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Heterozygosity under genetic drift

Heterozygosity under genetic drift


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The wright-Fisher model of genetic drift is:

$$p_{ij} = inom{2N}{j}left(frac{i}{2N} ight)^j left(1- frac{i}{2N} ight)^{2N-j} $$

,where $inom{2N}{j}$ is a binomial coefficient.

From this equation one can infer that the expected heterozygostiy should decrease by $1-frac{1}{2N}$ at each time step because:

$$E[x_{t+1}(1-x_{t+1}) space|space x_t] = (1-frac{1}{2N})x_t(1-x_t)$$

I don't understand this equality. That might be very simple though! Can you help me making sense of it?

source


To derive it, first use that $E[x(1-x)]= E[x-x^2]=E[x]-E[x^2]$ and that $E[x^2]= ext{Var}[x]+E[x]^2$ to rewrite the left-hand side: $$Eleft[x_{t+1}(1-x_{t+1}) ight] = Eleft[x_{t+1} ight](1-Eleft[x_{t+1} ight])- ext{Var}left[x_{t+1} ight].$$ The equation for $p_{ij}$ is just saying that $2Nx_{t+1}$ is binomially distributed with $2N$ trials with success probability $x_t$, so $E[x_{t+1}]=x_t$ and $ ext{Var}[x_{t+1}]=x_t(1-x_t)/(2N)$. (Note that $ ext{Var}[alpha x]=alpha^2 ext{Var}[x]$.) Plugging these values into the equation above gives the form you're looking for.


The notation at this site resembles that in your question but preserves the $frac{x_t}{2N}$ notation for probability of selecting an allele.

$$E[frac{x_{t+1}}{2N})(1 - frac{x_{t+1}}{2N} )|x_t] = (frac{x_{t}}{2N})(1 - frac{x_{t}}{2N}) (1 - frac{1}{2N}) $$

The expression $(frac{x_{t}}{2N})(1 - frac{x_{t}}{2N}) $ is the probability of heterozygosity at step $t.$

Your expression says that the expected heterozygosity at step $(t+1)$ is equal to the heterozygosity at step at $t$ multiplied by $(1-frac{1}{2N}).$

For a population of size $2N,~(1-frac{1}{2N})^{2N}approx 1/e,$ so large populations are insulated from this effect and small ones may lose genetic diversity rapidly.

A typical explanation can be found here. The notation at this site condenses the expression $(frac{x_{t}}{2N})(1 - frac{x_{t}}{2N}) $ to $H_n$ and explicitly says it is $1 - F_n, $ in which $F_n$ is the probability of homozygous at step $n.$


Previously, we have looked at the arguments of Vern Poythress as they pertain to human common ancestry, population genetics, and locating Adam and Eve in human prehistory. A second well-known apologist who has also interacted with these data is Dr. William Lane Craig. Like Poythress, Craig is an Old-Earth, progressive creationist who holds to the view that all humans descend uniquely from Adam and Eve – though Craig is more open than Poythress to the possibility of humans sharing ancestry with other forms of life. To his credit, Craig is aware that he is not an expert in this area, and often assumes a humble posture when discussing these matters:

So some sort of a progressive creationist view, I think, would explain the evidence quite well. It would allow you to affirm or deny if you wish the thesis of common ancestry and it would supplement the mechanisms of genetic mutation and natural selection with divine intervention. I find some sort of progressive creationism to be an attractive view.

Again, I want to reiterate that on these issues I am like many of you a scientific layperson. I am someone who has an interest in these subjects, I want to learn and to study them further, and explore them more deeply. So these opinions are held tentatively and lightly and are subject to revision.

I have had the opportunity to meet Dr. Craig in person, and found him to be thoughtful, congenial and interested in learning more about how modern genetics plays into the conversation about the historical Adam and Eve. While Craig has learned about— and has correctly understood the impact of—the genetic evidence for human common ancestry, his understanding of the evidence from human population genetics is lacking in certain respects. These misunderstandings, unfortunately, lead him to make some basic errors. As he sees it, holding to a historical Adam – in the sense that all humans descend uniquely from an original ancestral couple – remains defensible, since the conclusions of human population genetics are based on assumptions open to critique:

[Geneticists] look at the amount of variability in the genetic structure and then you calculate how this could have arisen based upon mutation rates and the amount of time available. That will then give you these population estimates. But there are quite a number of assumptions that go into this kind of modeling that the defender of the historical Adam, I think, could challenge.

Craig’s defense of a historic Adam and human descent from an ancestral pair (i.e. genetic monogenesis) thus rests on the idea that there is reasonable uncertainty in population genetics measures:

What we need to understand is that these are genetic estimates based upon mathematical modeling and projections into the past. We know that that kind of mathematical modeling is based upon certain assumptions that may or may not be true, and can sometimes be wildly incorrect in their projections… It could well be the case that these mathematical models are simply incorrect.

Moreover, Craig advances the opinion that this uncertainty is great enough to allow one to hold to genetic monogenesis:

When you think about it, it is really quite remarkable, it seems to me, that with these models they are able to get the minimum human population size down to a couple thousand people. I mean, that in itself is astonishing. It wouldn’t take a great error to go from two thousand to two, I think.

So, what “assumptions” does Craig have in mind? Two major themes in Craig’s interaction with population genetics evidence are (a) that population geneticists assume a constant mutation rate for the human lineage, and that (b) population genetics models have been shown to overestimate real populations whose actual demographics are known. We will deal with these issues in turn.

Accelerated mutation rates?

Craig posits that mutation rates may not have been constant over human history, and that in the past mutation rates may have been much higher. Such accelerated rates, he argues, could indeed produce the genetic diversity we see in the present day starting from only two people:

The problem is the population size. In order to get this amount of genetic diversity, the claim is you needed to have at least 2,000 people originally to result in this. One of the assumptions that is based upon is that the rate of mutation doesn’t change. But if the mutation rates are changing then they could accelerate and that could produce greater diversity than one might expect. You might say that increasing diversity would have a selective advantage so this perhaps would be a kind of accelerating process. Again, we just don’t know that these mutation rates have been constant over all of these thousands of years.

There are, of course, several problems with this line of argument (not least that genetics indicates that we descend from a population of about 10,000, not 2,000). First, it is an ad hoc line of argumentation – the argument is made only because the evidence does not fit with Craig’s prior expectation that we all descend from an ancestral couple. Second, there is indeed good evidence that the mutation rate our lineage experienced has not changed appreciably in the last several million years.

There are several independent ways to estimate human mutation rates. An excellent overview of the various methods can be found at this series of blog posts by Larry Moran, a biochemist at the University of Toronto: the biochemical method the phylogenetic method, and the direct method. For our question at hand, the phylogenetic method is of particular interest: it estimates the mutation rate on the human lineage since our separation from chimpanzees. In brief, we can compare the differences we see in the present-day human genome, the present-day chimpanzee genome, and using reasonable estimates of generation times, infer the number of mutations per generation in our lineage, as well as in the lineage leading to chimpanzees, with both lineages descending from a common ancestral population. This estimate is an average for our lineage over the last several million years – and it agrees well with the estimates we see using the biochemical and direct methods. Even the best-case scenario for Craig – that no mutations occurred at all in the lineage leading to chimpanzees, and every difference between the genomes of our two species resulted from mutations in the human lineage only – does not provide a mutation rate high enough to account for the diversity we see in present-day humans assuming they descend from an original pair.

Finally, only some techniques used to measure human population dynamics over time employ estimates of mutation rates. Other methods – which do not use estimates of mutation rates – return the same results as mutation rate-based methods. One such example is one we have discussed: estimating human population size over time using linkage disequilibrium. Craig’s argument thus needs to explain why these methods agree with mutation-based methods, since speculating about mutation rates does not affect these measurements. Craig also needs to address why the various independent methods used to measure the population size of our lineage agree with one another: if indeed we all descend uniquely from an ancestral pair, why is it that these independent methods all return the same values? The reasonable conclusion is that these methods are telling us something valid about our evolutionary history. These methods and their conclusions are subject to revision and refinement, of course – but unlike Craig hopes for, it is not reasonable to expect that that refinement will reduce our ancestry from 10,000 to two.


