel catfish make use of this fishway. However, the lack of genetic structure resulting from the dam does not signify the absence of major ecological and habitat effects on the aquatic organisms in the river (Nilsson & Berggren, 2000).
We designed and implemented a forward time model in order to better interpret a lack of population structure across a dispersal barrier.
Across all tested population sizes, and with all three analyses of genetic structure, genetic assessment of the simulated populations was more likely to identify two distinct populations with increasing time in isolation. This result fits population genetic theory; noninfinite populations no longer experiencing gene flow will drift apart genetically with time (Wright, 1966). census population size was estimated to number between 10,000 and 30,000 individuals (Zanatta & Murphy, 2007), and it is likely that Q. quadrula follow similar trends. As we found in the simulation, the likelihood of observing population divergence within a set time frame decreases with increasing population size. Thus, the combination of our inability to detect divergence at a small, simulated population size and a high probability of a large actual population size means that identifying genetic divergence between Grand River Q. quadrula is unlikely. This simulation then casts doubt on our inability to detect divergence in the empirical data.
Our model simulated a situation in which the Q. quadrula population was split by the dam, eliminating upstream and downstream gene flow, but downstream gene flow likely occurs in our case study; the Dunnville Dam, a low-head dam, is probably not an impassable barrier to fishes (with encysted glochidia) traveling down river, nor the sperm of unionids (Ferguson, Blum, Raymer, Eackles, & Krane, 2013), resulting in unidirectional gene flow. This gene flow would slow the development of structure among fragmented populations and increase the time to detectable divergence. However, the downstream Q. quadrula population, encompassing Lake Erie and many of its tributaries (Hoffman, 2016), is so large that the influx of immigrants from the smaller, upstream population is unlikely to dramatically influence the divergence time between the two populations.
The results of this case study illustrate the importance of understanding and considering biology and life history characteristics of the organisms studied in addition to population genetic data. While population genetic analyses can be useful and informative tools for conservation biologists and managers, these data inherently require time to show changes in patterns of dispersal and gene flow. Organisms with long lifespans and generation times require more time to show genetic response to ecological changes (Willoughby, Sundaram, Lewis, & Swanson, 2013), and large population sizes can reduce response time even further (Marsack & Swanson, 2009). Thus, similar to difficulties found in detecting bottlenecks (Davy & Murphy, 2014) The results of our simulation illustrate the potential for error when species life history characteristics are not considered in conjunction with population genetics when assessing the impacts of fragmentation. Inherent delays in responses to connectivity changes, particularly in large populations and long-lived species, make analyses of population structure unlikely be informative at timescales meaningful to conservation management. Additionally, the genetic consequences of intensive fragmentation, such as genetic divergence between populations and inbreeding within them, can be overshadowed by the more immediate population demographic and ecological effects, such as interference in migration, spawning or mating behavior (Liermann, Nilsson, Robertson, & Ng, 2012). As observed in this study, depending on the lifespan and generation time of the species, the negative demographic and ecological impacts from fragmentation need to be addressed sometimes centuries before the genetic impacts will become relevant. As such, we suggest that long-term impacts on genetic structure alone should not be an argument for or against the removal of a barrier, save for special cases of short-lived species or species existing at small N e . Rather, the negative effects of barriers on system functions, including discharge and flow regimes (Dynesius & Nilsson, 1994;Graf, 1999), dispersal (Pess, McHenry, Beechie, & Davies, 2008;Watters, 1996), and community assemblage structure (Dean et al., 2002;Helms, Werneke, Gangloff, Hartfield, & Feminella, 2011) should be favored as grounds for remediation or removal.
Recognizing when population genetic data does not reveal the current state of connectivity among populations, and when it is necessary to rely on alternative data, is critical to avoiding incorrect conclusions and recommendations for management of species or populations of conservation concern. As such, predictive models that anticipate when fragmentation and isolation are likely to be detected, like the case study described here, are useful tools for conservation managers. These models only require estimates of basic demographic information (population size, lifespan, age of reproductive maturity, and annual survival rate) and have the potential to assess the reliability of structure analyses and lend additional support to the conclusions and recommendations of managers. Furthermore, predictive models can be easily modified to permit evaluation of situationspecific variables, such as migration rate between the source and iso- Assisted and advised on case-study data collection and population genetic analysis, advised throughout the project.