Step 1: Theoretical Foundation of the Law of Dominance

The Law of Dominance, proposed by Gregor Mendel, states that in a heterozygous organism, one allele (the dominant allele) masks the presence of another allele (the recessive allele) for the same character. This ensures that the phenotype of the $F_1$ generation reflects only the dominant trait [Per Mendel’s First Law].

Step 2: Experimental Design (Monohybrid Cross)

To demonstrate this, we perform a cross between two true-breeding parents differing by a single trait: Stem Height.

• Parental Generation ($P$): Homozygous Tall ($TT$) $\times$ Homozygous Dwarf ($tt$).
• Gametes: The $TT$ parent produces only $T$ gametes; the $tt$ parent produces only $t$ gametes [Per the Principle of Segregation].
• $F_1$ Generation: The fusion of gametes results in $Tt$ (Heterozygous Tall).

Step 3: Visual Representation of the Monohybrid Cross

Step 4: Analytical Justification

In the $F_1$ generation, the presence of the $T$ allele dominates the $t$ allele. The dwarf trait is suppressed not because it is lost, but because it is masked. This is empirically evidenced when the $F_1$ progeny are self-pollinated to produce the $F_2$ generation, where the dwarf trait reappears in a $3:1$ phenotypic ratio [Per the observation of recessive traits in $F_2$].

Step 5: Conclusion on Allelic Interaction

The Law of Dominance confirms that factors (genes) occur in pairs. When factors are dissimilar, one member of the pair dominates the other. This cross confirms that the $F_1$ genotype $Tt$ displays the tall phenotype, confirming $T > t$ as the dominance hierarchy.

Final Solution: The monohybrid cross between homozygous tall ($TT$) and dwarf ($tt$) parents results in $F_1$ offspring that are all heterozygous tall ($Tt$), proving that the allele for tallness is dominant over the allele for dwarfness.

The Chromosomal Theory of Inheritance: Proponents and Theoretical Basis

The Chromosomal Theory of Inheritance was independently proposed by two prominent biologists in the early 20th century. This theory serves as the fundamental bridge between Mendelian genetics and cytological observations.

Step 1: Identifying the Proponents

The theory was proposed by Walter Sutton and Theodor Boveri in the year 1902. Consequently, it is formally known as the Sutton-Boveri Theory.

Step 2: Analytical Core of the Theory

Sutton and Boveri observed that the behavior of chromosomes during meiosis parallels the behavior of Mendel’s factors (genes). The core tenets established by this theory include:

• Pairing: Like genes, chromosomes are found in homologous pairs in diploid cells [Per the Law of Segregation].
• Segregation: During the formation of gametes, members of a chromosome pair segregate into different gametes, ensuring that each gamete contains only one allele for each trait.
• Independent Assortment: Different chromosome pairs migrate to the gametes independently of one another during Anaphase I of meiosis [Per the Law of Independent Assortment].

Step 3: Comparative Analysis

Visualization of Chromosomal Segregation

Step 4: Theoretical Synthesis

By identifying that chromosomes represent the physical vehicles of heredity, Sutton and Boveri provided the cytological basis for Mendel's laws. This unification of the particulate theory of inheritance and the physical structure of the cell is known as the Sutton-Boveri Chromosomal Theory of Inheritance.

Final Solution: The Chromosomal Theory of Inheritance was proposed by Walter Sutton and Theodor Boveri in 1902.

1. Biological and Physical Attributes of Pisum sativum

Gregor Johann Mendel’s selection of the garden pea (Pisum sativum) was a strategic decision based on both biological convenience and statistical advantages. The plant offered several distinct traits that made it the ideal model organism for observing patterns of inheritance.

2. Key Advantages: An Analytical Breakdown

• Distinctive and Contrasting Characters: The pea plant exhibits several sharply defined, non-blending traits. Mendel focused on seven pairs of contrasting characters (e.g., Tall vs. Dwarf, Violet vs. White flowers). This binary nature [per Mendel’s Law of Segregation] allowed for clear observation of dominance and recessiveness.
• Short Life Cycle: Pisum sativum is an annual plant with a short life span. This allowed Mendel to study multiple generations (up to the $F_2$ and $F_3$ generations) within a single growing season, providing a large data set in a relatively short timeframe.
• High Productivity and Large Sample Size: Pea plants produce a large number of seeds per pod. A larger sample size statistically minimizes the margin of error, allowing Mendel to apply probability and ratio-based analysis (e.g., the $3:1$ phenotypic ratio) to his findings.
• Ease of Controlled Cross-Pollination: The flower structure of the pea plant is hermaphroditic (bisexual), containing both male and female reproductive organs. However, the structure is designed such that self-pollination is natural, allowing for the creation of "pure-breeding" (homozygous) lines. Furthermore, the large size of the flowers made manual emasculation and artificial cross-pollination technically straightforward.