Somatic drift and rapid loss of heterozygosity suggest small effective population size of stem cells and high somatic mutation rate in asexual planaria

Planarian flatworms have emerged as highly promising models of body regeneration due to the many stem cells scattered through their bodies. Currently, there is no consensus as to the number of stem cells active in each cycle of regeneration or the equality of their relative contributions. We approached this problem with a population genetic model of somatic genetic drift. We modeled the fissiparous life cycle of asexual planarians as an asexual population of cells that goes through repeated events of splitting into two subpopulations followed by population growth to restore the original size. We sampled a pedigree of obligate asexual clones of Girardia cf. tigrina at multiple time points encompassing 14 generations. Effective population size of stem cells was inferred from the magnitude of temporal fluctuations in the frequency of somatic variants and under most of the examined scenarios was estimated to be in the range of a few hundreds. Average genomic nucleotide diversity was 0.00398. Assuming neutral evolution and mutation-drift equilibrium, the somatic mutation rate was estimated in the 10 −5 − 10 −7 range. Alternatively, we estimated Ne and somatic μ from temporal changes in nucleotide diversity π without the assumption of equilibrium. This second method suggested even smaller Ne and larger μ. A key unknown parameter in our model on which estimates of Ne and μ depend is g, the ratio of cellular to organismal generations determined by tissue turnover rate. Small effective number of propagating stem cells might contribute to reducing reproductive conflicts in clonal organisms.


Materials and Methods

Isle Royale researchers have collected moose carcasses from Isle Royale National Park since 1958. Approximate age, likely cause of death, and sex are known for Ϥ,500 carcasses. We haphazardly selected 55 moose born within each of five sampling periods: 1960�, 1970�, 1980�, 1990�, and 2000� for analysis. Sampling periods were chosen to minimize overlap of generations and because they included the extremes of the moose population fluctuations ( Fig. 2 ).

DNA was obtained from the pulp cavity of teeth by dissecting the teeth into ρ.5 cm sections of the root cavity using a Dremel 300 (Robert Bosch Tool Corporation, Racine, WI, USA) as well as any tissue attached to the sides of the teeth. The DNA was extracted using Qiagen DNeasy kits following published protocol (Qiagen Valencia, CA, USA), quantified with a NanoDrop Spectrophotometer ND-1000 (Thermo Scientific, Wilmington, DE, USA), and diluted to a concentration of 15 ng/µl.

All moose samples were amplified at nine microsatellite loci, using primers BM757, BM4513, BM848 (Bishop et al., 1994), MAF70 (Buchanan & Crawford, 1992), MAF46 (Swarbrick et al., 1992), McM58 (Hulme et al., 1994), RT5, RT9, RT30 (Wilson et al., 1997) via polymerase chain reaction (PCR) using an Eppendorf Mastercycler Gradient (Eppendorf, Westbury, NY, USA). PCRs were completed using Qiagen Multiplex Kit (Qiagen, Valencia, CA, USA). Primers MAF70, McM58 and BM848 had an optimized annealing temperature of 60 ଌ and comprised one multiplex grouping. Primers BM757, RT9 and BM4513 comprised a second multiplex grouping with an annealing temperature of 57 ଌ. The final multiplex grouping of primers MAF46, RT5 and RT30 had an optimal annealing temperature of 58 ଌ. We used 50 µl reactions containing 50� ng of DNA, 0.2 µM of each forward and reverse primer, 0.5x Q-Solution and 1X Qiagen Multiplex PCR Master Mix with a 3 mM concentration of MgCl2 and 15 µl RNase-free water. PCR conditions consisted of 15 min at 95 ଌ followed by 30� cycles of 94 ଌ for 30 s, 57 °� ଌ for 90 s, and 60 s at 72 ଌ with a final extension period of 10 min at 72 ଌ. Amplified DNA was analyzed using an ABI Prism 310 Genetic Analyzer (Applied Biosystems, Foster City, CA, USA). Allele size was determined using GENESCAN v. 3.1.2 and Genotyper v. 2.0 with TAMRA 500 base-pair size standard (Applied Biosystems, Foster City, CA, USA). Samples that were not successfully genotyped using multiplex kits were reamplified in 10 µl single primer PCR’s containing 50� ng of template DNA, 125 µM dNTP’s, 0.16 µM each of forward and reverse primer, 1x Buffer, and 0.375 units Hotmaster Taq Polymerase (5 PRIME, Gaithersburg, MD, USA). PCR cycles were performed as follows: denaturing of DNA for 2 min at 94 ଌ, followed by 30� cycles at 94 ଌ for 45 s for denaturing primer specific annealing at 57°� ଌ for 45 s and a final extension at 65 ଌ for 10 min (Broders et al., 1999). A sample negative was included in all PCR plates for quality control.

Prior to statistical analysis, we used MICRO-CHECKER to test microsatellite genotypes for the presence of null alleles, repeat motif consistency, scoring errors and allelic dropout (Van Oosterhout et al., 2004). We used GENEPOP on the web (Raymond & Rousset, 1995 Rousset, 2008) to test all microsatellite loci for deviations from Hardy Weinberg Equilibrium using probability testing with a Bonferroni corrected alpha (α =਀.0055 Rice, 1989). Additionally, all pairs of loci in each population were tested for linkage disequilibrium using log-likelihood ratio statistics with a Bonferroni corrected alpha (α =਀.0014). Markov chain parameters were set to 1,000 dememorizations, 100 batches with 1,000 iterations per batch.

To better understand how genetic diversity changed over time, we measured estimates of heterozygosity and inbreeding at each time period and determined the pattern of change observed over the study period. First, using GENALEX 6.3 (Peakall & Smouse, 2005), we estimated the number of alleles and observed heterozygosity for each sample period. We also estimated inbreeding coefficients (FIS) for each locus and across each sample period using the R Demerelate package (Kraemer & Gerlach, 2017 R Core Team, 2016) following Weir & Cockerham (1984). Finally, we used simple linear regressions to determine if our estimates of genetic diversity (number of alleles, observed heterozygosity, FIS) changed over time (five, 5-year time periods beginning in 1960� and ending in 2000�). In all three linear regressions, significant models were denoted by a slope that was significantly different from zero.

Population substructuring was assessed within and across sample periods using non-spatially explicit program, STRUCTURE (Pritchard, Stephens & Donnelly, 2000), and a spatially explicit program, BAPS (Corander et al., 2008). Given the recently highlighted complications of estimating population structure using microsatellites (Putman & Carbone, 2014), we required both model-based clustering analyses to indicate population substructure to have confidence in its presence. We ran five independent STRUCTURE runs using K =ਁ −ਉ for each sample period and for all sample periods combined, assuming correlated allele frequencies and admixture. Structure results were visualized using Structure Harvester (Earl & vonHoldt, 2012). Similarly, BAPS was run for five replicates of K =ਁ −ਉ for each sample period and the combined time period dataset, using the clustering of individuals option and no spatial prior. For both analyses, we used a burn-in of 100,000 steps and 100,000 replicates (Falush, Stephens & Pritchard, 2003), which allowed convergence.

We looked for evidence of migration onto Isle Royale using two genetic markers: (1) mtDNA and (2) microsatellite genotypes. First, 134 samples partitioned across our sample periods were selected for mitochondrial sequencing. DNA was extracted and run on a 1% agarose gel prior to PCR to check for DNA degradation. Gels were stained with SybrGold (Molecular Probes, Eugene, OR, USA) and examined on a UV transilluminator. The portion of DNA closest to the well was excised and cleaned using Qiagen MinElute Kits (Qiagen Valencia, CA, USA). DNA was amplified at the left hyper-variable domain of the control region via polymerase chain reaction (PCR) using an Eppendorf Mastercycler Gradient (Eppendorf, Westbury, New York) and primers LGL283 and ISM015 (Hundertmark et al., 2002). We used 20 µl PCR reactions containing 25� ng of DNA, 125 µM dNTP’s, 0.2 uM of forward and reverse primer, 1 × Buffer containing 1.5 mM MgCl2, 1 × Flexi Buffer, 1unit of GoTaq Polymerase (Promega Corporation, Madison, WI, USA) and 6.45 µl ultrapure water. Polymerase chain reaction conditions consisted of 2 min at 94 ଌ followed by 35 cycles of 94 ଌ for 45 s, 50 ଌ for 45 s, and 45 s at 72 ଌ with a final extension period of 10 min at 72 ଌ. Resulting sequences were aligned to known moose haplotypes from multiple populations across North America and Europe using BLASTn from the National Center for Biotechnology Information (http://blast.ncbi.nlm.nih.gov/Blast.cgi, last accessed 05/2017) to ensure our sequences were moose. All 134 sequences were verified as moose, and were subsequently aligned using the program MACVECTOR 7.2.3 (Cary, NC, USA http://www.macvector.com). Prior to analyses, the first and last 30 base pairs of each sequence were removed to reduce false identification of mutations. We used the ClustalW Alignment tool in MACVECTOR for multiple alignments to identify mutations and calculate the number of sequences per sample period. ClustalW parameters were set to a 10.0 open gap penalty, 5.0 extended gap penalty, 40% delay divergent, and weighted transitions.

