GENETIC DRIFT & EFFECTIVE POPULATION SIZE

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1 Instructor: Dr. Martha B. Reiskind AEC 450/550: Conservation Genetics Spring 2018 Lecture Notes for Lectures 3a & b: In the past students have expressed concern about the inbreeding coefficient, so please spend some time looking through these lecture notes and hopefully this will help clarify. As you can imagine spending time in class deriving equations, which we will do at critical points, is not efficient and doesn t allow enough time to bring in examples that you can hang these concepts on or help illustrate how we use these measures. For this reason, you ll need to use these lecture notes, the book, the lecture slides, your lecture notes, and me to help further clarify anything you are confused about. It s been my experience both as a student and as an instructor that clarity comes best from sitting with and working through these equations first (this is why those problem sets are important!!!). I can show you, tell you, demonstrate to the end of time, but until you actively embrace the equations, follow the derivations (which requires a wee bit of algebra) and/or use them for real examples they will continue to elude you. Sometimes it s effective to play with made up numbers. For example, I will demonstrate in class the effect of different population sizes, while keeping allele frequencies equal, to help show the relationship between population size and the effect of genetic drift on allele frequencies and genetic diversity. This is something you could do as well. In some respects, there will be a bit of acceptance required to get through. I ve found that if you let go of the confusion and move forward with these equations, it often becomes clearer with a little time and elbow grease. Please look over this graph for lecture, the relationships between allele and genotype frequencies in HW proportions: This figure helps illustrated a couple critical points that we will talk about as the course continues. First, I will talk about the absorbing boundaries of fixation or loss for alleles, in a one locus, two allele situation above, or with many alleles per locus, this is the end point for diversity or heterozygosity (the left and right hand sides of the graph). The peak in heterozygosity is when all alleles are at equal frequency, 1

2 which in the example above is 0.5, but could be 0.25 if there are 4 alleles, or if there are 8 alleles, etc. etc. As alleles frequencies move either to the left or right along either x-axis from the equal frequency point, there is a loss of heterozygosity at this locus. At an individual allele, it s not until the allele reaches a high enough frequency that the frequency of the homozygous genotype will be greater than the heterozygous genotype. For example, when the p allele is at a frequency of 0.7 the p 2 genotype frequency begins to exceed the 2pq frequency in the graph above. Genetic Drift is an evolutionary force present in all finite populations and the magnitude of the effects of genetic drift on allele frequencies or diversity is related to the size of the population. In population genetics we often talk about sampling error, meaning that when we sample a population to create a new one, if that sample is small we will not capture the same frequency of genotypes and alleles as the source population. Alleles may lose or gain in frequency at random. We will first focus on how we measure the magnitude of change in allele frequencies and then in genetic diversity. Change in allele frequency range due to genetic drift: This gives the 95% confidence range of allele frequency, the range declines as N increases (put some numbers in and you will see what I mean). We will run an example in class, but if you plugged in different numbers for N you ll see the 95% confidence range of that allele frequency in the next generation will contract. The current allele frequency is equal to the previous generations allele frequency times plus or minus 2 standard deviations. Inbreeding coefficient: Lower case f is often referred to as the inbreeding coefficient or the probability of autozygosity (you will see it interchanged with cap F often, don t let that freak out). It s a measure of the probability of two alleles being identical by descent (IBD) within an individual at time t. Yes, it s a theoretical measure at the individual level based on the relationship of gametes in the t-1 population (population one generation before the current one). We will extend this measure to population level as well without changing the equation to the right of the equal sign (okay, accept the confusion, maybe curse at those crazy geneticists a little ) 2