3. Comparative Table of Traits Observed

4. Visual Representation of Mendel's Monohybrid Cross

5. Statistical Justification

By using self-pollinating organisms, Mendel ensured that he started with homozygous lines ($AA$ or $aa$). Through the deliberate crossing of these lines, he could track the transmission of alleles across generations. [By the Law of Independent Assortment], he was able to conclude that traits are governed by particulate units (genes), a fundamental shift from the then-prevalent "blending inheritance" hypothesis.

Final Solution: The selection of Pisum sativum was advantageous due to its short generation time, high seed yield, ease of manual cross-pollination, and the presence of seven distinct, clearly contrasting pairs of inherited traits which facilitated rigorous statistical analysis.

1. Definition of Point Mutation

A point mutation is a category of genetic mutation characterized by the substitution, insertion, or deletion of a single nucleotide base pair within a DNA sequence. This alteration occurs at a specific, discrete locus on the chromosome [Per the molecular definition of genetic variation].

2. Mechanisms of Point Mutation

Point mutations generally manifest through the following biochemical processes:

• Transitions: A purine is replaced by another purine ($A \leftrightarrow G$) or a pyrimidine by another pyrimidine ($C \leftrightarrow T$).
• Transversions: A purine is replaced by a pyrimidine, or vice versa (e.g., $A \rightarrow C$ or $G \rightarrow T$).
• Frameshift Consequences: Insertions or deletions of a single base pair disrupt the triplet reading frame of the mRNA during translation [Per the Central Dogma of Molecular Biology].

3. Exemplar: Sickle Cell Anemia

The most clinically significant example of a point mutation is the substitution mutation causing Sickle Cell Anemia in humans.

In this instance, a single base pair substitution ($A \rightarrow T$) in the $\beta$-globin gene changes the sixth codon from $GAG$ to $GTG$. This results in the replacement of negatively charged Glutamic acid with hydrophobic Valine [Per the structural biology of proteins], causing the hemoglobin molecules to polymerize under low oxygen tension.

4. Visual Representation of Point Mutation

Final Solution: A point mutation is a mutation resulting from a change in a single base pair of DNA. A classical example is the substitution of the 6th codon of the $\beta$-globin gene from $GAG$ to $GTG$, which leads to the synthesis of abnormal hemoglobin responsible for Sickle Cell Anemia.

Step 1: Theoretical Foundation of the Law of Dominance

The Law of Dominance, proposed by Gregor Mendel, states that in a heterozygous organism (possessing two different alleles for a trait), one allele expresses itself in the phenotype (the dominant allele), while the other is masked (the recessive allele). This law is based on the postulate that characters are controlled by discrete units called "factors," which occur in pairs [Per Mendel’s Postulate of Unit Factors].

Step 2: Defining the Monohybrid Cross Parameters

To demonstrate this law, consider the height of pea plants ($T$ for tall, $t$ for dwarf):

• Parental Generation ($P$): Pure-breeding Tall ($TT$) $\times$ Pure-breeding Dwarf ($tt$).
• Gametes: $TT$ produces only $T$ gametes; $tt$ produces only $t$ gametes.
• $F_1$ Generation: The fusion of gametes results in $Tt$ genotype.

Step 3: Visual Representation (Monohybrid Cross)

Step 4: Analytical Breakdown of the $F_1$ Phenotype

Upon crossing homozygous parents ($TT$ and $tt$), the resulting offspring ($F_1$ generation) all exhibit the genotype $Tt$. Despite the presence of the recessive allele ($t$), all $F_1$ individuals display the "Tall" phenotype. [By Law of Dominance, the $T$ allele masks the phenotypic expression of the $t$ allele].

Step 5: Comparative Analysis Table

Conclusion of Principles

The Law of Dominance is validated here because the phenotype of the $F_1$ hybrid is identical to one of the parents, not an intermediate blend. The recessive trait ($t$) is not lost but remains latent, as it reappears in the $F_2$ generation when self-pollinated, proving the particulate nature of inheritance [Per Law of Segregation].

