CHAPTER 3 Basic Principles of Heredity (Lecture 2.1–2.5)

Mendel and Monohybrid Crosses (Lecture 2.1)

Learning objectives (end of lecture): explain how Gregor Mendel discovered principles of heredity, and predict progeny in simple crosses.
- Identify factors that led to Mendel’s success.
- Explain how the principle of segregation and dominance account for results from one-gene crosses.
- Explain how chromosome separation in meiosis produces inheritance of alleles.
- Predict progeny in genetic crosses using a Punnett square.
Chapter opener (p. 47): Blond hair in Solomon Islanders is recessive and has a different genetic basis from blond hair in Europeans, illustrating difference in genetic basis across populations.
Concepts interwoven throughout the chapter:
- Mendel’s principles of segregation and independent assortment.
- Probability.
- The behavior of chromosomes. These concepts are interconnected views of the same phenomenon.
Mendel’s choice of subject and approach (Page 5–6):
- Experimental subject: Pisum sativum (pea plant) – easy to grow, short generation time, many offspring (seeds), many pure-breeding varieties, chosen seven characters each with two contrasting forms.
- Good experimental methodology and accurate records.
- Interpreted results with mathematics; formulated and tested hypotheses.
- Mendel’s success rooted in careful observations, quantitative analysis, and a scientific method approach.
The seven pea traits Mendel examined (Page 6):
- Seed shape: Round vs Wrinkled
- Seed color: Yellow vs Green
- Seed coat color: Gray vs White
- Flower position: Axial vs Terminal (along stem vs at tip of stem)
- Stem length: Tall vs Short
- Pod color: Yellow vs Green
- Pod shape: Inflated vs Constricted
Genetic terminology (TABLE 3.1) – definitions:
- Gene: An inherited factor (encoded in DNA) that helps determine a characteristic.
- Allele: One of two or more alternative forms of a gene.
- Locus: The specific place on a chromosome occupied by an allele.
- Genotype: Set of alleles possessed by an individual.
- Heterozygote: An organism with two different alleles at a locus.
- Homozygote: An organism with two identical alleles at a locus.
- Phenotype (trait): The appearance or manifestation of a characteristic.
- Characteristic (or trait): An attribute or feature possessed by an organism.
Example genetics terminology: an individual can be heterozygous at the R-locus with genotype $Rr$ ; other individuals could be $RR$ or $rr$ at that locus.
Multiple loci on a chromosome: definitions when considering more than one gene:
- Homozygous at a locus: two identical alleles (e.g., $BB$ ).
- Heterozygous at a locus: two different alleles (e.g., $Gg$ , $Aa$ ).
- Loci for characteristic 1 (e.g., G), characteristic 2 (A), etc.
Genetic model: genotype vs phenotype and environment:
- Inheritance: An individual inherits only the alleles of the genotype.
- Phenotype is determined by the genotype (interaction of alleles at a locus) plus environmental factors (enviro effects can be large or small depending on the trait).
- Representation: $P = G + E$
- Mendel observed phenotypes to deduce genotypes and the rules of inheritance; not always a direct genotype–phenotype relationship.
Monohybrid crosses: revealing segregation and dominance (Lecture 2.1, 3.1):
- Monohybrid cross: cross between two parents differing in a single characteristic (e.g., male round seeds × female wrinkled seeds). Pure-breeding lines are homozygous.
- Reciprocal cross: opposite phenotypes in the parents (e.g., male wrinkled × female round).
- Mendel’s scientific method in action.
Mendel’s question and approach (Pages 12–13):
- Experimental strategy: cross peas with round and wrinkled seeds to determine whether progeny show one trait, both traits, or an intermediate trait.
- Observed results led to the conclusion that parent plant traits do not blend but that both traits are inherited and can appear in later generations.
- F1 plants show phenotype of one parent; F2 shows a 3:1 ratio (round:wrinkled) for the trait in question.
Mendel’s conclusions from monohybrid crosses (Page 14):
