Mendel's Laws of Inheritance

Before we define anything, consider a paradox that puzzled naturalists for centuries: if inheritance worked by blending (as was assumed before Mendel), then after many generations every population should converge on a gray average, and variation would disappear. But variation persists. In fact, a trait can skip a generation entirely and reappear.

Mendel’s genius was to count carefully rather than describe qualitatively. By crossing true-breeding pea plants and tracking thousands of offspring, he saw that traits appear in discrete, reproducible ratios — not a smooth spectrum.

Check your understanding

Mendel observed a 2.96:1 ratio of round to wrinkled seeds, not exactly 3:1. What does this deviation tell us?

• Mendel's model is slightly wrong — the true ratio isn't 3:1
• The deviation is expected sampling variation around a true 3:1 probability
• Dominant traits are slightly more common than the model predicts

The mechanism that implements this law is meiosis I, when homologous chromosomes are pulled to opposite poles of the dividing cell. Each homolog carries one allele; they are physically separated. ^{[Molecular Biology of the Cell]}

Unpacking the vocabulary precisely

Term	Precise definition	Common confusion
Allele	One variant of a gene at a given locus	Often conflated with “gene” itself
Locus	The chromosomal address (position) of a gene	Not the gene’s function
Homozygous	Both alleles at a locus are identical (AA or aa)	“Pure-breeding” is a functional consequence, not a definition
Heterozygous	The two alleles differ (Aa)	Does not imply which phenotype results
Genotype	The full allelic constitution (AA, Aa, aa)	Genotype ≠ phenotype (complete dominance makes Aa look like AA)
Phenotype	The observable trait expressed	Same phenotype can arise from different genotypes

From genotype to gametes: a worked derivation

An organism with genotype Aa produces two classes of gametes:

• Gamete carrying A — probability 1/2
• Gamete carrying a — probability 1/2

The probability that a diploid offspring receives a specific genotype is the product of independent gamete draws:

• P(AA) = P(A from parent 1) × P(A from parent 2) = 1/2 × 1/2 = 1/4
• P(Aa) = P(A from P1)×P(a from P2) + P(a from P1)×P(A from P2) = 2/4 = 1/2
• P(aa) = 1/2 × 1/2 = 1/4

This is the 1:2:1 genotype ratio. With complete dominance (A masks a), genotypes AA and Aa give the same phenotype, collapsing to a 3:1 phenotype ratio.

Show the formal probability derivation (binomial model)

For a cross Aa × Aa, the offspring genotype distribution follows a binomial model. Let p = P(A gamete) = 0.5, q = P(a gamete) = 0.5. In a diploid offspring:

$P(k \text{ copies of A}) = \binom{2}{k} p^k q^{2-k}$

• k=2 (AA): $\binom{2}{2}(0.5)^2(0.5)^0 = 0.25$
• k=1 (Aa): $\binom{2}{1}(0.5)^1(0.5)^1 = 0.50$
• k=0 (aa): $\binom{2}{0}(0.5)^0(0.5)^2 = 0.25$

This is precisely the binomial distribution $B(2, 0.5)$, confirming that Mendel’s First Law is a statement about fair Bernoulli trials at the gamete level.

Now explore the Punnett square — a 2D tool for visualizing all gamete combination outcomes:

Check your understanding

A plant homozygous dominant (AA) is crossed with a plant homozygous recessive (aa). What fraction of the F1 offspring will show the dominant phenotype under complete dominance?

• 1/4
• 1/2
• All of them (100%)
• 3/4

Mendel’s pea traits happened to show complete dominance — one allele fully masks the other. This is the exception, not the rule.

The pedagogical consequence: the 3:1 ratio is not universal. It follows from complete dominance in a monohybrid Aa × Aa cross. Change the dominance mode and the phenotype ratio changes even though the genotype ratio (1:2:1) does not.

Check your understanding

In an incomplete dominance cross (Aa × Aa), what is the expected phenotype ratio?

• 3:1 (dominant : recessive)
• 1:2:1 (full A : intermediate : full a)
• 1:1 (dominant : recessive)
• All offspring are intermediate

The mechanism: when two loci are on different chromosomes (or far apart on the same chromosome), the orientation of homologous pairs at metaphase I is random and independent for each bivalent. ^{[Molecular Biology of the Cell]}

For a dihybrid parent (AaBb), this means four gamete types are produced with equal 25% probability each: AB, Ab, aB, ab.

A dihybrid cross (AaBb × AaBb) then produces a 4×4 Punnett square with 16 equally probable cells. Under complete dominance at both loci, the phenotype classes are:

$\frac{9 \text{ A\_B\_} : 3 \text{ A\_bb} : 3 \text{ aaB\_} : 1 \text{ aabb}}{16}$

This multiplicative rule works only because the two loci are independent — the product rule of probability applies.

Check your understanding

Two loci are on different chromosomes. Parent 1 has genotype AaBb; parent 2 has genotype aabb. What fraction of offspring are expected to have genotype aabb?

• 1/16
• 1/4
• 1/2
• 0 — you need two heterozygous parents to get double recessives

Mendel’s ratios are probabilistic predictions. How do you know whether deviations from 3:1 (or 9:3:3:1) are just sampling noise or evidence that the model is wrong?

The answer is the chi-square goodness-of-fit test — the same test Mendel himself did not formally run, though his data imply he understood its logic.

Under the null hypothesis that the data follow the expected Mendelian ratio, χ² follows a chi-square distribution with degrees of freedom = (number of classes − 1). A high χ² and low p-value means the data deviate more than chance alone predicts.

Check your understanding

A student counts 280 dominant and 70 recessive offspring in an Aa × Aa cross (N=350). She computes χ² = 2.67 (df=1). What should she conclude?

• Reject the 3:1 model — the counts differ from 262.5 and 87.5
• Fail to reject — χ²=2.67 < 3.841 critical value, so the data are consistent with 3:1
• The data prove the 3:1 model is correct
• The sample size is too small to conclude anything

Mendel’s framework is powerful but not universal. Understanding its limits is what separates a beginner’s grasp from a working geneticist’s toolkit.

	Mendelian assumption	When it breaks down	Mechanism
Complete dominance	Universal 3:1 phenotype ratio	Incomplete dominance, codominance, haploinsufficiency	Gene dosage; both allele products matter
Independent assortment	Genes on different chromosomes assort independently	Linked genes — same chromosome, close proximity	Recombination frequency < 50%
One gene, one phenotype	Each locus controls one trait	Epistasis: gene A masks/modifies gene B's expression	Regulatory and pathway interactions
Constant allele effects	Allele always produces the same phenotype	Penetrance < 100%; variable expressivity	Modifier genes, environment, stochasticity
Equal gamete viability	All gamete types survive equally	Meiotic drive / segregation distortion	Selfish genetic elements bias transmission

Check your understanding

In a dihybrid cross (AaBb × AaBb) with duplicate recessive epistasis — where the phenotype requires at least one dominant allele at EITHER locus — what is the expected phenotype ratio?

• 9:3:3:1
• 15:1 (any dominant phenotype : double recessive)
• 3:1
• 9:7