3 Some things to consider: You could have two alleles that are homozygous, but not IBD, they are identical by kind. That is to say that they are the same alleles but do not share a recent ancestor. If we go back in time to the origin of that allele it would be IBD. This concept is something that I found confusing way back in ancient times when I was a student, because time seemed arbitrary, how recent, how far back? It becomes a little clearer when we think about subpopulations. In this case we use F, that is the population level homozygosity or IBD. Before the subpopulations formed at t0 there was no IBD. In this case, at time zero, F = 0 and IBD, by definition, can only occur after 1 generation at the split of the populations. Here s some of the formulas related to the inbreeding coefficient: f = 1/2N this is the probability of being IBD within an individual per generation (left-hand side of the diagram above) f = 1-h; and it follows that h = 1-f Looking at the figure above, the two alleles that are IBD have a probability of f and the rest of the alleles are 1-f or h. Or, you could say, the IBD alleles have a probability of 1/2N and the non-ibd alleles probability of 1-(1/2N), which is the right-hand side of the diagram above. Here s the full equation for probability of IBD in generation t (adding the probability of the two mutually exclusive events): Ft = 1/2N(1) + (1-1/2N)Ft-1 (equation 3.1) Here the 1 that the first term is multiplied by indicates that the alleles that are drawn at this probability are IBD, therefore F = 1, while the alleles that are not IBD, allozygous alleles, had a certain probability of IBD in the previous generation which is Ft-1. Through some substitutions and rearrangements and remembering that F0 = 0 (again, this is an arbitrary time in the past hint: first multiply by -1, then add 1, then do some rearranging remembering that F initially is 0), you get the following general equation: Ft = 1 (1-1/2N) t (equation 3.2) In this case the expected increase of homozygosity, the probability of autozygosity or IBD in generation t is equal to 1 minus the heterozygosity due to drift over t generations [(1-1/2N) t ]. 3

4 Changes in genetic diversity due to genetic drift: As we will talk about in lecture, we generally want to think about this in terms of heterozygosity or diversity, and therefore we convert the above equation to h: per generation equation (3.3) general equation (3.4) Again, there is some great algebra and substitutions that allow us to get from the above equation (3.1) to this equation (3.3). Here is how we get to this equation from the above one if you are interested, work it out yourself as well: From equation 3.1: Ft = 1/2N + (1-1/2N)Ft-1 We convert this to: Ft = 1/2N + Ft-1 (1/2N)Ft-1 We rearrange to: Ft = 1/2N (1/2N)Ft-1 + Ft-1 = 1/2N(1- Ft-1 ) + Ft-1 Multiple both sides by -1: -Ft = -1/2N(1- Ft-1 ) - Ft-1 Add 1 to both sides: 1-Ft = (1 -Ft-1)- 1/2N(1- Ft-1 ) Substituting Ht which is = 1 Ft we get: Ht = Ht-1-1/2N(Ht-1) OR Ht = (1-1/2N)Ht-1 (which is equation 3.3 above) To help find a connection between these equations (which are the ones we will use) to the h = -1/2N in your book you can see from equation 3.3 that heterozygosity declines by -1/2N each generation on average per locus. Note we will extend these individual based equations to make estimates at the population level, which works pretty well. That s for later, so stay tuned Bottlenecks & Founder Events: Couple important points, the equations above are pretty good at giving a sense of the magnitude of change in diversity. Loss of diversity or heterozygosity is used widely and luckily is insensitive to sample size, which helps in situations where you have low numbers for one population versus another. This becomes increasingly important in conservation genetics because when comparing species or populations within the species we often have very small population sizes. In addition, it is a measure that is independent of the number of alleles present (genetic drift is also independent of the number of alleles present). For the reasons above, it is not as sensitive to the effects of bottlenecks as other measures such as Allelic Diversity (A). I think this helps clarify a potentially confusing point: IF the population reduces down to 2 individuals there is a 25% increase in homozygosity, but 75% of the heterozygosity is retained, yet, with 2 individuals at a given locus there is only 4 alleles. What does this mean for those loci with more alleles? This is in part why allelic diversity is a better estimate of the effects of an extreme loss of population numbers. Going back to an example before, if you have 6 alleles, 15 of the 21 4

