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Can someone try to explain me why allelic diversity falls faster than heterozygosity, reminding you that we're talking about bottleneck or a founder effect? Look at this graphic:
It's clear to me that both are related, so both have to decrease.
It is clear that allelic diversity has a huge fall because there was lost of alleles (specially the rare and uncommon ones that, despite contributing less to allelle frequencies, each one has the same weight to allelic diversity, take a look at the other graphic to understand what I'm talking about).
Now, heterozygosity is specifically related to allele frequencies, and since the most rare/uncommon don't contribute much to this, the only alleles that will have a large impact on the decrease of heterozygosity are the more/most common.
Since that after a bottleneck, (probably) the alleles remaining are the most common, heterozigosity will decrease less than allelic diversity.
I'm aiming for some complementary ideas about this, specially when it comes to statistical models that describe heterozygosity. It's hard for me to understand the relation between allele frequencies and heterozygosity when it comes to this case - in what manner do allele frequencies influence H?
The effects of founding bottlenecks on genetic variation in the European starling (Sturnus vulgaris) in North America
Genetic variation in European starlings in North America was examined using enzyme electrophoresis and compared to that in their home range. The effect of the founding bottleneck matched theoretical predictions. Heterozygosity was unaffected, whereas allelic diversity may have decreased. Results from this study and others suggest that theoretical predictions of bottlenecks are robust for allozyme data, and applicable under a wide variety of conditions.
Aquatic invertebrates that disperse passively via an encysted embryo use a variety of transport methods to colonize new habitats. Abiotic factors, such as water and wind, , , , ,  and biotic vectors, such as birds, mammals, insects, amphibians and human activity can disperse invertebrates large distances , , , , , , , , , . Colonization of new habitats by a combination of these factors can be relatively quick, especially if ponds are located in close proximity , . The potential for dispersal, however, does not always equate to the actual immigration into ponds that is occurring by aquatic invertebrates . It is commonly observed that populations of many aquatic invertebrates can have a high degree of genetic differentiation despite being located in close proximity , , , a result not expected if contemporary dispersal is frequently occurring between populations.
In cyclically parthenogenetic zooplankton, De Meester et al.  emphasized the importance of local adaptation for monopolizing resources, thereby creating genetic differentiation between ponds in close proximity. Boileau et al.  concluded that founder events, not contemporary gene flow, have a pronounced effect on the population genetic structure of aquatic invertebrates that produce resting eggs. To demonstrate that genetic rriers” are formed to inhibit immigration into populations, Boileau et al.  used simulations to show that F ST does not decay for at least 2000 generations in a large population established by a few founders and subsequently experiencing migrant influx.
In addition to founder events, the mode of reproduction can also influence the amount of genetic structure and diversity in large Branchiopods . A population with individuals that reproduce via selfing experience a heterozygote deficit and decreased diversity due to small effective population sizes , . In addition, compared to species that outcross, populations of selfing individuals are genetically more isolated because of limited gene flow and often experience demographic fluctuations , .
The tadpole shrimp (Triops sp.) is a passively dispersing aquatic crustacean that has been said to use several forms of reproductive modes including parthenogenesis, hermaphroditism, androdioecy (a mix between outcrossing and hermaphrodites) and gonochorism (males and females that outcross) , , . Within Triops populations, low genetic diversity, deviations from Hardy-Weinberg equilibrium, large inbreeding values (F IS) and large population differentiation have been observed and has been attributed to founder events and the degree of outcrossing between individuals , , , , , .
Many of the previous studies have focused on Triops populations that are separated by distances of hundreds or thousands of kilometers between sampled ponds , , . Large geographic distances between populations makes it difficult to determine if it is the mating system influencing the genetic structure and diversity of Triops populations or if dispersal of encysted embryos is simply limited over long distances. The current study is designed to aid in differentiating between the influence of founding events, dispersal and mating systems by using nine Triops populations located within 30 km and encompassing two putative species with different presumed reproductive modes. Two of the species of Triops in the northern Chihuahuan Desert are T. longicaudatus “short” and T. newberryi , . Different reproductive modes are presumed for T. l. “short” and T. newberryi based on the male (absence of a brood pouch) to female (presence of a brood pouch) ratio within populations T. l. “short” is comprised of all females and is assumed to reproduce via parthenogenesis or hermaphroditism whereas T. newberryi is thought to be androdioecious, with populations comprised of hermaphrodites that outcross with males . A recent phylogeny of Triops showed that T. l. “short” and T. newberryi are not monophyletic, calling into question whether species status is warranted .
