7 Allelic Frequencies, Genotypic Frequencies and Hardy Weinberg Law

Ms. Gangaina Kameih

LEARNING OBJECTIVES:

It aims to understand how genotypic and allelic frequencies affect the principles and reproduction in Mendelian population.
It aims to understand the segregation of alleles in gamete formation and combination of allele influencing the gene pool
It aims to understand the mathematical model of Hardy Weinberg equilibrium.
It aims to understand the factors affecting the allelic frequency.

TABLE OF CONTENTS–

1. Introduction

2. Genotypic and allelic frequency

3. Calculating allelic frequency

4. Hardy Weinberg equilibrium

5. Genotypic frequencies at hardy Weinberg Equilibrium

6. Implications of Hardy Weinberg Law

7. Extension of Hardy Weinberg Equilibrium

8. Validity for Hardy Weinberg Equilibrium

9. Assuming hardy–Weinberg to test alternative models of inheritance

10. Factors affecting the Allele Frequency

Introduction:

Population genetics is the field of biology that studies the changes in the genetic composition of biological population that result from the operation of various factors, including natural selection and the various processes shaping its gene pool. A population evolves through changes in its gene pool; therefore, population genetics is also the study of evolution. There is a quiet evident contribution of mendelian genetics, chromosomal abnormalities and inheritance pattern to the medical practice where the disorders are analysed by physicians working with patients and families. Genetic Polymorphisms are maintained at the species level so that individuals can survive, more inclination is seen towards the homozygotes as they allow species to survive. Before the medical technicalities, population lacking sickle cell anaemia trait could not survive in the malarial regions of West Africa. The incidence of some human diseases like sickle cell anaemia, cystic fibrosis, Tay-Sachs disease, thalasemmia are affected by environmental factors which cause change in the gene frequency in large population which in turn causes gives some advantage to the heterozygotes carrying deleterious allele. The goals of Population geneticists are pursued by formulating mathematical models of gene frequency dynamics, extracting conclusions to study genetic variation in actual populations, and testing the conclusions against empirical data. Just for instance asthma suffers were in a minority among 15 ancestors , eight males and seven females in an island named Tristan Da Cuna in S. Atlantic roughly half way between Cape town & Buenos Aries but now the current population is 297. Here 23% of people are asthmatic whereas 50% were partially asthmatic. This illustrated another problem of small populations and population experienced founder effect and genetic drift were the asthma suffers reproduced more offspring‘s than average. As a result where populations stayed small, even mildly deleterious genes lead to ―drift which lead to fixation of genes‖, here the normal, beneficial gene going extinct.

Genotypic and allelic frequencies

An obvious and persuasive feature of life is variability. Variability can seen in a group of students as no two students in the class are similar in a typical college class, the members of which vary in eye colour, hair colour, skin pigmentation, height, weight, facial features, blood type and susceptibility to numerous diseases and disorders. Humans are unique in their extensive variability and much of this phenotypic variation is hereditary. Recognition by this phenotypic variation and its extent to to potentially adapt to environmental change led Charles Darwin to deduce the idea of evolution through natural selection in Galapagos Island. In fact, even more genetic variation exists in population than is visible in the phenotype. Much variation exists at the molecular level owing, in part, to the redundancy of the genetic code, which allows different codons to specify the same amino acid. Thus two members of a population can produce the same protein even if their DNA sequence are different. An important, but frequently misunderstood, tool used in population genetics is the mathematical model where they are simplified representation of a process and effects between the factors are understood.

Calculating genotypic frequencies:

A frequency is defined as a proportion or a percentage, usually expressed as a decimal fraction. For example, if 20% of the alleles at a particular locus in a population are A, we would say that the frequency of the A allele in the population is 0.20. In a large population, a sample of frequencies can be calculated for this sample. The genotypic and allelic frequency of the sample are then used to represent the gene pool of the population. The genotypic frequencies are calculated by simply adding up the number of individuals possessing the genotype and dividing it by the total number of individuals in the sample (N). Thus, for the locus with three genotypes AA, Aa, aa, the frequency (f) of each genotype is

The sum of all the genotypic frequencies is always equals 1.

