Natural Selection and Adaptation

Darwin's Theory of Natural Selection

Part 1 of 7

In 1859, Charles Darwin published On the Origin of Species, proposing a single, elegant mechanism to explain the diversity of life: natural selection. The power of Darwin's argument is that it is not a vague claim about "survival of the fittest" — it is a tight, logical syllogism. If a handful of observable facts about populations are true, then evolutionary change is the unavoidable consequence.

In modern AP Biology, evolution is defined as a change in the heritable allele frequencies of a population over generations. Natural selection is one of several mechanisms (alongside genetic drift, gene flow, mutation, and non-random mating) that can drive that change. This part builds the logical foundation; later parts add the quantitative machinery (Hardy-Weinberg) that lets you measure evolution.

Anchor idea: Individuals do not evolve. Populations evolve. A single beetle is born with whatever alleles it has and dies with them. What changes across generations is the proportion of alleles in the population as a whole.

The Logic of Natural Selection — Darwin's Four Postulates

Natural selection follows from four observations. If all four hold, selection must occur. Memorize this chain — AP free-response questions frequently ask you to lay it out.

The inevitable result: the favorable heritable traits become more common in the next generation. Repeated over many generations, this produces descent with modification — the gradual transformation of populations and, ultimately, the origin of new species.

Critical distinction: Postulate 2 is the linchpin. Variation that is not heritable (e.g., bigger muscles from exercise) cannot fuel evolution, because it is not transmitted to offspring. Selection can only act on the phenotype, but it only produces evolution when phenotypic differences have a genetic basis.

Fitness Means Reproductive Success — Not Strength

In everyday English, "fittest" suggests the strongest or fastest. In biology, fitness has a precise, narrow meaning:

Fitness = an individual's relative contribution of offspring to the next generation's gene pool.

It is relative (measured against other individuals in the same population) and it is fundamentally about reproduction, not survival for its own sake. Survival matters only insofar as it lets an organism reproduce.

Consider two implications that trap students:

• A massive, powerful male elephant seal that wins fights but sires zero pups has a fitness of 0. A smaller "sneaker" male that fathers several pups has higher fitness, despite being weaker.
• A salmon that spawns thousands of eggs and then dies immediately can have enormous fitness. Living a long time contributes nothing to fitness if it does not translate into offspring.

Relative fitness ($w$) is often scaled so the most successful genotype = 1. If genotype AA averages 10 offspring and aa averages 6, then $w_{AA} = 1.0$ and $w_{aa} = 0.6$. The selection coefficient $s = 1 - w$ measures the strength of selection against a genotype; here $s_{aa} = 0.4$.

Misconception repair: "Survival of the fittest" really means "reproduction of those whose heritable traits, in this environment, yield the most surviving offspring." Strength, size, and longevity matter only through their effect on reproductive output.

Checkpoint — The Logic and the Definition of Fitness

Evidence for Evolution

Evolution is supported by multiple independent lines of evidence that converge on the same conclusion. AP questions often ask you to identify which type of evidence a scenario illustrates.

Direct observation — evolution happening now:

• Peppered moths (Biston betularia): Before the Industrial Revolution, light moths camouflaged on lichen-covered trees; dark moths were eaten by birds. Industrial soot blackened trees, reversing the selective pressure — dark moths now survived better, and the dark allele frequency rose sharply. When pollution controls cleaned the air, the light form rebounded. This is directional selection driven by predation.
• Galápagos finches (Grants' study): During a drought, only large, tough seeds remained. Finches with deeper, stronger beaks could crack them and survived to reproduce; average beak depth in the population increased measurably within a single generation.
• Antibiotic resistance: In a bacterial population, rare pre-existing resistant cells survive antibiotic treatment and reproduce, so the resistance allele frequency climbs. The antibiotic does not create resistance — it selects for variants that already exist.

Misconception repair (very common AP trap): Bacteria do not "try" to become resistant, and antibiotics do not induce the mutation. The variation exists first (by random mutation); the environment then selects among existing variants. Evolution has no foresight and no goal — organisms never evolve a trait "in order to" meet a future need.

Checkpoint — Evidence and Misconceptions

Populations Evolve, Individuals Do Not

This is one of the most heavily tested conceptual points in the entire unit. Hold these two columns apart:

An individual giraffe does not stretch its neck and pass on a longer neck. Instead, in an ancestral population, giraffes varied in neck length; longer-necked individuals (for heritable reasons) reached more food, survived, and reproduced more; the frequency of long-neck alleles rose in the population over generations.

AP framing: When you write or evaluate evolutionary explanations, the subject of the sentence should almost always be the population or the allele frequency, never the individual "deciding," "needing," or "trying." Natural selection is the differential survival and reproduction of individuals, but evolution is the resulting change in the population's gene pool.

