Learning Objectives:

The Hardy Weinberg Model
Factors that cause deviations from the Hardy Weinberg Model:

non random mating
finite population size
natural selection
mutation
migration/immigration
generation time

The effects of inbreeding

How to calculate the inbreeding coefficient, F.

Griffiths et al 603-607; 613-619; 631-633.

Populations of species (individuals that can successfully mate with each other) are often subdivided into partially isolated groups called DEMES or LOCAL POPULATIONS. Such subdivided populations are often called METAPOPULATIONS. Genetic changes within metapopulations occur as a function of  the effects of the changes in each individual deme (some populations are not metapopulations, and and the entire population constitutes one deme). A deme is a group of individuals that can (but don't necessarily) mate randomly. Random mating is generally prohibited by geographic barriers or simply distance. For example, parakeets in Jackson cannot mate with native parakeets in Australia because of geographic barriers.

Models describing the distribution of alleles in populations that include the entire metapopulation are too complex for this course. We will thus look only at the situation where the entire population is one deme.

The model used to describe the manner in which genes are distributed in sexually reproducing (diploid dominant) populations is the Hardy-Weinberg model. This model is an idealization that has the following assumptions:
 

1. Allele frequencies in populations do not change

no genetic drift
no selection
no migration
no immigration
infinite population size

2. Genotype frequencies are determined by allele frequencies

random mating

3. When allele frequencies are changed, the new equilibrium is established in one generation.

The number of alleles in a population is equal to 2X the number of individuals. The number of "A" alleles in a population is equal to 2X the number of AA individuals plus the number of Aa individuals.

The number of "a" alleles in a population is equal to 2X the number of aa individuals plus the number of Aa individuals.

The Hardy Weinberg model uses ALLELE FREQUENCIES.

The frequency of an allele is the number of alleles of a type divided by the total number of alleles. The frequency of "A" alleles in a population is the number of A alleles divided by the number of A + a alleles in a population, or:

The frequency of "a" is the number of a alleles divided by the number of A alleles plus the number of a alleles, or:

f(A) is generally referred to as "p" and f(a) is generally referred to as "q".

One of the two equations central to this model is:

Which basically states the obvious; all the alleles are either A or a.

We also need to look at genotypes. The frequencies are:

The derivation of the genotypes is simple. The chance of an individual having an "A" allele (getting it from a parent) is equal to its frequency (p). The chance of having 2 "A" alleles is the square of its frequency (p²). The same holds for the "a" allele.

For Aa, one has to consider the two ways of becoming Aa. An individual can get A from its father and a from its mother or a from its father and A from its mother. The chance of each is the frequency of A multiplied by the frequency of a, or f(A) X f(a) or p X q, and the chance of Aa becomes 2pq.

The other basic equation is:

which again states the obvious, all individuals are either AA, Aa or aa. You need to know these two equations and where they come from (what do the p's and q's mean).

The key thing to remember about this model is that the relationships between p, q and genotype and allele frequencies is fixed. If you know p or q or p² or q² you can use the equations to calculate any allele or genotype frequency, and if you know the population size, you can calculate numbers of alleles and genotypes.

Real populations do not conform to the assumptions of the Hardy Weinberg model, and thus the predicted allele and genotype frequencies do not actually occur, but are often close enough to use to predict genotype and allele frequencies in populations.

To understand the general use of the Hardy Weinberg model, we need to review thje limitations of the assumpitions of th Hardy Weinberg model.
 
 
 

Random Mating

The random mating assumption is necessary for us to assume that the probability of an allele occurring in a given individual is equal to the frequency of the allele.

Deviations from random mating are of two sorts POSITIVE ASSORTATIVE MATING and NEGATIVE ASSORTATIVE MATING. Positive assortative mating occurs when individuals with similar phenotypes mate more often that random mating would predict and negative assortative mating occurs when similar phenotypes mate less often than random mating would predict.

Genetic consequences of deviations from random mating are also fo two types. INBREEDING is when the frequency of mating among close relatives is greater than expected under random mating and OUTBREEDING is less mating of close relatives than expected under random mating.

The effects of inbreeding and positive assortative mating are usually the same (positive assortative mating often IS inbreeding). It increases the number of homozygotes (p² or q²), reduces the number of heterozygotes, (2pq), but not the overall allele frequencies (p or q).

Outbreeding and negative assortative mating usually cause the opposite; an increase in the number of heterozygotes and a decrease in the number of homozygotes, again without changing allele frequencies.

Infinite Population Size

The Hardy Weinberg model requires an infinite population size so as to eliminate SAMPLING ERROR. Sampling error causes allele frequencies to change in models, and allele frequency changes due to sampling error are called GENETIC DRIFT. It is generally assumed, and quite obvious, that sampling error must occur in natural populations.
 
 
 
 

Natural Selection

Selection (in the narrow genetic sense) occurs when an allele causes a phenotype to leave more or less offspring than average. If individuals with an allele leave more offspring than average, that allele will increase in frequency through generations. If individuals with an allele leave fewer offspring than average, that allele will decrease in frequency through generations. Selection predicts that alleles are lost due to selection, and one of the main consequences of selection to population genetics is that it causes a reduction in genetic variation. Genetic drift also can and does cause losses of genetic variation by chance. As a consequence of knowledge of the effects of selection and drift, geneticists once believed that there was very little genetic variation in wild populations. This lead to concepts like the "wild type" that we have used previously. In reality, there is usually little phenotypic variation in wild populations, but there is a great deal of molecular genetic variation, and when phenotypes of populations are followed through time, they can be seen to change dramatically in some cases, even though there is very little phenotypic variation within any one population at any one time. This has been a particular problem with insects, where variants can come and go and come back again very quickly.

