Why do we use an equation for a population that is not evolving?

Because it gives a baseline. If you know what genotype numbers to expect when nothing is acting on a population, then any large mismatch between observed and expected numbers is evidence that something — drift, gene flow, selection, non-random mating, or mutation — is at work. The equation is a measuring stick, not a claim about how nature usually behaves.

What is the difference between q and q^2?

q is the frequency of the recessive allele among all allele copies in the gene pool. q^2 is the frequency of individuals who carry two recessive alleles, that is, the ones showing the recessive phenotype. Since q is less than 1, q^2 is always smaller than q. When a question hands you a percentage of individuals with the recessive trait, that number is q^2.

Can genetic drift make a population better adapted?

Not reliably. Drift is random, so it can increase a helpful allele, a harmful one, or a neutral one purely by chance. Only natural selection consistently increases alleles that improve survival and reproduction. In very small populations, drift can even override selection and push a harmful allele toward fixation.

Does gene flow always increase genetic variation?

It increases variation in the population receiving migrants, because new alleles arrive. But it decreases the differences *between* populations, making them more alike. That is why gene flow tends to oppose the divergence that eventually produces separate species.

BIO-7.3

Population Genetics & Allele Frequencies

Learn to calculate allele and genotype frequencies with p + q = 1 and p² + 2pq + q² = 1, and see how drift, gene flow, and selection reshape a gene pool.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Population Genetics & Allele Frequencies, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Population Genetics & Allele Frequencies free

What this lesson covers

Evolution is often described as "change over time," but biologists need something they can actually measure. That something is allele frequency: the proportion of a particular allele in a population's gene pool. If the frequency of an allele shifts from one generation to the next, the population has evolved — even if no single organism changed at all.

In this lesson you will learn to count alleles, convert those counts into frequencies, and use the Hardy-Weinberg equations

p + q = 1

and

p^2 + 2pq + q^2 = 1

to predict how many individuals should carry each genotype. Then you will use those numbers as a baseline: when real frequencies drift away from what the equations predict, something is acting on the population. Genetic drift, gene flow, and natural selection each leave a different fingerprint, and by the end you should be able to read those fingerprints from data.

The Gene Pool and How to Count Alleles

A population is a group of organisms of the same species living in the same area and interbreeding. All of the alleles carried by every member of that population make up the gene pool. Population genetics zooms out from the individual: instead of asking "what genotype is this rabbit?" it asks "what fraction of all the fur-color alleles in this meadow are the dark allele?"

For a gene with two alleles, biologists use

p

for the frequency of one allele (usually the dominant) and

q

for the other. Because those are the only two options,

p + q = 1

, and each is a decimal between 0 and 1.

To count alleles from genotype data, remember that each diploid individual carries two alleles. A population of 200 organisms holds 400 alleles for that gene. Homozygotes contribute two copies of the same allele; heterozygotes contribute one of each.

Suppose 200 plants are genotyped: 98 are

RR

, 84 are

Rr

, and 18 are

rr

. The count of

R

alleles is

2(98) + 84 = 280

, so

p = \frac{280}{400} = 0.70

The count of

r

alleles is

2(18) + 84 = 120

, giving

q = 0.30

. Notice the check:

0.70 + 0.30 = 1

.

A very common error is dividing by the number of individuals instead of the number of alleles, which produces frequencies larger than 1. Another is forgetting the heterozygotes entirely. If your two frequencies do not sum to exactly 1, you have made an arithmetic mistake — go back before doing anything else.

The Hardy-Weinberg Equation and Its Conditions

Once you know

p

and

q

, you can predict the genotype frequencies expected if mating is random. Think of gametes drawn at random from the gene pool: the chance of drawing two

R

gametes is

p \times p = p^2

, two

r

gametes is

q^2

, and one of each is

2pq

(the 2 accounts for either parent supplying either allele). Since these are the only possibilities,

p^2 + 2pq + q^2 = 1

Here

p^2

is the expected frequency of homozygous dominant individuals,

2pq

the heterozygotes, and

q^2

the homozygous recessives.

This equilibrium holds only under five conditions: a very large population, no migration in or out, no mutation, random mating, and no natural selection. Real populations violate at least one of these almost always. That is the point — Hardy-Weinberg is a null model, a description of a population that is not evolving. It gives you numbers to compare reality against.