- Conclusion 1: One character is encoded by two genetic factors (two alleles per gene).
- Conclusion 2: The two alleles separate during gamete formation, one per gamete (segregation).
- Conclusion 3: Concept of dominance and recessiveness.
- Conclusion 4: Two alleles separate with equal probability into gametes (equal segregation).
- These are Mendel’s laws of segregation and dominance.
Example: monohybrid cross (seed shape: round vs wrinkled):
- Parental genotypes: RR × rr produce F1 = Rr; self-fertilization (Rr × Rr) yields F2 with phenotypes 3/4 Round : 1/4 Wrinkled and genotypes 1/4 RR : 1/2 Rr : 1/4 rr.
- Interpretation: no blending; dominance; 3:1 phenotypic ratio; 1:2:1 genotypic ratio.
- Punnett square illustration (P, F1, F2 generations) demonstrates 3:1 and 1:2:1 ratios.
Principles in context of meiosis and DNA (Pages 17–24):
- Principle of segregation: two alleles for a locus segregate into gametes during meiosis (Anaphase I); gametes receive one allele each in equal proportions.
- Dominance: the phenotype observed depends on the presence of a dominant allele in a heterozygote.
- Role of DNA: chromosome is a linear DNA molecule; a gene is a DNA sequence on a chromosome that encodes a product (RNA or protein).
- Alleles are alternative forms of a gene; different alleles encode different variants of the trait.
- Molecular basis example: R locus encodes enzyme SBEI; R allele yields normal enzyme; r allele is a mutation resulting in a nonfunctional enzyme, leading to wrinkled phenotype due to disrupted starch and water balance in the seed.
- Genotype-phenotype correlation: RR and Rr produce sufficient enzyme for normal phenotype; rr lacks functional enzyme leading to wrinkled seeds.
Relating crosses to meiosis (Page 23–24):
- Chromosome theory of heredity: genes located on chromosomes; behavior of chromosomes during meiosis explains inheritance patterns.
- Without crossing over, segregation follows predictable patterns; crossing over can complicate but still underlies independent assortment for distant loci.
Predicting outcomes with the Punnett square (Chapter 3, 2.5):
- Punnett square as a tool to predict genotypic and phenotypic outcomes.
- Example: tall (T) vs short (t). If crossing tall × short with heterozygotes (Tt × TT or Tt × Tt), use Punnett or probability rules to deduce ratios.
- For the simple Aa × Aa cross, probability of AA: 1/4, Aa: 1/2, aa: 1/4; dominant phenotype probability is 3/4.
Worked examples and additional cross-pairs (Pages 26–28):
- Foxes: silver coat color (recessive r) vs red (dominant R). Expected genotypic and phenotypic ratios in carrier red × silver and pure red × silver crosses demonstrate classic Mendelian ratios.
- Rabbits: short hair (dominant H) vs long hair (recessive h). A cross between a short-haired female and a long-haired male produced 1 long-haired : 5 short-haired in observed results; consider genotype combinations and expected ratios; explain deviations.
- Punnett square and branch diagram methods can be used to illustrate these crosses.
Summary of key formulas and ratios:
- Mendelian phenotypic ratio for a monohybrid cross (dominant trait): $3:1$ (dominant:recessive).
- Mendelian genotypic ratio for a monohybrid cross (Aa × Aa): $1:2:1$ (AA:Aa:aa).
- Dihybrid phenotypic ratio for independent assortment: $9:3:3:1$ (round yellow : round green : wrinkled yellow : wrinkled green).
- Punnett square as a predictive tool; probability-based approaches can replace or complement Punnett squares for more complex crosses.
- Probability basics: $P = rac{n{ ext{times event occurs}}}{n{ ext{total outcomes}}}$ , and product rules for independent events: $P( ext{A and B}) = P(A) imes P(B)$ ; addition rule for mutually exclusive events: $P(A ext{ or } B) = P(A) + P(B)$ .
- Binomial expansion: $(p+q)^n = \binom{n}{0} p^{n} q^{0} + \binom{n}{1} p^{n-1} q^{1} + \binom{n}{2} p^{n-2} q^{2} + \dots + \binom{n}{n} p^{0} q^{n}$ ; for two outcomes, the common simplified form for five children: $p^5 + 5p^4 q + 10p^3 q^2 + 10p^2 q^3 + 5p q^4 + q^5$ where p and q are the probabilities of the two outcomes.

