Need an expert in Genetics to help me with my population genetics course assignment. The assignment documents has been attached with 3 questions. Also, a few other documents about sample calculations
Population Genetics 4303 Chapter 5: evolutionary Forces Prof. Sara V. Good Winter 2021 Population Genetics 4303 Winter 2021 1 MIDTERM changed till March 2nd MIDTERM changed till March 2 nd ! Guest lectures March 4 & 9 Group Presentations March 11 16, 18 23, 25 30th Probing the effect of violations of HWE on genetic variation in populations: the effect of Non - random mating • Instead of everyone having the same chance of mating, what happens in non -random mating? Two kinds: • Assortative mating • Positive assortative mating • Quite common in humans. Can measure by using correlation coefficient (range -1 to +1) • Positive correlation among husbands and wives common (+0.3 for height, +0.4 for political preference, +0.5 for education, Schwartz, 2013) • Negative assortative mating • Not common in humans but in some plants, fungi and some other taxa. • Can cause an excess of heterozygosity, typically at just a few loci (those linked to the dissoratative trait). • Inbreeding (or consanguinity) • Mating with relatives Identity by state vs identity by descent of alleles Panel A, identity by descent Panel B, identity by state The inbreeding coefficient, F • F = the probability that two alleles at a locus are identical by descent • Typically 0• Frequency (AA) = p2 + pqF • Frequency (Aa) = 2pq(1 -F) • Frequency (aa) = q2 + pqF • What happens to the allele frequencies after one generation of inbreeding?
Remember that the frequency of A in the next generation is equal to the frequency of the Aa homozygotes plus ½ of the Aa heterozygotes • p’ = p 2 + pqF + 0.5(2pq(1 -F)) • = p 2 + pqF + pq -pqF • = p 2 + pq = p( p+q )= p • So inbreeding does not change the allele frequency! If inbreeding doesn’t change the allele frequencies how does it influence genetic diversity • The allele frequencies do not change with inbreeding, just the genotype frequencies. You progressively lose heterozygotes . • Inbreeding will decrease heterozygosity in a population. But because different populations lose heterozygosity independently, you will lose heterozygosity at different loci. This will increase the divergence among populations. • Furthermore, many harmful recessive alleles may now become exposed in homozygous state. • Even a small inbreeding coefficient (for example, mating between 1 st cousins for which F=1/16) is typically sufficient to cause decreased viability and fertility Inbreeding depression • The reduction in fitness of an individual whose parents were related compared to an individual that is randomly outcrossed from the same population • The most widespread cause of inbreeding depression is the exposure of disadvantageous alleles that are normally kept at low frequency, and therefore normally found in heterozygous state, but become homozygous due to a reduction in population size and increase in inbreeding (see figure on the right). • Important concepts: • Inbreeding depression compares the relative fitness of individuals from the SAME population, one has related parents, the other does not. Our genomes have many deleterious variants, that were they to be in homozygous form, would be lethal or potentially have mild to severe effects on fitness In plants, researchers frequently find evidence of a “sweet spot” for optimizing fitness of progeny: matings should be with individuals that are too close (inbreeding depression) or too far (outbreeding depression) From Wikipedia page on inbreeding depression Inbreeding depression • Can happen because of non - random mating (individuals by choice or force reproduce with related individuals). • It typically arises when populations have reduced population size, such as often happens with habitat fragmentation or endangered species (Suarez and Good, 2015) • One of the perils of endangered populations From Suarez and Good, 2015 Evidence that forest fragmentation increases genetic relatedness among individuals in a shrub. In this study, we compared the spatial genetic structure of a flowering shrub (related to the saskatoon berry) in a continuous (red -Grand Beach) and forest fragment (blue -BP) populations, and found that in the forest fragment, there was more spatial structure – i.e. plants within 60 meters of each other were related, while in GB they were not. Small population size: loss of genetic variation due to random genetic drift • Whenever the effective population size ~ < 500, populations are at risk of losing genetic diversity due to random genetic drift. • Under small population size, alleles become lost (p=0) or fixed (p=1). • H t= (1 -1/2N) tH 0 NOTE BIEN : Time is measured in Generations!! – Why? What is a average length of a generation in humans? Models of the loss of genetic variation due to random genetic drift: random fluctuations in allele frequencies over time.
