Mendelian Inheritance: Pedigrees, Probability, and Chi Square

Studying Inheritance Patterns in Humans

Unlike model organisms, controlled crosses in humans are not ethical or feasible. Instead, geneticists use pedigree analysis to study inheritance patterns of traits and diseases in families.

• Pedigree: A family tree diagram that traces the inheritance of a particular trait across generations.

• Alleles: Genes may exist as a normal (non-disease) allele or a mutant (disease-causing) allele.

• Mendelian diseases: Traits that follow simple dominant or recessive inheritance patterns.

Pedigree Symbols for Human Phenotypes

Standardized symbols are used in pedigrees to represent individuals and their phenotypes:

• Circle: Female

• Square: Male

• Diamond: Sex unspecified or unknown

• Filled symbol: Affected individual

• Unfilled symbol: Unaffected individual

• Half-filled symbol: Carrier (for some pedigrees)

• Diagonal line through symbol: Deceased

• Double horizontal line: Consanguineous mating (between relatives)

• Twin lines: Diverging lines from a single point; horizontal bar for identical (monozygotic) twins

Pedigree Analysis: Dominant vs. Recessive Patterns

Pedigree analysis helps distinguish between dominant and recessive inheritance:

• Recessive inheritance:

  • Two unaffected heterozygotes have a 25% chance of producing an affected child.

  • Two affected individuals produce 100% affected offspring.

• Dominant inheritance:

  • An affected individual usually has at least one affected parent.

  • Rarely, a new mutation may cause the disease in a child with unaffected parents.

Example: Cystic Fibrosis Pedigree

Cystic fibrosis (CF) is a classic example of an autosomal recessive disorder:

• Caused by mutations in the CFTR gene, which encodes a protein regulating ion transport.

• Mutant CFTR leads to ion imbalance and affects organs such as the pancreas, intestine, sweat glands, and lungs.

• Pedigree analysis shows that two unaffected carrier parents can have affected children, consistent with recessive inheritance.

Probability and Statistics in Genetics

Geneticists use probability to predict the outcomes of genetic crosses and assess the likelihood of inheriting certain traits.

• Probability: The chance that a specific event will occur in the future.

• Formula:

• Example: Probability of flipping heads on a coin: or 50%.

Random Sampling Error

Observed outcomes may deviate from expected probabilities due to random sampling error, especially in small samples.

• Large samples: Smaller random error; observed ratios approach expected values.

• Small samples: Greater random error; more deviation from expected ratios.

Product Rule: Probability of Independent Events

The product rule is used to calculate the probability of two or more independent events occurring together.

• Rule: Multiply the probabilities of each independent event.

• Example: For two heterozygous parents (Pp x Pp), probability that first three children all have congenital analgesia (pp):

  • Probability for each child:

  • Combined probability: (1.6%)

Binomial Expansion Equation

The binomial expansion equation calculates the probability of a specific combination of unordered outcomes in a series of independent events.

• Formula:

• Variables:

  • = probability of the outcome

  • = total number of events

  • = number of events in one category

  • = probability of category

  • = probability of the other category ()

• Factorial notation: ;

Example: Two heterozygous brown-eyed parents (Bb x Bb) have five children. Probability that two have blue eyes (bb):

• , , (blue eyes), (brown eyes)

• Calculation: (26%)

Multinomial Expansion Equation

The multinomial expansion equation generalizes the binomial equation for three or more categories.

• Formula:

• Variables:

  • = total number of events

  • = number of events in each category (sum to )

  • = probabilities for each category (sum to 1)

The Chi Square Test: Goodness of Fit

The chi square (χ²) test is a statistical method to evaluate how well observed data fit expected outcomes based on a hypothesis.

• Purpose: To assess whether deviations between observed and expected results are due to random chance.

• Note: The test does not prove a hypothesis correct; it only evaluates the fit.

Chi Square Formula

• = observed value for each category

• = expected value for each category

Chi Square Test: Drosophila Example

Consider a cross involving two traits in Drosophila (fruit flies): wing shape (c+ = straight, c = curved) and body color (e+ = gray, e = ebony).

• F1 generation: all straight wings, gray bodies

• F2 generation (observed):

  • Straight wings, gray body: 193

  • Straight wings, ebony body: 69

  • Curved wings, gray body: 64

  • Curved wings, ebony body: 26

  • Total: 352

Steps in the Chi Square Test

1. Propose a hypothesis: Traits assort independently (Mendel's law).

2. List observed data: As above.

3. Calculate expected values:

   • Straight wings, gray body:

   • Straight wings, ebony body:

   • Curved wings, gray body:

   • Curved wings, ebony body:

4. Apply the chi square formula:

   • Calculates to

5. Determine degrees of freedom (df):

6. Determine the P value: Use a chi square table to find the probability associated with and .

7. Interpret the P value:

   • For , falls between 1.005 (P = 0.80) and 2.366 (P = 0.50).

   • P value is close to 0.80 (80%).

   • High P value (>0.05) means deviations are likely due to random chance; hypothesis is supported.

   • If P < 0.05, the hypothesis is rejected.

Chi Square Critical Values Table (Excerpt)

Summary

• Pedigree analysis is essential for studying human inheritance patterns.

• Probability rules (product, binomial, multinomial) allow prediction of genetic outcomes.

• The chi square test is a key statistical tool for evaluating the fit between observed and expected genetic data.