Exit Ticket — Part 1 Synthesis

Types of Natural Selection

Part 2 of 7

When a trait varies continuously across a population — height, beak depth, birth weight, running speed — we can plot its phenotype distribution as a bell-shaped curve. Natural selection reshapes that curve by favoring some phenotypes over others. There are three modes of selection on a quantitative trait, and each transforms the curve in a characteristic way.

The key analytical tools are the mean (where the peak of the curve sits) and the variance / standard deviation (how spread out the curve is). For each mode you must be able to state:

1. Which phenotypes are favored?
2. What happens to the mean?
3. What happens to the variance (spread)?

Anchor idea: Selection acts on phenotypes, but it changes the population by altering the allele (and genotype) frequencies that underlie those phenotypes. The curve is a phenotype distribution; the evolutionary consequence is a shift in the gene pool.

The Three Modes — Comparison Table

How to read each mode:

• Directional selection favors phenotypes at one end of the range. The mean moves in that direction. This is the mode behind most observed rapid evolution (drought, pollution, antibiotic exposure, predation pressure).
• Stabilizing selection favors the intermediate phenotype and selects against both extremes. The mean stays put, but the distribution gets narrower — variance drops. This is actually the most common mode in stable environments; it maintains the status quo. Human birth weight is the textbook case: very small babies face survival risks, very large babies face delivery complications, so intermediate weights have the highest fitness.
• Disruptive selection favors both extremes and selects against the intermediate. Variance increases, and the single peak can split into two peaks (a bimodal distribution). Over long periods, disruptive selection can contribute to the formation of new species.

Checkpoint — Identifying the Mode

Worked Example — Directional Selection Shifts Allele Frequency

Directional selection at the phenotype level produces a measurable change in allele frequency at the genetic level. Let's trace one generation of selection in a population where a dominant allele A produces a dark coat that camouflages well against a newly darkened (polluted) habitat, and the recessive a produces a light coat that predators spot easily.

Setup — starting allele frequencies:

Let the frequency of the dark allele be $p = 0.40$ and the light allele be $q = 0.60$, with $p + q = 1$. Assume the starting genotype frequencies follow $p^2 + 2pq + q^2 = 1$.

So 64% of the population is dark (AA + Aa) and 36% is light (aa).

Apply directional selection against light coats. Suppose predation means light (aa) individuals have relative fitness $w_{aa} = 0.5$, while both dark genotypes have $w_{AA} = w_{Aa} = 1.0$. Start with a population of 1000 individuals.

Recompute allele frequencies after selection using allele-counting. Each survivor carries 2 alleles, so survivors carry $2 \times 820 = 1640$ alleles.

• Count of A alleles: each AA contributes 2, each Aa contributes 1 → $2(160) + 1(480) = 320 + 480 = 800$.
• Count of a alleles: each aa contributes 2, each Aa contributes 1 → $2(180) + 1(480) = 360 + 480 = 840$.

New frequencies:

$p' = \frac{800}{1640} \approx 0.488$ and $q' = \frac{840}{1640} \approx 0.512$

Result: In a single generation, the dark allele frequency rose from $p = 0.40$ to $p' \approx 0.49$, while the light allele fell from $q = 0.60$ to $q' \approx 0.51$. The phenotype distribution shifted toward the dark extreme — the signature of directional selection — and we have quantified the underlying gene-pool change. Repeated over many generations, $p$ continues climbing toward fixation.

Skill to master: Going from genotype counts back to allele frequencies via allele-counting — $p = \frac{2(\text{AA}) + (\text{Aa})}{2N}$ — is the single most important computation in this unit. Practice it until it is automatic.

Reading the Effect on the MEAN and VARIANCE Numerically

Because the three modes are defined by their effect on the mean and variance of a phenotype distribution, it helps to see the numbers move. Consider a population of plants whose height (cm) is recorded before and after one generation of selection under each mode. The same starting distribution is used in every case (mean = 50 cm, with a wide spread).

How to interpret each row:

• Directional: the mean moves (50 → 58) because one tail of the distribution is favored; variance often drops as the disfavored tail is trimmed.
• Stabilizing: the mean is unchanged (50 → 50) but the variance falls (90 → 45) because both tails are removed and individuals cluster near the optimum.
• Disruptive: the mean can stay the same, but the variance rises (90 → 150) as the middle empties out and the two extremes fill in — the curve goes bimodal.

Exam shortcut: To classify a mode from a graph or data table, ask two questions in order: (1) Did the mean shift? If yes → directional. (2) If the mean held still, did the spread shrink or grow? Shrink → stabilizing; grow/bimodal → disruptive. These two checks resolve nearly every "which mode?" item.