In addition to examining mtDNA for a signal of immigration, we also examined our microsatellite data for evidence of recent migration onto Isle Royale. Specifically, we used an assignment test in GENECLASS2 (Piry et al., 2004), which is able to identify the probability an individual within a dataset originated from the genotyped population(s), even if the source population was not sampled or genotyped. We analyzed each sample period using the Baysian framework (Rannala & Mountain, 1997), with 10,000 iterations and the default threshold p value (type I error) of 0.01. Assignment probability was calculated using the Monte Carlo resampling method of Paetkau et al. (2004) which limits simulated populations to the same sample size as reference populations thereby more accurately reflecting sampling variance.

Finally, we simulated the loss of microsatellite genetic diversity in the Isle Royale moose population to provide a theoretical baseline for comparison against the observed change in diversity. The R simulations began by simulating 564 individuals, equal to the population size of moose on Isle Royale in 1960, and by assigning 2 alleles per locus for each individual, using the empirical allele frequencies from the 1960� dataset. We used this initial population as a starting point, and allowed the microsatellite allele frequencies to evolve in future years by assigning newly created (i.e., born) individuals alleles that were randomly selected from the parental genotypes of that individual. To simulate reproduction, we selected breeders by first limiting female breeding age to between two and 15 years (Schwartz & Hundertmark, 1993) and male breeding age between five and 12 years (Mysterud, Solberg & Yoccoz, 2005) and then randomly selecting from the remaining pools. We incorporated mutation into the models, assuming a mutation rate of 10 𢄤 (Schlötterer, 2000 Bulut et al., 2009), by allowing step-wise mutations during allele assignment. For all individuals, we assigned male/female with a probability of 0.5 for both sexes and assigned age (1� years) with equal probability in the first simulated year and incrementally increased age of individuals in subsequent years. Finally, we limited population growth by adhering to the known Isle Royale moose population census data and removing individuals over age 15 from the simulated population. In years of population increase, we increased the population from the current number to the next year census size with new offspring. In years of population decline, we randomly selected individuals from the population for input into the next simulated year. The simulation was run for 100 iterations to quantify the variability associated with our estimates. We calculated the mean observed heterozygosity and number of alleles in each year across all replicates.


The Mutation Model

The mutation model was developed according to the theory formulated by Cockerham (1984) , whereby the number of allelic states per locus (n) is assumed to be finite. For simplicity, I also assumed an equal mutation rate (v) of any allele to any other specific allele so that the overall mutation rate is u = (n − 1)v. Cockerham's (1984) gene alike index is used for the derivation. The gene alike index within population X is defined as the probability of a random pair of alleles being alike (identical by state) and is estimated by QX = 1 − DXX = Σi=1 n xi 2 . Similarly, the gene alike index between populations X and Y is defined as qXY = 1 − DXY = Σi=1 n xiyi. The phylogeny is assumed to have started with an equilibrium gene alike value in the ancestral population before the taxa diverged. This assumption implies that the gene alike index within each population and each internal node is a constant i.e., Qi = Q* for all i's, so that only qXY is informative for phylogenetic inference. Cockerham (1984) provided the equilibrium gene alike value Q* ≈ (1 + 4Nev)/(1 + 4Nenv). The assumption of equilibrium QX does not imply constant allelic frequencies over time. In fact, the allelic frequencies must change from time to time so that the gene alike index between populations also varies overtime.

Having expressed yij as a linear function of the lengths of branches, we are ready to evaluate a given phylogeny.

The same least-squares method is applied here to evaluate the tree and estimate the lengths of the branches.


Genetic drift

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Genetic drift, also called genetic sampling error or Sewall Wright effect, a change in the gene pool of a small population that takes place strictly by chance. Genetic drift can result in genetic traits being lost from a population or becoming widespread in a population without respect to the survival or reproductive value of the alleles involved. A random statistical effect, genetic drift can occur only in small, isolated populations in which the gene pool is small enough that chance events can change its makeup substantially. In larger populations, any specific allele is carried by so many individuals that it is almost certain to be transmitted by some of them unless it is biologically unfavourable.

Genetic drift is based on the fact that a subsample (i.e., small, isolated population) that is derived from a large sample set (i.e., large population) is not necessarily representative of the larger set. As might be expected, the smaller the population, the greater the chance of sampling error (or misrepresentation of the larger population) and hence of significant levels of drift in any one generation. In extreme cases, drift over the generations can result in the complete loss of one allele in an allele pair the remaining allele is then said to be fixed.

The Editors of Encyclopaedia Britannica This article was most recently revised and updated by Kara Rogers, Senior Editor.


Genetic stochasticity, mean fitness of individuals and population dynamics

Keller, Biebach & Hoeck (2007) and McGowan, Wright & Hunt (2007) both emphasize the need for further research aimed at gaining a greater understanding of the relationship between inbreeding levels and population dynamics, as well as the implications of that relationship for conservation efforts. We fully agree with them that further study is needed and we thank them for their kind words suggesting that our research ( Reed, Nicholas & Stratton, 2007a ) is an excellent step towards a better understanding of how genetic factors interact with the environment to influence the risk of extinction in wild populations. Our data suggest that inbreeding stress interactions lower the mean fitness of individuals in more inbred populations, exacerbating the dangerous part of population fluctuations and increasing the risk of extinction.

It has been demonstrated that smaller populations, as indicated through estimates of decreased genetic variation or through direct counts of individuals, have decreased values for numerous components of fitness relative to larger populations ( Reed & Frankham, 2003 Reed, 2005, 2007 ). This is not surprising as population size and breeding structure influence fitness through several mechanisms: efficiency of selection, inbreeding depression, fixation of deleterious alleles and the loss of potentially adaptive alleles through drift, and the input of beneficial mutations ( Reed, 2005, 2007 ). In the populations of spiders used in this study, we demonstrated directly that fecundity is lower in smaller populations ( Reed, Nicholas & Stratton, 2007b ) and, therefore, the potential growth rate is lower. We purposely shied away from using the terms inbreeding depression and random genetic drift, even though it seems obvious that is what is going on. Instead we used the term genetic quality. The term is not nebulous and does not rely on comparing individuals to some local optimum as stated by Brodie (2007) . We define genetic quality precisely as a decrease in the mean fitness of an individual and fitness is measured relative to the other populations in the study. We chose this term specifically because, in practical terms, that is what matters to the central question we sought to answer: Does reduced mean fitness of individuals lead to changes in population dynamics that make those populations more vulnerable to extinction despite density-dependent mortality? The fact that the fate of alleles is more stochastic in smaller populations means that, all else being equal, smaller populations will be composed of individuals of lower average genetic quality.

Brodie (2007) suggests that it is important to distinguish inbreeding from loss of genetic diversity. In our data, expected and observed heterozygosity are both excellent predictors of population fitness and the correlation between them is extremely high (r=0.99). It makes no difference which of these two variables is used in the models. Thus, whatever nonrandom mating is occurring within populations, it is almost identical among populations. The statistical techniques we used can only differentiate among independent variables that differ among populations hence, we cannot comment on the effects of nonrandom mating on fitness in these populations. However, we can say that the differences in expected heterozygosity among these populations are almost certainly due to differential amounts of random genetic drift and therefore the differences in fitness are also due mostly to drift.