The gene pool of a population can also be described in terms of the allele frequencies. There are always fewer alleles than genotypes; so the gene pool of a population can be described in fewer terms when the allele frequencies are used. In sexually reproducing population, the genotypes are only temporary assemblages of the allele as genotypes break down in each generation when individual alleles are passed to the next generation through gametes and make up the gene pool.

Allelic frequencies thus can be calculated from (1) the numbers or (2) the frequencies of the genotypes. To calculate the allelic frequency from the number of genotypes, we count the number of copies of a particular allele present in a sample and divide by the total number of all alleles in the sample.

For a locus with only two alleles ( A and a), the frequencies of the alleles are usually represented by the symbols p and , and can be calculated as follows:

Where n_AA, n_Aa and n_aa represent the numbers of Aa, Aa . aa individuals , and N represents the total number of individuals in the sample. We divide by 2N because each diploid individual has two alleles at a locus. The sum of the allelic frequency from the genotypic frequencies, we add the frequency of the homozygote for each allele to half the frequency of the heterozygote‘s alleles are of each type

p= f(A)= f(AA)+1/2f(Aa)

p= f(a)= f(aa)+1/2f(Aa)

we obtain the same values of p and q whether we calculate the allelic frequencies from the numbers of genotypes or from the genotypic frequency.

Hardy Weinberg Equilibrium

The Hardy-Weinberg principle was discovered independently by both G.H. Hardy and W. Weinberg in 1908. It is one of the simplest and most important principles in population genetics. The law is a mathematical model that evaluates the effects of reproduction and Mendelian population principles on allelic and genotypic frequencies. This relationship is of basic importance to population genetics because it enables us to describe the genetic content in diploid populations in terms of allele not and not in terms of genotype frequencies. With recent documentation of loci with many alleles or genes or genetic regions with haplotype, this principle has become very important.

The rule has three aspects.

I. The allelic frequencies at autosomal loci in equilibrium in a population will not change from one generation to the next.

II. The genotypic frequencies of the population are predicted by the allelic frequency.

III. The equilibrium is neutral and will be re-established within one generation of random mating at the new allelic frequencies.

The Hardy Weinberg Equilibrium makes several simplifying assumptions about the population and provides key predictions if these assumptions are met.

Assumptions: If the population is large, random mating, and not affected by mutation, natural selection, then:

Predictions: The genotypic frequencies stabilize after one generation in the proportions p2 (the frequency of AA) , 2pq ( the frequency of Aa), q2 ( the frequency of aa), where p equals the frequency of allele A and q equals the frequency of allele a.

A large population of sexually reproducing organisms is considered where the organisms are assumed to be diploids (two copies of each chromosome, one received from each parent). The gametes produced by them are haploid (only one of each chromosome pair). Two haploid gametes fuse to form a diploid zygote during sexual fusion and then it grows and develops into an adult organism. For an autosomal locus with two alleles there are two possible alleles, A1 and A2.Organisms with the A1A1 and A2A2 genotypes are called homozygote; those with the A1A2 genotype are heterozygote. The relative frequencies, of the three genotypes in the overall population may be denoted as f(A1A1), f(A1A2) and f(A2A2) respectively, where f(A1A1) + f(A1A2) + f(A2A2) = 1 and are same for both males and females. The relative frequencies of the A and B alleles in the population may be denoted p and q, where p + q = 1. For a locus with five alleles, there is threshold between five allele frequencies and 15 genotypic frequencies. This simplification is particularly useful for a locus with two alleles because it allows us to follow changes in the frequency of one allele instead of the frequency of two genotypes.

An important graphical tool to depict genotype and allele frequencies simultaneously for a single locus with two alleles is the De Finetti. These diagrams are helpful when we examine how population genetic processes dictate allele and genotype frequencies. In both the graph it is apparent that heterozygotes are most frequent when frequency of the two alleles is equal to 0.5.It can be easily deduced from the diagram that when an allele is rare, the corresponding homozygote genotype is even rarer since the genotype frequency is the square of the allele frequency.

Figure: Hardy-weinberg expected genotype frequencies for AA,Aa, aa genotypes( y axis) for any given value of the allele frequency ( x axis). Note that the value of the allele frequency not graphed can be determined by q= 1-p.

