Mendelian Genetics: Rules, Crosses, and Extensions

Product Rule and Sum Rule

• Key idea: probability laws help predict genetic outcomes when combining independent events.
• Product rule (for independent events A and B):
  • Formula: $P(A \text{ and } B) = P(A) \times P(B)$
  • Example: the probability of rolling a 2 on a die and getting heads on a penny:
  • Die: 12 outcomes, probability of 2 = $P(D2) = \frac{1}{6}$
  • Penny: 2 outcomes, probability of heads = $P(PH) = \frac{1}{2}$
  • Combined outcome (2 and heads): $P(D2 \text{ and } PH) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
• Sum rule (for mutually exclusive alternative outcomes A and B):
  • Formula: $P(A \text{ or } B) = P(A) + P(B)$
  • Example (two pathways to one head and one tail when flipping a penny and a quarter):
  • Pathway 1: PH and QT → $P(PH) \times P(QT) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
  • Pathway 2: QH and PT → $P(QH) \times P(PT) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
  • Combined probability: $[(PH) \times (QT)] + [(QH) \times (PT)] = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$
  • Note: we used the product rule to compute each pathway (PH × QT and QH × PT) before summing them.
• Application to dihybrid cross:
  • Probability of having just one dominant trait in the F2 generation:
  • For each trait, probability of dominant expression is $\frac{3}{4}$ under heterozygous cross assumptions.
  • Sum rule yields: $\left(\frac{3}{4}\right) + \left(\frac{3}{4}\right) = \frac{15}{16}$ (written as a sum of pathway probabilities: $\left(\frac{3}{4} \right)\times\left(\frac{3}{4} \right) = \frac{9}{16}$ for both dominant together, and other pathway combinations etc.).
• Table reference: Table 12.3 summarizes the Product Rule and Sum Rule.
• Practical note: large sample sizes are necessary because small samples are prone to chance deviations; Mendel’s large numbers of pea plants allowed precise probability calculations and predictions.

12.2 Characteristics and Traits

• Relationship between genotypes and phenotypes in dominant/recessive gene systems:
  • Phenotype: observable trait expression.
  • Genotype: underlying genetic makeup (two alleles per gene in diploids).
• Diploid organisms:
  • Two homologous chromosomes carry two copies of each gene (alleles).
  • Alleles: different versions of a gene arising by mutation.
• Alleles and dominance:
  • One allele may be dominant (expressed in phenotype) and the other recessive (hidden in heterozygotes).
  • Homozygous dominant (AA) and heterozygous (Aa) appear the same phenotype for a given trait; recessive allele only phenotypically observed in homozygous recessive (aa).
• Gene nomenclature conventions (italicization and case):
  • Use the first letter of the dominant trait as the gene symbol (e.g., violet flower color → V).
  • Dominant alleles: uppercase (V); recessive alleles: lowercase (v).
  • Genotypes: VV (homozygous dominant), vv (homozygous recessive), Vv (heterozygous).
• Mendel’s hybridization and genotype/phenotype separation:
  • True-breeding lines (homozygous) crossed to produce F1 that are phenotypically identical to one parent.
  • F1 individuals are often heterozygous; F2 shows segregation and a 3:1 phenotypic ratio for dominant vs recessive (for single trait) when selfed or crossed with another heterozygote.
• Dominant vs recessive alleles (historical terminology):
  • Dominant allele: expressed; sometimes called the expressed unit factor.
  • Recessive allele: latent unit factor; expressed only when homozygous recessive.
• Human traits table (example): many dominant and recessive human conditions (e.g., Achondroplasia, Huntington’s disease, Cystic fibrosis, Sickle-cell anemia).
• Punnett square concept for monohybrid crosses:
  • Checks all possible allele combinations from two true-breeding parents differing at one characteristic.
  • Parental genotypes example: YY (yellow seeds) × yy (green seeds).
  • Resulting F1 genotype: all Yy; phenotype: yellow.
• Gene and allele notation in cross contexts:
  • Example for pea flower color: V for violet (dominant); v for white (recessive).
  • Homozygous dominant: VV; homozygous recessive: vv; heterozygous: Vv.