Allelic richness, on the other hand, is a rather poor predictor of population fitness relative to expected heterozygosity levels (Table 1). Some believe that allelic richness is very important for the evolutionary potential of populations because the limit of selection response, over time frames where mutation is negligible, may be determined by the initial number of alleles ( James, 1971 Hill & Rasbash, 1986 ). Nonetheless, because the rare alleles are mostly deleterious, it is not surprising that allelic richness is strongly correlated with expected heterozygosity (r=0.95), but that expected heterozygosity has much stronger support than models using allelic richness.

Model AICc Δi W i
Rabidosa rabida 2003–2004
ΔPCR+HE+ΔPCR ×HE −26.97 0.00 0.676
ΔPCR+N+ΔPCR ×N −25.49 1.47 0.321
ΔPCR+AR+ΔPCR ×A −18.11 8.85 0.013
Rabidosa rabida 2004–2005
ΔPCR+N −30.88 0.00 0.615
ΔPCR+HE −29.38 1.50 0.291
ΔPCR+A −27.14 3.74 0.094
  • The Akaike weights (Wi) provide the relative support for each model. Expected heterozygosity provides the best overall fit across both years, but population size provides a nearly identical fit. There is far less support for a model including allelic richness.

Brodie (2007) also suggests that the most important aspect of many real-world conservation problems is the relative impact of demographic, ecological and genetic factors. On this point we must disagree. (1) There is no fixed relative importance among the factors. Their relative importance will depend on the carrying capacity of the habitat, the specifics of the environment, etc. ( Reed et al., 2007b ). (2) It is likely that the interactions among the different factors are very important and looking at factors individually may lead to poor management decisions ( Reed, 2007 ). (3) Just because one factor is more ‘important’ than another does not mean that it is the best target for management action. What is the best target for management action will also depend on to what extent the various factors can be manipulated and at what cost.

Practically speaking, the solution to conservation problems is to keep carrying capacities large enough to allow for evolutionary processes to continue and to ameliorate any environmental (including biotic) factors leading to deterministic decline. This reduces demographic and genetic stochasticity and it may reduce environmental stochasticity as well ( Reed, 2007 ). The fact that improving habitat quality (or increases in available habitat) can increase the persistence times of species has received attention in the literature ( Reed et al., 2003 Reed, 2007 ) and we addressed this issue in another paper ( Reed et al., 2007b ). However, genetic intervention can at times be necessary (e.g. Westemeier et al., 1998 Pimm, Dollar & Bass, 2006 ).

Keller et al. (2007) examine our results in the context of hard and soft selection. As they suggest, our study does not contain sufficient information on all the variables of importance. More comprehensive studies, using wild populations such as we have used, are needed. However, systems amendable to such evaluation may be difficult to find and we were unable to complete our study as comprehensibly as originally planned due to a lack of funding.

We can, however, offer some possible answers to what Keller et al. (2007) seem to view as a bit of a conundrum. They state that inbreeding depression in individual fitness is expected to translate more strongly into reduced population growth when populations exhibit low levels of density dependence. We suspect that the mechanism leading to the correlation between individual fitness and population fitness may occur through sources of mortality that vary with body condition. In years of negative population growth, the populations of both species decrease in number, and individuals decrease significantly in mean body size at maturity as well. Thus, in years of low prey availability, spiders may be more susceptible to diseases, heat stress, predators, etc. because a higher proportion of them are malnourished. This could push these populations, under these environmental conditions, further towards the hard end of the soft–hard selection continuum.

We thank our colleagues for their insightful commentaries on our paper. We also thank the editors of Animal Conservation for choosing our work as the featured paper.


Heterozygosity under genetic drift - Biology

For November 4 and 6, please read two of three articles each day closely, but you only need to critique one of the six articles. The critique is due on November 6.

  • Sork et al 2002
  • Dick et al 2003
  • Godoy and Jordano 2002 (New choice)
  • Heuertz et al. 2003 (No longer included for class discussion)

I. Major Evolutionary Forces (Review)

A. Random genetic drift

1. Chance fluctuations in allele frequency which occur as a results of random sampling among gametes.

2. Important in small populations

3. Loss of genetic variation

B. Gene flow or migration

1. Movement of genes among populations

2. Homogenizes differences among populations

3. Source of variation for single populations

C. Mutation

1. Changes in genetic material

2. Includes single nucleotide insertions, deletions, changes, chromosomal changes, and spontaneous polyploidy.

3. Source of genetic variation

D. Natural selection

1. Process by which genotypes with greater fitness leave, on the average, more offspring than do less fit genotypes.

2. Genetic composition gradually changes to promote greater adaptation to the environment

3. Usually results in a loss of genetic variation (but not always)

II. Genetic Drift

A. Binomial probability

1. For a population of diploid individuals (2N), the probability that a population will contain a specific number, i, of one type of allele .

2. Equation:

3. Probability of fixation is the probability that a given allele will have 100% frequency.

4. Probability of loss of allele is probability that it will have 0% frequency.

5. For example, a plant population of N=4 and 6 A alleles.

6. Note that example shows high likelihood of loss of allele in this small population.

(Figure taken from Felsensetein, NOAA Tech Memo NMFS NWFSC-30: Genetic Effects of Straying (http://research.nwfsc.noaa.gov/pubs/tm/tm30/felsenstein.html)

B. Opportunities for Genetic drift

1. Continuous drift random effects are often the most important factor contributing to evolutionary change in populations that are always small.

a. Endangered species, e.g. California condors

b. Insular species (small islands, fragmented habitats

c. Skewed mating systems - many individuals but few breeders.

2. Intermittent drift: large fluctuations in population size.

3. Bottleneck effects - e.g. northern elephant seals, cheetahs (or metapopulation)

a. if a small group of individuals becomes geographically isolated from the remainder of the population or a small group of individuals colonize a new site.

b. random effects will significantly determine the frequencies of genes in the new population. e.g. Hawaiian Drosophila

C. Simulations of genetic drift and natural selection over time

See Felsenstein's PopG Genetic Simulation Program

ftp://evolution.gs.washington.edu/pub/popgen/popg.html

1. As time goes on, more and more populations become fixed

2. Population show the effects of "inbreeding", that is an overall deficiency of heterozygotes.

3. Within subpopulations, allele frequencies fit HW expectations.

4. For large populations that become subdivided due to restricted gene flow, genetic drift will influence loss of local genetic variation but, if sufficient number of subpopulations, global genetic variation will remain constant.

D. Discussion: Gene flow and selection

Kirkpatrick and Barton 1997

III. Population genetic structure

A. Measures of heterozygosity (Sewall Wright)

1. H I = heterozygosity for an individual in some subpopulation

=average heterozygosity of all loci for individual

(usually, we calculate the average observed heterozygosity across all individuals in a subpopulation.

2. H S =heterozygosity of a randomly mating subpopulation

3. H T = heterozygosity of a randomly mating total population

B. Levels of genetic structure

1. Inbreeding coefficient

a. measures reduction in heterozygosity of an individual due to nonrandom mating within a subpopulation.

b. equation:

a. measures reduction in heterozygosity due to genetic drift within subpopulations

b. measures amount of genetic differentiation among populations

equation:

3. Overall inbreeding coefficient

a. measure of reduction in heterozygosity of an individual relative to total population

b. reflects of the effect of inbreeding and genetic drift

c. equation

C. Example from Hartl: Levin 1978

1. Phlox cuspidata , single locus, PGM-, with two alleles, a and b.

2. 43 subpopulations

3. results

population freq(b) H
1-40 1.0 0
41 .49 .17
42 .83 .06
43 .91 .06
means: .9821 .067

Note: the high degree of both inbreeding and population subdivision

Be careful of studies with only one locus.

