Table of Contents
Overview: What Mendel Wanted to Find Out
Gregor Mendel was interested in how traits (such as flower color or seed shape in pea plants) are passed from parents to offspring. He did controlled crossbreeding experiments and counted large numbers of offspring. From regular patterns in his results he formulated simple rules of inheritance that now bear his name: the Mendelian laws.
These laws describe how individual hereditary factors (today: alleles of genes) are distributed during sexual reproduction and how they appear in offspring as observable traits (phenotypes).
In this chapter we focus on:
- What the Mendelian laws say in simple terms
- How they appear in example crosses
- Where and why they apply (and where they do not)
The technical terms (gene, allele, genotype, phenotype, dominant, recessive, etc.) and Mendel’s specific experiments are covered in separate chapters and are only used here as tools.
Mendel’s First Law: Law of Uniformity
Statement of the Law
If two individuals of the same species that differ in only one trait and are both pure-breeding (homozygous) for that trait are crossed, then all offspring of this first filial generation (F₁) are:
- Genetically identical (with respect to that gene), and
- Phenotypically uniform (they all show the same appearance for that trait).
This is Mendel’s law of uniformity (or law of the F₁ generation).
Simple Example: One Trait, Two Pure Lines
Consider a gene with two alleles:
A– dominanta– recessive
Two pure-breeding parents:
- Parent 1:
AA - Parent 2:
aa
Their gametes:
- Parent 1 produces only
A - Parent 2 produces only
a
Possible combinations in the F₁ generation:
- All offspring receive
Afrom one parent andafrom the other → all areAa.
Result:
- Genotype F₁: all
Aa(uniform) - Phenotype F₁: all show the dominant trait (because
Ais dominant overa)
Using a Punnett square:
$$
\begin{array}{c|cc}
& A & A \\
\hline
a & Aa & Aa \\
a & Aa & Aa \\
\end{array}
$$
All four combinations are Aa → complete uniformity.
Scope and Limitations
The law of uniformity holds under these conditions:
- The parents are homozygous (pure-breeding) for the trait in question.
- The trait is controlled by a single gene with clear dominant–recessive or intermediate action.
- The gene is inherited according to Mendel’s rules (no linkage complications, no mitochondrial inheritance, etc.).
If any of these conditions is not fulfilled (for example, the parents are not pure or the trait is influenced by several genes), the F₁ generation will not be completely uniform.
Mendel’s Second Law: Law of Segregation
Statement of the Law
In the formation of gametes, the two alleles of a gene in an individual separate from each other. Each gamete receives only one allele. During fertilization, gametes from two parents combine randomly, restoring the pair of alleles in the offspring.
Consequences:
- Even if the F₁ generation appears uniform, their offspring (F₂) can show different trait combinations.
- Recessive traits that “disappear” in F₁ can reappear in F₂.
Example: Monohybrid Cross (One Trait)
Starting from the F₁ generation of the previous example:
- F₁ individuals: all
Aa - Each F₁ produces two types of gametes in equal frequency:
Aanda
Crossing Aa × Aa:
Punnett square:
$$
\begin{array}{c|cc}
& A & a \\
\hline
A & AA & Aa \\
a & Aa & aa \\
\end{array}
$$
Genotypes in F₂ (ratio 1 : 2 : 1):
AA:Aa:aa= 1 : 2 : 1
Phenotypes in F₂ (dominant–recessive case):
AAandAa: show the dominant traitaa: shows the recessive trait
Phenotypic ratio:
- Dominant : recessive = 3 : 1
So from two uniform F₁ parents, the F₂ generation splits into:
- 3/4 with the dominant phenotype
- 1/4 with the recessive phenotype
This 3 : 1 ratio is typical of a monohybrid cross when:
- One gene controls the trait
- There are two alleles
- One allele is completely dominant over the other
Genotype vs Phenotype Ratios
For a monohybrid cross Aa × Aa:
- Genotype ratio (F₂): 1
AA: 2Aa: 1aa - Phenotype ratio (F₂): 3 dominant : 1 recessive (if complete dominance)
This distinction is important: phenotypes can be the same even if genotypes are different.
The Law and Meiosis
Biologically, segregation corresponds to:
- The separation of homologous chromosomes (and thus alleles) into different gametes during meiosis.
- Each gamete gets only one of the two alleles from the parent.
Mendel’s Third Law: Law of Independent Assortment
Statement of the Law
If two (or more) traits are each controlled by different genes that:
- Are located on different chromosomes or
- Are far apart on the same chromosome,
then the alleles of these genes are assorted into gametes independently of each other. In other words, how one gene is inherited does not affect how the other is inherited.
This gives rise to characteristic ratios in crosses involving two traits (dihybrid crosses).
Dihybrid Cross: Two Traits at Once
Consider two genes:
- Gene 1:
A(dominant) /a(recessive) - Gene 2:
B(dominant) /b(recessive)
Two pure-breeding parents:
- Parent 1:
AA BB - Parent 2:
aa bb
F₁ offspring:
- All
Aa Bb(uniform for both traits)
Now cross F₁ with F₁: Aa Bb × Aa Bb.