Monohybrid Cross, Punnett Squares, and Self-Crosses

• Punnett square procedure for a monohybrid cross:
  • True-breeding parents differ at one gene locus; one parent contributes one type of allele, the other contributes the other type.
  • For YY × yy:
  • All offspring in F1 are Yy (yellow).
• Self-cross of F1 heterozygotes (Yy × Yy):
  • Offspring genotypes: YY, Yy, yY, yy.
  • Genotypic ratio: $YY : Yy : yY : yy = 1 : 2 : 1$
  • Two heterozygous genotypes (Yy and yY) are phenotypically identical (both yellow) and are grouped together.
  • Phenotypic ratio (dominant phenotype yellow vs recessive green): $3 : 1$
• F1 cross validation and F3 follow-up:
  • Self-cross of green-seeded plants (yy): all offspring green (yy).
  • Self-cross of yellow-seeded plants (Yy): one-third true-breeding (YY), two-thirds segregating (Yy); when selfed, results align with the F1 self-cross pattern (YY, Yy, yy in 1:2:1 genotype and 3:1 phenotype).
• Test cross (to distinguish genotype of a dominant phenotype):
  • Cross dominant-expressing individual with a homozygous recessive (aa) tester.
  • If the dominant parent is homozygous (AA): all offspring will be Aa (dominant phenotype).
  • If the dominant parent is heterozygous (Aa): offspring segregate 1:1 Aa : aa.
  • Test crosses validate the principle of segregation and reveal whether the dominant phenotype individual is AA or Aa.
• Example (pea peas): round (R) is dominant to wrinkled (r): test cross between rr and unknown round parent.
• Practical takeaway: test crosses help determine zygosity of individuals expressing a dominant trait.

Pedigree Analysis and Human Genetics

• Human genetic diseases often require pedigree analysis to infer inheritance patterns when direct crosses are unethical.
• Alkaptonuria example (recessive): blue affected individuals with genotype aa; unaffected have AA or Aa.
• Inference from offspring can reveal parental genotypes (e.g., if unaffected parents have an affected child, they must be Aa).
• Diagrammatic pedigrees help deduce who carries alleles and the likelihood of offspring being affected.

Alternatives to Simple Dominance and Recessiveness (Non-Mendelian Inheritance)

• Incomplete dominance:
  • Heterozygote phenotype is intermediate between the two homozygotes.
  • Snapdragon example: white (CCWCW) × red (CRCR) yields pink offspring (CRCW).
  • Genotype ratio in self-cross of a heterozygote: $1:2:1$ (CRCR : CRCW : CWCW).
  • Phenotypic ratio: $1:2:1$ (red : pink : white).
• Codominance:
  • Both alleles are expressed in the heterozygote.
  • Example: MN blood groups; M and N antigens on red blood cells.
  • Genotypic ratio in a self-cross of codominant heterozygotes remains $1:2:1$, but phenotypes show both alleles simultaneously.
• Multiple alleles (population-level diversity):
  • A gene may have more than two alleles in a population, though an individual has at most two.
  • Example: rabbit coat color gene (C): four alleles (C+ > cch > ch > c).
  • Wild-type is often denoted with +; dominance hierarchy can be complex (allelic series).
  • Phenotypes reflect dominance hierarchy and gene dosage effects; Himalayan allele can be temperature-sensitive.
  • Dosage effect: wild-type allele often produces the necessary gene product amount; mutant alleles produce reduced or no product.
• Antennapedia mutation (Drosophila): a dominant mutation that expands the gene product distribution, causing legs to grow in place of antennae.

Malaria Drug Resistance and X-Linked Traits (Case Studies)

• Malaria parasite drug resistance (example of population genetics and mutation):
  • Malaria parasite Plasmodium falciparum has dhps gene mutations conferring sulfadoxine resistance.
  • Resistance alleles arise and spread in regions with drug use; haploid parasite stage expresses resistance with a single allele.
  • Geographic localization of resistance alleles occurs due to regional mutation and interbreeding.
• X-linked traits (sex-linked inheritance) in humans and other species:
  • Sex chromosomes: X and Y; females XX, males XY (humans); Drosophila: females XX, males XY.
  • Genes on X chromosome can be X-linked; Y-linked genes are few and mostly affect male traits.
  • Hemizygosity in XY males makes dominance/recessiveness concept less relevant for X-linked traits.
• Drosophila eye color as classic X-linked trait (XW dominant to Xw):
  • Reciprocal crosses yield different offspring ratios due to X-linkage.
  • In a cross between a white-eyed male (XwY) and a red-eyed female (XW XW), all daughters are red-eyed (XWXw or XW XW) and sons are red-eyed (XWY) or white-eyed (XwY) depending on maternal allele.
  • When crossing XW Xw females with XwY males, F2 results: half of the females red-eyed (XWXw) and half white-eyed (XwXw); half of the males red-eyed (XWY) and half white-eyed (XwY).
• Sex-linked disorders in humans:
  • Color blindness, hemophilia, muscular dystrophy are X-linked.
  • Females heterozygous for X-linked recessive traits are carriers and may be unaffected.
  • Carrier females can pass the trait to sons (50% chance) and can pass the allele to daughters (50% as carriers).
• Sex determination variations across species (birds): in birds, females are the heterogametic sex (ZW), not males (ZZ); sex-linked inheritance patterns differ accordingly.