Term	Meaning	Read as
$p$	frequency of dominant allele	fraction of alleles
$q$	frequency of recessive allele	fraction of alleles
$p^2$	homozygous dominant	fraction of individuals
$2pq$	heterozygous	fraction of individuals
$q^2$	homozygous recessive	fraction of individuals

Students most often confuse the two kinds of frequency.

p

and

q

describe alleles;

p^2

2pq

, and

q^2

describe individuals. Also note that a dominant phenotype includes both

p^2

and

2pq

, so the frequency of the dominant phenotype is

p^2 + 2pq

, or equivalently

1 - q^2

Working Backward from Phenotypes

In most real data sets you cannot see heterozygotes — a carrier of a recessive allele looks exactly like a homozygous dominant individual. The one genotype you can identify by sight is the homozygous recessive, because only

rr

shows the recessive phenotype.

That gives the standard entry point: the observed frequency of the recessive phenotype equals

q^2

. Take its square root to get

q

, subtract from 1 to get

p

, then compute

p^2

and

2pq

.

For example, if 160 of 1,000 snails have the recessive banded shell, then

q^2 = 0.16

, so

q = 0.4

and

p = 0.6

. The expected heterozygote frequency is

2(0.6)(0.4) = 0.48

, meaning about 480 snails are carriers — three times as many as show the trait. This result surprises students, but it explains why rare recessive alleles persist: most copies are hidden in heterozygotes where selection cannot reach them.

Two traps show up constantly. First, do not take the square root of the dominant phenotype frequency;

p^2 + 2pq

is not a perfect square. Get

q

first, always. Second, watch whether a question asks for a frequency (a decimal) or a number of individuals (multiply the frequency by the population size). Reporting 0.48 snails or 480 percent both signal that the units got lost.

One more caution: these calculations assume the population is at equilibrium. Using

q^2

to find

q

when strong selection is acting gives only an approximation, which is exactly why the next section matters.

Drift, Gene Flow, and Selection as Frequency Changers

When observed frequencies drift away from Hardy-Weinberg predictions across generations, one of the model's assumptions has broken. Three mechanisms do most of the work.

Genetic drift is random change in allele frequencies caused by chance sampling in small populations. Which individuals happen to reproduce, and which gametes happen to fuse, is partly luck. In a population of ten frogs, a single accidental death can wipe out an allele; in a population of a million, it barely registers. Drift is non-directional — it can raise or lower any allele — and it tends to reduce variation, sometimes fixing an allele at frequency 1 or losing it at 0. A bottleneck (a crash in numbers) and the founder effect (a few individuals colonizing a new site) are drift events.

Gene flow is the movement of alleles between populations by migration or pollen transfer. It adds alleles the receiving population may have lacked and makes two populations more genetically similar, working against the divergence that leads to speciation.

Natural selection is non-random: alleles that raise survival and reproduction increase in frequency. Selection is directional and predictable, which distinguishes it from drift.

Mechanism	Random?	Typical effect on variation	Strongest when
Genetic drift	Yes	Decreases within a population	Population is small
Gene flow	Somewhat	Increases in the receiver	Migration is frequent
Natural selection	No	Depends on selection type	Environment pressures are strong

A frequent misconception: drift and selection are not opposites in kind, and both can operate at once. Another: mutation is the ultimate source of new alleles, but it changes frequencies far too slowly on its own to explain rapid shifts.

Key terms

Gene pool.: The complete set of alleles present in all individuals of a population at a given time.
Allele frequency.: The proportion of all copies of a gene in a population that are one particular allele, expressed as a decimal between 0 and 1.
Hardy-Weinberg equilibrium.: The state of a population in which allele and genotype frequencies stay constant across generations because no evolutionary mechanism is acting; a null model for comparison.
Genetic drift.: Random change in allele frequencies from generation to generation due to chance sampling; strongest in small populations.
Founder effect.: A form of genetic drift in which a small group establishes a new population whose allele frequencies differ from the original population by chance.
Bottleneck effect.: A form of genetic drift in which a sharp reduction in population size removes alleles at random and lowers genetic variation.
Gene flow.: The transfer of alleles between populations through migration of individuals or movement of gametes such as pollen.
Fixation.: The condition in which an allele reaches a frequency of 1 in a population, meaning all other alleles at that locus have been lost.

Worked example

In a field of 1,000 pea plants, 90 plants have wrinkled seeds, a recessive trait. Assume the population is in Hardy-Weinberg equilibrium. Find

p

and

q

, then predict how many plants are homozygous dominant and how many are heterozygous. Finally, state what fraction of the recessive alleles in the population are hidden in heterozygotes.

Step 1: Identify what you can see. Only homozygous recessive plants show wrinkled seeds, so the observed frequency of that phenotype equals

q^2

. Here

q^2 = \frac{90}{1000} = 0.09

.

Step 2: Solve for

q

by taking the square root.

q = \sqrt{0.09} = 0.30

.

Step 3: Use

p + q = 1

to get the other allele frequency.

p = 1 - 0.30 = 0.70

.