Checkpoint — Quantitative Directional Selection

Balancing Selection — How Variation Is Maintained

Directional and stabilizing selection tend to remove variation. Yet real populations stay genetically diverse. Balancing selection is an umbrella for mechanisms that actively maintain multiple alleles. Two AP-relevant forms:

• Heterozygote advantage (overdominance): The heterozygote has higher fitness than either homozygote. The classic example is the sickle-cell allele. In regions where malaria is endemic, individuals who are heterozygous (carriers) are resistant to severe malaria, while one homozygote suffers sickle-cell disease and the other lacks malaria resistance. Because the heterozygote is fittest, both alleles are maintained in the population — neither goes extinct.
• Frequency-dependent selection: The fitness of a phenotype depends on how common it is. In negative frequency-dependence, rare phenotypes have an advantage (e.g., predators form a "search image" for the common prey type, so rare morphs are overlooked), which preserves multiple forms.

Why this matters for Hardy-Weinberg (Part 5): Heterozygote advantage is a powerful reason a real population's genotype frequencies deviate from simple expectations — it keeps a "harmful" recessive allele at much higher frequency than selection against the homozygote alone would predict.

Exit Ticket — Part 2 Synthesis

Sexual Selection

Part 3 of 7

Darwin himself was puzzled by traits that seemed to reduce an organism's chance of survival — the peacock's enormous, conspicuous tail being the famous example. Why would natural selection produce a structure that makes its bearer slower, more visible to predators, and metabolically expensive? His answer was a second mode of selection: sexual selection, the differential reproductive success that arises from variation in the ability to obtain mates.

Sexual selection is a subset of natural selection in the broad sense (it acts through differential reproduction), but AP Biology treats it as a distinct concept because the selective agent is mating success, not survival. A trait can spread through a population even while lowering survival, as long as it raises mating success enough to more than compensate.

Anchor idea: Natural (survival) selection asks "Can you survive long enough to reproduce?" Sexual selection asks "Once you've survived, can you actually secure a mate?" The two can pull in opposite directions.

Two Forms of Sexual Selection

Intrasexual selection ("same-sex competition") favors traits that help win direct contests — these are often weapons and body size. The losers may be excluded from mating entirely.

Intersexual selection ("between-sex choice") favors traits that the choosing sex finds attractive — these are often ornaments and displays. The peacock's tail evolved because peahens preferentially mate with males bearing larger, more symmetric, more eyespot-rich tails.

Memory aid: Intra- means "within" (within one sex → male–male combat). Inter- means "between" (between the sexes → one sex chooses the other). Mixing these up is a frequent AP error.

Checkpoint — Intrasexual vs. Intersexual

Secondary Sexual Characteristics and Sexual Dimorphism

Secondary sexual characteristics are traits that differ between the sexes but are not directly part of the reproductive organs — bright plumage, antlers, manes, larger body size, elaborate songs. They develop around sexual maturity and function in competition or attraction.

Sexual dimorphism is the systematic difference in appearance between males and females of the same species (size, color, ornamentation). Pronounced sexual dimorphism is a strong signature of sexual selection:

• Intrasexual selection tends to produce dimorphism in size and weaponry (e.g., male elephant seals are several times larger than females).
• Intersexual selection tends to produce dimorphism in ornamentation and color (e.g., the brilliant male mallard vs. the drab, camouflaged female).

Why is the female usually the choosier, more cryptic sex and the male the ornamented one? The deepest cause is differential parental investment (anisogamy): eggs are large and costly and often paired with greater parental care, while sperm are cheap and abundant. The sex that invests more per offspring (usually females) becomes a limiting resource, so members of the other sex (males) compete and the choosy sex evolves discriminating preferences.

Connection: Sexual dimorphism is the observable evidence of sexual selection in the same way that homologous structures are observable evidence of common descent. Seeing strong dimorphism on an exam should prompt you to consider sexual selection.

"Good Genes," the Handicap Principle, and the Survival Trade-off

Why should a female "prefer" an exaggerated ornament? Two leading hypotheses explain how mate choice can be adaptive rather than arbitrary:

• "Good genes" hypothesis: An elaborate ornament is an honest signal of heritable quality. A male that can grow a large, symmetric tail despite the cost must have, e.g., a robust immune system and efficient metabolism. By choosing him, a female obtains good genes for her offspring, who inherit both his quality and (in sons) the attractive trait.
• Handicap principle (Zahavi): The ornament is reliable precisely because it is costly. Only a genuinely high-quality male can afford to "waste" resources on a giant tail and still survive. A low-quality male cannot fake the signal, so the handicap guarantees the signal's honesty. The cost is the point.

The survival trade-off. This is the crux of the part: sexual selection can directly oppose survival (natural) selection.

