Population Genetics 1: Introduction and Hardy-Weinberg equilibrium

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1 Population Genetics 1: Introduction and Hardy-Weinberg equilibrium Population genetics: discipline devoted to the study of the genetic basis of microevolution Population genetics: the practice of using the information obtained from a sample of a natural population to make inferences about the evolutionary processes affecting that population (i) to measure the extent of genetic variation in natural populations (ii) to explain natural genetic variation in terms of its origin, maintenance and evolutionary significance 1

2 Population genetics is based on statistical models: Parametric inference: use of models to test hypotheses about the evolutionary processes that generated the sample of the data and to estimate values of model parameters. Some important questions: What is the relationship between genetic variation and a disease phenotype What can the gene sequences tell us about the evolutionary history of a species; e.g., humans What has been the role of natural selection during the evolution of a species. Population genetics is based on statistical models: A model is an intentional simplification of a complex situation designed to eliminate extraneous detail in order to focus attention on the essentials of the situation (Daniel L. Hartl).

3 Statistical modeling and inference: Concerns: Define a model Rules / parameters / quantities Explore properties Summary stats / graphical data exploration / simulation Estimate model parameters from the data Moments / maximum likelihood / Bayesian methods Test goodness of fit Compare estimators / heterogeneity / outliers Refine Model Update parameters Forces of evolution Natural populations Sample Mutation Stochastic evolutionary process Stochastic sampling process Migration Recombination Selection ACTTAGGACTTATAA ACAAAGGACTTATAA ACTTAGCACTTATAA ACTTAGGACAAATAA ACCCAGGACTTATAA Genetic drift Inference 3

4 Population: a subgroup of individuals of the same species living within some set of restrictions, usually in a restricted geographic area. also called a local population or deme. the practical implementation of the definition will vary among researchers the evolving unit of the species the unit within which the evolution of adaptive characteristics occurs In this section: Evolution: the change in the genetic constitution of a population over time. More simply, the change in allele frequencies in a population over time. [microevolution] In all sections of this course: Evolution Natural selection [alone] Allele frequencies in populations Eskimo MN blood group data: Genotypes MM MN NN Total (n) count Frequency of M = p p + q = 1 Frequency of N = q ( MM) 1( MN) + p = q = 1 p n ( 33) + ( 385) p = = 0.57 q = =

5 Allele frequencies in populations p and q are parameters of a population p and q were estimated p and q have error Var ( p) = ( 1 p) p n Assuming that repeated estimates would be normally distributed we can use the variance to predict how close our estimate of p is to the population value. AA = 8 / Aa = 6 / aa = p = var(p) = StdDev = (x = 0.164) 95%CI = (0.53, 0.851) width = AA = 80 / Aa = 60 / aa = 0 p = var(p) = StdDev = (x = 0.05) 95%CI = (0.636, 0.739) width = Allele frequencies differ among populations Genotypes Total (n) MM MN NN Iceland Greenland Evolution has occurred! What is the origin and evolutionary significance of such change? 5

6 Allele frequencies differ among populations What are the possible causes for microevolution: 1. Finite population size. Mutation 3. Non-random mating 4. Natural selection 5. Migration / gene flow Our null model: Nothing interesting ever happens in the population (or Hardy-Weinberg equilibrium; G.H. Hardy and W. Weinberg, 1908) Assumption of the HW model 1. The organism is diploid. Reproduction is sexual 3. Mating is random 4. Generations are discrete 5. Population size is infinite (or very large) 6. No migration 7. No mutation 8. No natural selection Idealized population 6

7 Hardy-Weinberg equilibrium M or N and M or N (p + q) x (p + q) (p + q) x (p + q) = 1 p + pq + q = 1 So, under HW conditions, the frequency of the blood group genotypes in the next generation are: f MM = p f MN = pq f NN = q Same thing, but by using the traditional cross-multiplication table Male gametes Female gametes M (p) AA (p ) M(p) N(q) Aa (pq) N (q) Aa (pq) aa (q ) Note 1: here we are mixing gametes at random among all members of the population! (Not, as in transmission genetics, mixing gametes of just two parents at random Note : these are the expected frequencies of alleles at the same locus when they are randomly associated with each other. 7

8 Hardy-Weinberg equilibrium Keynotes of the HW model: HW model specifies the relationship between allele frequencies and gene frequencies Natural populations can be tested for HW Mendelian inheritance means that frequencies do not change unless some external pressure is acting. No matter what the initial frequencies, just one generation of random mating will result in HW frequencies. Note: HW is not very sensitive to certain kinds of violations power issue Rare recessive alleles can hide in the heterozygotes q pq : q : : :1 Cystic Fibrosis (CF) example: CF: about 1 in 1700 newborn Caucasians ASSUMING HW: q = 1/1700 q = (1/1700) 1/ = 0.04 pq = x 0.04 x (1-0.04) = Note we assumed HW without testing the assumption. Clearly it is subject to natural selection. But, with just one generation of random mating we see 1 in 1 individuals are carriers, although only 1 in 1700 exhibit the disease. 8