Source:http://www.nature.com/scitable/content/ne0000/ne0000/ne0000/ne0000/1333861 5/andrews_figure4_ksm.jpg

Random mating means the absence of a genotypic correlation between mating partners, i.e. the probability that a given organism mates with an A1A1 partner, for example, does not depend on the organism’s own genotype, and similarly for the probability of mating with a partner of one of the other two types.

The Hardy Weinberg law indicates that, when the assumptions are met, reproduction alone does not alter allelic or genotypic frequencies and the allele frequencies determine the frequencies of genotypes.

Genotypic frequencies at hardy Weinberg equilibrium

That random mating will lead the genotypes to be in the above proportions (so-called Hardy-Weinberg proportions) is a consequence of Mendel’s law of segregation. To see this, note that random mating is in effect equivalent to offspring being formed by randomly picking pairs of gametes from a large ‗gamete pool‘ and fusing them into a zygote. The gamete pool contains all the successful gametes of the parent organisms. Since we are assuming the absence of selection, all parents contribute equal numbers of gametes to the pool. By the law of segregation, an A1A2heterozygote produces gametes bearing the A1 and A2 alleles in equal proportion. Therefore, the relative frequencies of the A and B alleles in the gamete pool will be the same as in the parental population, namely p and q respectively.

The gamete pool is very large, when we pick pairs of gametes from the pool at random, we will get the ordered genotypic pairs {A1A1}, {A1A2}, {A2A1}, {A2A2} in the proportions p2:pq:qp:q2. But order does not matter, so we can regard the {A1A2} and {A2A1} pairs as equivalent, giving the Hardy-Weinberg proportions for the unordered offspring genotypes. Importantly, whatever the initial genotypic proportions, random mating will automatically produce offspring in Hardy-Weinberg proportions (for one-locus genotypes). So if generations are non-overlapping, i.e. parents die as soon as they have reproduced, just one round of random mating is needed to bring about Hardy-Weinberg proportions in the whole population; if generations overlap, more than one round of random mating is needed. Once Hardy-Weinberg proportions have been achieved, they will be maintained in subsequent generations so long as the population continues to mate at random and is unaffected by evolutionary forces such as selection, mutation etc. The population is then said to be in Hardy-Weinberg equilibrium—meaning that the genotypic proportions are constant from generation to generation

The relationship well know for Hardy Weinberg equation is p2+2pq+q2=1, where p and q are allele frequencies for a genetic locus with two alleles. In most organisms, random union of gametes is quite unlikely because it is the parental genotypes that pair and then produce gametes that unite. Therefore, let us consider the situation in which reproductive individuals randomly pair. If we consider a diploid organism , there are three possible genotypes in the population- A1A1, A1A2, A2A2-that are present present in frequencies in the P,H and Q respectively(P+H+Q=1). Because all of the alleles in the two homozygotes A1A1 and A2A2 are A1 and A2 respectively, and half the alleles in the heterozygote are A1 and half are A2, the allele frequencies in terms of the genotype frequencies are then:

p= P+1/2H

q= Q+1/2H.

Let us assume that there is random mating in the population, which yields nine possible combinations of mating between the male and female genotype as given in table below.

TABLE1-The frequency of different mating types for two alleles at an autosomal locus when there is random mating.

These six mating types, their frequencies and the expected frequencies of their offspring genotypes assuming that gametes segregate in Mendelian proportions. For example, the mating A1A1× A1A1 produces 1/2A1A1 and 1/2A1A2 and so on.

TABLE 2-Demonstration of the Hardy Weinberg Principle Assuming Random Mating In the Parents and Mendelian Segregation To Produce The Progeny

Hardy Weinberg law holds true for any frequencies of A and a, as long as the frequencies add to 1 and all the assumptions are evoked. A population in which the allele frequencies remain constant from generation to generation and in which genotype frequencies can be predicted from the allele frequencies can be predicted from the allele frequencies is said to be in a state of Hardy Weinberg equilibrium for that locus. Genotype proportions may deviate from Hardy Weinberg expectations for several different reasons. The most significant evolutionary factors are selection, inbreeding and gene flow and thus it is often said that hardy Weinberg proportions are expected only in situations in which there is no selection, random mating, and gene flow.