Final Solution: The Law of Dominance is demonstrated by the observation that all $F_1$ progeny of a cross between homozygous tall ($TT$) and dwarf ($tt$) plants are phenotypically tall ($Tt$), confirming that the tall allele is dominant over the dwarf allele.

Step 1: Definition of Genotypes and Alleles

In a single-locus trait, we assign the dominant allele as $A$ and the recessive allele as $a$. Based on the problem description:

• Homozygous Female: Assuming the homozygous recessive state ($aa$) for standard Mendelian analysis.
• Heterozygous Male: Represented as $Aa$.

[Justification: Mendelian genetics defines homozygosity as possessing identical alleles ($AA$ or $aa$) and heterozygosity as possessing different alleles ($Aa$).]

Step 2: Gamete Formation

Gametes are formed through meiotic division, where the alleles segregate according to the Law of Segregation:

• Male parent ($Aa$) produces two types of gametes: $A$ and $a$ in equal proportion ($50\%:50\%$).
• Female parent ($aa$) produces only one type of gamete: $a$ ($100\%$).

Step 3: Punnett Square Construction

The Punnett square facilitates the determination of all possible fertilization events between the parental gametes.

Step 4: Analytical Representation (SVG)

Step 5: Analysis of Phenotypic Distribution

From the Punnett square, the genotypic ratio is $2Aa : 2aa$ (or $1:1$). Based on the Law of Dominance:

• $Aa$ results in the Dominant phenotype.
• $aa$ results in the Recessive phenotype.

Therefore, the phenotypic ratio in the $F_1$ generation is $1$ Dominant : $1$ Recessive, which is equivalent to $50\%$ dominant features and $50\%$ recessive features.

Final Solution: The phenotypic distribution of the $F_1$ generation results in a $1:1$ ratio, where 50% of the progeny exhibit the dominant phenotype and 50% exhibit the recessive phenotype.

1. Theoretical Foundation: Linkage vs. Independent Assortment

In a standard Mendelian dihybrid cross involving two unlinked genes, we expect a $9:3:3:1$ phenotypic ratio [Per Mendel’s Law of Independent Assortment]. However, when two loci are linked (located on the same chromosome), they tend to be inherited together during gamete formation. The degree of deviation from the $9:3:3:1$ ratio depends on the distance between the two loci and the frequency of crossing over during meiosis.

2. Genetic Configuration and Setup

Consider two genes, $A$ and $B$, with alleles $A, a$ and $B, b$. The heterozygous parents are in the cis (coupling) phase configuration: $\frac{AB}{ab} \times \frac{AB}{ab}$.

If the genes are linked, the gametes produced are not in equal proportions ($1:1:1:1$). Instead, the parental combinations ($AB$ and $ab$) will appear in higher frequency than the recombinant combinations ($Ab$ and $aB$).

3. Mathematical Representation of Gamete Frequency

Let $p$ be the map distance (recombination frequency) between the two loci. The gametes produced by each parent are as follows:

• Parental Gametes: $AB = \frac{1-p}{2}$; $ab = \frac{1-p}{2}$
• Recombinant Gametes: $Ab = \frac{p}{2}$; $aB = \frac{p}{2}$

4. Phenotypic Distribution Analysis

The distribution of phenotypes in the $F_1$ generation (often represented as $F_2$ in traditional genetics problems) is derived from the square of the gametic probabilities. The phenotypes observed will be:

5. Visualizing the Linkage Impact

6. Logical Conclusion

Because the genes are linked, the distribution will show a significant deviation from the Mendelian $9:3:3:1$ ratio. The parental phenotypes ($A-B-$ and $aabb$) will be over-represented (frequency $> 9/16$ and $> 1/16$ respectively), while the recombinant phenotypes ($A-bb$ and $aaB-$) will be under-represented (frequency $< 3/16$). If the linkage is complete ($p=0$), the ratio becomes $3:1$ (phenotypically $3$ Dominant both : $1$ Recessive both), effectively behaving as a single-gene trait.

Final Solution: The phenotypic distribution will deviate from the 9:3:3:1 ratio, showing an excess of parental phenotypes and a deficiency of recombinant phenotypes, with the exact ratio governed by the recombination frequency ($p$).