Lethality and Its Effects on Inheritance Patterns

• Lethal alleles can be recessive or dominant and may kill individuals at various life stages.
• Recessive lethal alleles:
  • Example cross of two heterozygotes (Aa) for a recessive lethal gene yields a 1:2:1 genotypic ratio among offspring, but only two genotypes survive (AA and Aa); aa die before birth or early development.
  • Phenotypic ratio among survivors is 2:1 (wild-type : heterozygotes).
• Dominant lethal alleles:
  • Lethality occurs in both heterozygotes and homozygotes; such alleles are rare because affected individuals typically die before reproduction.
  • They can still be transmitted if lethality occurs after reproductive age or if some individuals survive long enough to reproduce.
• Huntington’s disease (dominant lethal example):
  • Heterozygotes (Hh) develop a fatal neurodegenerative disease but may not show symptoms until later in life.
  • Because onset is later than peak reproductive age, the allele can be passed to offspring before death, ensuring transmission.
  • In this case, the genotype is Hh, and affected individuals eventually manifest disease, while HH individuals are usually affected earlier and may have been selected against prior to reproduction.

Mendel's Laws and Postulates (Foundations of Inheritance)

• Four postulates (often called laws) that summarize classical Mendelian genetics:
  • Pairs of unit factors (genes) exist in individuals and are passed faithfully from generation to generation through gametogenesis and fertilization (dissociation and reassociation of paired factors).
  • For each gene, there are two variants (alleles) in a diploid organism; these alleles may be identical or different.
  • If two alleles differ, one is dominant and the other is recessive; the dominant allele determines the phenotype in heterozygotes.
  • Genes for different traits assort independently (independent assortment) during gamete formation, leading to the various possible allele combinations in offspring.
• Methods and concepts to apply these laws:
  • Forked-line method: a probability technique to track the transmission of alleles across multiple loci and generations.
  • Probability rules (product and sum rules) used to calculate genotype and phenotype proportions in multi-gene crosses.
  • Consideration of linkage and recombination can modify the expected ratios when genes reside on the same chromosome.
  • Epistasis and interactions between genes can modify phenotypic outcomes beyond simple dominance/recessiveness.
• Practical conventions and notation:
  • Dominant trait initials used as gene symbol (e.g., violet flower color → V).
  • Uppercase indicates dominant allele, lowercase indicates recessive allele (VV, Vv, vv).
  • Population-level concepts like wild type (+) denote the most common phenotype/genotype in a population.
• Relevance and limitations:
  • Mendel’s laws explain many traits in peas and other organisms but fewer cross patterns in more complex systems (incomplete dominance, codominance, multiple alleles, X-linked inheritance, lethality, etc.).
  • Modern genetics integrates these extensions to explain broader patterns of inheritance in real-world populations.

Summary of Key Concepts to Remember

• Probability rules are essential for predicting genetic outcomes across generations:
  • Product rule: $P(A \text{ and } B) = P(A) \times P(B)$
  • Sum rule: $P(A \text{ or } B) = P(A) + P(B)\quad\text{(if A and B are mutually exclusive)}$
• Monohybrid crosses yield a classic 1:2:1 genotype ratio and a 3:1 phenotype ratio in the F2 generation.
• The Punnett square is a planning tool to visualize all possible offspring genotypes in a cross.
• The test cross distinguishes homozygous vs heterozygous dominant phenotypes.
• Non-Mendelian inheritance patterns (incomplete dominance, codominance, multiple alleles) add complexity to genotype–phenotype mapping.
• X-linked inheritance introduces sex-specific patterns due to different chromosome compositions in males (hemizygous X in males) vs females (two X chromosomes).
• Lethal alleles (recessive or dominant) can alter expected genotype and phenotype ratios and have implications for survival and reproduction.
• Pedigree analysis is a practical tool in human genetics to infer inheritance patterns when controlled crosses are not possible.
• The core Mendelian postulates explain the basic transmission of genes but must be extended to understand real-world genetic complexity (epistasis, linkage, recombination, and population genetics).