2.3 Probability and Pedigree Analysis

Probability in Genetic Crosses

Probability in Genetic Inheritance

Mendel's two laws form the foundation for predicting how traits pass from parents to offspring.

• Law of Segregation: During gamete formation, the two alleles for a gene separate so each gamete carries only one allele (either A or a, not both).
• Law of Independent Assortment: Alleles of different genes sort independently during gamete formation, producing new allele combinations in the gametes. This applies when genes are on different chromosomes (or far apart on the same chromosome).

These laws let you calculate the probability of specific genotypes in offspring. The numbers below assume a cross between two heterozygous parents:

Monohybrid cross (Aa × Aa):

• $P(AA) = 1/4$
• $P(Aa) = 1/2$
• $P(aa) = 1/4$

Dihybrid cross (AaBb × AaBb):

• $P(AABB) = 1/16$
• $P(AaBb) = 4/16$
• $P(aabb) = 1/16$

Phenotype probabilities depend on the dominance relationship between alleles. With complete dominance in a monohybrid cross of two heterozygotes:

• Dominant phenotype $P(A\_) = 3/4$ (e.g., purple flowers in pea plants, where both AA and Aa look the same)
• Recessive phenotype $P(aa) = 1/4$ (e.g., white flowers, expressed only when homozygous recessive)

Combining Probabilities Across Genes

Two rules let you handle multi-gene problems without drawing enormous Punnett squares:

• Multiplication rule (AND): The probability of two independent events both occurring equals the product of their individual probabilities. For example, in an AaBb × AaBb cross, $P(aabb) = P(aa) \times P(bb) = 1/4 \times 1/4 = 1/16$.
• Addition rule (OR): The probability of either of two mutually exclusive events occurring equals the sum of their individual probabilities. For example, the probability of a heterozygote from Aa × Aa is $P(Aa) = P(A \text{ from mom, a from dad}) + P(a \text{ from mom, A from dad}) = 1/4 + 1/4 = 1/2$.

These rules are especially useful for dihybrid and trihybrid crosses, where Punnett squares get unwieldy fast.

Punnett Squares for Inheritance Visualization

Punnett squares give you a visual way to map out all possible offspring genotypes and their probabilities.

Setting up a monohybrid Punnett square (e.g., Aa × Aa):

1. List the gametes from one parent across the top (A, a).
2. List the gametes from the other parent down the left side (A, a).
3. Fill in each cell by combining the column allele with the row allele.
4. Count the resulting genotypes to get your ratios: 1 AA : 2 Aa : 1 aa.

Setting up a dihybrid Punnett square (e.g., AaBb × AaBb):

1. Determine all possible gamete types for each parent. An AaBb parent produces four gamete types: AB, Ab, aB, ab.
2. Arrange one parent's gametes across the top and the other's down the left side of a 4×4 grid.
3. Fill in each of the 16 cells by combining alleles from both parents.
4. Count phenotype classes to get the classic 9:3:3:1 ratio (assuming complete dominance at both loci).

Pedigree Analysis

Pedigree Symbols and Construction

A pedigree is a diagram that maps a trait through a family across generations. Knowing the standard symbols is essential for reading and constructing them.

• Squares represent males; circles represent females.
• Shaded (filled-in) symbols indicate affected individuals; unshaded symbols indicate unaffected individuals.
• Half-shaded symbols represent known carriers of a recessive disorder.
• A horizontal line connecting a male and female represents a mating pair.
• A vertical line descending from a mating pair connects to their offspring.
• Siblings are shown as branches off the same horizontal sibship line.

Generations are labeled with Roman numerals (I, II, III) from top to bottom. Generation I is the oldest generation shown. Individual members within a generation are numbered left to right (e.g., II-3 is the third person in the second generation).

Identifying Inheritance Patterns

The most important skill in pedigree analysis is figuring out which inheritance pattern fits the data. Here are the hallmarks of each.