The tail size we observe represents an equilibrium between these opposing pressures. The trait stops growing when the marginal mating benefit of a bigger tail no longer outweighs the marginal survival cost. This is why sexual selection can produce traits that, viewed through survival alone, look maladaptive — they are favored because fitness is reproductive success, and getting a mate is half of reproducing.

AP framing: When a trait seems to hurt survival yet persists or spreads, suspect sexual selection. The fitness payoff comes from increased mating, which can outweigh a survival cost.

Checkpoint — Good Genes and the Trade-off

Quantifying the Trade-off — When Does a Costly Ornament Spread?

A costly sexual ornament spreads only if its mating benefit outweighs its survival cost in terms of total reproductive output. We can make this concrete with relative fitness.

Suppose in a bird population we compare long-tailed males to short-tailed males over a breeding season:

Even though the long tail cuts survival from 0.80 to 0.50 (a real survival cost imposed by natural selection), the long-tailed male's expected reproductive output (1.00) exceeds the short-tailed male's (0.80). So the long-tail allele has higher overall fitness and will spread — sexual selection wins the tug-of-war here.

Now flip the mating advantage to a smaller value — say long-tailed males average only 1.4 mates:

$0.50 \times 1.4 = 0.70 < 0.80$

Now the survival cost dominates: the long tail yields fewer total offspring than the short tail, so natural selection wins and the ornament does not spread. The trait's evolution depends entirely on whether the mating gain exceeds the survival loss, which is exactly the equilibrium described by the peacock-tail trade-off.

Runaway sexual selection (Fisherian). When a female preference and a male trait are both heritable and become genetically correlated, choosing the trait also propagates the preference. This positive feedback can drive the ornament to ever-more-exaggerated extremes far beyond the survival optimum — until the survival cost finally halts the runaway. This is one proposed explanation for why some ornaments (like the peacock's tail) become so spectacularly elaborate.

AP framing: Sexual selection is not a violation of "survival of the fittest" — it is a clarification of it. Fitness = total reproductive success, and a trait that trades some survival for a larger mating gain can have the highest net fitness of all, which is why it spreads.

Exit Ticket — Natural vs. Sexual Selection

Adaptation Mechanisms — Sources of Variation and Forces of Evolution

Part 4 of 7

Natural selection can only sculpt variation that already exists; it does not create it. So we must answer two questions:

1. Where does heritable variation come from? (the raw material of evolution)
2. What forces actually change allele frequencies? (the mechanisms of evolution — selection is only one of five)

An adaptation is a heritable trait that increases an organism's fitness in its environment — but crucially, adaptation is a population-level outcome of these forces acting over generations, not something an individual does on demand.

Anchor idea: Mutation + recombination + gene flow generate and reshuffle variation. Natural selection, genetic drift, gene flow, mutation, and non-random mating then change allele frequencies. Evolution = the net result.

Sources of Genetic Variation

Three points students must keep straight:

• Mutation is the only source of brand-new alleles. Recombination and gene flow rearrange or import alleles but cannot invent a sequence that did not exist somewhere.
• Mutations are random with respect to need. A bacterium does not mutate toward resistance because an antibiotic is present; mutations occur regardless, and selection then acts on whichever happen to be beneficial.
• Most evolutionarily important variation in sexual species at any given moment comes from recombination shuffling the standing pool of alleles, but that pool was ultimately built by mutation.

The Five Forces That Change Allele Frequencies

A population stays in Hardy-Weinberg equilibrium (allele frequencies constant) only if none of these five forces act. Each force is therefore a violation of a Hardy-Weinberg condition (see Part 5). Memorize this table — it is the conceptual core of the entire unit.

Two subtleties that are heavily tested:

• Non-random mating changes GENOTYPE frequencies, not ALLELE frequencies. Inbreeding, for instance, raises the frequency of homozygotes and lowers heterozygotes, but the underlying $p$ and $q$ are unchanged. (It can, however, expose recessive alleles to selection, indirectly enabling evolution.)
• Mutation alone is far too slow to change allele frequencies appreciably over a few generations; its evolutionary importance is as the source of variation, not as a frequency-changing force on its own.

Genetic Drift: Bottlenecks and Founder Effects

Genetic drift is the change in allele frequencies due to random sampling from one generation to the next. Like flipping a coin only a few times, small samples deviate from expectation by chance. The smaller the population, the stronger the drift. Two special cases:

• Bottleneck effect: A drastic, often disaster-driven reduction in population size (disease, hunting, habitat loss). The few survivors carry only a random subset of the original gene pool, so allele frequencies shift by chance and overall genetic diversity drops. Example: northern elephant seals were hunted to ~20 individuals; the recovered population (now > 100,000) has extremely low genetic variation. Cheetahs show a similar genetic signature.
• Founder effect: A few individuals colonize a new, isolated area (e.g., an island). The new population's allele frequencies reflect only the founders' alleles, not the source population's, and can differ markedly by chance. Example: the high frequency of certain rare alleles in genetically isolated human populations such as the Amish or settlers of remote islands.