The first objective of this study is to assess the genetic structure of each Triops species and determine what factors (founding events or contemporary dispersal) influence population differentiation. Secondly, we compare the effect of different presumed reproductive modes and the degree of inbreeding to the genetic diversity and structure of the Triops populations. We hypothesize, based on population genetic theory, that the androdioecious species will have more alleles, higher allelic richness, fewer private alleles, higher observed heterozygosity, lower F IS and F ST, and relatively greater genetic variance within as opposed to between populations. The last objective is to evaluate whether the two putative species of Triops in southern New Mexico are reproductively isolated in the ponds in which they co-occur.
The extent to which the different initial genetic-diversity measures are correlated with the rate of adaptation was investigated by carrying out multiple simulation runs with a range of initial demographic and genetic parameter values corresponding to different effective population sizes and mutation or migration rates, thus implying a substantial variation in responses to selection across runs (see Supporting Information, Figure S1 for an example of selection responses in a particular case). We first present the results for single undivided populations and then for structured populations.
Correlation between diversity measures and response to selection in single undivided populations
To ascertain to what extent each variability measure (quantitative genetic variance, gene-frequency, or allelic-diversity variables) accounts for the response to selection, we carried out a correlation analysis of each variable with response to selection. The variables in this scenario are the initial additive genetic variance (VA), the average initial heterozygosity for the QTL (H*), the average initial number of segregating alleles for the QTL (K*), and the corresponding variables for the markers (H, K). The values of the squared correlation coefficient (R 2 ) with selection response are presented in Figure 1. Figure 1, top, shows that VA is the best predictor of short-term response (R10), whereas the number of alleles (K*) is the best predictor of long-term (R50–100) and total (RT) responses. Diversity for genetic markers (Figure 1, bottom) predicts long-term and total response better than short-term response, correlations being marginally but consistently larger when based on allelic number. However, for all periods, these correlations were much smaller than those for the quantitative trait (QT) or QTL (note the different scale between the top and bottom of Figure 1).
Squared correlation coefficients (R 2 ) between initial population genetic-diversity variables and response to selection (R10, response to selection until generation 10 R10–50, response from generations 10–50 R50–100, response from generations 50–100 RT, total response until generation 100) for a single undivided population. Population size and mutation rate (u) was varied randomly among simulations (N between 100 and 1000, and u between 0.00001 and 0.0004). VA: initial additive genetic variance. H: initial average expected heterozygosity. K: initial average number of segregating alleles per locus. Terms with an asterisk refer to QTL whereas terms without one refer to neutral markers. The results are based on 10 sets of 2000 simulation runs. Error bars indicate two standard errors of the mean at each side.
Correlation between diversity measures and response to selection in subdivided populations
Constant demographic parameters:
We ran a set of simulations for each of several specific combinations of demographic parameter values (fixed N and m). For each combination of parameters, ordinary correlations were computed between all diversity measures and the short-term (R10), long-term (R10–100), and total response (RT). Table 1 gives the largest ordinary correlations (irrespective of sign) for each combination of parameter values. Regarding neutral genetic marker variables, correlations between diversity variables for genetic markers and response to selection were always very small and nonsignificant, so they are not included in the table. This suggests that when demographic parameters, such as N and m, are invariable, genetic markers do not convey any information on response to selection for a quantitative trait.
For QT and QTL variables (Table 1), the largest correlations with short-term response (R10) were for different gene-frequency measures or the genetic variance for the trait. This holds for the long-term (R10–100) or total (RT) response in the cases with Nm < 0.5. However, for the cases with Nm > 0.5, the largest correlations mostly involved allelic measures (underlined), suggesting that these convey more information on long-term response than gene-frequency measures in this scenario.
