Mendelian Inheritance and Probability Laws

Probability Laws and Mendelian Inheritance

• Probability laws govern Mendelian inheritance, following the same rules as tossing coins, rolling dice, and drawing cards from a deck.
• The probability scale ranges from 0 to 1, where 0 represents an event that is certain not to occur, and 1 represents an event that is certain to occur.

Probability Rules

• The probability of an event is the likelihood of it occurring, with a probability of 1 indicating certainty and a probability of 0 indicating impossibility.
• The probability of tossing heads with a coin that has heads on both sides is 1, and the probability of tossing tails is 0.
• With a normal coin, the probability of tossing heads is 1/2, and the probability of tossing tails is 1/2.
• The probability of drawing the ace of spades from a 52-card deck is 1/52.
• The probabilities of all possible outcomes for an event must add up to 1.

Independent Events

• Coin tosses are independent events, meaning that the outcome of one toss is unaffected by previous tosses.
• Each toss of a coin, whether done sequentially or simultaneously, is independent of every other toss.
• The alleles of one gene segregate into gametes independently of another gene's alleles, following the law of independent assortment.

Probability Rules in Genetics

• The multiplication rule is used to determine the probability of two or more independent events occurring together in a specific combination.
• The rule states that the probability of multiple events occurring is found by multiplying the probability of each individual event.

Probability in Monohybrid Crosses

• In an F1 monohybrid cross, the genotype of F1 plants is Rr, with segregation in heterozygous plants resembling a coin flip in calculating probability.
• Each egg produced has a 1⁄2 chance of carrying the dominant allele (R) and a 1⁄2 chance of carrying the recessive allele (r).
• The same odds apply to each sperm cell produced.

Probability of Traits in F2 Plants

• The probability of an F2 plant having wrinkled seeds (rr) is 1⁄4, found by multiplying the probability of the egg having an r allele (1⁄2) and the sperm having an r allele (1⁄2).
• The probability of an F2 plant carrying both dominant alleles for seed shape (RR) is 1⁄4.

Addition Rule in Genetics

• The addition rule is used to calculate the probability of any one of two or more mutually exclusive events occurring.
• The probability of a heterozygous F2 plant (Rr) is found by adding the individual probabilities of the dominant allele coming from the egg and the recessive allele from the sperm, and vice versa.
• The probability of one possible way of obtaining an F2 heterozygote is 1⁄4, and the probability of the other possible way is also 1⁄4, so the total probability of a heterozygous F2 plant is 1⁄2.

The Multiplication Rule

• The multiplication rule states that to determine the probability of two or more independent events occurring together, we multiply the probability of each event.
• Example: The probability of two coins tossed simultaneously landing heads up is 1/2 * 1/2 = 1/4.

Applying the Multiplication Rule to Monohybrid Crosses

• In an F1 monohybrid cross, the genotype of F1 plants is Rr.
• Segregation in a heterozygous plant is like flipping a coin, with each egg or sperm cell having a 1/2 chance of carrying the dominant allele (R) and a 1/2 chance of carrying the recessive allele (r).
• The probability of an F2 plant having wrinkled seeds (rr) is found by multiplying the probabilities of each gamete carrying the r allele: 1/2 * 1/2 = 1/4.
• The probability of an F2 plant carrying both dominant alleles for seed shape (RR) is also 1/4.

The Addition Rule

• The addition rule states that the probability of any one of two or more mutually exclusive events occurring is calculated by adding their individual probabilities.
• In a monohybrid cross, the probability of an F2 heterozygote can occur in two mutually exclusive ways: the dominant allele from the egg and the recessive allele from the sperm, or vice versa.
• The probability of each way is 1/4, so the total probability of an F2 heterozygote is 1/4 + 1/4 = 1/2.

Solving Complex Genetics Problems with Probability

• The rules of probability can be applied to predict the outcome of crosses involving multiple characters.
• Each allelic pair segregates independently during gamete formation, following the law of independent assortment.
• A dihybrid or multicharacter cross is equivalent to two or more independent monohybrid crosses occurring simultaneously.

Predicting Genotypes and Probabilities

• In a monohybrid cross, the probability of offspring genotypes can be determined using a simple Punnett square:
  • 1⁄4 for YY, 1⁄2 for Yy, and 1⁄4 for yy
• The same probabilities apply to seed shape: 1⁄4 for RR, 1⁄2 for Rr, and 1⁄4 for rr
• The multiplication rule can be used to determine the probability of each genotype in the F2 generation:
  • Probability of YYRR = (probability of YY) × (probability of RR) = 1⁄4 × 1⁄4 = 1⁄16
  • Probability of YyRR = (probability of Yy) × (probability of RR) = 1⁄2 × 1⁄4 = 1⁄8

Combining Rules to Solve Complex Problems

• The multiplication and addition rules can be combined to solve complex problems in Mendelian genetics.
• Example: a cross of two pea varieties tracking the inheritance of three characters (flower color, seed color, and seed shape).
• The cross is PpYyRr * Ppyyrr, and the goal is to find the fraction of offspring exhibiting the recessive phenotypes for at least two of the three characters.
• List all possible genotypes that fulfill the condition: ppyyRr, ppYyrr, Ppyyrr, PPyyrr, and ppyyrr.
• Calculate the probability of each genotype using the multiplication rule.
• Use the addition rule to add the probabilities of all genotypes that fulfill the condition: 6⁄16 or 3⁄8.

