Determining whether or not evolution is occurring within a particular gene (i.e. whether or not a given allele is spreading or diminishing) can be surprisingly difficult. One way to determine if evolution is acting on the gene is to determine mathematically what the genetic makeup of a population would be if evolution were not occurring.
A population that is not evolving is said to be at Hardy-Weinberg equilibrium, named for the mathematician and physician who independently worked it out.
Godfrey Hardy & Wilhelm Weinberg
For a population to be at Hardy-Weinberg equilibrium, and, thus, not evolving, there are a number of conditions that must be met. These conditions are very unrealistic for a real population, and that should be no surprise. Evolution can occur via natural selection, genetic drift, or gene flow, so it is difficult for a real population to elude these processes.
Here is a table from your textbook that summarizes the conditions that a population must meet in order to be considered to be at Hardy-Weinberg equilibrium. The right column demonstrates why each condition needs to be met to avoid evolution - essentially it explains how each of those causes evolution when the criterion is not met.
Again, the utility of this is to establish a null hypothesis - one against which we can compare a real population. So the unrealistic nature of these conditions is acceptable for our purposes.
Urry, L., Cain, M., Wasserman, S., Orr, R., & Minorsky, P. (2020). Campbell Biology in Focus, AP Edition (3rd ed.). Pearson Education.
So what exactly are allele frequencies? Alleles are, of course, versions of a gene that code for a protein. Frequencies in science and math refer to how often something appears in a sample or population. Thus, an allele frequency is simple the proportion of one allele compared to all the alleles for that gene in the population.
Remember that in diploid organisms, each individual will have two alleles, so you will need to account for both of the individual's alleles. For example, a heterozygous individual will contribute one dominant allele (A) and one recessive allele (a), whereas a homozygous dominant individual would contribute two dominant alleles (A).
If we do this for the entire population, we can see what the gene pool contains. A gene pool is essentially the genetic makeup of an entire population. Imagine all those alleles just floating around in a big pool. If most of those alleles are one type, that allele frequency is much higher than the other.
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Evolution is typically described simply as "change over time". That is not incorrect, but a more precise and useful definition for biologists is "any change in allele frequencies over time". If a population is evolving (whether through selection, drift, or gene flow) that means that one allele is becoming more or less common within that population. If one allele goes up in frequency, the other must go down to compensate. Remember, these are proportions, or percentages, so they must always add up to 100%.
When representing these allele frequencies, we use p to represent the allele frequency for the dominant allele and q to represent the frequency of the recessive allele. Remember that frequencies are essentially proportions (or percentages). This means that they should always be parts of a whole, or 100%.
When there are only two alleles for a given gene (which will always be the case when using Hardy-Weinberg for us), that means that p+q=1. This is essentially the first equation to consider when using Hardy-Weinberg equilibrium. If you are able to find either p or q, you can find the other by simply plugging into that equation.
To calculate p and q, you have to consider what information you have about the populations. If you are given genotypes for all the individuals, it is easy: count up the alleles! Every homozygous dominant individual will contribute two dominant alleles to the gene pool; every homozygous recessive individual will contribute two recessive alleles; and every heterozygote will contribute one of each allele.
The total number of alleles should be equal to 2 x N, where N is your population size because each individual has two alleles.
p = # dominant alleles / total number of alleles
q = # recessive alleles / total number of alleles
https://www.expii.com/t/hardy-weinberg-principle-overview-equation-10992
Hardy-Weinberg equilibrium has those unrealistic criteria in order to be met, but if we assume those to be true (or if we are told they are true), we can assume that all alleles can intermingle with all others when producing offspring. As a result, imagine that pool of alleles. There is a nexample of this in this image from your textbook with p=0.8 and q=0.2.
Each gamete has an 80% chance of receiving a dominant allele (CR) and a 20% chance of receiving a recessive allele (CW) . So if a given offspring was to receive a homozygous dominant genotype (CRCR) , it must receive a dominant allele for the egg (80% chance) AND a dominant allele for the sperm (also an 80% chance).
According to the rules of probability, if two independent events need to occur, we multiply their probabilities to find the probability of both occurring.
So the chances of an individual having the homozygous dominant genotype (CRCR) are (0.8)(0.8)=64%
Thus, we would expect a population with these allele frequencies to produce 64% homozygous dominant individuals. Remember that these are expectations, so we can compare the observed data to determine if the population is at Hardy-Weinberg equilibrium or not. If the population is NOT at equilibrium, then it is evolving because the genotypes are not meeting expectations and allele frequencies are changing.
Urry, L., Cain, M., Wasserman, S., Orr, R., & Minorsky, P. (2020). Campbell Biology in Focus, AP Edition (3rd ed.). Pearson Education.
We can apply these rules of probability to determine the proportions for each genotype if a population is at Hardy-Weinberg equilibrium. These proportions are represented in this image from your book.
All of these proportions are a continuation of the previous example. If the allele frequencies were different, we would see different expected proportions.
These proportions can be summarized by the following equation:
p2 + 2pq + q2 =1
Because p refers to the dominant allele and q to the recessive allele, p2 pertains to individuals that have received two dominant alleles and q2 to homozygous recessive individuals.
However, heterozygotes can be made in two different ways: the sperm can receive the dominant (80% chance) and the egg recessive (20% chance) OR the sperm can receive the recessive (20% chance) and the egg dominant (80% chance). So the probability of each of these is pq= (0.8)(0.2)=0.16
Because either option results in a heterozygote, we can add their probabilities together according to the rules of probability. The probability for either of these scenarios occurring is therefore pq+pq=2pq.
Urry, L., Cain, M., Wasserman, S., Orr, R., & Minorsky, P. (2020). Campbell Biology in Focus, AP Edition (3rd ed.). Pearson Education.
https://images.app.goo.gl/gZ2gBohQ4mjKCsCR6
Keep in mind that each of these is an expected proportion assuming the population is at Hardy-Weinberg equilibrium. Because they are proportions, or percentages, they need to add up to 1, or 100%.
We can use these calculations to determine whether or not this population accurately represents these predictions with a chi-square test.
Essentially, Hardy-Weinberg equilibrium is a null hypothesis that states evolution is not occurring. Then a scientist can collect data and compare the actual data to these expectations and accept or reject the null hypothesis.
Comparing observed data to expected data necessitates a chi-square test. When using the chi-square test, the expected values come from the Hardy-Weinberg equilibrium equation calculations.
The p2 value is the expected proportion of homozygous dominant individuals, the 2pq value is the expected proportion of heterozygous individuals, and the q2 value is the expected proportion of homozygous recessive individuals. Keep in mind that these are proportions, so be sure to multiply it by the sample size to get the expected number for your calculations!
If it is determined that evolution is occurring at that gene locus, further investigation can explore why evolution is occurring.