Implications of Hardy Weinberg law

When considering the genetic structure of a population, the populations maintaining the Hardy Weinberg Equilibrium has several implications. A significant implication of the Hardy Weinberg relationship is that the frequency of the dominant and recessive alleles will remain unchanged from one generation to the next under given certain conditions.

IMPLICATION 1: a population cannot evolve if it meets the hardy Weinberg assumptions, as evolution consists of changes in the allelic frequencies of a population. Therefore it can be concluded that reproduction alone cannot alone bring evolution. Other evolutionary processes like natural selection, migration, mutation are required for populations to evolve.

IMPLICATION 2: the genotypic frequencies are determined by the allelic frequencies for the populations in hardy Weinberg equilibrium. Considering a locus with two alleles, the frequency greatest for the heterozygote is when the allelic frequencies are between 0.33 and 0.66 and is maximum when the allelic frequencies are 0.5. when the frequency of one allele is low, homozygotes for that allele will be rare and will be present in heterozygote

IMPLICATION 3: In a single generation of random mating produces equilibrium frequencies of p2,2pq and q2 and this fact doesn‘t proves that the population is free from natural selection‘

Extension of Hardy Weinberg law

The Hardy Weinberg principle is also applied to X- linked genes and to genes with multiple alleles. For an X-linked gene such as the one that controls colour vision, the allele frequencies are estimated from the frequencies of the genotype is males, and the frequencies of the genotypes in females are obtained by hardy Weinberg principle to these estimated allele frequencies. In one generation, the genotypic frequencies are at equilibrium when random mating is occurring. If the alleles are X linked and sexes differ in allele frequency, the equilibrium frequencies are approached over several generations as males receive their X chromosome from their mother only, whereas female receive an X chromosome from both mother and father. For genes with multiple alleles, the Hardy Weinberg genotype proportions are obtained by expanding a multinominal expression. To calculate the allelic frequencies from the number of genotypes, we count up the number of copies of an allele by adding twice the number of homozygotes to the number of heterozygotes that possess the allele and divide this sum by twice the number of individuals in the sample.

Validity for Hardy Weinberg Equilibrium:

It is possible to establish whether a population is in Hardy Weinberg equilibrium for a particular trait. Consider a system with two alleles A and a, with three resulting genotypes AA, Aa/aA, aa. Amongst 1000 selected at random, the following genotype distributions are observed.

AA 800

Aa/aA185

Aa 15 From this data,

The frequency of the allele A (p) = [(2×800)+185]/2000=0.8925 The frequency of the allele a (q) = [185+(2×15]/2000=0.1075

Now consider what the expected genotype frequencies would be if the population is in Hardy-Weinberg equilibrium and compare these with observed values.

With observed and expected genotypic frequencies we can compute statistical analysis with a ᵡ2 test to confirm and determine probability that the difference between observed and expected value is matter of chance. The chi-square is computed by the formula

Thus p˃0.05,(p=3.84) the deviation of expected from the observed is not statistically significant.

We can always find allele frequency from Hardy Weinberg law, if we assume that the genotypes in the population are found in Hardy Weinberg equilibrium.

Assuming Hardy–Weinberg to test alternative models of inheritance

Landsteiner observed the presence of four blood phenotypes A, B, AB, and O and it led to the anticipation of a logical question was, ―What is the genetic basis of these four blood group phenotypes?‖ We will test two hypothesis (or models) to explain the inheritance of ABO blood groups. The approach to test the hypothesis uses the frequency of genotypes in a sample population. The two hypotheses are as follows:

1) The four blood group phenotypes are explained by either two independent loci with two alleles each with one allele completely dominant at each locus or a single locus with three alleles where two of the alleles show no dominance with each other but both are completely dominant over a third allele.

2) Hypothesis 2 requires A and B to have no dominance with each other but complete dominance when paired with the O allele.

The O blood group under hypothesis 1 is the frequency of a homozygous genotype at two loci (aa bb). The frequency of one homozygote is the square of the allele frequency: fa2 and fb2 if we use fx to indicate the frequency of allele x. The genotype A_ means AA or Aa: in other words, any genotype but aa. Since the frequencies of the three genotypes at one locus must sum to one, we can write fA_ as 1 − faa or 1 − fa2. Then the frequency of the A_ bb genotype is (1 − fa2)fb2

Table 3: expected genotype frequencies for the abo blood groups under the hypotheses.

