*Greg Chin*

# Hardy-Weinberg Equilibrium II: The Equation

## Conditions for Hardy-Weinberg Equilibrium

I think our research project is going to be great. I mean, think about it - we have all these great flying hamster traits we've been studying with Adrian. It's going to be really interesting to see how these traits affect the flying hamster populations. Do the coat color, tail color or fire-breath traits improve or hinder the ability of the hamsters to survive and mate?

Recall that flying hamsters are diploid, meaning they have two homologous copies of each gene. We've identified two versions, or alleles, of each gene we're going to study. That means that three different genotypes are possible in any flying hamster population. For instance, at the coat color gene, hamsters can be *BB*, *Bb*, or *bb*. To make our research project easier, let's also only consider traits where the distribution in males and females is equal and generations don't overlap. Hardy-Weinberg equilibrium can be calculated for genes with more than two alleles and where the generations overlap, but the math is much more complicated.

Speaking of math, let's see what the **Hardy-Weinberg equilibrium equation** looks like: (** p2 + 2pq + q2 = 1**). Oh, boy. It's a polynomial equation. Well, let's push up our sleeves, take a deep breath and see if we can figure out how all these crazy letters are supposed to describe genotypic frequency.

## Allelic Frequency

I think the first thing to try to understand, especially when dealing with a science equation, is the variables. At least the Hardy-Weinberg equation only has two variables, ** p** and

**. These stand for**

*q**the allelic frequency of the two possible alleles for the gene in question*.

Let's analyze the coat color gene in the context of the Hardy-Weinberg equation. We can arbitrarily assign *p* to the dominant brown allele (*B*) and *q* to the recessive white allele (*b*). In this example, our gene pool consists of the *B* and *b* alleles, and *p* and *q* represent the frequency at which those alleles appear in the population. But, how do we calculate allele frequency?

Allele frequency is basically the proportion the allele occupies in the gene pool. The sum of all of the alleles found in a population is known as the **gene pool**. To calculate *p*, we simply need to divide the number of *B* alleles by the sum of all of the alleles associated with the gene. That is *B* and *b*. Similarly, *q* can be calculated by dividing the number of *b* alleles by the sum of *B* + *b*. But, how can we determine how many *B* and *b* alleles are in a population?

Well, we should be able to infer the number of *B* and *b* alleles based on the genotype of the organisms in the population, right? Let's see if that works.

Let's examine the coat color gene in a population of 1,000 hamsters. A homozygous dominant brown hamster has two *B* alleles, so if we saw 360 of them, that would account for 720 *B* alleles. A heterozygous brown hamster has one *B* and one *b* allele, so if we saw 480 of them, that would translate to 480 *B* and 480 *b* alleles. A homozygous recessive white hamster has two *b* alleles, so if we saw 160 white hamsters, they'd contribute 320 *b* alleles to our calculations. The white hamsters are obviously pretty easy to identify, but the brown ones are a little trickier since some are homozygous and some are heterozygous. Luckily, we can distinguish the homozygotes from the heterozygotes if we genetically type the brown hamsters with a method like DNA sequencing.

Alright, let's plug all those numbers into a frequency formula. The frequency of *B* is equal to *p*, and that is equal to 720 + 480 and all that is divided by the gene pool (the sum of) 720 + 480 + 480 + 320. All of that comes out to 0.6. That means that 60% of the alleles in the gene pool are *B*.

Now that we've got that calculation under our belt, calculating the frequency of *b* should be a little bit easier now, right? The frequency of *b* is equal to *q*, and that's equal to the sum of 480 + 320 divided by the sum of 720 + 480 + 480 + 320. That's all equal to 0.4. That means that 40% of the alleles in the gene pool are *b*.

Note that the sum of the frequencies of the *B* and *b* alleles is one. That makes sense because together they make up 100% of the alleles in the gene pool. Let's restate the relationship between the two frequencies as a new equation that we can use in our population genetic studies, we can say that ** p + q = 1**.

