Mendelian rules for inheritance of single genes

Mendelian inheritance refers to the patterns of trait transmission discovered by Gregor Mendel through experiments with pea plants.

Key terms in Mendelian inheritance

• Allele - One of two or more versions of a gene that determine a trait; for example, a gene for flower color might have a purple allele and a white allele
• Genotype - The combination of alleles an organism has for a particular gene, such as homozygous (two identical alleles) or heterozygous (two different alleles)
• Phenotype - The observable trait resulting from the genotype, such as purple flowers from a dominant allele

For single genes, inheritance follows the law of segregation, which states that each individual has two alleles for a trait, and these alleles separate during gamete (sex cell) formation so that each gamete receives only one allele.

This law leads to predictable patterns in offspring when parents with known genotypes are crossed. For example, in a monohybrid cross (involving one gene with two alleles), the genotypes of offspring can be determined using tools like Punnett squares, which show all possible combinations of parental alleles.

The product rule of probability

Probability is the likelihood of a particular event occurring, expressed as a fraction between 0 and 1. In genetics, probability helps predict the chances of specific genotypes or phenotypes in offspring. The product rule applies when calculating the probability of two or more independent events happening together.

Formula for the product rule

$P(A\text{ and }B)=P(A)\times P(B)$

Where:

• P(A and B) = Probability of both events A and B occurring together
• P(A) = Probability of event A
• P(B) = Probability of event B (assuming A and B are independent, meaning one does not affect the other)

This rule is useful in genetics for combining probabilities from separate parental contributions, such as the chance of inheriting specific alleles from each parent.

The sum rule of probability

The sum rule is used to calculate the probability of one event or another occurring, when the events are mutually exclusive (they cannot happen at the same time).

Formula for the sum rule

$P(A\text{ or }B)=P(A)+P(B)$

Where:

• P(A or B) = Probability of either event A or event B occurring
• P(A) = Probability of event A
• P(B) = Probability of event B (assuming A and B are mutually exclusive)

In genetics, this rule helps calculate the total probability of different paths leading to the same phenotype, such as multiple genotypes that result in the same trait.

Applying probability rules to predict trait ratios in genetic crosses

Probability rules can replace or complement Punnett squares to predict ratios of genotypes and phenotypes in offspring for single-gene crosses. This approach is especially efficient for calculating expected ratios without listing all possibilities.

Steps for applying probability to monohybrid crosses:

1. Identify the parental genotypes and determine the probability of each allele being passed to offspring (based on the law of segregation, each allele has a 50% chance if heterozygous).
2. Use the product rule to find the probability of specific genotype combinations in offspring.
3. If multiple genotypes lead to the same phenotype, use the sum rule to add their probabilities.
4. Convert probabilities to ratios by multiplying by the total number of offspring or expressing as fractions.

These steps assume independent events, aligning with Mendelian rules.

Worked example - Predicting genotype ratios using probability

In a monohybrid cross, both parents are heterozygous for a gene with alleles A (dominant) and a (recessive). Calculate the probability of each possible offspring genotype.

Step 1: Identify probabilities

• Probability of inheriting A from mother: 0.5
• Probability of inheriting a from mother: 0.5
• Probability of inheriting A from father: 0.5
• Probability of inheriting a from father: 0.5

Step 2: Calculate using product rule for each genotype

• AA: (0.5 from mother) × (0.5 from father) = 0.25
• Aa: Two ways - (A from mother × a from father) + (a from mother × A from father) = (0.5 × 0.5) + (0.5 × 0.5) = 0.25 + 0.25 = 0.5 (using sum rule for the two paths)
• aa: (0.5 from mother) × (0.5 from father) = 0.25

Step 3: Interpret ratios

The genotype ratios are 1:2:1 (AA:Aa:aa), or probabilities of 0.25:0.5:0.25.

Worked example - Predicting phenotype ratios using probability

Using the same cross as above (Aa × Aa), where A produces a dominant phenotype (e.g., tall plants) and aa produces recessive (short plants), calculate phenotype probabilities.

Step 1: Identify genotype probabilities

From previous example: AA = 0.25, Aa = 0.5, aa = 0.25

Step 2: Group by phenotype

• Dominant phenotype (AA or Aa): Use sum rule = 0.25 + 0.5 = 0.75
• Recessive phenotype (aa): 0.25

Step 3: Interpret ratios

The phenotype ratio is 3:1 (dominant:recessive), or probabilities of 0.75:0.25.

Assumptions of Mendelian inheritance models

Mendelian models simplify inheritance to make predictions, but they rely on specific assumptions. These ensure the rules apply accurately to idealized scenarios.

Key assumptions:

• Independent segregation - Alleles for a single gene separate independently during gamete formation, with each gamete equally likely to receive either allele
• No linkage - The gene is considered in isolation, without influence from other genes on the same chromosome
• Random mating - Parents mate randomly, with no selection pressures affecting allele frequencies
• Large population size - Predictions assume infinite population size to avoid random fluctuations
• No mutations or gene flow - Allele frequencies remain constant without external changes

These assumptions allow for straightforward probability calculations but may not hold in nature.

Deviations from Mendelian predictions in real populations due to linkage and selection

In real populations, inheritance often deviates from Mendelian ratios because the assumptions do not always apply. Two key factors causing deviations are linkage and selection, which alter expected trait ratios.

Deviation due to linkage

• Linkage occurs when genes are located close together on the same chromosome and tend to be inherited together, violating the assumption of independent assortment.
• This leads to fewer recombinant (new combination) offspring than predicted.
• In a dihybrid cross (two genes), linked genes produce more parental-type offspring and fewer recombinants, deviating from the expected 9:3:3:1 ratio
• During meiosis (cell division for gametes), crossing over can break linkage, but if genes are very close, crossing over is rare, preserving linked combinations

Deviation due to selection

• Selection refers to natural or artificial pressures that favor certain traits, changing allele frequencies over generations and deviating from Mendelian predictions.
• If a trait confers a survival advantage, its frequency increases, leading to non-Mendelian ratios in offspring populations (e.g., fewer recessive phenotypes if they are disadvantageous)
• Selection acts on phenotypes, indirectly affecting genotypes.
• For example, in natural selection, organisms with favorable traits reproduce more, shifting population ratios away from simple probability predictions

These deviations highlight why Mendelian models are useful approximations but require adjustments for real-world genetics.