Step 1: Identifying Parental Genotypes and Gametic Potential

The parents involved in the dihybrid cross are $TtYy$ (Tall, Yellow) and $Ttyy$ (Tall, Green). To determine the offspring phenotypes, we must first identify the potential gametes produced by each parent based on the Principle of Segregation and Independent Assortment [Mendel's Law of Independent Assortment].

• Parent 1 ($TtYy$): Can produce four types of gametes: $TY$, $Ty$, $tY$, and $ty$ in a $1:1:1:1$ ratio.
• Parent 2 ($Ttyy$): Can produce two types of gametes: $Ty$ and $ty$ in a $1:1$ ratio.

Step 2: Punnett Square Analysis

We construct a $4 \times 2$ matrix to evaluate the recombination frequencies of the resulting progeny.

Step 3: Calculating Phenotypic Ratios for "Tall and Green"

For an offspring to be "Tall and Green," it must possess at least one dominant allele for height ($T$) and be homozygous recessive for seed color ($yy$). The genotype must be $T\_yy$.

Scanning the Punnett Square outcomes:

• 1. $TTyy$ (Tall, Green)
• 2. $Ttyy$ (Tall, Green)
• 3. $Ttyy$ (Tall, Green)

Total possible combinations = $8$. Number of Tall and Green offspring = $3$.

Step 4: Visualizing the Genetic Distribution

The probability is calculated as the product of independent probabilities for each trait:

• Probability of Tall ($Tt \times Tt \rightarrow TT, Tt, Tt, tt$): $3/4$
• Probability of Green ($Yy \times yy \rightarrow Yy, yy$): $1/2$
• Result: $\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$

Final Solution: The proportion of tall and green offspring expected is 3/8 (or 37.5%).

1. Foundation of Chromosomal Theory of Inheritance

Thomas Hunt Morgan (1866–1945) utilized the fruit fly, Drosophila melanogaster, to provide experimental evidence for the Chromosomal Theory of Inheritance. His work transitioned genetics from purely mathematical probability to a physically observable science linked to cellular structures [Per the Sutton-Boveri Chromosomal Theory].

2. The Drosophila melanogaster Model System

Morgan selected Drosophila for several analytical advantages:

• Life Cycle: Short life cycle (approximately 12–14 days), allowing for multiple generations within weeks.
• High Fecundity: A single mating produces hundreds of offspring.
• Cytological Advantage: Clear differentiation between male and female and a low diploid chromosome number ($2n = 8$).
• Variability: Existence of distinct, heritable morphological variations.

3. The Discovery of Sex-Linked Inheritance

Morgan observed a white-eyed male mutation in a population of red-eyed flies. Upon crossing, he noted that the inheritance pattern of the white-eye trait correlated specifically with the $X$ chromosome, effectively proving that genes are located on specific chromosomes [By application of Mendelian principles to cytological observation].

4. Linkage and Recombination

Morgan’s most significant contributions involve the behavior of genes located on the same chromosome:

• Linkage: The physical association of genes on the same chromosome. He observed that traits like "body color" and "wing size" did not assort independently, contradicting Mendel’s Law of Independent Assortment for linked genes.
• Recombination: He identified that the frequency of recombination between linked genes is proportional to the physical distance between them on the chromosome.

5. Analytical Framework: Genetic Mapping

Morgan’s student, Alfred Sturtevant, used Morgan's recombination frequencies to map the relative positions of genes on chromosomes. This methodology established the concept of the Map Unit or Centimorgan (cM), where $1$ cM is equivalent to a $1\%$ recombination frequency [By the Principle of Genetic Linkage Mapping].

Final Solution: Thomas Hunt Morgan established the field of experimental genetics by proving that genes are physically located on chromosomes, identifying the phenomenon of sex-linkage, demonstrating the principles of genetic linkage and recombination, and providing the framework for the construction of genetic maps.

Step 1: Identifying Parental Genotypes and Gametic Potential

The cross involves two dihybrid parents with the following genotypes:

• Parent 1: $TtYy$ (Tall and Yellow)
• Parent 2: $Ttyy$ (Tall and Green)

To determine the phenotypic ratios, we apply the Law of Independent Assortment [Per Mendel's Second Law]. Each parent produces gametes based on the segregation of alleles:

• Gametes from $TtYy$: $TY, Ty, tY, ty$ (Each with probability $1/4$)
• Gametes from $Ttyy$: $Ty, ty$ (Each with probability $1/2$)

Step 2: Punnett Square Construction for Probability Analysis

We use a Punnett square to map all possible genotypic combinations of the $F_1$ generation:

Step 3: Calculating Phenotypic Proportions for "Tall and Green"

A plant is phenotypically "Tall and Green" if it possesses at least one dominant allele for height ($T$) and is homozygous recessive for seed color ($yy$). The required genotype is $T\_yy$.