Variable demographic parameters:
We carried out simulations where the values of N and m were randomly changed across runs. The above results with fixed Nm values (Table 1) suggest different outcomes for highly isolated subpopulations (Nm < 0.5, i.e., FST > ∼0.3) or less isolated subpopulations (Nm > 0.5, i.e., FST < ∼0.3). In fact, an inspection to the response to selection achieved for different values of the number of migrants (Nm) per generation and subpopulations (see Figure S2) shows that the two scenarios should be analyzed separately. In the very highly isolated subpopulation scenario (Nm < ∼0.5), an increase in migration implies higher short- and long-term response. In the less isolated subpopulations scenario (Nm > ∼0.5), however, an increase in migration implies higher short-term response but lower long-term response. Thus, in what follows, analyses are made separately for these two levels of migration.
To see which type of variables predicts better the response to selection, we carried out four sets of regression analyses, each including the four main diversity measures. Thus, we performed separate analyses for the quantitative genetic parameters (VW, VB, VT, and QST), the gene-frequency variables for QTL (HS*, DG*, HT*, and GST*), the allelic number variables for QTL (AS*, DA*, AT*, and AST*), and the corresponding sets for marker variables (HS, DG, HT, and GST for gene-frequency variables or AS, DA, AT, and AST for allelic variables). Figure 2 shows the values of R 2 for each of these regressions. All five sets of variables explain a relatively large proportion of the variability for selection response (i.e., show large R 2 values), except for the total response for the scenario with Nm > 0.5.
Squared correlation coefficients (R 2 ) between initial population genetic-diversity variables and response to selection (R10, response to selection until generation 10 R10–100, response from generations 10 to 100 RT, total response until generation 100) for a structured population. The scenario refers to a subdivided population with n = 10 subpopulations, Nm migrants per generation and subpopulation, mutation rate u = 0.00001, and strength of stabilizing selection given by ω 2 = 25. The variables included in the model are quantitative trait (QT) variables (VW, VB, VT, QST: black bars), gene-frequency variables (HS, DG, HT, GST: blue bars), allelic-diversity variables (AS, DA, AT, AST: red bars). The results are based on five sets of 2000 simulation runs varying the subpopulation size (N) randomly between 100 and 1000 and the migration rate (m) between 0.0001 and 0.1. Error bars indicate two standard errors of the mean at each side.
In contrast with the results obtained for the undivided population scenario or the subdivided population scenario with Nm fixed, measures based on neutral marker loci do not account for less response than those based on QTL. Thus, when demographic parameters are variable, diversity measures based on neutral markers are substantially correlated with response to selection for a quantitative trait. The results also clearly show that allelic-diversity variables (Figure 2, red bars) contain more information about long-term or total response than gene-frequency (blue) or quantitative trait (black) variables.
The squared correlations presented in Figure 2 involve five diversity measures. To see more specifically which diversity variables are more correlated with response, Figure 3 gives the correlation for each of these diversity measures with short-term and total response. Correlation coefficients for all variables are presented in Table S1. For the strongly subdivided scenario (Nm < 0.5 Figure 3A), the correlations with the largest magnitude correspond to measures of internal diversity (Figure 3A, top, with r > 0) and of genetic differentiation between subpopulations (Figure 3A, bottom, with r < 0). This holds for all five sets of variables and for both short-term and total response. The measure showing the largest correlation with short-term response is the within-subpopulation additive variance (VW), a correlation that can be ascribed to causality, since short-term response depends directly on this parameter. For total response, however, the best predictors are the allelic measures of genetic differentiation (AST and AST*).
Ordinary correlation coefficients (r) between initial population genetic-diversity variables and response to selection (R10, response to selection until generation 10 RT, total response until generation 100) for a structured population. The scenario refers to a subdivided population with n = 10 subpopulations, Nm < 0.5 (A), or Nm > 0.5 (B) migrants per generation and subpopulation, mutation rate u = 0.00001, and strength of stabilizing selection given by ω 2 = 25. Black bars, quantitative trait variables blue bars, gene-frequency-diversity variables and red bars, allelic-diversity variables. Top row: within-subpopulation-diversity measures (VW, HS, AS). Second row: between-subpopulation-diversity variables (VB, DG, DA). Third row: global-diversity measures (VT, HT, AT). Bottom row: differentiation statistics (QST, GST, AST). The results are based on five sets of 2000 simulation runs varying the subpopulation size (N) randomly between 100 and 1000 and the migration rate (m) between 0.0001 and 0.1. Error bars indicate two standard errors of the mean at each side.