Table 4: CALCULATION FOR THE EXPECTED NO. OF EACH GENOTYPES UNDER BOTH THE HYPOTHESES.

Chi-square analysis can be obtained on the basis of difference between the observed and expected genotype. For hypothesis 1 chi-square is 43.52, whereas for hypothesis 2 chi- square =1.60. Clearly the chi-square depicts that the hypothesis of three alleles at one locus is a better hypothesis than two alleles each on two loci .Thus genotypic frequency sampled from a population was used to distinguish between two hypotheses for the genetic basis of blood groups.

Factors affecting allele frequencies.

The Hardy Weinberg law indicates that there is no change in the allelic frequency of a population until and unless evolutionary forces don‘t act on it. For an evolving population genetic variation must exists within and between population with the help of processes like mutation, migration, natural selection and genetic drift.

1. Mutation:

Mutation is one of the processes that generate genetic variation as new combination of existing alleles arises through recombination in meiosis. Mutation can influence the rate at which one genetic variant increases at the expense of another. Consider a locus with 25 diploid individuals, so the gene pool of a population consists of 50 allelic copies. Let us assume two different alleles designated G1 and G2 with frequencies p and q respectively.

As we can notice mutation has changed the allele frequency by increasing the frequency of G2 from 0.10 to 0.12. If the copies of G1 continue to mutate to G2 , a time will come when the frequency of G1 decreases and G2 increases. The change in the G2 as a result of mutation equals the mutation rate times the allelic frequency:

Δp= Δq

Only the effects of G1 G2 is called forward mutation and reverse mutation G2 G1. The rate of forward as well as reverse mutation will be equal to u and v respectively. Whenever a reverse mutation occurs , the frequency of G2 decreases and frequency of G1 increases. The overall change in allelic frequency is a balance between the opposing forces of forward mutation and reverse mutation.

2. Migration

Migration is the influx of the genes from other populations and is one of the assumptions of Hardy Weinberg law to take place which demands no gene pool. But for a natural population the overall effect of migration can be observed as it prevents the genetic divergence between populations and increases the genetic variance between populations. Let us consider a unidirectional model of migration between two populations that differed in the frequency of allele a. The frequency of allele in population 1 is assumed to be q1 and in population II is q11. In each generation a representative sample migrates from population I to population II and reproduces by adding its gene. After migration, population II consists of original residents and migrants. If the migrants make up proportion m of

population II, then the residents make up 1-m., because the residents originated in population II, the frequency of allele a in this group is q11. After migration, the frequency of allele a in the merged population II (q‘11) is

(q‘11)= q1(m)+q11(1-m)

Where q1(m) is the contribution to q made by the copies of allele a in the migrants and q11(1-m) is the contribution to q made by copies of allele a in the residents. The change in the allele frequency is equal to new allele frequency of allele a (q‘11) minus the original frequency of the allele (q 11)

Δq11= q‘11-q11

The overall effect of migration, causes the gene pools of population to become more similar. For an instance when q1-q11=0, there will be no further change in the allelic frequency of population II inspite of the fact that migration continues.

3. Genetic drift

The Hardy Weinberg law assumes random mating in an infinitely large population, but no real population is large and when the population size is limited, the gametes that unite in the second generation carry some traces of alleles of the parental generation. Sampling error occurs when gametes unite to produce progeny. Many organisms produce a large number of gametes to produce individual of the next generation. Chance influences which alleles are present in this limited sample and further sampling error might lead to changes in the allele frequencies. The deviations and directions of change are random and unpredictable. The amount of change resulting from genetic drift is determined by two parameters of allele frequency p and q and the population size N. In a large number of segregated populations , each with N individual and allelic frequencies p and q. After one generation of random mating , the genetic drift expressed in terms of variance on allelic frequencies among population is sP2= pq/2N.

Sex ratio also influences the gene pool as half the genes come from males and the other half comes from females. When one sex is present in low number, genetic drift increases because half the genes is coming from small number of individuals. In population of 100 individuals, there are 10 males and 90 females, but only 10 males will be able to contribute half the genes to the next generation. Other factors like fluctuations in population size and age structure of the population leads to genetic drift. As a result of genetic drift, allelic frequencies in different populations diverged and became fixed for one allele.