D. Example from Schaal 1975

1. Liatris cylandacea , herbaceous perrenial

2. study site: 18 m x 13 m plot in sand prairie, Illinois, 66 quadrats of 3 m 2

3. 27 loci, 15 polymorphic

4. Results:

Locus F IS F ST F IT
GOT .3773 .1084 .3885
MDH .4853 .0903 .5318
ADH .4508 .0452 .4755
AP-1 .4669 .2240 .5863
AP-2 .5050 .0438 .5267
Est-1 .5020 .0464 .5249
Est-2 .5059 .2190 .4110
Pep .3025 .0256 .3203
G-6PGDH .4013 .0677 .4419
6-PGDH .3629 .0756 .4110
Per-- .1004 .0139 .1121
Per+ .2579 .0395 .1004
PGI .4401 .0767 .4830
AlkP .4148 .0361 .4358
Est-3 .4289 .0091 .4344
MEANS: .4070 .0687 .4257

High inbreeding, and moderate population differentiation

Careful: both measures are a function of her quadrat size.

Question: what would happen if quadrat size was too large relative to area of

E. Other models to estimate genetic structure

IV. Gene flow estimated from population structure

A. Wright's island model

      Assuming equibrium between gene flow and genetic drift

      (Equation 1)

    B. Slatkin's isolation by distance model

    1. Uses equation 1 to estimate pairwise gene exchange,

    2. Based on Kimura's stepping stone model that individuals are dispersed from neighboring populations.

    3. Isolation by distance predicts that gene exchange should decrease with interpopulation distance.

    4. The ibd approach can incorporate landscape features by estimating various interpopulation pathways.

    C. Advantages and limitations of Indirect Methods

    1. Provide insight about evolutionary equilibrium of gene flow and genetic drift
    2. Genetic structure may be confounded by other evolutionary forces, which would mislead us about extent of gene flow

    D. Discussion

    V . Other types of genetic structure

    A. Fine scale genetic structure

    1. Measures distribution of genotypes within a population

    2. Spatially explicit

    3. Most plant populations show clusters of individuals with shared alleles

    a. presumed to be due to restricted gene flow

    b. affects estimates of mating system.

    See Rousett 1997 for sophisticated use of spatial autocorrelation statistics to examine gene flow

    B. Metapopulation genetic structure (source: Hedrick and Gilpin 1997)

    1. Dynamics of colonization and extinction create different genetic structure than predicted by conventional population genetic models (e.g. island, mainland-continent, stepping stone)

    2. Hedrick and Gilpin estimated the effective size of a metapopulation and show that metapopulation dynamics have the following impact effects:

    a. heterozygosity declines to some lower level

    b. F ST increases to some lower level

    c. Metapopulation effective size is reduced

    d. Number of subpopulations increases effective population size and F ST

    Table IV. Estimated effective population sizes for different numbers of subpopulations when local patches have infinite size, c=.2 and e=.05.
    N P N e(S) N e(T) N e(S) / N e(T) F ST
    5 43.3 57.2 .756 .166
    10 40.7 766 .531 .312
    20 39.4 148.8 .265 .386
    40 39.5 306.1 .124 .418

    e. Gene flow increases metapopulation effective population size and reduces F ST

    Table V. The estimated effective population sizes for different levels of gene flow between patches when K is infinite, c=.2, e=.05, and N P =10.
    m N e(S) N e(T) N e(S) / N e(T) F ST
    .00 40.7 76.6 .531 .312
    .00125 66.5 95.1 .699 .224
    .0025 88.9 115.7 .769 .167
    .005 125.8 140.1 .898 .114
    .01 156.7 172.4 .909 .069
    .02 217.7 219.7 .991 .040

    3. Metapopulation dynamics may cause loss of heterozygosity can be lost more quickly than predicted by traditional estimates of effective population size.

    4. Metapopulation dynamics may be a better explanation for low genetic variation in cheetah than the bottleneck hypothesis.

    C. Topics not covered that should be

    Coalescent theory

    Phylogenetic approach

    Phylogeography

    VI. Contemporary gene flow (Direct measures)

    A. Background

    1. Estimates gene movement per reproductive episode

    2. Estimates are not the same as Nm more similar to Wright's Neighborhood model:

    3. Early approaches tracked animal movement or documented pollen and seed movement

    4. Most genetically based plant population studies quantify pollen-mediated gene movement, but seed-mediated gene flow is possible.

    B. Parentage model

    1. Usually focusses on pollen-mediated gene flow

    2. Example, Streiff et al. 1999

    B. Genetic structure appoach to estimate contemporary pollen-mediated gene flow

    1. TwoGener: Two generation model to estimate effective number of pollen donors and average distance of pollen movement.

    2. Example: Discuss Sork et al 2002: Gene flow in wind-pollinated Valley oak in California oak-savanna

      • The TwoGener model suggest that the genetic consequences of pollen-mediated gene flow may result in much more restricted gene flow than suggested by paternity analysis.

      3. Example 2: Discuss Dick et al 2003: pollen movement in Brazilian forest fragments


      Genetic drift

      Genetic drift is a process that causes a population's allele frequencies to change from one generation to the next simply as a result of chance. This happens because reproductive success within a population is variable, with some individuals producing more offspring than others. As a result, not all alleles will be reproduced to the same extent, and therefore allele frequencies will fluctuate from one generation to the next. Because genetic drift alters allele frequencies in a purely random manner, it results in non-adaptive evolutionary change. The effects of drift are most profound in small populations where, in the absence of selection, drift will drive each allele to either fixation or extinction within a relatively short period of time, and therefore its overall effect is to decrease genetic diversity. Genetic drift will also have an impact on relatively large populations but, as we shall see later in this chapter, a correspondingly longer time period is required before the effects become pronounced. Genetic drift is an extremely influential force in population genetics and forms the basis of one of the most important theoretical measures of a population's genetic structure: effective population size (Ne). Because genetic drift and Ne are inextricably linked, we will now spend some time looking at how Ne differs from census population size (Nc), how it is linked to genetic drift, and what this ultimately means for the genetic diversity of populations.

      What is effective population size?

      A fundamental measure of a population is its size. The importance of population size cannot be overstated because, as we shall see throughout this text, it can influence virtually all other aspects of population genetics. From a practical point of view, relatively large populations are, all else being equal, more likely to survive than small populations. This is why the World Conservation Union (IUCN) uses population size as a key variable, considering a species to be critically endangered if it consists of a population that numbers fewer than 50 mature individuals. Taken in its simplest form, population size refers simply to the number of individuals that are in a particular population -- this is the census population size (Nc). From the point of view of population genetics, however, a more relevant measure is the effective population size (Ne).

      The Ne of a population reflects the rate at which genetic diversity will be lost following genetic drift, and this rate is inversely proportional to a population's Ne. In an ideal population Ne = Nc, but in reality this is seldom the case. If an actual population of 500 individuals is losing genetic variation through drift at a rate that would be found in an ideal population of 100 individuals, then this population would have Nc = 500 but Ne = 100, in other words it will be losing diversity much more rapidly than would be expected in an ideal population of 500. An Ne/Nc ratio of 100/500 = 0.2 would not be considered unusually low one review calculated the average ratio of Ne/Nc in wild populations, based on the results of nearly 200 published results, as approximately 0.1 (Figure 3.3 Frankham, 1995). We will now look at three of the most common reasons why Ne is often much smaller than Nc: uneven sex ratios, variation in reproductive success, and fluctuating population size. At the end of this section we will return to an explicit discussion of the relationship between Ne, genetic drift, and genetic diversity.