For the mild subdivision scenario (Nm > 0.5 Figure 3B), short-term responses are positively correlated with all measures of within subpopulations variability (Figure 3B, top), the largest correlation corresponding to the within-subpopulation additive variance (VW), and negatively correlated with all measures of genetic differentiation (Figure 3B, bottom) or of between-subpopulation genetic distances (VB, DG*, DA*, DG, DA). However, regarding total response, the best predictors are the allelic-diversity variables, both for QT-QTL and markers.
Neutral diffusion predictions of allelic-diversity statistics
In what follows we use diffusion approximations to derive predictions for allelic-diversity measures under an infinite-island neutral model. Let us assume a neutral locus under infinite-allele mutation with rate u per generation, in a population subdivided in n ideal subpopulations of size N, following an island model of migration among subpopulations with rate m. Therefore, the expected gene-frequency differentiation is GST ≈ 1/[1 + M] with M = 4Nm[n/(n – 1)] 2 + 4Nu[n/(n – 1)] (Takahata 1983), and the expected effective size of the population (Wright 1943) is (10) The expected number of alleles whose frequency lies within the range p to p + dp in the equilibrium population is φ(p)dp, where (11) (Ewens 1964 Kimura and Crow 1964 Crow and Kimura 1970), where θ = 4Neu. Although Equation 11 strictly applies to a random mating population, we show that it provides good approximations regarding several properties of a subdivided population under a wide range of conditions.
Under the infinite-island model, the distribution of allele frequencies within subpopulations (ps) is given by the beta distribution with parameters α = Mp and β = M(1− p), i.e., (12) (Wright 1937, 1940), where Γ denotes the gamma function, and p is the whole population allele frequency. Thus, the total number of alleles segregating in the population is (13) (Ewens 1964, 1972 Crow and Kimura 1970). This latter expectation can also be approximated with a generally lower precision by Ewens (1972) formula, (14) and by with an even lower precision.
By considering expressions (11) and (12) jointly it is possible to obtain predictions of diversity measures in a subdivided population. This approach has been previously followed by Barton and Slatkin (1986) to obtain the distribution of rare alleles in a subdivided population. The expected number of alleles segregating in each subpopulation is (15) where (16) is the cumulative distribution function of between 0 and 1/2N, which gives the probability that a given subpopulation lacks an allele that has frequency p in the overall population. Likewise, the expected number of alleles common to two subpopulations is (17) The expected allelic diversity within subpopulations (AS) and the expected average allelic difference between subpopulations (DA) are then (18) (19) Using Equations 7 and 8, this gives which can be computed as (20) All the above expressions can be modified to account for sampling of g genes within each subpopulation (gn over the whole population). Thus, the total number of segregating alleles in the overall sample of gn copies is (21) This can also be approximated by Equation 14, replacing 2Nn by gn.
Accordingly, the expected values of KS and KcS would be obtained as above (Equations 15 and 17, respectively) replacing expression (16) with
Precision of the diffusion approximations
Figure 4 plots predicted and simulated values of the allelic-diversity measures (AS, DA, KT, and AST) against Nm for a range of m values. They are computed for samples of g = 100 neutral genes from each subpopulation for two different mutation rates (a more comprehensive list of results is shown in Table S2 and Table S3). In general, predictions for AS and DA are rather accurate, although those for DA slightly underestimate the simulation values for the large mutation rate scenario. Predictions for KT, however, are well above the values obtained through simulations for low values of Nm. The predictions of AST are very precise in all cases.
Comparison between computer simulations (lines) and diffusion approximations (symbols) for different allelic-diversity variables. The scenario considered refers to a subdivided population with n = 10 subpopulations, each of size N = 1000 individuals, mutation rate u = 0.00001 (A–D) and 0.0002 (E–H), variable migration rate (m), and g = 100 sampled genes per subpopulation. AS: average allelic diversity within subpopulations. DA: average pairwise allelic distance between subpopulations. KT: average total number of alleles segregating in the whole population. AST: allelic differentiation.