Gametes and Independent Assortment
Each F₁ individual forms four types of gametes, all equally likely:
AB,Ab,aB,ab
Because assortment is independent, each combination has probability:
- $P(AB) = P(A \text{ and } B) = P(A)\cdot P(B) = \tfrac{1}{2}\cdot\tfrac{1}{2} = \tfrac{1}{4}$
(and analogously for the others)
Punnett Square for a Dihybrid Cross
All combinations of the four gamete types from each parent (4 × 4 = 16 possibilities):
F₁ gametes (top and side): AB, Ab, aB, ab.
Resulting genotypes (16 combinations) lead to the classic 9 : 3 : 3 : 1 phenotypic ratio:
- 9 show both dominant traits (A– B–)
- 3 show dominant A, recessive b (A– bb)
- 3 show recessive a, dominant B (aa B–)
- 1 shows both recessive traits (aa bb)
(“–” means either allele at that position, e.g. A– means AA or Aa.)
This 9 : 3 : 3 : 1 ratio is characteristic of:
- A dihybrid cross
Aa Bb × Aa Bb - Independent assortment
- Complete dominance at each gene
When Independent Assortment Does Not Hold
Independent assortment fails if:
- The genes lie close together on the same chromosome (they are linked).
- Certain chromosomal mechanisms (e.g. non-random segregation) interfere.
Then the combinations of traits do not follow the simple 9 : 3 : 3 : 1 pattern, and some combinations appear more often than others. The details of gene linkage and chromosome theory are treated in a separate chapter; here it is enough to note that Mendel’s law strictly applies only to independently assorting genes.
Dominance Patterns and Their Effect on Ratios
Mendel’s laws describe the distribution of alleles, not the exact appearance of the traits. How the alleles are expressed (dominance pattern) can modify the phenotypic ratios:
- Complete dominance (Mendel’s classic case):
AAandAalook the same (dominant phenotype).- Monohybrid F₂ phenotype ratio: 3 : 1.
- Incomplete (intermediate) dominance:
- Heterozygote (
Aa) has an intermediate phenotype between the two homozygotes. - Monohybrid F₂ phenotype ratio: 1 : 2 : 1 (each genotype has its own visible phenotype).
- Codominance:
- Both alleles are expressed side by side in the heterozygote.
- Again, monohybrid F₂ phenotype ratio: 1 : 2 : 1.
Crucially, the genotypic ratios from Mendel’s segregation (1 : 2 : 1 in F₂) remain the same; only the mapping from genotype to phenotype changes.
Using Mendelian Laws in Practice
Predicting Offspring Ratios
Mendelian laws allow:
- Calculation of expected genotype and phenotype ratios in offspring.
- Use of Punnett squares or probability rules to predict outcomes.
For a monohybrid cross Aa × Aa:
- Probability of each genotype:
P(AA) = \tfrac{1}{4},P(Aa) = \tfrac{1}{2},P(aa) = \tfrac{1}{4}.
For a dihybrid cross with independent assortment:
- Probability of each trait combination can be calculated by multiplying single-gene probabilities.
Example:
- Probability of double recessive (
aa bb) inAa Bb × Aa Bb: - $P(aa) = \tfrac{1}{4}$ and $P(bb) = \tfrac{1}{4}$
- $P(aa\ bb) = \tfrac{1}{4} \cdot \tfrac{1}{4} = \tfrac{1}{16}$.
Test Crosses
A test cross uses Mendelian principles to determine an unknown genotype that shows the dominant phenotype.
Idea:
- Cross the individual with unknown genotype (e.g.
A?) to a homozygous recessive (aa).
If the unknown is:
AA: all offspring show the dominant phenotype (allAa).Aa: offspring show a 1 : 1 ratio of dominant : recessive phenotypes.
Thus Mendel’s uniformity and segregation patterns allow inference of hidden genotypes from observed offspring.
When Mendelian Ratios Do Not Appear
Mendel’s laws are a simplified model. Real inheritance can deviate from Mendelian expectations when, for example:
- A trait is influenced by many genes (polygenic inheritance).
- One gene affects multiple traits (pleiotropy).
- Genes are linked on the same chromosome.
- There are more than two alleles in the population (multiple alleles).
- Environmental factors strongly modify the phenotype.
- There is non-Mendelian transmission (e.g. mitochondrial inheritance).
These exceptions do not contradict the basic idea that hereditary factors segregate and assort, but they need more complex models than Mendel’s original rules.
Summary of the Mendelian Laws
- Law of Uniformity
- Crossing two pure-breeding parents that differ in one (or more) traits gives a uniform F₁ generation for those traits.
- Law of Segregation
- In heterozygotes, the two alleles of a gene separate during gamete formation.
- F₂ generation from a monohybrid
Aa × Aacross shows genotypic ratio 1 : 2 : 1 and, with complete dominance, phenotypic ratio 3 : 1. - Law of Independent Assortment
- Alleles of different genes assort independently into gametes (if the genes are unlinked).
- A dihybrid cross (
Aa Bb × Aa Bb) yields a characteristic 9 : 3 : 3 : 1 phenotypic ratio under complete dominance and independent assortment.
These laws form the classical foundation of genetics and are the starting point for more advanced topics like gene linkage, chromosome theory, and complex inheritance patterns.