## Genotypic Frequency

Well, that's all fine and dandy, but when you're studying a population of organisms, you're interested in the organisms. The Hardy-Weinberg equation describes the ratio of different genotypes in a population. Before we tackle genotypic frequencies, let's consider a simpler example first.

If we flip two coins, what are the odds that both come up heads? There's a one-in-two chance to get heads, so that means there's a one-in-four chance of getting heads twice in a row. Since flying hamsters are diploid, a hamster with a *BB* genotype got one *B* allele from its mom and one *B* allele from its dad. Just like coin flipping, the chance of getting two *B* alleles is merely the chance of getting one *B* allele squared (*B*Ë†2). Since we know the frequency of the *B* allele in the gene pool is *p*, the chance of getting a homozygous dominant individual is ** p2**.

Now, consider the chances of getting heads and tails if we flip two coins. Since the chance of getting heads is the same as getting tails, there's still a one-in-four chance of getting one of each. However, since it doesn't matter if we get heads in the first or second coin flip, there's twice as many ways to achieve the desired result, meaning there's a one-in-two chance to get one of each coin.

If we apply this thinking to the heterozygotes in our population, a brown coat heterozygote can receive the *B* allele from either mom or dad, so there's two ways to make a heterozygote just like there's two ways to get heads and tails. And, since we said the frequency that the *B* and *b* alleles occur in the gene pool is *p* and *q*, respectively, that means that the proportion of heterozygous individuals in the population can be represented by **2 pq**.

Finally, as we determined for the *BB* hamsters, the frequency of *bb* hamsters in the population can be represented by the frequency of the *b* allele squared, or ** q2**. Since these are the only three possible genotypes, the sum of the frequencies of each should be one - since they represent 100% of the possible genotypes. Now, if we put all of this information together, we arrive at the Hardy-Weinberg equilibrium equation of

*p*2 + 2

*pq*+

*q*2 = 1.

## Using the Hardy-Weinberg Equation

Before we go any further, keep in mind as we learn to use the Hardy-Weinberg equilibrium equation that we will be considering simple examples that would likely be much more complex in the wild.

Now, let's see how our coat color data fits into the Hardy-Weinberg equation. If we know that the frequency of the *B* allele is 0.6 and the frequency of the *b* allele is 0.4, we can insert those numbers into the Hardy-Weinberg equation and determine if the population is in equilibrium.

So, *p*2 + 2*pq* + *q*2 simplifies to 0.36 + 0.48 + 0.16, which adds up to one. That means that if the population is in equilibrium, we predict that we would see 360 *BB*, 480 *Bb* and 160 *bb* hamsters among a population of 1,000 hamsters. In fact, those are the exact numbers we observed in the population.

Since the genotypic frequencies predicted by the Hardy-Weinberg equilibrium equation exactly match the observed frequencies, we can conclude that this flying hamster population is in a state of equilibrium with respect to the coat color gene. That is, the coat color alleles are perfectly balanced, such that the same allelic, genotypic and phenotypic ratios were observed in every generation.

## Lesson Summary

In summary, the Hardy-Weinberg equilibrium equation describes genotypic frequency in a population. For a sexually reproducing, diploid organism where there are only two allelic possibilities for the gene in question, the equation can be written as *p*2 + 2*pq* + *q*2 = 1.

Typically, *p* and *q* represent the allelic frequency of the two possible alleles for the gene in question - *p* is often used to represent the dominant allele, and *q* usually represents the recessive allele.

A gene pool is the sum of all of the alleles found in a population. The equation which describes the allele frequency in the population is *p* + *q* = 1. So, *p*2 represents the proportion of individuals in the population homozygous for the first allele. In the case of a trait governed by simple dominance, this will be the homozygous dominant genotype. Also, 2*pq* represents the proportion of individuals in the population with a heterozygous genotype. And, *q*2 represents the proportion of individuals in the population homozygous for the second allele. In the case of a trait governed by simple dominance, this will be the homozygous recessive genotype.

## Learning Outcomes

After watching this lesson, you should be able to:

- Explain what the variables in the Hardy-Weinberg equilibrium equation represent and how to find them
- Discuss the significance of allelic frequency and genotypic frequency

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