4. Natural selection

A final process that brings about changes in allelic frequencies is natural selection, the differential selection takes place when individuals with adaptive traits produce more offspring, adaptive traits have a genetic basis as the characters. A trait that provides a reproductive advantage and increases overtime, enabling the population to become better suited to their environment. The effect of natural selection on the gene pool of a population depends on the fitness value of the genotype of a population. Natural selection changes allelic frequencies; the direction and magnitude which changes the selection intensity. There six forms of natural selection which involves selection against the recessive allele, selection against the dominant allele, selection against the heterozygote, selection against the incompletely dominant allele, overdominance and underdominance. The result of selection depends on the relative fitness of the genotypes. For instance the three genotypes are A1A1 , A1A2 and A 2A2.and their relative fitness of these genotypes is W11 , W 12 and W22.the various types with their fitness relation is illustrated in Table 5.

TABLE 5 Types of natural Selection

Thus the above discussed evolutionary force affects the real population by influencing the genetic divergence between the populations .Natural selection increases the divergence between the populations if different alleles are favoured in different direction and by favouring the same allele it decreases the divergence. Mutation being a rare phenomenon always increases divergence between populations. Migration and Genetic drift act in opposite directions, migration increases genetic variation within populations, whereas genetic drift decreases genetic drift within populations. Mutation increases both variance and divergence within the population. Natural selection tends to decrease the allelic frequency eventually producing equilibrium.

Summary

The gene pool of a population can be described in terms of allele frequency and genetic frequency. In a sexually reproducing population are only temporary assemblages of the allele, the genotypes break down each generation when the alleles are passed from one generation to another through gametes in order to maintain continuity from one generation to another.
The primary goal of population genetics is to understand the various processes that shape gene pool i.e. collective group of alleles of all individuals in a population and the hardy Weinberg Law evaluates the effect of reproduction on allelic and genotypic frequencies.
The Hardy Weinberg law requires a population which is infinitely large, non random mating and is not affected by natural selection, migration where the chance deviations from expected genotypes do not hamper the allelic frequencies significantly.
The Hardy Weinberg law can be used to estimate the allelic frequencies if the population is in hardy Weinberg equilibrium for that locus. The frequency of the recessive allele will be equal to the square root of the frequency of the recessive trait.
Inbreeding increases the percentage of homozygous individuals homozygous in a population but close inbreeding is harmful because it increases homozygosity and boosts the probability of deleterious gene. The lethal recessive gene will continue to combine and produce harmful homozygous lethal gene and decreases the. Although inbreeding is generally harmful, but a number of inbreeding organisms are successful.
The amount of change in allelic frequency due to mitigation between populations depends on the difference in the allelic frequency and extent of migration. Migration has two effects in the first effect it causes the gene pool to look exactly similar and in the second effect it shows how natural selection and genetic drift leads to genetic difference between the populations.
The direction and magnitude of change in the allelic frequency is dependent on the selection intensity and direction of selection related to the dominance of alleles. Directional selection leads to favouring of one allele over another. Overdominance leads to maintainence of both the alleles and in under- dominance the heterozygote has lowest fitness compared to homozygotes.
It is important to keep in mind that the real populations are effected by evolutionary forces and thus evolution results from a combination of complex forces. The micro-evolutionary forces increasing the genetic variation within and between populations is mutation and migration and the genetic variation within and between the population is decreased by genetic drift and natural selection.

you can view video on Allelic Frequencies, Genotypic Frequencies and Hardy Weinberg Law

Suggested readings

Futuyma, D. J. (2009). Evolution. Sunderland: Sinauer Associates Inc.
Hamilton, M. B. (2009). Population Genetics. Chischester: Wiley- blackwell.
Pierce, B. A. (2013). Genetics: A Conceptual Approach. W.H.Freeman.
Relethford, J. H. (2012). Human Population Genetics. new york: Willey Blackwell.
Russell, p. J. (1997). genetics. Benjamin-Cummings Publishing Company; 5 Sub edition.
Sforza, W. &. (1975). Genetics Evolution and Man. W H Freeman & Co
Strickberger, M. W. (2006). Genetics. South Asia: Pearson Prentice Hall.