Points:
i) The loss of genetic diversity depends on N e ii) The smaller the N e, the more genetic variation is lost and variation is lost sooner.
iii) The probability that a new mutation will reach fixation = frequency of the allele = 1/2N e H t= (1 -1/2N) tH 0 Useful equation! In the absence of selection, the decay in heterozygosity declines at a rate inverse to population size N.B. In general, because population geneticist know that whenever we use the parameter N, we really mean Ne, You will often see it written as just N, assume that is in Ne unless you are told that N= census size. Ft = 1 -(1-1/2N) tH 0 OR re -arranged and in one generation 1-Ft= (1 – 1/2N) (1 – 1F t-1) These equations are closely related: as heterozygosity decays the inbreeding coefficient (see earlier slide, the probably that two alleles are identical by descent in an individual increases) increases. Compare to previous slide So what number of individuals is enough to sustain genetic diversity in populations?
• Scientists estimate that about 1000 nesting Kemp's Ridley sea turtles, 300 right whales, and 65 northern hairy -nosed wombats survive in the wild, to name just a few of the world's endangered species. Are these enough? • According to evolutionary theory and population genetics, very small populations face two dangers — inbreeding depression and low genetic variation — that might keep them from recovering, despite our best efforts to preserve them. • Minimum Viable Population size : Models have shown that a population size of 500 is considered a minimum number for population recovery. Below 50 individuals is critically low. • Anytime that the number of individuals falls below 500, populations are in serious risk of losing genetic variation and suffering from inbreeding depression.
Unless action is put in place to secure habitat and augment population size, they have a high probability of extinction. Even when the actual population size starts to recover, the increase in genetic variation, and thus the change in N e is much slower to rebound (but can). There is a lag (see bottom right) Two related concepts: bottleneck and founder event: both result in a loss of genetic variation due to a crash in population size (bottleneck) or founding of a new population (plants moving to an island, Ahmish moving to Pennsylvania) Can cause a random increase in frequency in traits or genetic variants that were rare in the original population.
May also cause new traits to evolve because of small population effects in the new populations, e.g. many island populations of plants lose self - incompatibility systems which prevent self - fertilization. Note: Different parts of the genome have different effective population sizes • A Wright -Fisher population is a simple idealized population used in population genetics (more than the HW one). It has constant population size, equal sex ration, non -overlapping generations, random mating and all individuals have the same probability of having offspring. Unlike the HW population, it has a FINITE size and MUTATION is allowed (next section). Living on the edge: Was demographic weakness the cause of Neanderthal demise?, by Anna Degioanni , Christophe Bonenfant , Sandrine Cabut , Silvana Condemi , 2019, Plos One • The causes of disappearance of the Neanderthals, the only human population living in Europe before the arrival of Homo sapiens , have been debated for decades by the scientific community • The Neanderthals, a human metapopulation that lived between 250,000 and 40,000 yrs ago (OIS 7 – 3), but went extinct ~ 10,000 years after arrival of Homo sapiens circa 50,000 years ago in Europe. • Primary hypotheses about their extinction are that a) Homo sapiens had superior skills etc , and outcompeted them or b) we outfought them c) climatic change in Europe + competition from Homo sapiens • In this paper, they investigated demographic models to test whether the disappearance of Neanderthals over a period of 10,000 years ( yrs ) could have led extinction. Fig 1. Spatial distribution and location of the 3 Neanderthal subpopulations. Degioanni A, Bonenfant C, Cabut S, Condemi S (2019) Living on the edge: Was demographic weakness the cause of Neanderthal demise?. PLOS ONE 14(5): e0216742.