Step 4: Compute the expected genotype frequencies. Homozygous dominant:

p^2 = (0.70)^2 = 0.49

. Heterozygous:

2pq = 2(0.70)(0.30) = 0.42

. Check the total:

0.49 + 0.42 + 0.09 = 1.00

, so nothing is missing.

Step 5: Convert frequencies to counts by multiplying by 1,000. Homozygous dominant:

0.49 \times 1000 = 490

plants. Heterozygous:

0.42 \times 1000 = 420

plants. Homozygous recessive: 90 plants, matching the given data.

Step 6: Count recessive alleles. Each of the 420 heterozygotes carries one recessive allele, for 420 copies. Each of the 90 wrinkled plants carries two, for 180 copies. Total recessive copies

= 600

. The fraction hidden in heterozygotes is

\frac{420}{600} = 0.70

, or 70 percent.

That last number is the biological payoff: even though only 9 percent of plants show the recessive trait, most recessive alleles sit invisibly in carriers, which is why selection against a recessive phenotype removes the allele very slowly.

Practice questions

In a large population at Hardy-Weinberg equilibrium, 4 percent of individuals show a recessive disorder. What percentage of the population is expected to be heterozygous carriers?

4 percent
16 percent
32 percent
64 percent

Answer: 32 percent

The recessive phenotype frequency is

q^2 = 0.04

, so

q = 0.2

and

p = 1 - 0.2 = 0.8

. Heterozygotes are

2pq = 2(0.8)(0.2) = 0.32

, or 32 percent. The choice of 16 percent is

pq = (0.8)(0.2)

with the factor of 2 left out, and 4 percent is just

q^2

handed back, as if the affected frequency were the carrier frequency. Note that carriers outnumber affected individuals eight to one.

A biologist genotypes 250 beetles and finds 90

BB

, 120

Bb

, and 40

bb

. Calculate

p

and

q

, then state whether the genotype counts match Hardy-Weinberg expectations.

Answer: $p = 0.60$ and $q = 0.40$ ; expected counts are 90 $BB$ , 120 $Bb$ , and 40 $bb$ , so the population matches the expectation.

There are

2 \times 250 = 500

alleles total. Count

B

2(90) + 120 = 300

, so

p = \frac{300}{500} = 0.60

. Count

b

2(40) + 120 = 200

, so

q = 0.40

, and the two sum to 1. Expected frequencies are

p^2 = 0.36

2pq = 0.48

, and

q^2 = 0.16

; multiplied by 250 these give 90, 120, and 40. Because observed equals expected, there is no evidence from these data that drift, gene flow, or selection is currently shifting this locus.

Two neighboring lizard populations live on separate rock outcrops. Population A has 200 lizards; Population B has 15 lizards. Explain why the allele frequencies in Population B are likely to change more from one generation to the next, and describe one way that a highway built between the outcrops could affect both populations.

Answer: Genetic drift acts more strongly on the small population, so chance sampling of which lizards reproduce will swing Population B's frequencies more; a highway would block gene flow, allowing the two populations to diverge and intensifying drift in B.

Drift is a sampling effect. With only 15 individuals, the alleles passed to the next generation are a small, luck-dependent sample of the gene pool, so frequencies can jump sharply and alleles can be lost or fixed. With 200 individuals, chance deviations largely cancel out. A highway removes migration, and gene flow is what normally replenishes lost alleles and keeps populations genetically similar. Without it, Population B loses variation faster and the two populations accumulate different frequencies — the starting conditions for divergence.

FAQ

Why do we use an equation for a population that is not evolving?: Because it gives a baseline. If you know what genotype numbers to expect when nothing is acting on a population, then any large mismatch between observed and expected numbers is evidence that something — drift, gene flow, selection, non-random mating, or mutation — is at work. The equation is a measuring stick, not a claim about how nature usually behaves.
What is the difference between $q$ and $q^2$ ?: $q$ is the frequency of the recessive allele among all allele copies in the gene pool. $q^2$ is the frequency of individuals who carry two recessive alleles, that is, the ones showing the recessive phenotype. Since $q$ is less than 1, $q^2$ is always smaller than $q$ . When a question hands you a percentage of individuals with the recessive trait, that number is $q^2$ .
Can genetic drift make a population better adapted?: Not reliably. Drift is random, so it can increase a helpful allele, a harmful one, or a neutral one purely by chance. Only natural selection consistently increases alleles that improve survival and reproduction. In very small populations, drift can even override selection and push a harmful allele toward fixation.
Does gene flow always increase genetic variation?: It increases variation in the population receiving migrants, because new alleles arrive. But it decreases the differences between populations, making them more alike. That is why gene flow tends to oppose the divergence that eventually produces separate species.

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