From the Punnett square analysis:

• $TtYy$ occurrences: $1/8 + 1/8 = 2/8$ (Tall, Yellow)
• $Ttyy$ occurrences: $1/8 + 1/8 = 2/8$ (Tall, Green)
• $ttYy$ occurrences: $1/8 + 1/8 = 2/8$ (Dwarf, Yellow)
• $ttyy$ occurrences: $1/8 + 1/8 = 2/8$ (Dwarf, Green)

Summing the probabilities for the target phenotype (Tall and Green):

Probability $= P(Ttyy) + P(TTyy) = 1/8 + 1/8 = 2/8 = 1/4$

Visual Representation of Segregation

Final Solution: The proportion of offspring expected to be tall and green is $1/4$ or $25\%$.

1. Definition of Pedigree Analysis

Pedigree analysis is the systematic study of an inherited trait or disorder across multiple generations of a family, represented as a visual chart or diagram known as a pedigree or family tree. [Per Principles of Mendelian Inheritance]. It serves as a tool to determine the mode of inheritance—whether autosomal dominant, autosomal recessive, X-linked dominant, or X-linked recessive—by observing the distribution of phenotypes within related individuals.

2. Symbolic Notation in Pedigree Analysis

To ensure universal clinical and academic communication, pedigree charts utilize standardized symbols. The following SVG illustrates the basic constituents of a pedigree.

3. Utility and Application of Pedigree Analysis

Pedigree analysis provides essential data for genetic counseling and medical diagnostics. Its utility is categorized as follows:

4. Analytical Procedure

To analyze a pedigree effectively, the following steps are performed:

• Step 1: Identify Presence/Absence: Determine if the trait appears in every generation. If it does not, the trait is likely recessive.
• Step 2: Evaluate Sex Distribution: If the trait appears predominantly in males, it suggests an X-linked recessive mode of inheritance [Per Sex-Linked Inheritance theory].
• Step 3: Test Cross Verification: Assign genotypes to individuals in the pedigree (e.g., $AA, Aa, aa$) and check for logical consistency across parental and $F_1$ generations.

Final Solution: Pedigree analysis is the study of a family history regarding a particular trait to determine its inheritance pattern. It is invaluable for diagnosing hereditary genetic disorders, providing risk assessment for prospective parents, and mapping the transmission of alleles through generations, thereby acting as a critical tool in clinical genetics and medical diagnosis.

Step 1: Identifying Parental Genotypes and Gametic Potential

The cross involves two dihybrid parents:

• Parent 1: Tall with yellow seeds ($TtYy$)
• Parent 2: Tall with green seeds ($Ttyy$)

Gametes produced by Parent 1 ($TtYy$): The possible combinations are $TY$, $Ty$, $tY$, and $ty$. Each occurs with a probability of $0.25$.

Gametes produced by Parent 2 ($Ttyy$): Since the plant is homozygous recessive for color, the only allele for color is $y$. The possible combinations are $Ty$ and $ty$. Each occurs with a probability of $0.5$.

Step 2: Punnett Square Construction for Phenotypic Probability

Step 3: Calculating the Proportion of Dwarf and Green Offspring

The phenotype "dwarf and green" requires the homozygous recessive genotype ($ttyy$). From the Punnett Square calculation:

Mathematical derivation:

• Probability of $tt$ (dwarf) from $Tt \times Tt$: $1/4$
• Probability of $yy$ (green) from $Yy \times yy$: $1/2$
• Combined probability (Product Rule of Probability): $P(tt) \times P(yy) = 1/4 \times 1/2 = 1/8$

Visual Representation of Genetic Segregation

Final Solution: The proportion of offspring expected to be dwarf and green is $1/8$ (or $0.125$, which is $12.5\%$).

Definition of Test Cross

A test cross is a specialized genetic cross used to determine the zygosity (genotype) of an organism displaying a dominant phenotype. In this procedure, an individual of unknown genotype (exhibiting the dominant trait, denoted as $A\_$) is crossed with a homozygous recessive individual (denoted as $aa$). By analyzing the phenotypic ratio of the offspring, the unknown genotype can be definitively identified [Per Mendel’s Law of Segregation].