Probability as a Tool in Genetics (Lecture 2.2)

Probability (P) in genetics:
- Definition: P expresses the likelihood of an event; $P = rac{ ext{number of times event occurs}}{ ext{number of all possible outcomes}}$
- Examples: card draws, dice rolls, etc. (e.g., for an Ace in a deck: $rac{4}{52} = rac{1}{13}$ ).
- Use in genetics: predicting offspring outcomes using rules (multiplication, addition), and conditional probability.
The multiplication rule (independent events):
- If two events occur together, multiply their probabilities: e.g., rolling a die twice and getting a 4 then a 6: $P(4 ext{ then } 6) = rac{1}{6} imes rac{1}{6} = rac{1}{36}$
- In genetics, applying the multiplication rule to independent loci (e.g., two unlinked genes) to predict joint genotype/phenotype probabilities.
The addition rule (mutually exclusive events):
- If only one of two mutually exclusive outcomes occurs (e.g., rolling a 3 or a 4): $P(3 ext{ or } 4) = P(3) + P(4) = rac{1}{6} + rac{1}{6} = rac{1}{3}$
Using probability rules in genetics (beyond Punnett squares):
- For Aa × Aa, probability of AA is $rac{1}{4}$ ; probability dominant phenotype (AA or Aa or aA) is $rac{3}{4}$ .
- For more complex crosses or multi-locus problems, binomial expansion is often faster than drawing large Punnett squares.
Conditional probability in genetics (Lecture 2.2):
- Example: Tt × Tt (tall plants). Among tall progeny, what fraction are heterozygous (Tt)?
- Cross: tall phenotypes include TT and Tt; genotype distribution for all offspring is $TT: Tt: tt = frac{1}{4} : frac{1}{2} : frac{1}{4}$ , but conditional on tall phenotype (T_), the distribution among tall plants is $rac{TT}{TT+Tt} : rac{Tt}{TT+Tt} = rac{1}{3} : rac{2}{3}$ , i.e. $ext{P}( ext{heterozygous} \big| ext{tall}) = rac{2}{3}$ .
The binomial expansion and probability in genetics (Lecture 2.2):
- When there are multiple births (e.g., children) with two outcomes (disease vs normal), and each event is independent with probabilities p and q (p + q = 1), the binomial expansion applies.
- Coefficients come from Pascal’s triangle; for n = 5, coefficients are 1, 5, 10, 10, 5, 1.
- Example: Aa × Aa → 3:1 phenotypic ratio; 5-child example: probability of exactly s occurrences of an outcome is given by the term
  $\binom{n}{s} p^{s} q^{n-s}$
- The general formula for any specific combination (order not specified) is:
  $P = rac{n!}{s! \, t!} p^{s} q^{t}$ where s + t = n.
Worked problem examples (Lecture 2.2):
- Five children, two possible outcomes (sickle cell anemia vs normal): for exactly 3 affected (sickle), probability is $\binom{5}{3} (1/4)^3 (3/4)^2 = 10 (1/4)^3 (3/4)^2 = 0.088.$
- If order matters, or multiple combinations are considered, use the product rule and/or binomial expansion to sum probabilities.
Practical exercise prompts and common checks:
- Practice with binomial coefficients from Pascal’s triangle (Table 3.3) to determine coefficients for (p + q)^n expansions.
- Understand when to apply multiplication vs addition rules in genetic crosses.