AP trap: Drift is random, not adaptive — it can increase the frequency of a harmful or neutral allele purely by chance. It is most powerful in small populations and negligible in very large ones. Do not describe drift as "the population adapting"; nothing is being optimized.

Checkpoint — Forces and Drift

Worked Example — Founder Effect Changes Allele Frequency

A large mainland beetle population has allele frequencies $p = 0.70$ (allele A) and $q = 0.30$ (allele a) at a color locus. A storm blows 10 beetles onto a remote island, where they establish a new population. By chance, the founders happen to consist of:

Compute the island (founder) allele frequencies by allele-counting. Total alleles $= 2N = 2(10) = 20$.

$p_{island} = \frac{10}{20} = 0.50$ and $q_{island} = \frac{10}{20} = 0.50$

Compare to the source population:

Interpretation: Nothing about fitness changed — the island beetles were not "better adapted." Purely by the chance composition of the 10 founders, the rare a allele jumped from 0.30 to 0.50, a change of $\Delta q = +0.20$ in a single colonization event. This is the founder effect: a small founding sample carries an unrepresentative slice of the source gene pool, so the new population's allele frequencies differ from the source by chance. With only 10 founders, sampling error is large; had 10,000 beetles colonized, $p_{island}$ would lie very close to 0.70.

Key takeaway: Both founder effects and bottlenecks are genetic drift — random allele-frequency change driven by small sample size. The smaller the founding group, the larger the expected deviation from the source population.

Coevolution, Convergent vs. Divergent Evolution

Coevolution is reciprocal evolutionary change between two interacting species, where each acts as a selective pressure on the other. Examples: predator–prey "arms races" (faster cheetahs select for faster gazelles, and vice versa); flowering plants and their specific pollinators; hosts and parasites.

Convergent vs. divergent evolution — and the structures they leave behind:

Distinguishing the structures is a high-frequency AP item:

• Homologous = shared ancestry, possibly different functions (evidence of common descent → divergent evolution).
• Analogous = shared function, independent origins (evidence of convergent evolution; NOT close relatedness).

Trap: Two species looking alike (e.g., a dolphin and a shark both being sleek and finned) does not mean they are closely related. Analogous structures arise from convergent evolution under similar selection pressures, and they mislead if used to infer ancestry. Use homologies and molecular data — not analogies — to build evolutionary relationships.

Worked Example — Gene Flow Homogenizes Two Populations

Gene flow has the opposite effect of drift: instead of making isolated populations diverge by chance, migration makes populations more genetically similar. A quick calculation shows how.

Setup. Two beetle populations are isolated at a color locus:

Now suppose the island population is, after a migration event, composed of 80% long-time residents and 20% new migrants from the mainland. What is the island's new $p$?

Step — Weighted average of the contributing allele frequencies:

$p_{new} = (0.80)(p_{island}) + (0.20)(p_{mainland})$

$p_{new} = (0.80)(0.30) + (0.20)(0.90) = 0.24 + 0.18 = 0.42$

Result: The island's frequency of A rose from 0.30 to 0.42 in one migration event, $\Delta p = +0.12$, moving it toward the mainland value of 0.90. The recessive allele correspondingly fell, $q_{new} = 1 - 0.42 = 0.58$. With continued gene flow, the two populations' allele frequencies converge.

Contrast with drift. Note the direction of each force:

• Gene flow makes populations more similar (homogenizing) and can introduce alleles that are new to a population.
• Genetic drift (founder effect, bottleneck) makes small, isolated populations diverge from one another by chance.

AP synthesis: This is why isolation (no gene flow) is a prerequisite for speciation — gene flow continually blends populations back together, preventing them from diverging into separate species. Cut off gene flow, and drift plus differing selection can drive two populations apart.

Exit Ticket — Part 4 Synthesis

The Hardy-Weinberg Principle

Part 5 of 7

The Hardy-Weinberg principle is the null model of population genetics. It describes the allele and genotype frequencies expected in a population that is NOT evolving. By comparing a real population's data to this idealized prediction, we can detect whether (and how much) evolution is occurring — much as a control group lets you detect the effect of a treatment.

Hardy-Weinberg gives us two equations:

$p + q = 1$

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

where $p$ is the frequency of the dominant allele and $q$ is the frequency of the recessive allele at a locus with two alleles.