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Figure 2 displays the allele frequency data for several model populations, averaged over 100 model runs (only 10 runs for n = 50,000). Population sizes were: a) 500, b) 1,000, c) 5,000, d) 50,000. A Flood-type bottleneck with random DILs occurred at year 1,600. The two wings that appear in later years among the smaller populations represent alleles that drifted to fixation (0% or 100%). There was almost no noticeable difference between the populations with 5,000 to 50,000 individuals, meaning we successfully captured the size range required to draw conclusions about any larger population size.
Figure 7. The relationship between heterozygosity and fixation for models with different population sizes and with related (open squares) and random (filled diamonds) DILs. Smaller populations are on the top left. Also included is a polynomial regression line for the random DILs. The degree of allele loss through fixation is inversely proportional to population heterozygosity. Thus, it may be possible in future models to estimate how many ‘created’ alleles were lost to drift based on modern human heterozygosity values (the three HapMap populations used in this study averaged 30.2% heterozygosity across the over 1 million sites included in the database).
Changes in average population-wide heterozygosity in models with different maximum population sizes are shown in figure 3. When the population size was ≥ 3,000, heterozygosity levels were consistent and similar, and most of the loss occurred when the population was rebounding from small numbers. The average loss of heterozygosity from the Flood year to the next measurement period (100 years after the Flood) in the smallest population was 9.4%. Loss of heterozygosity in all other populations was similar, averaging 7.5%. By the time the population reached 50,000 people, the average heterozygosity had levelled out to a value of 0.427 and had not changed (to three significant figures) in 1,300 years. The slope of that line over the final 2,000 years was –3 x 10 –7 , meaning we would expect a –0.03% change over the next 1,000 years.
We compare the allele frequency spectrum for multiple population sizes at model year 6,000 in figure 4. As above, the larger population sizes begin to converge. In this case, a normally distributed curve centred on 0.5 was obtained. Since all alleles started at a frequency of 0.5, and since drift was expected to create variation in the allele frequencies, this was a good demonstration that our methods were producing realistic results. Models that restricted the population to less than 1,000 people had appreciable allele loss (fixation). All other populations exhibited a more-or-less normal distribution, with only slight levels of fixation.
We also tested what would happen with two extreme models: one with the DILs pulled at random from the available females at the time of the Flood and one with the DILs as sisters of Shem, Ham, and Japheth. Data were taken from an average of 100 iterations for each population size of 100, 500 and 1,000 to 10,000 and 10 iterations for a population size of 50,000. We calculated the difference in mean heterozygosity between the 1,600 th year, just prior to the flood event, and the 1,700 th year, 100 years after the event (figure 5). Note that for the algorithm to work in small populations where it was rarely possible to find a ‘Noah’ with 6 children, some new sons and their wives had to be created during the Flood event. This figure shows an average reduction of 7.8% (for random DILs) or 16.1% (for sibling DILs) in mean heterozygosity, irrespective of population size. Figure 6 displays the allele frequency spectrum of the two modelled populations. The two ‘wings’ on each graph represent alleles that have gone to fixation (at 0% or 100% allele frequency). The models with random DILs lost 0.76% of the alleles, on average, due to fixation for population sizes between 4,000 and 50,000. In the model where the DILs were daughters of Noah, 3.07% of the alleles were lost to fixation for those same population sizes (400% higher, but still modest). We were also able to compare heterozygosity and fixation for these models (figure 7).
Figure 8. The effects of population growth rate. By varying the spacing of children (‘S’) from 1 to 10 years, and by varying the year of maturity for females (‘M’) from 15 to 25 years, we can affect the allele frequency distribution. Clearly, faster population growth slows genetic drift. Shown here are the average distributions after the first 500 years. There are only 10 model runs per variable, so the curves are not as smooth as in the other figures.
Since most of the loss in heterozygosity occurred when the population was small, we created models with varying population growth rates and tracked the allele frequency spectrum for 500 model years (figure 8). Fast growth led to less drift (a tighter allele frequency distribution). Slower growth created a flatter, wider curve, meaning more alleles had drifted away from their 50% starting point. In the slowest-growing population (S10/M25) it took a little less than 400 years to reach 10,000 people. This is slow compared to biological realities, so we feel the range of variables in these models span what we might expect to occur in the real world.