Theoretical Foundation

• Unknown Parent: Either homozygous dominant ($AA$) or heterozygous ($Aa$).
• Tester Parent: Always homozygous recessive ($aa$), which produces only recessive gametes ($a$).
• Diagnostic Logic: If the unknown is $AA$, 100% of offspring will show the dominant phenotype ($Aa$). If the unknown is $Aa$, the offspring will exhibit a 1:1 phenotypic ratio of dominant ($Aa$) to recessive ($aa$).

Visual Representation of the Test Cross

Analytical Tabulation of Genotypic Outcomes

Conclusion and Application

The test cross acts as a diagnostic tool in classical genetics. If the unknown dominant individual produces even one recessive offspring in a test cross, the unknown parent must be heterozygous ($Aa$). Conversely, if only dominant offspring are observed across a statistically significant sample size, the unknown parent is likely homozygous dominant ($AA$). [By the Principle of Dominance and Segregation].

Final Solution: A test cross is a cross between an individual of unknown genotype and a homozygous recessive individual. It yields a 1:1 phenotypic ratio if the unknown is heterozygous, and a 100% dominant phenotypic result if the unknown is homozygous dominant.

1. Foundation of Chromosomal Theory of Inheritance

Thomas Hunt Morgan (1866–1945) utilized the fruit fly, Drosophila melanogaster, as a model organism to provide experimental evidence for the chromosomal theory of inheritance [Per the Sutton-Boveri Chromosomal Theory]. His work moved genetics from abstract mathematical ratios (Mendelian) to physical entities located on chromosomes.

2. Key Experimental Contributions

• Linkage: Morgan observed that genes located on the same chromosome do not always assort independently. He defined linkage as the physical association of genes on a chromosome, which results in the parental combination of alleles being inherited together more frequently than expected by independent assortment [Per the Principle of Genetic Linkage].
• Recombination: He identified that linkage is not absolute. During meiosis, genetic material can be exchanged between homologous chromosomes via crossing over, leading to non-parental (recombinant) gene combinations [Per the Crossing-Over Mechanism].
• Sex-Linked Inheritance: Morgan discovered the phenomenon of X-linked inheritance by observing white-eyed male Drosophila. He proved that the gene for eye color is located specifically on the X-chromosome, establishing that inheritance patterns are influenced by sex-determining chromosomes [Per the Law of Segregation in Sex Chromosomes].

3. Chromosomal Mapping

Morgan’s student, Alfred Sturtevant, used the frequency of recombination between gene pairs on the same chromosome as a measure of the distance between them. This paved the way for the construction of Genetic Maps [Per the Map Unit (centimorgan) convention].

4. Summary Table of Morgan’s Discoveries

5. Visual Representation of Linkage vs. Crossing Over

Final Solution: T.H. Morgan is celebrated as the "Father of Experimental Genetics" for his discovery of linkage, sex-linkage, and crossing-over in Drosophila melanogaster, which provided the physical basis for genetic inheritance and the foundation for chromosomal mapping.

Step 1: Establishing the Genetic Foundation

Human ABO blood grouping is determined by a series of three alleles at a single locus on chromosome 9: $I^A$, $I^B$, and $i$. This system exhibits both co-dominance and multiple allelism [Per Mendel’s Law of Segregation and the codominance principle].

• Allele $I^A$ and $I^B$ are dominant over allele $i$.
• Allele $I^A$ and $I^B$ are codominant with each other.
• Genotype $ii$ results in Phenotype $O$.

Step 2: Determining Parental Genotypes

The child possesses blood group $O$, which corresponds to the homozygous recessive genotype $ii$. For a child to express the $ii$ genotype, they must inherit one $i$ allele from each parent [By the Law of Independent Assortment].

Given the phenotypes of the parents:

• Father (Blood Group $A$): Must have at least one $I^A$ allele. Since he must contribute an $i$ allele to the child, his genotype must be $I^A i$.
• Mother (Blood Group $B$): Must have at least one $I^B$ allele. Since she must contribute an $i$ allele to the child, her genotype must be $I^B i$.

Step 3: Punnett Square Analysis of Possible Offspring

Crossing the parental genotypes $I^A i \times I^B i$ reveals the potential genotypes and phenotypes of subsequent offspring.