9 A proof of the HW model Genotypes MM MN NN Genotype frequencies P 1 P P 3 Present generation P 1 P P 3 Next generation P 1 P P 3 If in HW equilibrium: P 1 = P 1 = p P = P = pq P 3 = P 3 = q A reminder of allele frequencies in populations Eskimo MN blood group data: Genotypes MM MN NN Total (n) count ( MM) 1( MN) + p = n ( MM) p = n 1(MN) + n q = 1 p p = P1 + ( P 1 ) 9

10 A proof of the HW model Genotype frequencies of offspring Mating Frequency MM MN NN MM MM P MM MN P1 P 1/ 1/ 0 MM NN P1 P MN MN P 1/4 1/ 1/4 MN NN P P3 0 1/ 1/ NN NN P Total in next generation: P1 P P3 Note: p = (P1) + (P 1/) and q = (P3) + (P 1/) ( If HWE ) ' P P1 = P1 + P1 P + ( 1/ 4) P = P 1 + = p P 1 = P 1 ' P P P = P1 P + P1 P P ' 3 ( 1/) P + P P = P + P + pq = = ( 1/ ) P + P P + P = P + = q P P = P P 3 = P 3 Nice proof: HW in 1 generation; hence, no changes once in HW Testing for HW equilibrium 3 steps: 1. Compute observed genotype frequencies. Compute expected genotype frequencies 3. Test goodness of fit Let s use some real data as an example: the following data are for the MN blood genotypes in Pueblo Indians: MN blood types in Pueblo Indians Genotypes MM MN NN Observed counts Total = n =

11 Testing for HW equilibrium PART 1: Observed Genotype frequencies: MM = 83/140 = 0.59 MN = 46/140 = 0.33 NN = 11/140 = 0.08 Observed allele frequencies: M = p = (1/) = N = q = ( ) = 0.45 Do NOT compute the allele frequencies at this step by assuming HW (i.e., p 0.59). Anyone who does this will automatically get an F in the class! Testing for HW equilibrium PART : Expected genotype frequencies: p = (0.755) = 0.57 pq = x x 0.45 = 0.37 q = (0.45) = 0.06 Expected genotype counts: p x n = = 79.8 pq x n = = 51.8 q x n = = 8.4 Compare these counts to the observed counts in the table above. 11

12 Testing for HW equilibrium PART 3: ( observed - expected ) ( ) ( ) ( ) χ = = + + expected χ = χ =1.58 d.f. = [(number of categories tested) (non-independent categories) (calculate p from data)] = [3 1 1] = 1 P = 0.0; i.e., there is a 0% chance that we would have observed a test statistic this large (or larger) under HW. Testing for HW equilibrium Testing HW is NOT possible under dominance Example: Rh + phenotype (DD or Dd) in North America: Genotype Phenotype DD Rh + Dd Rh + dd Rh - Rh + = (DD or Dd) Rh - = 0.14 (dd) q = (0.14) 1/ = p = ( ) = 0.63 d.f. = (for two classes of data) 1 1 (for estimating q) = 0 1

13 HW equilibrium with three alleles Alleles: A 1 A and A 3 Frequencies: p 1 p and p 3 p 1 + p + p 3 = 1 Female gametes The traditional cross-multiplication square A1 (p1) A (p) A3 (p3) Male gametes A1 (p1) A (p) A3 (p3) A1A1 p1 A1A p1 p A1A3 p1 p3 A1A p1 p AA p AA3 p p3 A1A3 p1 p3 AA3 p p3 A3A3 p3 Genotype frequencies in the next generation A1A1: p1 A1A: p1 p A1A3: p1 p3 AA: p AA3: p p3 A3A3: p3 Statistical modeling and inference: Concerns: Define a model Rules / parameters / quantities Explore properties Summary stats / graphical data exploration / simulation Estimate model parameters from the data Moments / maximum likelihood / Bayesian methods Test goodness of fit Compare estimators / heterogeneity / outliers Refine Model Update parameters 13

14 HW model: no change in frequencies Alt model; change in frequencies (molecular evolution) Change in frequencies Agency Genotype Allele Notes Linkage no no Creates disequilibrium among loci Inbreeding yes no Acts on all loci in genome; results in loss of heterozygosity Assortative Mating yes no Only acts on the locus subject to assortment, and those loci linked to it Migration a yes yes Depends of migration rate and frequency differences between populations Mutation yes yes Very very very slow Natural Selection yes yes Acts on the locus subject to selection, and those loci linked to it Genetic Drift yes yes Acts on all loci in the genome; results in loss of heterozygosity and loss of alleles 14