We also tested the effects of chromosome arm length on fixation and the retention of heterozygosity. No differences were found, to three significant figures, in either measure (data not shown). In order to assess the effects of recombination rate, we created models with a variable number of recombinations per arm per generation. With no recombination the allele frequency spectrum was quite erratic because there were essentially only 80 different alleles in the population, each at a different specific frequency (figure 9).
The allele frequency data for three major world populations is given in figure 10. In all three populations there are many alleles at both high and low frequencies, consistent with significant levels of drift. The average heterozygosity across the populations was 30.2%, consistent with the values generated at lower population sizes in our computer model (figure 3). It is not possible to measure fixation with these data, however, for HapMap would have skipped over any location that displays no allelic variation within, or among, contemporary populations. Figure 11 plots the relative difference for each of the 1.3 million HapMap alleles in two populations (CHB+JPT and YRI) compared to CEU. The difference in allele frequency between the European population and one of the other populations is shown along each axis. From this figure, we can see that the frequency of an allele in one population is an excellent predictor of the frequency in the other two populations. If these were created alleles, a significant amount of drift must have occurred to drive them from their expected starting frequency of 50%. Yet, since the frequency of an allele in one population is a general predictor of the frequency in another population, this is an indication that the alleles took on this frequency spectrum prior to the separation of the populations at Babel. Subsequent within-population drift caused the widening of the distribution, but note how minimal this is. The slight ridge along the JPT+CHB axis represents alleles that drifted in this population but not the other two. The dual ridge that lies on the diagonal represents alleles that drifted in the CEU population and not the others.
Genetic bottleneck and founder effect signatures in a captive population of common bottlenose dolphins Tursiops truncatus (Montagu 1821) in Mexico
Background. The captive cetacean industry is very profitable and popular worldwide, focusing mainly on leisure activities such as “Swim-with-dolphins” (SWD) programs. However, there is a concern for how captivity could affect the bottlenose dolphin Tursiops truncatus, which in nature is a highly social and widespread species. To date, there is little information regarding to the impact of restricted population size on their genetic structure and variability.
Methods. The aim of this study was to estimate the genetic diversity of a confined population of T. truncatus, composed of wild-born (n=25) from Cuba, Quintana Roo and Tabasco, and captive-born (n=24) dolphins in southern Mexico, using the hypervariable portion of the mitochondrial DNA and ten nuclear microsatellite markers: TexVet3, TexVet5, TexVet7, D18, D22, Ttr19, Tur4_80, Tur4_105, Tur4_141 and GATA098.
Results. Exclusive mtDNA haplotypes were found in at least one individual from each wild-born origin populations and in one captive-born individual total mean haplotype and nucleotide diversities were 0.912 (±0.016) and 0.025 (±0.013) respectively. At microsatellite loci, low levels of genetic diversity were found with a mean number of alleles per locus of 4 (±2.36), and an average expected heterozygosity over all loci of 0.544 (±0.163). Measures of allelic richness and effective number of alleles were similar between captive-born and wild-born dolphins. No significant genetic structure was found with microsatellite markers, whereas the mtDNA data revealed a significant differentiation between wild-born organisms from Cuba and Quintana ROO.
Discussion. Data analysis suggests the occurrence of a recent genetic bottleneck in the confined population probably because of a strong founder effect, given that only a small number of dolphins with a limited fraction of the total species genetic variation were selected at random to start this captive population. The results herein provide the first genetic baseline information on a captive bottlenose dolphin population in Mexico.
International Crops Research Institute for the Semi-Arid Tropics (ICRISAT), Patancheru PO, AP, 502324, India
Hari D Upadhyaya, Sangam L Dwivedi, Rajeev K Varshney, Cholenahalli LL Gowda, David Hoisington & Sube Singh
International Center for Agricultural Research in the Dry Areas (ICARDA), PO Box 5466, Aleppo, Syrian Arab Republic
Michael Baum & Sripada M Udupa
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