Step 4: Evaluating Probability and Genotypic Ratios

The cross $I^A i \times I^B i$ produces the following genotypic distribution in the offspring generation:

1. $I^A I^B$: Blood group AB (25%)
2. $I^A i$: Blood group A (25%)
3. $I^B i$: Blood group B (25%)
4. $ii$: Blood group O (25%)

Final Solution: The genotypes of the parents are $I^A i$ (Father) and $I^B i$ (Mother). The possible genotypes for other offspring are $I^A I^B$, $I^A i$, $I^B i$, and $ii$, resulting in phenotypes AB, A, B, and O respectively, each with a 25% probability.

Definition and Theoretical Foundation

In classical genetics, an organism's genotype is determined by the specific alleles present at a particular genetic locus on homologous chromosomes. The distinction between homozygous and heterozygous states depends on the identity of these alleles.

Step 1: Structural Differentiation

The fundamental difference lies in the composition of the allelic pair for a given trait. If an organism possesses two identical alleles for a specific gene, the condition is defined as homozygous. If the organism possesses two distinct alleles for the same gene, the condition is defined as heterozygous.

Step 2: Comparative Analysis

The following table delineates the biological and genetic implications of both states:

Step 3: Schematic Representation

Step 4: Theoretical Justification

[Per Mendel's Law of Segregation], during gamete formation, the alleles of a gene pair segregate from each other so that each gamete carries only one allele. In a homozygous individual, the segregation produces gametes that are genetically uniform. In a heterozygous individual, the segregation results in a $1:1$ ratio of gametes carrying the dominant allele versus the recessive allele, which is the mechanism facilitating genetic variation in subsequent generations [Ref: Law of Purity of Gametes].

Final Solution: A homozygous organism possesses identical alleles (e.g., $RR$ or $rr$) and is pure-breeding for a trait, whereas a heterozygous organism possesses contrasting alleles (e.g., $Rr$) and is termed a hybrid, producing diverse gametic combinations upon meiotic division.

Definition: Co-dominance

Co-dominance is a non-Mendelian inheritance pattern where both alleles of a gene pair in a heterozygote are fully expressed in the phenotype. Unlike incomplete dominance, where the phenotype is an intermediate blend, co-dominance results in the simultaneous presence of both parental traits without any masking or blending [Ref: Mendelian Genetics Deviations].

Step 1: The Molecular Basis of Co-dominance

In a heterozygous condition ($I^A I^B$), the two alleles code for different, functional protein products. Because neither allele is dominant or recessive, the cell machinery processes both, leading to the manifestation of both characteristic markers on the cell surface.

Step 2: Illustrative Example - ABO Blood Grouping in Humans

The human $ABO$ blood group system is governed by the $I$ gene, which has three alleles: $I^A$, $I^B$, and $i$. The alleles $I^A$ and $I^B$ produce distinct sugar polymers (antigens) on the surface of red blood cells (RBCs).

• Allele $I^A$: Directs the production of 'A' antigen.
• Allele $I^B$: Directs the production of 'B' antigen.
• Allele $i$: Does not produce any antigen (recessive).

When an individual inherits $I^A$ from one parent and $I^B$ from the other, the genotype is $I^A I^B$. In this state, both the 'A' and 'B' antigens are produced on the RBC surface, resulting in blood type $AB$ [Per the Principle of Codominant Allelic Expression].

Step 3: Comparative Analysis

Visual Representation of Antigen Expression

Final Solution: Co-dominance is a genetic phenomenon where both alleles of a gene pair express themselves equally in a heterozygote, such as in the $I^A I^B$ genotype of human blood group AB, where both A and B antigens are simultaneously present on the erythrocytes.

Definition and Theoretical Foundation

Incomplete dominance is a form of intermediate inheritance in which one allele for a specific trait is not completely expressed over its paired allele. This results in a third phenotype in which the expressed physical trait is a combination of the phenotypes of both alleles. [Per Mendelian Genetics, this violates the Law of Dominance, which states that the dominant allele should completely mask the recessive allele in the heterozygous condition.]

Mechanism of Action

In incomplete dominance, the phenotypic ratio in the $F_2$ generation deviates from the classic Mendelian ratio of $3:1$. Instead, it follows a genotypic and phenotypic ratio of $1:2:1$ ($1$ homozygous dominant : $2$ heterozygous : $1$ homozygous recessive).

