# EvoMath 1: The Hardy-Weinberg Principle

The Hardy-Weinberg Principle states that a population satisfying certain primary conditions will not evolve. This result is very important because any departure from these conditions will result in an evolving population. Three scientists in the early 20th century (G.H Hardy, Wilhelm Weinberg, and W.E. Castle) independently discovered this principle which is now used as the null model of population biology.

Consider a group of interbreeding organisms (a population)…

#### Initial assumptions about the population:

In order to construct any model, assumptions need to be made. These establish the “world” in which the model exists. How applicable the model is to the real world depends on how well the assumptions reflect reality. With this in mind we will make the following assumptions to establish the theoretical conditions for the model.

Primary:

• Random mating (panmixia)
• Random union of gametes (Mendelian inheritance)
• The size of the population is infinite.
• No Migration
• No Mutation
• No Selection

Secondary:

• Diploid
• Sexual, hermaphroditic (no difference between males and females)
• Nonoverlapping generations
• The gene under consideration has two forms (alleles). (A and a)
• The gene is not on a sex chromosome.

#### Some Quick Definitions:

• Allele: an alternate form of a gene
• Chromosome: a long molecule of double stranded DNA that is associated with proteins and contains many genes
• Evolution: the change in heritable characteristics of a group of organisms over time
• Gene: a segment of DNA responsible for producing an RNA transcript
• Genotype: the genetic constitution of an individual
• Locus: the place on a chromosome where a particular gene is located
• Migration: movement of genes from one population to another
• Mutation: a change in an organism’s genetic constitution
• Phenotype: the physiological traits of an individual
• Population: a group of interbreeding organisms
• Selection: the process where different alleles have different effects on the ability of an organism to produce offspring

#### Allele Frequencies:

There are two alleles in the population: A and a. Let the frequency of the A allele be represented by and the frequency of the a allele be represented by . Note: .

#### Genotype Frequencies:

Let , , and represent the genotype frequencies AA, Aa, aa respectively in the parental population and , , and represent the genotype frequencies in the daughter population. At Hardy-Weinberg equilibrium the frequency of the genotypes will be related to the allele frequencies as follows.

Genotype Frequency
AA
Aa
aa

#### Rationale:

Each individual in the population has two alleles. Homozygous individuals have two identical alleles; heterozygous individuals have one copy of each allele. Using these facts we can determine allele frequencies as the ratio of the number of each allele over the total number of alleles in the population. But first, to simplify things let be the number of individuals in a population, which means that the number of AA individuals is , the number of Aa individuals is , and the number of aa individuals is . Then the allele frequencies can be calculated from counts of individuals as follows.

Note: .

Because mating is random, the frequency of a mating is the frequency of the genotype of the first parent multiplied by the frequency of the genotype of the second. For example, the probability that an AA individual and an aa individual mate with one another is

Because random union of gametes (Mendelian inheritance) is occurring, the genotype of an offspring is derived by randomly choosing one allele from each parent. AA individuals will always produce A gametes, aa individuals will always produce a gametes, and Aa individuals will always produce both A gametes and a gametes in a 50:50 ratio. It is easy to see that an AA x aa mating will always produce Aa offspring. These frequencies can be calculated likewise for each of the six possible matings, and the results are displayed in the table below.

Mating Freq Offspring
AA Aa aa
AA x AA 1.0 0.0 0.0
AA x Aa 0.5 0.5 0.0
AA x aa 0.0 1.0 0.0
Aa x Aa 0.25 0.5 0.25
Aa x aa 0.0 0.5 0.5
aa x aa 0.0 0.0 1.0
Total 1.0

#### Therefore:

Using this table, we can calculate the genotype frequencies in the next generation as the average of the proportion of offspring of each genotype that each mating produces, weighted by the frequency of the mating.

#### Equilibrium:

The process of reproduction described above does not affect the allele frequency in the population: . Because the allele frequency does not change from one generation to the next, the population is considered to be at “equilibrium” with respect to its alleles. The only things that have changed in the population are the frequencies of genotypes, and that only happens if they are not already at . After the first round of reproduction, the genotype frequencies also reach equilibrium.

