Hardy Weinberg Equilibrium

1 Hardy Weinberg EquilibriumWilhem Weinberg (1862 1937)Gregor MendelG. H. Hardy (1877 -1947)(1822-1884)Carol E. Lee, University of Wisconsin-MadisonCopyright 2020; do not upload without permissionLectures 4-12: Mechanisms of Evolution (Microevolution) Hardy Weinberg Principle (MendelianInheritance) Genetic Drift Mutation Sex: Recombination and Random Mating Epigenetic inheritance Natural SelectionThese are mechanisms acting WITHIN populations, hence called population genetics EXCEPT for epigenetic modifications, which act on individuals in a Lamarckian mannerEvolution acts through changes in allele frequencyat each generationLeads to averagechange in characteristic of the populationRecall from Previous LecturesDarwin s ObservationHOWEVER, Darwin did not understand how genetic variation was passed on from generation to generationRecall from Lecture on History of Evolutionary ThoughtDarwin s ObservationGregor Mendel, Father of Modern Genetics Mendel presented a mechanism for how traits got passed on Individuals pass alleles on to their offspring intact (the idea of particulate (genes) inheritance )Gregor Mendel(1822-1884)

2 #synopsisGregor Mendel, Father of Modern Genetics #synopsis Law of Independent Assortment different pairs of alleles are passed to offspring independently of each otherMendel s Laws of inheritance Law of Segregationonly one allele passes from each parent on to an offspring In cross-pollinating plants with either yellow or green peas, Mendel found that the first generation (f1) always had yellow seeds (dominance). However, the following generation (f2) consistently had a 3:1 ratio of yellow to 29,000 pea plants, Mendel discovered the 1:3 ratio of phenotypes, due to dominant alleles Mendel uncovered the underlying mechanism, that there are dominant and recessive allelesMathematical description of mendelian inheritanceHardy- Weinberg PrincipleGodfrey H. Hardy (1877-1947)English MathematicianWilhemWeinberg(1862 1937)German obstetrician-gynecologistJeremy Irons playing GH Hardy in the film The Man Who Knew Infinity Testing for Hardy - Weinberg Equilibrium can be used to assess whether a population is evolvingThe Hardy - Weinberg Principle A population that is not evolving shows allele and genotypic frequencies that are in Hardy Weinberg Equilibrium If a population is not in Hardy - Weinberg Equilibrium , it can be concluded that the population is evolvingEvolutionary Mechanisms (will put population out of HW Equilibrium ): Genetic Drift Natural Selection Mutation Migration*Epigenetic modificationschange expression of alleles but not the frequency of alleles themselves, so they won t affect the actual inheritance of allelesHowever, if you count the phenotype frequencies,and not the genotype frequencies, you might see phenotypic frequencies out of HW Equilibrium due to epigenetic silencing of alleles.

3 (epigenetic modifications can change phenotype, not genotype)Requirements of HWEvolutionLarge population size Genetic driftRandom MatingInbreeding & otherNo MutationsMutationsNo Natural SelectionNatural SelectionNo Migration MigrationAn evolving population is one that violates Hardy - Weinberg AssumptionsViolationFig. 23-5aPorcupineherd rangeBeaufort SeaNORTHWESTTERRITORIESMAPAREAALASKACANA DAF ortymileherd rangeALASKAYUKON What is a population? A group of individuals within a species that is capable of interbreeding and producing fertile offspring(definition for sexual species)Patterns of inheritance should always be in Hardy Weinberg Equilibrium Following the transmission rules of MendelIn the absence of Equilibrium According to the Hardy - Weinberg principle, frequencies of allelesand genotypesin a population remain constant from generation to generation Also, the genotype frequencies you see in a population should be the Hardy - Weinberg expectations, given the allele frequencies Null Model for No Evolution A population in Hardy - Weinberg Equilibrium serves as the Null Model (for no evolution) to test if evolution is happening Example: Is this population in Hardy Weinberg Equilibrium ?

4 AAAaaaGeneration TheoremIn a non-evolving population, frequency of alleles and genotypes remain constantover generationsYou should be able to predict the genotype frequencies, given the allele frequenciesimportant concepts gene:A region of genome sequence (DNA or RNA), that is the unit of inheritance , the product of which contributes to phenotype locus:Location in a genome (used interchangeably with gene, if the location is at a but, locus can be anywhere, so meaning is broader than gene) loci: Plural of locus allele:Variant forms of a gene ( alleles for different eye colors, BRCA1 breast cancer allele, etc.) genotype:The combination of alleles at a locus (gene) phenotype:The expression of a trait, as a result of the genotype and regulation of genes (green eyes, brown hair, body size, finger length, cystic fibrosis, etc.)important concepts allele:Variant forms of a gene ( alleles for different eye colors, BRCA1 breast cancer allele, etc.) We are diploid (2 chromosomes), so we have 2 alleles at a locus (any location in the genome) However, there can be many alleles at a locus in a population.