Autosomal Dominant:

1. The trait appears in every generation (no skipping).
2. Every affected individual has at least one affected parent.
3. Male-to-male transmission can occur (which rules out X-linkage).
4. Unaffected individuals who mate with affected heterozygotes have a 50% chance of affected offspring.

• Example: Huntington's disease.

Autosomal Recessive:

1. The trait can skip generations because carriers (heterozygotes) are phenotypically normal.
2. Two unaffected parents can have affected children if both are carriers.
3. Affected individuals are more common in families with consanguinity (mating between relatives), because related parents are more likely to both carry the same recessive allele.

• Example: Cystic fibrosis.

X-linked Recessive:

1. Far more affected males than females, because males need only one copy of the recessive allele (they're hemizygous for X-linked genes).
2. No male-to-male transmission. Fathers pass their Y chromosome to sons, so an affected father cannot give his X-linked allele to a son.
3. Affected males typically inherit the allele from carrier mothers. All daughters of an affected father will be carriers (at minimum).

• Example: Hemophilia A.

Y-linked:

1. Only males are ever affected.
2. Every affected male has an affected father, and all sons of an affected male are affected.
3. These traits are rare and pass strictly through the paternal line.

• Example: Some forms of male infertility.

Carrier Probability in Recessive Disorders

Determining carrier status is a common pedigree problem. Here's how to think through it:

Autosomal recessive:

• If a couple has an affected child (aa), both parents must be carriers (Aa). They are called obligate carriers because there's no other explanation.
• For unaffected siblings of an affected individual: given that both parents are Aa, the probability of being a carrier among the unaffected offspring is $2/3$, not $1/2$. That's because you already know the sibling is not aa, so you're choosing from the remaining three outcomes (1 AA : 2 Aa), and two out of three of those are carriers.
• Population carrier frequencies vary by disorder and population. For example, roughly 1 in 25 people of European descent are carriers of cystic fibrosis.

X-linked recessive:

• If a woman has an affected son, she is an obligate carrier (100% probability).
• A woman whose brother is affected but whose parents are unaffected has a $1/2$ probability of being a carrier (her mother is an obligate carrier, and each daughter has a 50% chance of inheriting the X chromosome with the recessive allele).
• Carrier frequency in females is higher than the disease frequency in males. For Duchenne muscular dystrophy, about 1 in 2,500 females are carriers, while about 1 in 5,000 males are affected.

Complications in Pedigree Analysis

Real pedigrees don't always match textbook patterns perfectly. Several factors can muddy the picture.

Incomplete penetrance: Not everyone with a disease-causing genotype actually shows the phenotype. For example, BRCA1 mutations significantly increase breast cancer risk, but not every carrier develops cancer. This can make a dominant trait appear to "skip" a generation.

Variable expressivity: Individuals with the same genotype can show different severity of the phenotype. Neurofibromatosis type 1 is a classic example: some individuals have only mild skin findings, while others develop serious tumors.

Genetic heterogeneity: The same phenotype can result from mutations in different genes. This comes in two forms:

• Locus heterogeneity: Mutations in different genes produce the same disorder (e.g., hereditary hearing loss can be caused by mutations in dozens of different genes).
• Allelic heterogeneity: Different mutations within the same gene cause the same disorder.

Environmental factors: Gene expression can be modified by the environment. Phenylketonuria (PKU) is a genetic disorder, but its severity depends heavily on dietary phenylalanine intake. Gene-environment interactions like smoking increasing lung cancer risk in genetically predisposed individuals also complicate simple inheritance predictions.

Genetic Testing for Inheritance Confirmation

When pedigree analysis alone can't give a definitive answer, genetic testing fills the gap. Testing can:

1. Confirm specific genetic variants in an individual (e.g., identifying CFTR mutations to diagnose cystic fibrosis).
2. Determine carrier status for recessive disorders in at-risk family members.
3. Estimate the likelihood of developing a late-onset condition (e.g., presymptomatic testing for Huntington's disease).
4. Inform family planning decisions and guide genetic counseling.