Anchor idea: Hardy-Weinberg is a baseline for comparison, not a claim that real populations are static. If observed genotype frequencies deviate from the predicted $p^2$, $2pq$, $q^2$, at least one evolutionary force (selection, drift, gene flow, mutation, or non-random mating) is acting.

The Five Conditions for Hardy-Weinberg Equilibrium

A population remains in equilibrium (no change in allele frequencies) only if ALL FIVE conditions hold. Each condition is the absence of one of the five evolutionary forces from Part 4.

A handy way to remember the spirit of these: a Hardy-Weinberg population is large, isolated, non-mutating, randomly mating, and selectively neutral. In reality no natural population perfectly meets all five — which is exactly why the model is useful as a comparison point.

AP trap: Hardy-Weinberg equilibrium does not require that genotype frequencies be equal or that the dominant phenotype be most common. It requires that the frequencies match the predicted $p^2 : 2pq : q^2$ ratios and stay constant across generations. A population can be in equilibrium with $q$ much larger than $p$.

What Each Term Means

For a locus with a dominant allele (A) and recessive allele (a):

Two relationships you will use constantly:

• Carrier frequency = frequency of heterozygotes = $2pq$. Carriers show the dominant phenotype but secretly carry one recessive allele.
• Dominant phenotype frequency = $p^2 + 2pq$ (both AA and Aa look dominant). This is why you can almost never read $p$ directly off the phenotypes — but you can read $q$ off the recessive phenotype, because only aa shows it.

The master move: The recessive phenotype frequency equals $q^2$ (only aa shows it). So $q = \sqrt{q^2}$ lets you back-calculate the recessive allele frequency from the observed fraction of recessive individuals, then $p = 1 - q$, and finally all genotype frequencies. Almost every Hardy-Weinberg problem starts here.

Worked Example (a) — From 16% Recessive Phenotype to All Frequencies

Problem: In a population in Hardy-Weinberg equilibrium, 16% of individuals show the recessive phenotype (genotype aa). Find $q$, $p$, and the frequencies of all three genotypes, including carriers.

Step 1 — Recessive phenotype frequency = $q^2$. Only aa individuals show the recessive phenotype, so

$q^2 = 0.16$

Step 2 — Solve for $q$.

$q = \sqrt{q^2} = \sqrt{0.16} = 0.4$

Step 3 — Solve for $p$ using $p + q = 1$.

$p = 1 - q = 1 - 0.4 = 0.6$

Step 4 — Compute genotype frequencies.

Check: $0.36 + 0.48 + 0.16 = 1.00$. ✓

Answers: $q = 0.4$, $p = 0.6$; AA = 36%, carriers (Aa) = 48%, aa = 16%. Note that the carrier frequency (48%) is far higher than the affected (16%) frequency — a recessive condition that is rare in appearance can be carried by a large fraction of the population. That insight is a favorite exam point.

Checkpoint — Back-Calculating from the Recessive Phenotype

Worked Example (b) — From Allele Frequencies to Next-Generation Genotype COUNTS

Problem: A randomly mating population of 2000 mice is in Hardy-Weinberg equilibrium at a coat-color locus with $p = 0.8$ (allele B) and $q = 0.2$ (allele b). Predict the number of individuals of each genotype expected in the next generation.

Step 1 — Compute the genotype frequencies.

Check: $0.64 + 0.32 + 0.04 = 1.00$. ✓

Step 2 — Multiply each frequency by the population size $N = 2000$ to get counts.

Answers: about 1280 BB, 640 Bb, 80 bb. Because the population meets the equilibrium conditions, these expected counts also represent the next generation — under Hardy-Weinberg, frequencies do not change from one generation to the next. This is the "no evolution" prediction against which real data are compared.

Note the workflow direction: Example (a) went backward (phenotype → allele frequencies). Example (b) goes forward (allele frequencies → genotype counts). AP problems travel in both directions, so practice each.

Worked Example (c) — "Is This Population in Equilibrium?"

Problem: A biologist samples 1000 snails and records the genotypes directly: 490 RR, 420 Rr, 90 rr. Is this population in Hardy-Weinberg equilibrium?

Step 1 — Find the actual allele frequencies by allele-counting. Total alleles $= 2N = 2000$.

• R count $= 2(\text{RR}) + 1(\text{Rr}) = 2(490) + 420 = 980 + 420 = 1400$
• r count $= 2(\text{rr}) + 1(\text{Rr}) = 2(90) + 420 = 180 + 420 = 600$

$p = \frac{1400}{2000} = 0.70 \qquad q = \frac{600}{2000} = 0.30$

Step 2 — Predict the genotype counts IF the population were in equilibrium (using $p = 0.70$, $q = 0.30$, $N = 1000$).

Step 3 — Compare observed vs. expected.