5 genotypes will be heterozygous, this is a tremendous amount of diversity that would be lost if you go down to 2 individuals, even if those two individuals have 4 different alleles at one locus. If you are concerned about a bottleneck and its affects on genetic diversity for a population of interest, typically we report both measures if we have a marker with multiple alleles. Allelic diversity. First we want to know what is the probability of an individual allele loss as it relates to population size: equation 3.5 p is the frequency of the allele and N is the size of the population. Rare alleles less than 0.05 are more likely to be lost, but the loss of this low frequency allele won t affect heterozygosity as much as a higher frequency allele. For example, p = 0.01 has a 60% chance of being lost in a bottleneck of 25 individuals, while p = 0.1 has a 0.5% chance of being lost in a bottleneck of the same number. The estimate of allelic diversity is: equation 3.6 This estimates the allele number at time t compared to the initial number of alleles. pj is the frequency of the jth allele (from allele # 1 to the total number of alleles (A)). We will talk about a real example of the brown bear in the Brooks Range in class and how this measure was better at making estimates of the effect of the bottleneck. We will also look at this in the context of founder events as well. Absorption and fixation time for neutral alleles: For selectively neutral alleles, the probability of ultimate fixation is equal to its initial allele frequency. This figure shows the average times to fixation, loss, and persistence of a neutral allele in an ideal diploid population of size N, plotted against initial allele frequency. The unlabeled curve is the persistence curve. 5

6 Genetic drift and fitness: Both changes in allele frequencies and loss of allelic diversity cause fitness effect. A random increase in frequency of alleles that harbor harmful effects, such as deleterious alleles that are typically tamped down (if recessive) or removed by Natural Selection, can have a direct effect on fitness. We will look at a couple examples of this in natural populations and zoo populations. Effective population size: Effective population size (NE) is the size of the ideal population that will result in the same amount of drift as in the actual population being considered. Or, to say it in a different way, the effective population size of an actual population is the number of individuals in a theoretically ideal population having the same magnitude of random genetic drift as the actual population. We want to know this number as managers or in a conservation context. If the number of effective breeders is lower than the actual population size, this is critical for informing how we triage these populations. It helps us understand how drift is effecting the population. In an ideal population: (1) there will be equal frequencies of males and females (2) self-fertilization is an option (3) there is no variation in how many progeny each pairing produces (4) population size doesn t fluctuate (5) generations are discrete (no reproduction between offspring and parent generation). Sounds great, right? What we want to figure out is what the effective population size is given any of these violations, in particular variance in offspring contribution, unequal sex ration, and fluctuations in population size. There are three different NE based on how we choose to measure magnitude, and they are: (1) Inbreeding effective populations size This is the measure we mostly use and typically the one reported in most manuscripts. In this case we are looking at change in probability of identity by descent. We will focus on this one. (2) Variance effective size In this one we are looking at change in variance in allele frequencies (3) Eigenvalue effective size Here we are looking at the rate of loss of heterozygosity Inbreeding Effective Population Size (NE): In 1931, Sewell Wright first worked out the effective population size by looking at the increase in IBD in various situations. 6

7 Unequal sex ratio: Recall one of the assumptions above is that there are equal numbers of each sex and that individuals can self. This is ideal, but often not reality. Therefore, in most populations the total population N = Nm + Nf, which is the number of males and females respectively. In the case where there is no selfing, this means ½ the alleles at a locus come from the female and the other half come from the male. Therefore, we have to consider the amount of genetic drift separately. You can imagine all sorts of scenarios where you might have unequal sex ratios and can consider that this must impact probability of autozygosity or identity by descent. Here s the equation: equation 3.7 Your book does an okay job of explaining how you get to this equation, but here s the way I like to explain it. Here s a simple pedigree, I neither condone nor judge what this pedigree may imply! Here there are 6 unique alleles for this particular locus and what we want to know first is the probability that Fred gets two copies of one of Grandpa s two alleles. We will also consider the mutually exclusive probability that he gets the other allele. First let s consider the probability that Freddy gets two A3 from his Grandpa: Following what I ve drawn, there is a ½ probability that Grandpa passes his A3 to daughter Judy, and a ½ probability that Judy passes it to Fred, and on the other side, a ½ probability he passes A3 to Bob and ½ probability Bob passes it to Fred. This totals up to: ½ x ½ x ½ x ½ = 1/16 Probability that Fred gets A4 A4 from grandpa: 1/16 7