Mutation and Migration

A mutation changes allele frequencies by eliminating a copy of one allele and replacing it with the mutant. Migration either adds alleles of immigrants or subtracts alleles of emigrants, in both cases changing allele frequencies.

Generation Time

The Hardy-Weinberg model assumes discrete generations; no overlapping of generations; offspring and parents do not co-exist; one generation does not mate with another (this is not "incest", but any individual of a parental generation mating with any individual of any successive generation). This is not true in natural populations. The time it takes to go from "seed to seed" is the basic definition of GENERATION TIME. You could look at it as the time it would take for a given zygote to develop to the point where it would contribute a gamete to a new zygote. In humans, for example, the generation time is about 20 years. Very short generation times cause the problems with the application of the Hardy Weinberg model to organisms such as insects; you get many generations in a short time, and thus lots of overlapping of generations and mating between differen generations. In organisms like humans, with long generation times, we still get lots of matings between different generations.

Regardless of the fact that in reality the assumptions of the Hardy-Weinberg model do not hold in natural populations, it does predict genotype frequencies quite well in many cases, and cases where it does not predict genotype frequencies well usually indicate an unusually strong effect of one or other of the factors that cause deviations from the Hardy-Weinberg model. For example, rapid losses of genetic variation are indicative of a strong effect of natural selection and high frequencies of homozygotes are indicative of inbreeding.

Here we have data for the MN blood group types for some people in Ohio. f(N) = q = 0.24 and f(M) = p = 0.76. From this we calculate p² (0.5776), 2pq (0.3648) and q² (0.0576).

From these values and the total population size (200) we can get the expected number of individuals that are MM (0.5576 X 200 = 115.32), MN (0.3648 X 200 = 72.96) and NN (0.0576 X 200) = 11.52.

In the population wer observed that 114 were MM, 76 were MN and 10 were NN.

If we test our expected values with a chi-squared test, we get a chi-squared value of 0.348 with 2 degrees of freedom (critical value for p<0.05 = 3.841). Thus the Hardy Weinberg model predicted genotype frequencies well.

The Hardy Weinberg model does not necessarily have to work. In another sampling, there are no MN individuals, but the allele frequencies are the same. 152 individuals are MM and 48 are NN. The expected (as we calculated above for the same allele frequencies) does not match the observed (chi-squared value = 200). What do you think would cause such deviation?

The Hardy-Weinberg model is often used to calculate genotype frequencies from frequencies of recessive phenotypes. The thing to be aware of here is that the frequency of recessive phenotypes is q². From this we get q, then p and then all the other parameters. The main parameter of interest is 2pq, the frequency of heterozygotes, or the frequency of carriers of deleterious recessive alleles. The example we will use is PKU, where 1/10,000 are born with PKU. 1/10,000 = 0.0001= q². If q² = 0.0001, q = 0.01. If q = 0.01 the p = 0.99 and 2pq = 0.02, or 2% of the population are carriers of the allele that causes PKU.

Inbreeding

Inbreeding is a major parameter in population genetics. It is generally considered important to NOT breed with close relatives because each individual contains a number of LETHAL EQUIVALENT ALLELES (alleles that would be lethal if homozygous). We inherit these alleles from our parents, and the lethal equivalent alleles that we have are far more common in our close family than in society at large. Thus we are far more lilely to produce an offspring homozygous for lethal equivalent alleles when we mate with close relatives than when we mate with people who are not close relatives.

Inbreeding leads to reduced fertility due to increased frequencies of lethal equivalent alleles in families (many ancient societies, and some current societies frowned on matings outside the tribal group, and thus there was a reduction in fertility in such groups, which is recorded in many ancient accounts..i.e. a man marries his sister and cannot produce offspring.)

An individual can mate with an individual that is not closely related and still share a lethal equivalent with that individual. In reality, however, such an event has a very low probability in comparison with the chance of sharing a lethal equivalent with a close relative that in the calculation of inbreeding parameters, only the chance of getting a lethal equivalent for a close relative is considered. The parameter used to describe inbreeding is the INBREEDING COEFFICIENT (F).

The inbreeding coefficient is simply the chance of sampling an allele identical by descent given an allele. For example, if you have an "a", what is the chance of sampling another "a" that is identical by descent to the "a" that you are considering? F is calculated as the reduction in HETEROZYGOSITY (H) is a population due to inbreeding. For a given gene with 2 alleles, the expected heterozygosity is 2pq. The actual heterozygosity (H) is observed. so the inbreeding coefficient is:

Problems:
 

1. Solid white color is recessive to pigmentation in cattle. A herd 500 cattle has 9 white cattle. How many cattle are carriers (have the allele causeing the solid white color, but are pigmented...or are heterozygous!) of the allele causing solid white?

2. A population of Cheetahs are surveyed for a genetic marker with 2 polymorphisms, called "A" and "B". Of 114 Cheetahs, 100 are AA, 13 are BB and 1 is AB. Is this population in Hardy Weinburg equilibrium? What is the inbreeding coefficient?

3. Look at *ALL* of our PV 92 data posted on the website; what are the allele frequencies for ++, -- and +- ? Is the class in Hardy Weinburg equilibrium? What is the inbreeding coefficient?