Conclusion: Observed = expected for every genotype, so this population IS in Hardy-Weinberg equilibrium — there is no evidence of evolutionary change at this locus. (In Part 6 we make this comparison rigorous using a chi-square goodness-of-fit test when the observed and expected values differ.)

The general procedure for an "is it in equilibrium?" question: (1) compute actual allele frequencies from the genotype counts by allele-counting; (2) use those $p$ and $q$ to predict equilibrium genotype counts; (3) compare predicted to observed. A large discrepancy signals that an evolutionary force is acting.

Exit Ticket — Predicting and Testing Equilibrium

Problem-Solving Workshop

Part 6 of 7

This part is pure practice. We work three multi-step problems end to end, the kind that appear on AP free-response and the hardest multiple-choice items:

1. Carrier frequency in a human genetics context.
2. Selection against recessive homozygotes — how $q$ changes over a single generation.
3. Chi-square goodness-of-fit — a statistical test of whether a population is in Hardy-Weinberg equilibrium.

Keep two toolkits handy:

• Hardy-Weinberg: $p + q = 1$ and $p^2 + 2pq + q^2 = 1$; recessive phenotype $= q^2$, so $q = \sqrt{q^2}$; carriers $= 2pq$.
• Allele-counting: $p = \frac{2(\text{hom. dom.}) + (\text{het.})}{2N}$.

Strategy: Always (1) define what allele $p$ and $q$ stand for, (2) find $q$ first (usually from the recessive phenotype), (3) get $p = 1 - q$, then (4) compute whatever the question asks. Label every number with its meaning.

Problem 1 — Carrier Frequency (Cystic Fibrosis style)

Problem: Cystic fibrosis is an autosomal recessive condition. In a population in Hardy-Weinberg equilibrium, the disease (genotype aa) occurs in 1 out of every 2500 newborns. (a) What is the frequency of the recessive allele $q$? (b) What fraction of the population are unaffected carriers? (c) In a town of 50,000 people, about how many carriers are there?

Step 1 — Recessive phenotype frequency = $q^2$.

$q^2 = \frac{1}{2500} = 0.0004$

Step 2 — Solve for $q$.

$q = \sqrt{0.0004} = 0.02$

Step 3 — Solve for $p$.

$p = 1 - q = 1 - 0.02 = 0.98$

Step 4 (b) — Carrier frequency = $2pq$.

$2pq = 2(0.98)(0.02) = 0.0392 \approx 0.039$

So about 3.9% of the population are carriers — roughly 1 in 25 people.

Step 5 (c) — Number of carriers in 50,000 people.

$0.0392 \times 50{,}000 = 1960 \text{ carriers}$

Answers: $q = 0.02$; carrier frequency $\approx 3.9\%$; about 1960 carriers in the town.

Headline insight: Although only 1 in 2500 are affected, about 1 in 25 are carriers — roughly 100 times as many people carry the allele as express the disease. Rare recessive disorders hide an enormous reservoir of heterozygous carriers, because $2pq$ greatly exceeds $q^2$ when $q$ is small.

Problem 2 — Selection Against the Recessive Homozygote

Problem: A population starts with allele frequencies $p = 0.6$ (A) and $q = 0.4$ (a), in Hardy-Weinberg proportions. The recessive homozygote (aa) is lethal before reproduction (relative fitness $w_{aa} = 0$); AA and Aa both have fitness 1. After one generation of this selection, what is the new value of $q$?

Step 1 — Starting genotype frequencies (per 1.0, or imagine $N = 1000$).

Step 2 — Apply selection (remove all aa). Survivors: 360 AA + 480 Aa = 840 individuals; all 160 aa die.

Step 3 — Count alleles among survivors. Survivors carry $2 \times 840 = 1680$ alleles.

• A count $= 2(360) + 1(480) = 720 + 480 = 1200$
• a count $= 2(0) + 1(480) = 480$ (only the surviving Aa heterozygotes still carry a)

Step 4 — New allele frequencies.

$q' = \frac{480}{1680} \approx 0.286 \qquad p' = \frac{1200}{1680} \approx 0.714$

Shortcut formula (worth knowing). For complete selection against a lethal recessive ($w_{aa}=0$), the recessive allele frequency after one generation is

$q' = \frac{q}{1 + q}$

Check: $q' = \frac{0.4}{1 + 0.4} = \frac{0.4}{1.4} \approx 0.286$. ✓ — matches the long calculation.

Answer: $q$ falls from 0.40 to about 0.286 in one generation, $\Delta q \approx -0.114$.

Why selection against a recessive is SLOW at low $q$: the heterozygous Aa survivors still carry and transmit a alleles, "hiding" them from selection. Using $q' = \frac{q}{1+q}$: starting at $q = 0.01$ gives $q' \approx 0.0099$ — an almost imperceptible drop. This is precisely why deleterious recessive alleles persist in populations for many generations and can never be fully eliminated by selection against the homozygote alone.