Biological Example: Antirrhinum majus (Snapdragon)

In Antirrhinum majus, flower color is determined by a pair of alleles: $R$ (Red) and $r$ (White). When a pure-breeding red-flowered plant ($RR$) is crossed with a pure-breeding white-flowered plant ($rr$), the $F_1$ generation produces pink flowers ($Rr$).

Visual Representation of the Cross

Genetic Rationale

• Genotype $RR$: The dominant allele produces functional enzymes necessary for full red pigment synthesis.
• Genotype $rr$: The recessive allele produces a non-functional enzyme, resulting in no pigment (White).
• Genotype $Rr$: A single copy of the dominant allele is insufficient to produce the full intensity of the red pigment, leading to a diluted, intermediate pink color. This is an example of dosage-dependent gene expression.

Final Solution: Incomplete dominance is the phenomenon where the dominant allele does not completely mask the recessive allele, resulting in an intermediate phenotype. In the snapdragon (Antirrhinum majus), a cross between red-flowered ($RR$) and white-flowered ($rr$) plants results in pink ($Rr$) $F_1$ offspring, demonstrating this inheritance pattern.

1. Conceptual Definitions

In classical Mendelian genetics, the interaction between alleles of a single gene determines the phenotype of an organism. An allele is defined as a variant form of a given gene located at a specific locus on a chromosome [Per the Law of Segregation].

• Dominance: A relationship between alleles where one allele masks the expression of the other in a heterozygous state. The allele that expresses itself is termed the "dominant" allele.
• Recessive: A relationship where the phenotype of an allele is only expressed when the organism is homozygous for that trait, meaning it is masked in the presence of a dominant allele.

2. Comparative Analysis Table

3. Visual Representation of Allelic Interaction

4. Theoretical Justification

The distinction between dominant and recessive traits is grounded in the Law of Dominance. This law states that in a dissimilar pair of factors (alleles), one member of the pair dominates (dominant) the other (recessive). Biologically, this often occurs because the dominant allele produces a sufficient amount of the functional protein required for the trait's expression, whereas the recessive allele is often a "loss-of-function" mutation that produces either a defective protein or no protein at all [Per Molecular Biology of the Gene].

Final Solution: Dominance is the phenotypic expression of an allele in both heterozygous and homozygous states, whereas Recessivity is the phenotypic expression of an allele restricted strictly to the homozygous state where no dominant allele is present.

Definition: Pedigree Analysis

Pedigree analysis is the study of the inheritance of specific traits or genetic disorders over several generations within a family. It involves the construction and systematic examination of a pedigree chart (or family tree), which utilizes standardized symbols to represent individuals (males as squares, females as circles) and their ancestral lineages [Per Mendelian Genetics and human heredity principles].

Standardized Pedigree Symbols

Step 1: Establishing the Utility of Pedigree Analysis

Pedigree analysis is a crucial tool in human genetics because controlled crosses, which are possible in model organisms like Drosophila melanogaster, cannot be performed on humans for ethical and practical reasons. The analysis is useful for the following reasons:

• Determining Mode of Inheritance: It helps identify whether a trait is governed by a dominant or recessive allele. Furthermore, it clarifies if the trait is located on an autosome or a sex chromosome ($X$ or $Y$).
• Genetic Counseling: By analyzing the family history, geneticists can calculate the probability of a future offspring inheriting a specific genetic disorder [Per Probability Theory in Genetics].
• Tracing Genetic Disorders: It facilitates the tracing of hereditary diseases such as haemophilia, cystic fibrosis, sickle-cell anaemia, and phenylketonuria through multiple generations.
• Identifying Carriers: It allows for the identification of phenotypically normal individuals who carry a recessive allele for a disorder, which is essential for family planning.

Step 2: Analytical Framework for Inheritance Patterns

Step 3: Logical Application

To perform an analysis, the following logical operations are conducted:

1. Data Collection: Recording phenotypes of individuals across at least three generations.
2. Pattern Recognition: Assessing if the trait appears in every generation (suggesting dominance) or if there is a gender bias in inheritance (suggesting sex-linkage).
3. Genotypic Deduction: Using the phenotypes of parents and progeny to assign genotypes ($AA, Aa, aa$) based on the Law of Segregation [Per Mendel's First Law].

Final Solution: Pedigree analysis is the retrospective study of family history to trace the inheritance of traits. It is an indispensable tool in medical genetics used to determine the mode of transmission of genetic disorders, identify carriers within a family, and provide accurate genetic counseling for informed reproductive decision-making.