Hardy-Weinberg Equilibrium

Evolution can be defined simply as inherited change within a lineage; however, various mechanisms within a lineage can affect the genetic in a natural population.

Populations

A (a group of interbreeding individuals and their offspring) fluctuates in (N) over time. The genetic variation within that population is expressed by the frequencies of the various (gene variants); these frequencies vary over time and are affected by changes in population size. Because not all individuals in a population contribute equally to reproduction, (Ne) is usually less than the actual population size (N), so that Ne < N.

Four factors typically regulate effective population size (Ne):

1. Breeding sex ratio — If one sex is rarer than the other in the population, it is likely that all members of that rarer sex will mate and reproduce. Therefore, the rarer sex will dominate the and it is more likely that genes of the rarer sex will be passed on to the next generation.
2. Population size — If the population size (N) fluctuates, effective population size (Ne) will be dominated by the smallest gene pool.
3. Breeding groups — If the population forms small groups for breeding (for example, prides of lions), Ne will equal Ng (population size of a group). Ne is influenced by how long small groups last and by the amount of migration between groups.
4. Fertility — Individuals within population often vary in fertility; more fertile individuals will have a greater influence on the gene pool.

In practice, Ne is difficult to measure. Ne = N when none of the four factors listed above apply to a population. In this case, the sex ratio is 1:1, the population is of a constant large size, mating is random (no groups exist), and fertility is approximately equal among individuals. Under these ideal conditions (when Ne = N), after one generation (and assuming that there is no selection or mutation), the population ’s will attain equilibrium.

The Hardy-Weinberg Equation

The above mentioned equilibrium, known as the (or Principle or Law), states that when Ne = N, in the absence of selection or mutation, gene frequencies will acheive equilibrium after one generation. The term was named after English mathematician Godfrey Harold Hardy (1877-1947) and German physician Wilhelm Weinberg (1862-1937), who independently and nearly simultaneously published on the principle in 1908. [It was called Hardy ’s Law for a brief time, because Weinberg had published his results in an obscure German journal.]

These equilibrium values are expressed as simple functions of gene frequencies, where the frequency of the dominant allele (A) = p, the frequency of the recessive allele (a) = q, and p + q = 100%. Although the Hardy-Weinberg Equation in its simplest form (above) is calculated for two alleles, it is easily expanded for three or more alleles: Using the Hardy-Weinberg Equation

The Hardy-Weinberg equation can be used to calculate gene frequencies (under Hardy-Weinberg conditions) from any one of its components.

Example 1:

In one hypothetical Zebra Mussel (Dreissena polymorpha) population, most of the individuals have dark, zebra-striped shells (below left). However, solid light-colored shells (below right, caused by a homozygous recessive gene, aa) occur in 1 of every 10,000 individuals.

Problem:

Calculate gene frequencies and numbers of dominant homozygotes (AA, at left) and recessive homozygotes (aa, at right) in a population of 10,000 individuals.

Solution:

1. frequency of aa = q2 = 1/10,000 = 0.0001, so q = 0.01
2. number of aa = 0.0001 x 10,000 = 1 individual
3. p + q = 1, so p = 0.99
4. frequency of AA = p2 = 0.9801
5. number of AA = 0.9801 x 10,000 = 9,801 individuals
6. For extra credit: frequency of Aa = 2pq = 2 x 0.9801 x 0.01 = 0.0198 or 198 individuals

Example 2:

The Coquina Clam (Donax variabilis) is highly (with shells of many different colors. In a population of 2,000 clams, 1,920 are solid colored, whereas the remainder has radiating color bands. Solid color occurs in homozygous dominant (BB) and heterozygotes (Bb); color banding only occurs in homozygous recessive individuals (bb).

Problem:

Calculate gene frequencies and numbers of BB and Bb.

Solution:

1. 1,920 are solid (BB and Bb), so 80 banded are recessive (bb)
2. frequency of bb = q2 = 80/2000 = 0.04, so q = 0.20
3. p + q = 1, so p = 0.80
4. number of BB: p2 = 0.64, so BB in population of 2,000 = 0.64 x 2,000 = 1,280 individuals
5. number of Bb: 2pq [frequency of Bb] = 2 x 0.2 x 0.8 = 0.32, so Bb = 0.32 x 2,000 = 640 individuals

Deviation from the Hardy-Weinberg Equilibrium

The Hardy-Weinberg Equilibrium can be used as a null hypothesis, compared to values from a real population, to describe statistically significant deviations from the Equilibrium. If the deivation is significant, then the gene frequencies are changing and thus, evolution is occurring.

Example 3:

In a population of Zebra Mussels (Dreissena polymorpha), 1,469 individuals have the CC form of the enzyme phosphoglucose isomerase (PGI), an enzyme critical in the second step of glycolysis. Another 138 have the Cc form of PGI, and another 5 individuals have the cc form of the enzyme. All of these forms of the enzyme are fully functional.

Problem:

Is this population in Hardy-Weinberg Equilibrium?? [In other words, the null hypothesis is that the population is at H-W equilibrium.]

Solution:

1. p = frequency of dominant allele (C) = 2(CC) + 1(Cc) / 2n
= 2 x 1,469 + 138 / 2 (1,469 + 138 + 5) = 3,076 / 3,224 = 0.954
2. q = 1 - p = 0.046
3. For this example, HW expectations are:
• Exp(CC) = p2n = 0.9542 x 1,612 = 1,467.4
• Exp(Cc) = 2pqn = 2 x 0.954 x 0.046 x 1,612 = 141.2
• Exp(cc) = q2n = 0.0462 x 1,612 = 3.4
4. Determine deviation of actual from HW expectations: 5. Is the deviation significant? Use Pearson ’s Chi-square (χ2) test for significance:
 χ2 = ∑ (O - E)2 (square of deviation) E = (1469 - 1467.4)2 + (138 - 141.2)2 + (5 - 3.4)2 1467.4 141.2 3.4 = 0.001 + 0.073 + 0.756 = 0.823
6. Look up 5% significance level for 1 degree of freedom = 3.84 [look up in a table or use a significance calculator].
7. Because the calculated χ2 (0.823) < 3.84, the actual and expected values are not significantly different, the null hypothesis is not rejected, and we conclude that the population is (currently) at Hardy-Weinberg Equilibrium.