5 For example, you might have inherited a blue eye allele from your mom and a brown eye allele from your you can t have more alleles than that (only 2 chromosomes, one from each parent) BUT, there could be many alleles at this locus in the population, blue, green, grey, brown, etc. Alleles in a population of diploid organismsA1A2A3A4A1A1A2 SpermEggs GenotypesRandom Mating (Sex)ZygotesA1A3A1A1A1A1A2A4A3A1A1A1A1A2 A1A1A3A4So then can we predict the expected % of alleles and genotypes in the population at each generation?A1A2A3A4A1A1A2 SpermEggsZygotesA1A3A1A1A1A1A2A4A3A1A1A1 A1A2A1A1A3A4 Hardy - Weinberg TheoremIn a non-evolving population, frequency of alleles and genotypes remain constantover generationsFig. 23-6 Frequencies of allelesAlleles in the populationGametes producedEach egg:Each sperm:80%chance80%chance20%chance20%chan ceq= frequency ofp= frequency ofCRallele = = proportions indicate the expected allele and genotype frequencies, given the starting frequencies By convention, if there are 2 alleles at a locus, pand qare used to represent their frequencies The frequency of all alleles in a population will add up to 1 For example, p+ q= 1If pand qrepresent the relative frequencies of the only two possible alleles in a population at a particular locus, then for a diploid organism (2 chromosomes),(p+ q)2= 1= p2+ 2pq+ q2= 1 where p2and q2represent the frequencies of the homozygous genotypes and 2pqrepresents the frequency of the heterozygous genotypeWhat about for a triploid organism?

6 What about for a triploid organism? (p+ q)3= 1= p3+ 3p2q+ 3pq2 + q3 = 1 Potential offspring: ppp, ppq, pqp, qpp, qqp, pqq, qpq, qqqHow about tetraploid? You work it Weinberg TheoremALLELESP robability of A= pp+ q= 1 Probability of a= qGENOTYPESAA:pxp= p2Aa:pxq+ qxp= 2pqaa:qxq= q2p2+ 2pq+ q2= 1 More General HW Equations One locus three alleles: (p+ q+ r)2 = p2+ q2 + r2 + 2pq +2pr + 2qr One locus n# alleles: (p1+ p2+ p3+ p4 ..+ pn)2 = p12+ p22 + p32+ ..+ pn2+ 2p1p2+ 2p1p3+ 2p2p3+ 2p1p4+ 2p1p5+ .. + 2pn-1pn For a polyploid(more than two chromosomes): (p+ q)c, where c= number of chromosomes If multiple loci (genes) code for a trait, each locus follows the HW principle independently, and then the alleles at each loci contribute to and/or interact to influence the trait Calculating allele frequenciesWe can then define the relative frequencies of A1A1 and A1A2 genotypes as: Frequency of A1A1 = Number of A1A1 (N11)/Total Number N Frequency of A1A2 = Number of A1A2 (N12)/Total Number NFrequency of A2A2 = Number of A2A2 (N22)/Total Number NThen, the proportion of Allele A1:If there are two alleles, then the frequency of A2: 1 -pALLELE FrequenciesFrequency of A = p= of a = q= + q= 1 Expected GENOTYPE FrequenciesAA: pxp= p2 = = : pxq+ qxp= 2pq = 2 x( ) =.