#### Significance:

A population satisfying the initial assumptions does not evolve after the first generation. The allele frequencies never change, and the genotype frequencies only change in the first breeding cycle to the equilibrium distribution of . This is significant because, before the Hardy-Weinberg principle was proved, it was argued by some biologists that a gene will go extinct unless it was actively selected for. In fact the opposite is true; in the absence of evolutionary forces, alleles will be maintained at their present frequencies.

This result can be shown to be consistent when the secondary assumptions are changed, i.e. there is an analogous principle for multiallelic, continuous, and/or polyploid populations. Although populations that violate the primary conditions will technically evolve, in many instances these violations can be so slight that the rate of evolution is not significantly different than zero. For instance, a very large population might be effectively infinite, and a mutation or migration rate might be so small that it is effectively zero. The Hardy-Weinberg principle forms the null hypothesis for population biology, and most population-based models of evolution derive from modifying the assumptions of this model.

Back to EvoMath Introduction.

Slightly simpler version for the benefit of the Creationists: blue eyes are not going away even though the trait is recessive.

I love blog posts that explain basic principles and mathematics that form the core of a discipline - this is a classic, clear and important post on population genetics and the basis of the dynamics of neutral alleles. I hope you will elaborate further in future posts, so that people can see how, for example, one can see selective pressures at work in a population and so on.

Reed:

Bravo on your efforts; I agree whole-heartedly with Stirling. Forgive me for picking nits, but one definition could use a tweak: “Migration: movement of genes from one population to another”. In the spirit of keeping the model relevant to reality (real organisms in real populations), you might consider changing the definition of “migration” to ‘the movement of individuals from one population to another’. Organisms migrate, not genes. Add “gene flow” to your list with your definition and the observation that gene flow results from migration.

Cheers!

In the field where I work, population genetics, “migration” is synonymous with “gene flow.” Ecologists or others might disagree.

“Organisms migrate, not genes. Add “gene flow” to your list with your definition and the observation that gene flow results from migration.”

Actually, when you work on plants and fungi sometimes it is the genes that migrate. They migrate in gametes that travel distances. The individuals often aren’t the ones that migrate, so the definition Reed used is appropriate.

Jim

Reed:

It as a biologist trained in population genetics also that I offered my suggestion. We population geneticists will do well to keep in mind the ecological context in which population genetic and evolutionary processes transpire. My sense is that “gene flow” is something we can estimate from allele frequency data, etc. but “migration” is something we might need to observe directly. A quick look at Weir’s (1996) GDA II and Hartl and Clark (1997, 3rd ed.) did not clear it up for me. The former seems to use the terms in a distinct but not clearly defined manner (pp. 182-183) and the latter don’t use the term “gene flow” at all in a discussion of “migration” (pp. 189-198). If you have any good references on distinctions of use (allowing that you claim they are synonymous), I’d appreciate seeing them.

Jim, thank you for your helpful comment; I should have thought of that.

Cheers,

Shaggy

Shaggy,

My late advisor, Marjorie Asumssen, and Andy Clarke both came out of Marc Feldman’s lab at Stanford. It’s no suprise that I use migration and gene flow the same way that Clake does. In fact, I’d say that Hartl and Clarke is an authoritative source.

Can migration happen without gene flow? Sure. Can gene flow happen without migration? Sure. However, it is traditional in popgen theory to use the terms interchangably.

If I felt like being rigorous, I could have made some distinction between the two, but I didn’t want to introduce a new term like “gene flow,” when “migration” can and does serve quite well.

I liked the article, and look forward to more in the series. A clear exposition of the maths of genetic drift would be nice, as would a derivation of Fischer’s law about the probability of survival of a new beneficial mutation. However, being a stickler for accuracy, I must object to the claim that, “This result is very important because any departure from these conditions will result in an evolving population.”

This simply does not follow, and it is not hard to think of situations which violate the conditions, but which preserve gene frequencies in the population. The migrating population may have the same gene frequency, for example; or the selection may be stabilizing selection.

Tom Curtis

Tom,

You are correct. I should have said “no net migration.” However, stablizing selection is a little tricker as we shall see. Also it’s not just allele frequencies we are talking about, but genotype frequencies are also important.