      Figure 3.3 A review of published studies revealed a range of Ne/Nc values in insects, molluscs, fish, amphibians, reptiles, birds, mammals and plants. Note that although Ne is often much less than Nc, it is a theoretical measurement and under some conditions can be greater than Nc (data from Frankham, 1995, and references therein)

      Figure 3.3 A review of published studies revealed a range of Ne/Nc values in insects, molluscs, fish, amphibians, reptiles, birds, mammals and plants. Note that although Ne is often much less than Nc, it is a theoretical measurement and under some conditions can be greater than Nc (data from Frankham, 1995, and references therein)

      Sex ratios Unequal sex ratios generally will reduce the Ne of a population. An excess of one or the other sex may result from adaptive parental behaviour. Although the mechanisms behind this are not well understood, there is increasing evidence for parental manipulation of offspring sex ratios in a number of taxo-nomic groups, including some bird species, which may be responding to environmental constraints such as a limited food supply (Hasselquist and Kempenaers, 2002). Even if the overall sex ratio in a population is close to 1.0, the sex ratio of breeding adults may be unequal, and it is the relative proportion of reproductively successful males and females that ultimately will influence Ne. In elephant seal populations, for example, fighting between males for access to harems is fierce. This intense competition means that within a typical breeding season only a handful of dominant males in each population will contribute their genes to the next generation, whereas the majority of females reproduce. This disproportionate genetic contribution results in an effectively female-biased sex ratio. The effect of an unequal sex ratio on Ne is approximately equal to:

      where Nef is the effective number of breeding females and Nem is the effective number of breeding males. The importance of the sex ratio can be illustrated by a comparison of two hypothetical populations of house wrens (Troglodytes aedon), which tend to produce an excess of females when conditions are harsh (Albrecht, 2000). Each of these populations has 1000 breeding adults. In the first population, conditions have been favourable for several years and so the Nef of 480 was comparable to the Nem of 520. The Ne therefore would be:

      The second population, however, has been experiencing relatively harsh conditions for some time. As a result, the Nef is 650 but the Nem is only 350. The Ne in this population is:

      In this example, the Ne/Nc in the first population, which had almost the same number of males and females, was 998/1000 = 0.998. The Ne/Nc in the second population, with its disproportionately large number of females, was 910/1000 = 0.910. Although the Ne/Nc ratio was smaller in the second population, the reduction in Ne that is attributable to uneven sex ratios was actually relatively low in both of these hypothetical populations compared to what we would find in many wild populations. According to one survey of multiple taxa, uneven sex ratios cause effective population sizes to be an average of 36 per cent lower than census population sizes (Frankham, 1995), although not surprisingly there is considerable variation both within and among species.

      Variation in reproductive success Even if a population had an effective sex ratio of 1:1, not all individuals will produce the same number of viable offspring, and this variation in reproductive success ( VRS) will also decrease Ne relative to Nc. In some species the effects of this can be pronounced. Genetic and demographic data were obtained from a 17-year period for a steelhead trout (Oncorhynchus mykiss) population in Washington State, and variation in reproductive success was found to be the single most important factor in reducing Ne (Ardren and Kapuscinski, 2003). When this trout population is at high density, i.e. when Nc is large, females experience increased competition for males, spawning sites and other resources. The successful competitors will produce large numbers of offspring whereas the less successful individuals may fail to reproduce. In other species, variation in reproductive success may have relatively little influence on Ne. The relatively high Ne/Nc ratio in balsam fir (Abies balsamea Figure 3.4) has been attributed partly to overall high levels of reproductive success in this windpollinated species (Dodd and Silvertown, 2000).

      The effects of reproductive variation on Ne can be quantified if we know the VRS of a population. Reproductive success reflects the number of offspring that each individual produces throughout its lifetime and therefore can be determined from a single breeding season in short-lived species, although individuals with multiple breeding seasons must be monitored for the requisite number of years. Long-term monitoring of a population of Darwin's medium ground finch (Geospiza fortis) on the Galapagos archipelago provided an estimated VRS of 7.12 (Grant and Grant, 1992a). The effects of VRS on Ne can be calculated as follows:

      If the census population size of G. fortis is 500 on a particular island, then the influence of variation in reproductive success on Ne will be:

      Therefore, even if the sex ratio is equal, the variation in the number of chicks that each individual produces will cause Ne to be substantially smaller than Nc.

      VRS may be highest in clonal species. In the freshwater bryozoan (moss animal) Cristatella mucedo (Figure 3.5), clonal selection throughout the growing season means that some clones are eliminated whereas others reproduce so prolifically

      Figure 3.4 Balsam fir (Abies balsamea). Wind pollination in this species helps to maintain overall high levels of reproductive success, and this helps to keep the Ne/Nc ratios high within populations. Photograph provided by Mike Dodd and reproduced with permission

      that the Nc of a population may be in the tens of thousands by the end of the growing season (Freeland, Rimmer and Okamura, 2001). Because clonal selection is decreasing the proportion of unique genotypes throughout the summer (Figure 3.6), the VRS must be substantial, with some clones producing no offspring and others producing large numbers of young. In the most extreme scenario, some populations ofbryozoans and other clonal taxa may become dominated by a single clone that experiences all of the reproductive success within that population, and when this happens the effective population size is virtually one (Freeland, Noble and Okamura, 2000). If this occurs in a population with a large Nc, the Ne/Nc ratio will approach zero.

      Figure 3.5 A close-up photograph showing a portion of a colony of the freshwater bryozoan Cristatella mucedo. These extended tentacular crowns are approximately 0.8 mm wide and capture tiny suspended food particles. Photograph provided by Beth Okamura and reproduced with permission

      0 25 50 75 100 125 150 175 200 Relative date

      Figure 3.6 Linear regression of ln-relative date (sampling date represented as number of days after 1 January) versus total number of alleles in a UK population of the freshwater bryozoan Cristatella mucedo (redrawn from Freeland, Rimmer and Okamura, 2001). Clonal selection has reduced the genetic diversity of this population throughout the growing season, even though the number of colonies increased during this time. This leads to a reduction in the Ne/Nc ratio

      0 25 50 75 100 125 150 175 200 Relative date

      Figure 3.6 Linear regression of ln-relative date (sampling date represented as number of days after 1 January) versus total number of alleles in a UK population of the freshwater bryozoan Cristatella mucedo (redrawn from Freeland, Rimmer and Okamura, 2001). Clonal selection has reduced the genetic diversity of this population throughout the growing season, even though the number of colonies increased during this time. This leads to a reduction in the Ne/Nc ratio

      Fluctuating population size Regardless of a species' breeding biology, fluctuations in the census population size from one year to the next will have a lasting effect on Ne. A survey of multiple taxa suggested that fluctuating population sizes have reduced the Ne of wild populations by an average of 65 per cent, making this the most important driver of low Ne/Nc ratios (Frankham, 1995). This is because the long-term effective population size is determined not by the Ne averaged across multiple years, but by the harmonic mean of the Ne (Wright, 1969). The harmonic mean is the reciprocal of the average of the reciprocals, which means that low values have a lasting and disproportionate effect on the long-term Ne. A population crash in one year, therefore, may leave a lasting genetic legacy even if a population subsequently recovers its former abundance. A population crash of this sort is known as a bottleneck and it may result from a number of different factors, including environmental disasters, over-hunting or disease.

      Because fluctuations in population size have such lasting effects on genetic diversity, we will take a more detailed look at bottlenecks later in this chapter. For now, we will limit ourselves to looking at how fluctuating population sizes influence Ne, which can be calculated as follows:

      Ne = t/[(l/Nei) + (1/Nez) + (1/N*) ■■■ + (1/Net)] (3.10)

      where t is the total number of generations for which data are available, Ne1 is the effective population size in generation 1, Ne2 is the effective population size in generation 2, and so on.

      The fringed-orchid (Platanthera praeclara) is a globally rare plant that occurs in patches of tallgrass prairie in Canada. The Ne of most populations is substantially reduced by fluctuations in population size from one year to the next. If a population had a census size of 220, 70, 40 and 200 during each of the past four years, and we assume that Ne/Nc = 1.0, then the effects that these fluctuations would have had on the Ne can be calculated as:

      Ne = 4/[(1/220) + (1/70) + (1/40) + (1/200)] Ne = 82

      Even though this population rebounded from the bottleneck that it experienced in years 2 and 3, this temporary reduction in Nc means that the current Ne/Nc ratio is only 82/200 = 0.41. Note that we have limited our example to a 4-year period for the sake of simplicity, although a longer period is needed for an accurate estimation of Ne.

      So far we have looked at how individual factors -- sex ratios, VRS, and fluctuating population sizes - can influence Ne. In each of the preceding sections we calculated the effects of a single variable on Ne, but in reality all of these variables can simultaneously influence a population's Ne. We are highly unlikely to have enough information to calculate individually the reduction in Ne that is attributable to each relevant variable. In the next section, therefore, we will move away from examining the effects of single variables and instead look at how we can calculate a population's overall Ne regardless of which factors have caused the biggest reduction in Ne.