Testcrosses, Nomenclature, and Dihybrid Crosses (Lecture 2.3)

Testcross definition and purpose:
- Cross an individual with an unknown genotype to a homozygous recessive tester (tt) to reveal genotype.
- Example: tall phenotype plant: genotype could be TT or Tt. Cross TT × tt yields all Tt; Tt × tt yields 1/2 Tt : 1/2 tt.
Allele nomenclature and naming rules (Lecture 2.3, Section 3.4):
- A gene is on the same locus on homologous chromosomes, but different alleles exist.
- Use the same letter for alleles and distinguish alleles by case, superscripts, subscripts, or a combination.
- Dominant allele is uppercase; recessive allele is lowercase (A vs a).
- Wild type allele can be marked with a plus superscript (e.g., A+), while mutant/rare allele is without plus.
- Superscripts and subscripts help distinguish multiple alleles (e.g., Lfr1, Lfr2).
- Slash notation can distinguish two alleles in a genotype (El+/ElR or +/ElR).
- For multi-locus genotypes, spaces separate loci (e.g., El+/ElR G/g).
Simple genetic-crossover ratios (TABLE 3.5 and 3.6):
- Phenotypic ratios for a single locus with dominance:
- 3:1 (Aa × Aa; 3/4 A_ : 1/4 aa)
- Genotypic ratios for Aa × Aa:
- 1:2:1 (AA : Aa : aa)
- Genotypic/phenotypic outcomes depend on parental genotypes and dominance relationships.
The complexity of genetic traits (Lecture 2.3, Page 59):
- Not all traits follow simple Mendelian inheritance.
- Genetic variation at multiple loci can produce the same phenotype.
- Alleles at multiple loci can combine their effects to influence a trait.
- Environmental factors can influence trait expression (multifactorial inheritance).
Dihybrid cross and independent assortment (Chapter 3.3):
- Dihybrid cross: two parents differ at two loci (e.g., seed shape and color): Round, Yellow (RRYY) × Wrinkled, Green (rryy).
- F1: all Round, Yellow (RrYy).
- F2: expected phenotypic ratio for independently assorted loci: $9:3:3:1$ (Round Yellow : Round Green : Wrinkled Yellow : Wrinkled Green).
- Summary steps: P generation (RRYY × rryy) → gametes (RY) → F1 (RrYy) → self-fertilization → F2 genotypic/phenotypic outcomes.
Independent assortment and meiosis:
- Independent assortment applies to loci on different chromosomes (or far apart on the same chromosome).
- Genes on the same chromosome do not assort independently unless crossing over occurs between them (see later). The rule still holds for unlinked genes.
Branch diagrams and dihybrid test crosses (Lecture 2.4):
- Use branch diagrams to work crosses with two or more genes, predicting genotype and phenotype ratios.
- Use probability to predict progeny for crosses with more than two loci.
- Testcrosses are valuable for resolving genotype information in multi-locus scenarios.
Dihybrid testcross (example): RrYy × rryy cross using branch diagram:
- First characteristic (R locus): Rr × rr yields 1/2 Rr and 1/2 rr.
- Second characteristic (Y locus): YY × yy yields 1/2 Yy and 1/2 yy, etc.
- Combined expectations give 1/4 for each of the four phenotypic classes in a dihybrid testcross when both loci segregate independently.
Using probability with multiple loci (Lecture 2.4):
- When more than two loci are involved, it is efficient to multiply independent single-locus probabilities or to use branch diagrams to combine probabilities.
- Example: Aa Bb cc Dd Ee × Aa Bb Cc dd Ee: probability of aa bb cc dd ee = P(aa) × P(bb) × P(cc) × P(dd) × P(ee).
Worked cucumber example (Lecture 2.4): three genes on different chromosomes: dull vs glossy (D/d), orange vs cream (R/r), bitter vs nonbitter (B/b).
- Parental genotypes: DD RR BB × dd rr bb → F1: DdRrBb (all phenotypes intermediate).
- F1 × F1 intercross yields 9:3:3:1 dihybrid-type outcomes for three loci under independent assortment when all loci assort independently; note deviations can occur with linkage or epistasis.