Understanding Probability in Genetics

• The rules of probability provide the likelihood of various outcomes, but cannot predict with certainty the exact numbers of progeny.
• The larger the sample size, the closer the results will conform to predictions.
• Mendel understood the statistical feature of inheritance and had a keen sense of the rules of chance.

Inheritance Patterns Beyond Mendelian Genetics

• Inheritance patterns can be more complex than predicted by simple Mendelian genetics.
• Mendel's principles were extended to diverse organisms and more complex patterns of inheritance in the 20th century.

Limitations of Mendel's Work

• Mendel chose pea plant characters with a relatively simple genetic basis, where each character is determined by one gene with only two alleles.
• In reality, not all heritable characters are determined so simply, and the relationship between genotype and phenotype is rarely straightforward.

Extending Mendelian Genetics

• Alleles can show different degrees of dominance and recessiveness in relation to each other.
• Incomplete dominance occurs when neither allele is completely dominant, and the F1 hybrids have a phenotype somewhere between those of the two parental varieties.
• Examples of incomplete dominance include the cross between red and white snapdragons, resulting in pink-flowered F1 hybrids.

Incomplete Dominance

• F1 hybrids have a phenotype intermediate between those of the two parental varieties.
• The F2 generation produces a phenotypic ratio of 1:2:1, with segregation of the alleles in the gametes.
• This confirms that the alleles for flower color are heritable factors that maintain their identity in the hybrids.

Codominance

• Codominance occurs when two alleles each affect the phenotype in separate, distinguishable ways.
• Examples of codominance include the human MN blood group, where both M and N molecules are present on the red blood cells of individuals heterozygous for the M and N alleles.
• The MN phenotype is not intermediate between the M and N phenotypes, but rather exhibits both M and N phenotypes.

The Relationship Between Dominance and Phenotype

• An allele is considered dominant if it is seen in the phenotype, not because it interacts with a recessive allele.
• Alleles are variations in a gene's nucleotide sequence.

Dominance and Recessiveness in the Pathway from Genotype to Phenotype

• Dominance and recessiveness come into play in the pathway from genotype to phenotype.
• The dominant allele codes for an enzyme that converts an unbranched form of starch to a branched form in the seed.
• The recessive allele codes for a defective form of the enzyme, leading to an accumulation of unbranched starch and excess water in the seed.

The Relationship Between Dominance and Phenotype: Round vs. Wrinkled Pea Seed Shape

• One dominant allele results in enough enzyme to synthesize adequate amounts of branched starch, leading to a round seed phenotype.
• Dominant homozygotes and heterozygotes have the same phenotype: round seeds.

The Observed Dominant/Recessive Relationship of Alleles

• The observed dominant/recessive relationship of alleles depends on the level at which we examine phenotype.
• The Tay-Sachs disease allele is an example of this, where it appears recessive at the organismal level, but shows incomplete dominance at the biochemical level and codominance at the molecular level.

Tay-Sachs Disease

• Children with Tay-Sachs disease cannot metabolize certain lipids because a crucial enzyme does not work properly.
• Only children who inherit two copies of the Tay-Sachs allele (homozygotes) have the disease.
• The activity level of the lipid-metabolizing enzyme in heterozygotes is intermediate between the activity level in individuals homozygous for the normal allele and the activity level in individuals with Tay-Sachs disease.

The Level of Analysis and Dominance

• Whether alleles appear to be completely dominant, incompletely dominant, or codominant depends on the level at which the phenotype is analyzed.

Frequency of Dominant Alleles

• The dominant allele is not always the most common allele in a population.
• Example: polydactyly, a condition where a person is born with extra fingers or toes, is caused by a dominant allele, but it is relatively rare, occurring in about 1 in 400 babies in the US.
• This indicates that the recessive allele, which results in five digits per appendage, is much more prevalent in the population.

Multiple Alleles

• Most genes have more than two allelic forms.
• Example: the ABO blood groups in humans, which are determined by three alleles: IA, IB, and i.
• The combination of these alleles results in four different blood types: A, B, AB, and O.

Pleiotropy

• Most genes have multiple phenotypic effects, a property known as pleiotropy.
• Example: the gene that determines flower color in peas also affects the color of the coating on the outer surface of the seed.
• In humans, pleiotropic alleles are responsible for the multiple symptoms associated with certain hereditary diseases, such as cystic fibrosis and sickle-cell disease.

Epistasis

• Epistasis is the phenomenon where one gene affects the phenotype of another gene.
• Example: in Labrador retrievers, the gene that determines black or brown coat color interacts with a second gene that determines whether pigment is deposited in the hair.
• The combination of these genes affects the coat color of the dog, with the epistatic gene influencing the expression of the other gene.

Polygenic Inheritance

• Polygenic inheritance is the phenomenon where multiple genes independently affect a single trait.
• Example: human skin color, which is influenced by multiple genes, each contributing to the overall skin color.
• The combination of these genes results in a range of skin colors, with the majority of individuals having intermediate phenotypes.

Nature and Nurture

• The phenotype of an individual is influenced by both genetics and environment.
• Example: identical twins, who have the same genetic makeup, can accumulate phenotypic differences due to their unique experiences.
• Environmental factors, such as nutrition, exercise, and exposure to the sun, can affect the phenotype of an individual.

Mendelian View of Heredity and Variation

• Mendelian genetics explains heritable variations in terms of alternative forms of genes (alleles) that are passed along from parents to offspring according to simple rules of probability.
• The theory of Mendelian inheritance applies to all organisms with a sexual life cycle, including peas, flies, fishes, birds, and humans.
• The principles of segregation and independent assortment can be extended to explain more complex hereditary patterns, such as epistasis and quantitative characters.