8 And, these are two mutually exclusive ways that Fred can be autozygous for one of Grandpa s two alleles. Therefore: 1/16 + 1/16 = 2/16 = 1/8 This is the probability of being autozygous (IBD) for either of grandpa s two alleles This 1/8 is corrected for the frequency of Grandpa in the total number of males in the population: 1/8 x 1/Nm = 1/8Nm Now we can consider the probability of autozygosity from the female line: 1/8 x 1/Nf = 1/8Nf Now using the inbreeding coefficient F = 1/2NE 1/2NE = 1/8Nm + 1/8Nf and multiplying both sides by 2 we end up with 1/NE = 1/4Nm + 1/4Nf Finally, we rearrange to get the equation 3.7 above: NE = 4NmNf/(Nm + Nf) Note: play around with the numbers and you ll see that the effective population size is constrained by the sex with the smaller number. Try a few ones on your own. Nonrandom number of progeny: An ideal population each individual has an equal chance of contributing offspring to the next generation. Note: I will not derive this equation for you. In some cases, you ll see the numerator with a # subtracted from Nc this is to correct for population size, here I ve shown the ideal large population equation. N E = 4Nc 2 + 2(V k ) equation 3.8 Here Nc is the actual population size and variance in reproductive success Vk. If you have a population of size N that produce gametes (k) to the next generation, and there is an equal chance of contributing offspring to the next generation, and N is constant, the mean number of gametes produced is k = 1. Following a Poisson distribution, the variance and the mean would be equal. Using this, the effective population size would be: N E = 4Nc 2+2(1) = Nc 8

9 If N is reasonably large and every single member of the population reproduces the same number of offspring, there is no variance in reproduction (Vk = 0), then the effective population size would be: N E = 4Nc 2+2(0) = 2Nc This population is accumulating inbreeding as if there were 2 times as many individuals in the population. Fluctuating population size: Here we estimate the effective population size by using the mean of the reciprocal of the population size in successive generations, rather than the mean of N itself, this is the harmonic mean: With a little bit of manipulation: What s important here is that the effect of inbreeding increases with the smaller population sizes, more so than with large population sizes. For example, if we have a population that goes from N = 1000 to 10 to We can compare taking this harmonic mean versus the mean and plug it into the coefficient of inbreeding, Ft = 1 (1-1/2N) t (equation 3.2). NE = 3/((1/1000)+ (1/10)+(1/1000)) = Then F3 = 1 (1-1/(2x29.41)) 3 = 0.05 Mean N = ( )/3 = 670 Then F3 = 1 (1-1/(2x670)) 3 = Gene Genealogies, Coalescence, & Lineage Sorting: We can look at population-level genetic processes, such as effective population size and genetic diversity in a different way from the above. We can use a gene genealogy approach that traces genes back to their common ancestor. This is referred to as coalescence. We will spend a bit of time talking about this in the future, as we look at historic effective population sizes using coalescence theory. 9

10 Application of effective population size to management strategies: Important to think of the effective population size as a standard to help us make management decisions. We can use the equations above, depending on what the population looks like, to see how different management strategies might affect the effective population size. Usually it helps if we have a measure of the inbreeding coefficient. The often sited and published rule of an effective populations size of 50 is debatable and likely may be either an under or over-estimation. We will talk about this more in later lectures. One thing that is helpful to have is generation time (G). The measure of inbreeding is a per generation measure. A measure of generation time helps us convert to the correct generation time, and will help with management decisions. 10

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