Population Genetics

Notes on the Web - Unit Five- Part 5

Population Genetics: Quantifying Evolutionary Change

Bruce G. Stewart

General Objectives

Your objectives for these Notes on the Web and associated readings and exercises are:

To define microevolution;
To define population and gene pool;
To state the Hardy-Weinberg equation and describe the meaning of each variable;
To list, describe, and understand the assumptions and predictions of the Hardy-Weinberg Law;
To know common terms and facts associated with the topic of population genetics.

Related Textbook Readings:

For General Biology (BIO 1114) Class only: Johnson and Losos (2008) Pages 306-309of Chapter 17
For General Zoology (ZOO 1114) Class only: Hickman, et al. (2004) (to be assigned in class)

Characterizing the Gene Pool

Think of a group of sexually-reproducing organisms of the same biological species. Recall that individuals of the same species in a population can interbreed, and thus the genes that they possess and "flow" through time from parents to offspring. For a given locus (remember this is the location on a pair of homologous chromosomes where the gene occurs), any single organism has two alleles which could be the same or different forms of the gene. Thus, two individuals would have a total of four alleles, three individuals would have six and so on. This total number of alleles that exists in the whole population is referred to as the gene pool.

If we are to understand the genetic makeup of the whole population rather than just a single individual, we must describe how many of each allele of a gene is found in the whole gene pool. This can be done by giving the proportions that the numbers of each allele represents of the total. For example:

Five individuals would have --- 5 x 2 = 10 total alleles for a given locus.

If there was only one form of the gene at that locus, then all ten alleles in the gene pool would be the same, and thus the frequency of the allele would be 1.0 (or 100%). However, suppose there were two kinds of alleles for the trait (e.g. a dominant purple allele versus a white recessive form). If both kinds were found in our population of five individuals, then each would represent some proportion of the total of ten. For example, if:

Seven of the alleles were the white recessive form its proportion would = 0.7 (or 70%),
Three of the alleles were the purple dominant form its proportion would = 0.3 (or 30%)

These kinds of mathematical expressions of the proportions of each allele for a given locus in the total gene pool is referred to as their gene frequencies.

Tracking Gene Frequencies over Generations

Let's now see how some simple math can be used to track gene frequencies in a gene pool from one generation to the next. To do this let's first use some simple algebra and represent our frequencies with letter variables as follows:

p = the frequency of the dominant allele
q = the frequency of the recessive allele

Using the frequencies in our example above, p = 0.3 and q = 0.7, and p + q = 1.0. Make sure you understand how we got these numbers. If certain conditions in the population are met (you can read about these in detail in your text) then the chance (probability) of these alleles being in a gamete randomly draw from the population is equal to their gene frequencies. That is, the more there are in the population, the greater chance there will be of them occurring in gametes from that population. In our example above, the probability for p = 0.3 and for q = 0.7.

Now let's talk about calculating the probability of a combination occurring together, but let's use first use flipping a coin to illustrate. What is the probability of flipping a nickel and getting a "head"? Answer: h = 0.5. What is the probability of flipping the nickel and getting tails? Answer: t = 0.5. What is the chance of getting two heads in a row? Answer: h x h or 0.5 x 0.5 = 0.25. How about two tails in a row? Right! t x t or 0.5 x 0.5 = 0.25. What about getting a head and a tail? Answer: t x h = 0.5 x 0.5 = 0.25? Wrong! Remember that there are two ways to get a head and a tail: t x h and h x t. So we must add the chances together as follows: (t x h) + (h x t) = (0.5 x 0.5) + (0.5 x 0.5) = 0.5. Our total possible combinations could be expressed as h² + 2ht + t² = 1.0. Our numbers in the coin toss would be 0.25 + 0.5 + 0.25 = 1.0. That is, there is a 25% chance of getting two heads in a row, a 50% chance of getting a head and tail, and a 25% chance of getting two tails.