Chi-Square Goodness-of-Fit Test (Lecture 2.5)

Objective:
- Use the chi-square goodness-of-fit test to determine whether deviations between observed and expected progeny numbers can be attributed to chance.
Why observed ratios may deviate from expected:
- Random fluctuations, sampling error, small sample sizes increase deviation likelihood.
The chi-square test basics:
- Formula: $\chi^2 = \sum \frac{(Oi - Ei)^2}{Ei}$ where Oi are observed counts and E_i are expected counts for each category.
- Degrees of freedom: $df = ext{(number of classes)} - 1.$
- Compare the calculated chi-square value to a chi-square distribution table to obtain a probability (P-value).
- If P > 0.05, differences are not considered statistically significant (fail to reject H0); if P < 0.05, there is a significant difference beyond chance.
Example and interpretation (cockroaches):
- Cross Yy × yy (brown dominant to yellow). Expected 20 brown, 20 yellow out of 40 offspring (ratio 1:1).
- Observed data: various distributions; compute chi-square to determine if observed deviation could occur by chance.
- If x^2 calculation yields P > 0.05, no significant difference; if P ≤ 0.05, significant deviation.
Worked Mendelian dihybrid chi-square example (Lecture 2.5):
- Data: 315 round yellow, 108 round green, 101 wrinkled yellow, 32 wrinkled green in F2 of a dihybrid cross.
- Test whether this fits the expected 9:3:3:1 ratio using
 $\chi^2 = \sum \frac{(Oi - Ei)^2}{E_i}$
- Degrees of freedom: $df = 4 - 1 = 3$ (four phenotypic classes).
- Compare to critical values (Table 3.7) to determine P-value and significance.
Practical notes:
- When multiple classes are involved, it is common to group rare classes to maintain validity of the chi-square test.
- If the calculated P-value is less than 0.05, conclude a significant deviation exists.
End-of-unit study guidance:
- Revise using the textbook and lectures; utilize Achieve resources and problem sets; complete Achieve assignments; prepare for tutorials; seek help if needed.

Connections and Real-World Relevance

The chromosome theory of inheritance explains Mendelian patterns via behavior of chromosomes during meiosis.
The concept of alleles and their molecular basis shows how DNA sequence variation leads to phenotypic variation (e.g., SBEI enzyme in peas and the R/r alleles affecting seed shape).
Simple Mendelian patterns provide foundations for understanding more complex inheritance, including multifactorial traits and gene interactions.
Probability and statistics (binomial expansion, Pascal’s triangle, chi-square) are essential tools for predicting and testing genetic hypotheses in populations.

Key Takeaways (quick reference)

Mendel’s laws: segregation (first law) and independent assortment (second law) explain how alleles separate and how different loci assort.
Dominance and recessiveness describe how certain alleles mask others in heterozygotes.
Genotype vs phenotype: not always a direct genotype–phenotype mapping; environment can influence phenotype.
Punnett squares predict offspring ratios; probability rules provide a powerful alternative method, especially for multi-locus crosses.
Dihybrid crosses reveal independent assortment; phenotypic ratio 9:3:3:1 when two loci assort independently.
Testcrosses help determine unknown genotypes by crossing with homozygous recessives.
Nomenclature and notation for alleles follow consistent rules to distinguish multiple alleles across loci.
Chi-square test is used to assess whether observed deviations from expected Mendelian ratios can be attributed to chance; df = (# classes) - 1.

Worked Problems and Practice Themes (highlights)

Predicting offspring with single-locus crosses: 3:1 phenotypic, 1:2:1 genotypic.
Multi-locus crosses: 9:3:3:1 dihybrid ratio; 1/2 and 1/4 probabilities for various gamete outcomes.
Conditional probability can refine predictions when conditioning on a phenotype (e.g., tall plants that are heterozygous).
Binomial expansion is a powerful tool for calculating probabilities in families with many children or multiple independent events.

Notes on the Study Unit Structure

Lectures covered: 2.1 Mendel and monohybrid crosses; 2.2 Probability as a tool in genetics; 2.3 Testcross, dihybrid crosses; 2.4 Branch diagrams; 2.5 Chi-square test.
Each lecture built a scaffold from basic Mendelian genetics to probability methods and statistical testing, culminating in methods to analyze complex crosses and assess data against Mendelian expectations.