7 Qxq= q2 = = + 2pq + q2= + + = 1 Expected Allele Frequencies at 2nd Generationp= AA + Aa/2 = + ( ) = aa+ Aa/2 = + ( ) = Allele frequencies remain the same at next generationHardy Weinberg TheoremALLELE FrequencyFrequency of A = p= + q= 1 Frequency of a = q= GENOTYPE FrequencyAA: pxp= p2 = = : pxq+ qxp= 2pq = 2 x( ) = : qxq= q2 = = + 2pq + q2= + + = 1 Expected Allele Frequency at 2nd Generationp= AA + Aa/2 = + ( ) = aa+ Aa/2 = + ( ) = Similar example,But with different starting allele frequenciespqp22pqq2 The frequency of an allele in a population can be calculated from # of individuals: For diploid organisms, the total number of alleles at a locus is the total number of individuals x 2 The total number of dominant alleles at a locus is 2 alleles for each homozygous dominant individual plus 1 allele for each heterozygous individual; the same logic applies for recessive allelesCalculating Allele Frequencies from # of IndividualsAAAaaa1206035 (# of individuals)#A = (2 x AA) + Aa = 240 + 60 = 300#a = (2 x aa) + Aa = 70 + 60 = 130 Proportion A = 300/total = 300/430 = a = 130/total = 130/430 = + a = + = 1 Proportion AA = 120/215 = Aa = 60/215 = aa = 35/215 = + Aa + aa = + + = 1 Calculating Allele and Genotype Frequencies from # of IndividualsApplying the Hardy - Weinberg Principle Example: estimate frequency of a disease allele in a population Phenylketonuria (PKU) is a metabolic disorder that results from homozygosity for a recessive allele Individuals that are homozygous for the deleterious recessive allele cannot break down phenylalanine, results in build up mental retardation The occurrence of PKU is 1 per 10,000 births How many carriers of this disease in the population?

8 Rare deleterious recessives often remain in a population because they are hidden in the heterozygous state (the carriers ) Natural selection can only act on the homozygous individuals where the phenotype is exposed (individuals who show symptoms of PKU) We can assume HW Equilibrium if: There is no migration from a population with different allele frequency Random mating No genetic drift Etc The occurrence of PKU is 1 per 10,000 births(frequency of the disease allele):q2= sqrt(q2 ) = sqrt( ) = The frequency of wildtype (normal) alleles is:p= 1 q= 1 = The frequency of carriers (heterozygotes) of the deleterious allele is:2pq= 2 = approximately 2% of the populationSo, let s calculate HW frequenciesConditions for Hardy - Weinberg Equilibrium The Hardy - Weinberg theorem describes a hypothetical population The five conditions for nonevolving populations are rarely met in nature: No mutations Random mating No natural selection Extremely large population size No gene flow So, in real populations, allele and genotype frequencies do change over timeDEVIATION from Hardy - Weinberg EquilibriumIndicates that EVOLUTIONIs happening In natural populations, some loci might be out of HW Equilibrium , while being in Hardy - Weinberg Equilibrium at other loci For example, some loci might be undergoing natural selection and become out of HW Equilibrium , while the rest of the genome remains in HW equilibriumHardy- Weinberg across a GenomeAllele A1 DemoHow can you tell whether a population is out of HW Equilibrium ?

9 Perform HW calculations to see if it looks like the population is out of HW Equilibrium Then apply statistical tests to see if the deviation is significantly different from what you would expect by random chanceExample: Does this population remain in Hardy Weinberg Equilibrium across Generations?AAAaaaGeneration In this case, allele frequencies (of A and a) did not change. **However, the population did go out of HW Equilibrium because you can no longer predict genotypic frequencies from allele frequencies For example, p= , p2= , but in Generation 3, the observe p2= can you tell whether a population is out of HW Equilibrium ? allele frequencies are changing across you cannot predict genotype frequencies from allele frequencies (means there is an excess or deficit of genotypes than what would be expected given the allele frequencies)Example: Does this population remain in Hardy Weinberg Equilibrium across Generations?AAAaaaGeneration for Deviatonfrom Hardy - Weinberg Expectations A c2goodness-of-fit testcan be used to determine if a population is significantly different from the expectations of Hardy - Weinberg Equilibrium .

10 If we have a series of genotype counts from a population, then we can compare these counts to the ones predicted by the Hardy - Weinberg model. O= observed counts, E= expected counts, sum across genotypes Example Genotype Count: AA30 Aa55 aa15 Calculate the c2 value:GenotypeObserved Expected(O-E)2/E AA30 33 Aa55 49 aa15 18 Total 100 100 Since c2= < (from Chi-square table, alpha = ), we conclude that the genotype frequencies in this population are not significantly different than what would be expected if the population is in Hardy - Weinberg for Deviatonfrom Hardy - Weinberg Expectations A c2goodness-of-fit testcan be used to determine if a population is significantly different from the expectations of Hardy - Weinberg Equilibrium . If we have a series of genotype counts from a population, then we can compare these counts to the ones predicted by the Hardy - Weinberg model. O= observed counts, E= expected counts, sum across genotypes 56 Testing for Deviatonfrom Hardy - Weinberg Expectations O= observed counts, E= expected counts, sum across genotypes We test our c2value against the Chi-square distribution (sum of square of a normal distribution), which represents the theoretical distribution of sample values under HW Equilibrium And determine how likely it is to get our result simply by chance ( due to sampling error); , do our Observed values differ from our Expected values more than what we would expect by chance(= significantly different)?