      There are three general approaches for estimating Ne. The first of these, based on long-term ecological data, requires accurate census sizes and a thorough understanding of a population's breeding biology, neither or which are available for most species. A second approach is based on some aspect of a population's genetic structure at a single point in time, e.g. heterozygosity excess (Pudovkin, Zaykin and Hedgecock, 1996) or linkage disequilibrium (Hill, 1981). The application of mutation models to parameters such as these can provide estimates of Ne, although this approach is not used widely because it makes many assumptions about the source of genetic variation and can be influenced strongly by demographic processes such as immigration (Beaumont, 2003).

      The third approach, which is considered by many to be the most reliable, requires samples from two or more time periods that are separated by at least one generation. Several different methods can then be used to calculate Ne from the variation in allele frequencies over time. At this time, the most widely used method is based on Nei and Tajima's (1981) method for calculating the variance of allele frequency change (Fc) as follows:

      Fc = 1/K£(x - yO/[(x + yi)2/(2 - xyi)] (3-11)

      where K = the total number of alleles and i = the frequency of a particular allele at times x and y, respectively. This value then can be used to calculate Ne while correcting for sample size and Nc by using the following equation (after Waples, 1989):

      where t = generation time, S0 = sample size at time zero and St = sample size at time t.

      The temporal variance in allele frequencies was used to calculate the Ne of crested newt ( Triturus cristatus) populations that were sampled from ponds in western France. Researchers first were able to obtain an accurate census size of these populations using a standard mark--recapture method. As they were counted, individuals were marked by removing toes, which then were used as sources of DNA for deriving genetically based estimates of Ne. The census

      Table 3.5 Some estimates of Ne/Nc. In all these examples, Ne was calculated using a method based on the temporal variance in allele frequencies

      Steelhead trout (Oncorhynchus mykiss) Domestic cat (Felis catus)

      (Sciaenops ocellatus) Crested newt ( Triturus cristatus) Marbled newt ( T. marmoratus)

      ( Phalacrus substriatus) Carrot ( Daucus carota) Grizzly bear (Ursus arctos) Apache silverspot butterfly

      (Speyeria nokomis apacheana) Pacific oyster ( Crassostrea gigas)

      Ardren and Kapuscinski (2003) Kaeuffer, Pontier and Perrin (2004)

      Turner, Wares and Gold (2002)

      Ingvarsson and Olsson (1997)

      Le Clerc et al. (2003) Miller and Waits (2003) Britten et al. (2003)

      Hedgecock, Chow and Waples (1992)

      population size in one pond was approximately 77 newts in 1989 and 73 newts in 1998. The variance in allele frequencies between 1989 and 1998, based on eight microsatellite loci, provided an Ne estimate of approximately 12 and an Ne/Nc ratio of 0.16 (Jehle et al., 2001). Other examples of Ne/Nc ratios that have been calculated from temporal changes in allele frequencies are given in Table 3.5.

      Estimating Ne from the variance in allele frequencies can be logistically challenging because of the time and expense involved in sampling the same population in multiple years. Obtaining samples from museums is one answer to this, although museum specimens are a finite resource and not all species will have sufficient representation. Furthermore, some taxa such as soft-bodied invertebrates are not amenable to preservation in museums, and in many cases plants will be underrepresented. Practical limitations may also arise from the availability of markers because it is based on allele frequencies, the temporal method ideally should be done with data from co-dominant loci. Dominant data such as AFLPs can also be used, although, as noted earlier, accompanying estimates of allele frequencies will assume Hardy--Weinberg equilibrium, which may be unrealistic.

      Perhaps the biggest drawback to estimating Ne from the temporal variance in allele frequencies is the assumption that all changes in allele frequencies are a result of genetic drift. This does not allow for the possibility that immigrants from other populations are introducing new alleles and therefore altering allele frequencies through a process that is completely separate from genetic drift. As we will see in the next chapter, most populations receive immigrants with some regularity, and therefore this assumption is unlikely to be met. This problem has been partially addressed by a recently developed maximum likelihood (ML) approach that estimates Ne from temporal changes in allele frequencies in a way that partitions the effects of both immigration and genetic drift (Wang and Whitlock, 2003).

      Maximum likelihood is a general term for a statistical method that first specifies a set of conditions underlying a particular data set, and then determines the likelihood that these particular conditions would have given rise to the data in question. In the case of Ne, conditions may include a particular evolutionary history of the alleles in question, and maximum likelihood would be used to calculate the probability that different scenarios would have resulted in the observed variance in allele frequencies (Berthier et al., 2002). Maximum likelihood is a powerful approach, although it is computationally demanding and analytically complex. For these reasons it has avoided the mainstream so far, although its popularity is increasing as computers become more powerful and software becomes more user-friendly, and it may soon become the analytical method of choice for several aspects of molecular ecology including estimates of Ne.

      Wang and Whitlock's (2003) method is an extremely promising development in the quest for accurate estimates of Ne. However, it does require data from a sufficient number of variable markers to allow the detection of even relatively small changes in allele frequencies this may be particularly demanding when Ne is relatively large and migration rates are relatively small. In addition, it requires allele frequency data from both the population under investigation (focal population) and the populations from which immigrants may be originating (potential source populations). Assuming that the latter can be identified, one option is to pool data from all possible source populations and estimate the extent to which their collective contribution of migrants to the focal population has influenced the variance in allele frequencies that might otherwise be attributed entirely to drift. This method was applied to a metapopulation of newts ( Triturus cristatus and T. marmoratus) in France. The Ne/Nc ratios ranged from 0.07 to 0.51 when researchers assumed that changes in allele frequencies were solely a result of drift, and were 0.05 - 0.65 when they allowed for the effects of immigrants (Jehle et al., 2005). Because it aims to separate the effects that genetic drift and migration have on changing allele frequencies, this approach marks a significant step forward in the quantification of Ne. Although none is perfect, methods for estimating Ne have become increasingly refined in recent years, and this trend will undoubtedly continue because accurate estimates of Ne are crucial for understanding many different aspects of population genetics and evolution.

      Effective population size, genetic drift and genetic diversity

      We started this section by identifying genetic drift as one of the key processes that influences the genetic diversity of populations. We will now return to that concept by looking at the specific relationship between Ne, genetic drift and genetic diversity. The genetic diversity of a population will be reduced whenever an allele reaches fixation (attains a frequency of 1.0) because, when this occurs, the population has only one allele at that particular locus. The probability that a novel mutation will become fixed in a population as a result of genetic drift is 1/(2Ne) for diploid loci, in ohter words it is inversely proportional to the population's Ne (Figure 3.7). Since the rate at which alleles drift to fixation also represents the rate at which all other alleles at that locus will be lost, 1/(2Ne) can be considered as the rate at which genetic variation will be lost within a population as a result of genetic drift.

      The predictable relationship between Ne and genetic drift means that if we know the effective size of a population and its current genetic diversity (measured as expected heterozygosity), and if we assume that the population size remains essentially constant, we can calculate what the heterozygosity will become after a given time period as:

      where Ht and H0 represent heterozygosity at time t and time zero, respectively. Time intervals refer to generations, not years (although they will of course be the same if the generation time is 1 year). The predicted heterozygosity at time t is represented more commonly as a proportion of the heterozygosity at time zero:

      This tells us what proportion of the initial heterozygosity will be remaining after t generations. We can use this equation to compare the expected changes in hetero-zygosity in two hypothetical populations of crested newts that have a generation time of 1 year. The first population lives in a lake and retains an effective population size of approximately 200 for a period of 10 years. The second population inhabits a small pond and has an Ne of approximately 40 for the same time period. From Equation 3.14 we can estimate the proportional change in heterozygosity as:

      Ht/H0 =[1 - 1/(2 x 200)]10 = [0.9975]10 = 0.975 for the lake population, and as:

      Ht/H0 = [1 - 1/(2 x 40)]10 = [0.9875]10 = 0.882

      for the pond population. This means that the lake population will lose approximately 2.5 per cent of its initial heterozygosity in ten generations, whereas the smaller pond population will lose around 12 per cent of its heterozygosity.