Checkpoint — Carriers and Selection

Problem 3 — Chi-Square Goodness-of-Fit Test for Hardy-Weinberg

When observed genotype counts differ from the Hardy-Weinberg prediction, we need a statistical test to decide whether the difference is real or just sampling noise. The chi-square ($\chi^2$) goodness-of-fit test does this.

$\chi^2 = \sum \frac{(O - E)^2}{E}$

where $O$ = observed count and $E$ = expected count for each category, summed over all categories.

Problem: A biologist samples 1000 beetles and observes 600 AA, 280 Aa, 120 aa. Test whether the population is in Hardy-Weinberg equilibrium at $\alpha = 0.05$.

Step 1 — Actual allele frequencies (allele-counting). Total alleles $= 2000$.

• A count $= 2(600) + 280 = 1480 \Rightarrow p = \frac{1480}{2000} = 0.74$
• a count $= 2(120) + 280 = 520 \Rightarrow q = \frac{520}{2000} = 0.26$

Step 2 — Expected counts under equilibrium (using $p = 0.74$, $q = 0.26$, $N = 1000$).

Step 3 — Compute $\chi^2$ term by term.

$\chi^2 \approx 5.01 + 28.54 + 40.62 = 74.2$

Step 4 — Degrees of freedom. For a Hardy-Weinberg chi-square, $df = (\text{number of genotype classes}) - (\text{number of alleles}) = 3 - 2 = 1$. (We estimated one independent allele frequency from the data, since $p$ and $q$ are linked by $p + q = 1$.)

Step 5 — Compare to the critical value. At $df = 1$ and $\alpha = 0.05$, the critical $\chi^2 = 3.84$.

Since $\chi^2 = 74.2 \gg 3.84$, we REJECT the null hypothesis of Hardy-Weinberg equilibrium.

Interpretation: The observed genotypes deviate far more from expectation than chance allows — there is a striking deficit of heterozygotes (280 observed vs. ~385 expected) and an excess of both homozygotes. This pattern is consistent with non-random mating (inbreeding) or selection against heterozygotes. The population is evolving (or its mating is non-random) at this locus.

Decision rule: If $\chi^2 \geq$ critical value → reject the null (significant deviation; the population is NOT in equilibrium). If $\chi^2 <$ critical value → fail to reject (the deviation is within chance; consistent with equilibrium). A large $\chi^2$ means a poor fit to the "no-evolution" model.

Exit Ticket — Chi-Square Reasoning

AP Review — Natural Selection and Adaptation

Part 7 of 7

This review pulls the unit together and inoculates you against the most common AP traps. Use the synthesis map to see how every concept connects, then drill the traps and finish with AP-style application questions (including a Hardy-Weinberg computation).

Big-picture synthesis map

One-sentence summary of the unit: Heritable variation (ultimately from mutation) is acted on by selection and other forces to change a population's allele frequencies over generations, producing adaptation — and Hardy-Weinberg is the no-evolution baseline we measure that change against.

The AP Trap Table — Misconceptions That Lose Points

The two distinctions students confuse most:

• Allele frequency vs. genotype frequency. $p$ and $q$ are allele frequencies; $p^2$, $2pq$, $q^2$ are genotype frequencies. Selection and drift change allele frequencies; non-random mating changes only genotype frequencies.
• Phenotype vs. genotype as the target. Selection "sees" only the phenotype (what survives and reproduces), but evolution is the resulting change in the underlying genetic makeup.

Exam habit: When you read an answer choice, check it against this table. Choices that say an individual "evolved," that equate fitness with strength, or that describe evolution as goal-directed are almost always distractors.

Rapid Diagnosis — Match the Scenario to the Mechanism

On the AP exam you must read a short scenario and instantly name the right mechanism. Use these keyword triggers:

Quantitative trigger phrases:

• "____% show the recessive phenotype" → that percent = $q^2$; take $q = \sqrt{q^2}$, then $p = 1 - q$.
• "How many carriers?" → $2pq \times N$ (heterozygotes only).
• "Is it in equilibrium?" → compute actual $p, q$ by allele-counting, predict $p^2, 2pq, q^2$ counts, compare (chi-square if needed).
• "Recessive homozygote is lethal; find next-gen $q$" → $q' = \frac{q}{1+q}$.

FRQ tip: When a free-response question asks you to "explain how the population evolved," structure your answer as: (1) source of variation → (2) the specific force acting → (3) differential reproduction / chance change → (4) resulting shift in allele frequencies across generations. Name the population, never the individual, as the unit that evolves.

AP Application — Conceptual Synthesis

AP Application — Quantitative Hardy-Weinberg