Now let's use our allele probabilities to calculate this in the same way, only this time our results can show us what genotypes would be produced. Let's also use our variables "p" and "q". The binomial formula this time will be p² + 2pq + q² = 1.0. So here goes:

The chance of a gamete having the dominant allele is p = 0.3, so the chance of two dominant alleles combining in the zygote is

p² = 0.3 x 0.3 = 0.09 (note: this would be the frequency of homozygous dominant individuals)

The chance of getting a dominant and recessive allele combination in the zygote would be

2pq = 2(0.3 x 0.7) = 0.42 (note: this would be the frequency of the heterozygous individuals)

Finally, the chance of getting two recessive alleles together in the zygote would be

q² = 0.7 x 0.7 = 0.49 (note: this would be the frequency of homozygous recessive individuals)

Thus, we can see that we can calculate the expected frequencies of the genotypes in the next generation, where p² + 2pq + q² = 1.0 represents all possible genotypes. Suppose our original individuals produced 100 offspring. Then our expected numbers of individuals with each genotype along with the number of "p" versus "q" alleles would be:

Genotypes	Number of the 100 offspring that would have the genotype	Number of dominant alleles the offspring would have	Number of recessive alleles the offspring would have
Homozygous dominant - p² = 0.09	9 individuals	18 dominant alleles	0 recessive alleles
Heterozygous - 2pq = 0.42	42 individuals	42 dominant alleles	42 recessive alleles
Homozygous recessive - q² = 0.49	49 individuals	0 dominant alleles	98 recessive alleles
TOTALS - 1.0	100 individuals	60 dominant alleles	140 recessive alleles

Let's now see what the new gene frequencies would be for "p" and "q". There are a total of 200 alleles (60 dominant alleles and 140 recessive alleles) in the next generation gene pool so p = 60/200 = 0.3 and q = 140/200 = 0.7. These are exactly the original frequencies of the dominant and recessive alleles in our original gene pool! Thus, has evolution occurred? No!

What we have demonstrated is called the Hardy-Weinberg Law. This law expresses the observation that genetic makeup of a population does not change, if certain conditions are met that allow the probability of any allele being passed on in relation to its exact original frequency in the gene pool. This is a powerful mathematical method which allows us to see what would happen if

there were no mutations adding new alleles;
there was no natural selection;
there was no genetic drift;
there was no emigration or immigration;
every individual had an equal probability of mating.

The fact that we can mathematically demonstrate precisely what goes on in living systems under conditions of random mating, etc. allows use to measure very precisely the effects of any of these factors that can alter the gene pool! Thus, the study of evolutionary processes can, as in other sciences, be understood in elegant, quantifiable, mathematical terms.

Example when Certain Assumptions of the Hardy-Weinberg are not Met: a Lethal Recessive Allele

Let's try this with an extreme example in which the recessive allele which we described previously is lethal in the homozygous condition. We can start with our original population of 10 which has a gene pool with frequencies of p = 0.3 and q = 0.7. Just as before, 100 offspring would have the same expected genotype probabilities as shown in the previous table. However, what would happen to all the homozygous recessive offspring? They would die, thus producing the following results:

Genotypes	Number of the 100 offspring that would have the genotype	Number of dominant alleles the offspring would have	Number of recessive alleles the offspring would have
Homozygous dominant - p² = 0.09	9 individuals	18 dominant alleles	0 recessive alleles
Heterozygous - 2pq = 0.42	42 individuals	42 dominant alleles	42 recessive alleles
Homozygous recessive - q² = 0.49	0 (because all would die)	0 ((all zygotes would die)	0 (all zygotes would die)
TOTALS - 1.0	51 individuals	60 dominant alleles	42 recessive alleles

In this example, there would be 102 alleles in the next generation, 60 dominant alleles and 42 recessive alleles. Thus, the new gene frequencies would be p = 60/102 = 0.59 and q = 42/102 = 0.41. Evolution has occurred!

In nature, there are natural processes, such as natural selection, that can affect the survival and reproductive success of individuals with particular genotypes. These processes have been and continue to be studied in wonderful detail. Like Copernicus and Galileo observing evidence that the Earth moves about the Sun, so do biologists observe the evidence that species change over time. In the next unit we shall study some of the major areas of evidence that support the principle of evolution.

Reminder about Textbook Study

As with other topics, your textbooks have excellent presentations of the materials on the population genetics, including generous excellent illustrations. Check the general objectives above to make sure that you have covered all of the topics in the textbook readings. However, in this particular case, your "Notes on the Web" contain most of the material you will need to know for your exam.

The "Testing Yourself" questions will be helpful for general biology students, although many more detailed questions will be included in the lecture exam. Similarly, study questions in the zoology textbook will be helpful review for general zoology students, but again, they are not comprehensive.

As with all materials throughout the semester, you will have opportunities to ask questions or ask that any relevant material from your assignments be discussed in class and/or in threaded discussions on Internet.

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Work in progress.

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