https://doi.org/10.1371/journal.pone.0216742 https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0 216742 Three subpopulations:
A, southern Europe – Green B-northern Europe – yellow C. Eastern Europe – purple Include migration parameter ψ that allowed for migration between sub -populations Used demographic, age -structured models. Differential survival among age - classes:
i-Low survival rates as juveniles ii-High survival of prime -aged individuals iii -Decreasing survival with age Defined 5 age classes, <1 year 2-15 childhood 16 -18 sub -adults – only age that could migrate 19 -29 – prime adult >30 old Table 1. Demographic parameters entered in the stochastic Leslie matrix (mean and standard errors) to project population size of Neanderthals according to different scenarii of Neanderthal time of extinction in Western Europe. By changing just the fertility rate, slightly from 0.1376 to 0.1300 the population can go extinct in just 4,000 years. Fig 2. Simulated population trajectories of the Neanderthals over 10,000 yrs. Population Size Time Time MVP Eastern Europe Southern Europe Northern Europe Survival Demise in 10,000 years Demise in 6,000 years Demise in 4,000 years Table 2. Extinction probability and average time of extinction for the overall Neanderthal population and for each of the 3 subpopulations. Fig 3. Simulated trajectories for the Neanderthals overall population and for the 3 sub -populations with reduced survival. Parameters used in the simulation are shown in Table 1 “Survival”, reducing young infants survival by 0.4% (3A) or reducing adult survival by 10% (3B) Their conclusion:
“Our results lead us to the conclusion that the size of the Neanderthal population could have slowly and gradually decreased over time and that when it was already small and began to decline, Homo sapiens may well have simply taken advantage of an already low density of Neanderthals in order to settle into Europe.. Our model can make possible to better understand Neanderthal demise at the level of the entire territory and to identify the role of each demographic parameters in this process.” Note: Different parts of the genome have different effective population sizes • A Wright -Fisher population is a simple idealized population used in population genetics (more than the HW one). It has constant population size, equal sex ration, non -overlapping generations, random mating and all individuals have the same probability of having offspring. Unlike the HW population, it has a FINITE size and MUTATION is allowed (next section). What happens if we introduce mutation?
• The previous models, did not include mutation, but Wright -Fisher model can. • Mutations are the primary source of novel genetic variation in populations • Infinite alleles model : Let’s assume that every mutation results in a NEW allele, no back mutations (reversion to prior state), no recurrent mutations (same mutation occurring independently, in different lineages). Then, the frequency of any given allele will decrease over time because of mutation. Realistic for SNPs in humans. • Then, if we if we know the average mutation rate for the whole genome (say just for SNPs), assuming no back mutation, then the frequency of allele at the beginning (t=0) , p0, and the mutation rate = μ , then the frequency of allele p at time t (see page 137 text). • p t = p 0 e -ut • * If you plug in some values you will see that it changes very slowly • Equilibrium between mutation and drift • Genetic drift erodes genetic diversity, but mutation introduces it. Thus, we expect that the amount of genetic diversity in a population as measured by H, the mean level of heterozygosity, would reach an equilibrium point given just the reduction in diversity due to population size (N) and the increase in diversity due to mutation (μ) • Under the infinite alleles models this equilibrium point is: = 4 μ 4 μ + 1 • Where is the heterozygosity at equilibrium, N = effective population size and μ is the mutation rate. • There is no selection in this model, so it is sometimes referred to as the expected heterozygosity under neutral evolution The equilibrium heterozygosity for neutral alleles, quickly maximizes if when N or μ high enough. ϴ = 4N eμ ϴ is a very important parameter in population genetics because it tells you how much neutral variation you expect to find in a population What is the rate of neutral evolution?
• If we have a population with effective size N and μ is the neutral mutation rate (mutations are neither advantageous nor disadvantageous). Then, • The number of alleles in the population = 2N • Number of new mutations = 2N μ • Probability that a new mutation will reach fixation = 1/2N • Rate of fixation of new mutations = rate at which new mutations arise * probability of fixation = 2N μ * 1/2N = μ THUS, the rate of neutral evolution is equal to the neutral mutation rate!!! Rate of fixation of new mutations (i.e.
new alleles) when the alleles are neutral is dependent only on mutation rate and population size Figure 5.6 your text book Y axis is population size Population A bigger than b and c X axis shows differences in mutation rate B has a higher rate of mutation than c Blue lines – alleles that become fixed Gray lines – alleles that get lost The average time to fixation t in generations for a given mutation rate is: t = 4N e Evolutionary Force: Migration • Up until now, we have assumed that we are talking about a single population. What if we have more than one population? • What if there is migration between them? In the above figure, new alleles are introduced from the source population to the recipient population Even low rates of migration “quickly” equalizes allele frequencies between populations. m = proportion of population that are “immigrants” Only when m is very low, do populations retain different allele frequencies In reality population structures are of course complex. Diverse methods exist to estimate the migration rate between populations using molecular data.
The model to the right is called the n - island model, where the same rate of migration between all populations is assumed, or the stepping stone model, where there are equal migration rates between neighbouring populations Measuring population structure under the island model Sewall Wright developed a series of statistics to measure the degree of isolation between populations, called the F - statistics, or Wright’s F -statistics. The provide a simple measure of how “distinct” or separate populations have become due to the effect of drift, and a lack of migration.