      The rate of drift does not depend solely on a population's Ne it is also influenced by the population sizes of the genome in question (Table 3.6 and Figure 3.7). Since the population sizes of plastids and mitochondria are effectively

      Figure 3.7 The probability that a neutral mutation will reach fixation in any given generation reflects the rate at which genetic variation will be lost following genetic drift. In this figure, the probability of fixation is given for diploid loci, calculated as 1/(2Ne), and for mitochondrial haplotypes, which is calculated as 1/Nef (here we have assumed that half of the breeding population is female see Table 3.6). Note that the probability of fixation (and the accompanying loss of alleles) following genetic drift is inversely proportional to the effective population size a quarter those of nuclear genomes in diploid species, they will lose genetic variation at a faster rate than most nuclear genes (Figure 3.7). Returning to our example of crested newts, we know that the rate at which genetic variation is lost from diploid nuclear genes is 1/2Ne per generation, which is around 1/400 = 0.0025 in the lake population of 200 individuals. Table 3.6 tells us that in the same population, assuming that half of the Ne is female, mitochondrial variation will be lost at the much faster rate of approximately 1/100 = 0.01 per generation.


      Discussion

      Our key finding is that drift load was present at heightened levels in populations of small size (that is, low HE) and with low levels of intraspecific competition, whereas segregating load was not associated with either population size or local plant density. It is not clear whether this implies a demographic meltdown (sensu Lynch et al., 1995a) in small, sparse populations. However, if the reduction in performance due to drift load decreases population growth rate, then this would only reinforce the small population size and low local densities that exacerbated drift load in the first place. Given that the 13 populations assessed here were selected without prior knowledge of long-term population size or intraspecific competition, we suggest that many A. lyrata ssp. lyrata populations in nature harbor substantial genetic loads.

      The average segregating load of around 0.2 was low relative to values observed in other outcrossing plant populations ( ⩾ 0.5 Winn et al., 2011). Our study populations, or the ancestral populations that gave rise to them, may have purged part of the segregating load at the end of the last glacial maximum as they underwent multiple founder events during range expansion. It could also be argued that segregating load was low because of benign conditions in the experimental common-garden environment (Armbruster and Reed, 2005). But stressful conditions seem to increase load only moderately, and interspecific competition—a potential source of stress in nature for this species—does not generally interact with segregating load to enhance it (Willi et al., 2007a). Also, the high levels of drift load present in some populations indicate that common-garden conditions were not so benign as to prevent the expression of load.

      Our results are in line with theoretical predictions that segregating load should not strongly decrease in populations of long-term small size (Glémin, 2003). Other empirical studies confirm this. Comparisons of naturally large and small populations of plants and snails find that segregating load can be substantial, but is unrelated to population size (van Treuren et al., 1993 Willi et al., 2005 Escobar et al., 2008). One study of a self-compatible gentian reported reduced segregating load in smaller populations (Paland and Schmid, 2003), possibly because these populations have declined only in the last 50 years and purging via non-random mating may have occurred.

      Drift load in A. lyrata was substantial, particularly in long-term small populations and those with low plant density. Our estimates may even be low if heterosis was reduced by a certain level of outbreeding depression (Lynch, 1991). Other studies that compare a reasonable number of replicate populations (>5) also report that drift load is high and sometimes related to population size. The plant and snail studies mentioned earlier found that drift load was significantly greater in small populations in some cases (Willi and Fischer, 2005 Willi et al., 2007b Escobar et al., 2008) but not others (van Treuren et al., 1993). The experimental study of snails revealed significant heterosis for the smallest bottleneck size class compared with any of the three larger classes (Coutellec and Caquet, 2011). Here again, Paland and Schmid’s (2003) gentian study was an exception, with generally low drift load uncorrelated with population size. On balance, drift load seems to be important in populations that are small or have gone through a recent bottleneck, but further studies that include an adequate number of replicate populations would be valuable.

      Drift load declined with increasing intraspecific density. When plants from low-density populations were out-crossed, their seeds were more likely to germinate than those from within-population crosses (Figure 1b). This suggests that there was drift load for germination. In later life stages, drift load declined with density for flowering time in the second year and for cumulative flower production up to the fourth year of plants that had successfully germinated and established in the first year without being diseased (Figure 1d). These results confirm those of Pujol and McKey (2006), who observed that individuals with higher multilocus heterozygosity were favored in denser plant clusters, because purging of deleterious mutations appears to be more likely under conditions of high intraspecific competition.

      Both types of genetic load were generally expressed late in life. Segregating load was absent for seed size, germination, seedling establishment and early pathogen infection, but it strengthened after the onset of reproduction for both male and female performance components. Drift load affected earlier life history components as well as reproductive output. There was significant drift load in germination, and drift load for pathogen resistance was negatively correlated with long-term population size (Figure 1c). Load expression patterns found here agree with those observed in selfing species, for which there is very low segregating load for seed production, germination, and survival to reproduction, but substantial load in growth and reproduction (Husband and Schemske, 1996). Husband and Schemske concluded that most early-acting load must be due to recessive lethals and is purged through inbreeding, whereas much of the late-acting load is due to weakly deleterious mutations and is difficult to purge. If early-acting recessive lethals in our outcrossing A. lyrata were purged during postglacial recolonization, then population sizes must have remained low since then to prevent the recovery of segregating load (Kirkpatrick and Jarne, 2000). In contrast, purging of late-acting and presumably weakly deleterious mutations in our system seems hampered by long-term small size and low competition. This has led to relatively high drift load in small, sparse populations.

      Why is it important to distinguish segregating load and drift load? One reason is that the two kinds of load have different influences on important evolutionary transitions. For example, the evolution of selfing from outcrossing is expected only if segregating load is smaller than a certain threshold value (Lande and Schemske, 1985) the level of drift load is not directly relevant to this transition. Similarly, the evolution of asexuality from sexuality (and the long-term persistence of asexuality) is more likely under low segregating and particularly under low drift load, because the load in asexual lines can only increase (Muller, 1964 reviewed in Hartfield and Keightley, 2012). The two types of load also have different implications in applied fields such as conservation biology. Fitness decline caused by load that is fixed within relatively isolated populations is unlikely to be reversed without migration, whether natural or assisted. While segregating load remains mostly unaffected by long-term small population size, fixed drift load requires tens to hundreds of generations of large size to be overcome without genetic restoration. For example, Caenorhabditis elegans kept for 240 generations as single-individual bottlenecked lines accumulated substantial drift load, and required up to 80 generations of culture at large population size under highly competitive conditions to experience fitness recovery (Estes and Lynch, 2003). In natural systems with lower levels of competition, recovery may take a few hundred generations.

      Our results highlight an important point about neutral evolution and mutation accumulation that contradicts general wisdom in evolutionary biology and conservation. Many specialist species are poor dispersers and occur in habitats with patchy distribution. Small populations of such species undergo neutral evolution that is not actually neutral with respect to fitness, because mutation accumulation lowers mean fitness. Conservation biologists believe that mutation accumulation in small and isolated populations is a less important threat than inbreeding depression and the loss of genetic diversity (Frankham, 2005). The reasoning is that inbreeding depression acts immediately, and a loss of genetic variation is important whenever the environment changes, whereas it takes many generations for mutation accumulation and fixation to appreciably lower mean fitness. This makes good sense for species of conservation concern that have declined in population size recently not enough generations have passed for drift load to accumulate. However, many species have persisted in small and isolated populations for a long time, and under these conditions mutation accumulation can take its toll. This is exemplified by some populations of A. lyrata.

      More generally, this study questions the adaptationist perception that populations usually exist on or near local fitness optima. A well-known challenge to this perspective comes from studies of the genetic, developmental and selective constraints that bias the production of variation and response to selection in certain traits (Maynard Smith et al., 1985). But high levels of genetic load represent a more general challenge to adaptation because they cause a reduction in population mean fitness that limits adaptation in all traits at once (Lynch and Lande, 1993 Willi et al., 2006 Willi and Hoffmann, 2009). Species that have naturally fragmented distributions, for which drift load is especially important, may be badly positioned for long-term persistence. This is true even under ordinary conditions, but when the environment changes quickly drift load combines with low genetic diversity to sharply limit persistence time of small populations (Hoffmann and Willi, 2008 Willi and Hoffmann, 2009). Although bottlenecks and founder events might occasionally produce adaptive novelty (Carson, 1975 Templeton, 1980 Wright, 1982), the more typical fate of such populations is extinction. Many species exist in populations that are small or that have experienced repeated bottlenecks, and in this case neither genetic innovation nor healthy growth rate for long-term persistence can be expected.



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