Example, if a population of size 2000 becomes separated into 10 populations of size 200 (middle panel) at t=0, the allele frequencies will be ~ equal in all populations to start, but after 40 generations, the allele frequencies will have diverged.
If you POOLED all populations, and took the average allele frequency, p, at t=40 generations, it would still be ~ p=0.5, but the individual populations would have divergent allele frequencies ranging from say ~ p=0.18 to ~p=0.77. Measuring population structure: the equation • S = subpopulation • T = total population • I = individual • FST = H T – H S/H T • FST = the difference in mean heterozygosity within subpopulations compared to what you would expected if you pooled all individuals, treating them as the “total population” • Calculating H Syou calculated the oserved heterozygosity in each subpopulations and then take an average across subpopulations, • Calculating H T-you calculate the overall (pooling all populations) frequency of p, and then calculated the expected heterozygosity in the total population assuming there is no population subdivision. • Thus the Fst measures the relative proportion of the total diversity that is attributable to differences among subpopulations. • Typically 0< Fst <1, but like F, it can go slightly below 0 in unusual circumstances. • FIS measures the relative amount of inbreeding in subpopulations, this is a measure of nonrandom mating within subpopulations, that can occur due to selfing or mating with relatives. F ST values will depend on N e, the mutation rate and selection • In too small a population (1/4 N much greater than u and s ) there is nearly complete fixation, little variation, little effect of selection and thus a static condition modified occasionally by chance fixation of new mutations leading inevitably to degeneration and extinction” (Wright 1931 p. 157) [N = population size, u = mutation rate and s = selection co -efficient] Borrell et al., 2018, Heredity: Genetic diversity maintained among fragmented populations of a tree undergoing range contraction Isolation by Distance • : Pairwise population FST values are also expected to increase as the distance between the two populations being compared increases. Fig 1. A) Map of the Lough Feeagh and Furnace, Burrishoole Catchment, western Ireland. Ravinet M, Hynes R, Poole R, Cross TF, McGinnity P, et al. (2015) Where the Lake Meets the Sea: Strong Reproductive Isolation IsAssociated with Adaptive Divergence between Lake Resident and Anadromous Three -Spined Sticklebacks. PLOS ONE 10(4): e0122825. https://doi.org/10.1371/journal.pone.0122825 https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0122825 Fig 3. Lateral plate distributions in the Burrishoole system; A) Counts of individuals from each lateral plate morph and B) histograms of mean lateral plate number for all morphs in Lough Feeagh and Lough Furnace.
Ravinet M, Hynes R, Poole R, Cross TF, McGinnity P, et al. (2015) Where the Lake Meets the Sea: Strong Reproductive Isolation IsAssociated with Adaptive Divergence between Lake Resident and Anadromous Three -Spined Sticklebacks. PLOS ONE 10(4): e0122825. https://doi.org/10.1371/journal.pone.0122825 https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0122825 Fig 2. Morphological divergence amongst plate morphs in the Burrishoole catchment; A) divergence in geometric morphometric shape space, deformation grids represent shape change along PC1 at extremes of - 0.06 (left) and 0.06 (right); boxplots showing differences in lateral plate number (B) anti -predator traits (C), gill raker length (D) and gill raker number (E) between lateral plate location groupings. Ravinet M, Hynes R, Poole R, Cross TF, McGinnity P, et al. (2015) Where the Lake Meets the Sea: Strong Reproductive Isolation IsAssociated with Adaptive Divergence between Lake Resident and Anadromous Three -Spined Sticklebacks. PLOS ONE 10(4): e0122825. https://doi.org/10.1371/journal.pone.0122825 https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0122825 Fig 4. STRUCTURE and NewHybrids assignments for Burrishoole sticklebacks using all, neutral and QTL markers. Ravinet M, Hynes R, Poole R, Cross TF, McGinnity P, et al. (2015) Where the Lake Meets the Sea: Strong Reproductive Isolation IsAssociated with Adaptive Divergence between Lake Resident and Anadromous Three -Spined Sticklebacks. PLOS ONE 10(4): e0122825. https://doi.org/10.1371/journal.pone.0122825 https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0122825 What happens if we add selection? • Different models of selection, depending on which genotype is being selected against • See table 5.3 of your text • All depend on the strength of selection, s .