Question

Assuming the chromosomes in different individuals are genetically dissimilar because of different alleles, how many unique zygotic combinations are possible following fertilization in an organism where n = 3 (Assuming that no crossing over occurs)?

• A. 8
• B. 16
• C. 32
• D. 64

Answer 1

Answer: (D)

64

Answer 2

An organism with a haploid number of chromosomes n=3 has 3 pairs of homologous chromosomes in its diploid state (2n=6). Let's denote the three pairs as Pair 1, Pair 2, and Pair 3.

Consider a male and a female individual. For each homologous pair, the male has two chromosomes, say $M_i^A$ and $M_i^B$ for pair i. Similarly, the female has two chromosomes, $F_i^A$ and $F_i^B$ for pair i. The question states that chromosomes in different individuals are genetically dissimilar because of different alleles. This implies that the four chromosomes involved in forming a homologous pair in the zygote (two from the male's parents and two from the female's parents) are genetically distinct. So, for pair i, the chromosomes $M_i^A$, $M_i^B$, $F_i^A$, and $F_i^B$ are all genetically different from each other.

During meiosis, a male produces gametes containing one chromosome from each pair. Since no crossing over occurs, the chromosomes in the gametes are either $M_i^A$ or $M_i^B$ for each pair i. The number of unique male gametes is $2^n = 2^3 = 8$. A male gamete is a combination $(C_{1M}, C_{2M}, C_{3M})$, where $C_{iM} \in \{M_i^A, M_i^B\}$.

Similarly, a female produces gametes containing one chromosome from each pair. The number of unique female gametes is $2^n = 2^3 = 8$. A female gamete is a combination $(C_{1F}, C_{2F}, C_{3F})$, where $C_{iF} \in \{F_i^A, F_i^B\}$.

A zygote is formed by the fusion of a male gamete $(C_{1M}, C_{2M}, C_{3M})$ and a female gamete $(C_{1F}, C_{2F}, C_{3F})$. The zygote is diploid, with chromosome pairs $((C_{1M}, C_{1F}), (C_{2M}, C_{2F}), (C_{3M}, C_{3F}))$.

Let's consider the possible genetic constitutions for a single homologous pair (say, pair 1) in the zygote. The male contributes either $M_1^A$ or $M_1^B$, and the female contributes either $F_1^A$ or $F_1^B$. The possible combinations for the first pair in the zygote are $(M_1^A, F_1^A)$, $(M_1^A, F_1^B)$, $(M_1^B, F_1^A)$, and $(M_1^B, F_1^B)$. Since $M_1^A$, $M_1^B$, $F_1^A$, and $F_1^B$ are all genetically distinct, these four combinations represent four unique genetic constitutions for the first chromosome pair in the zygote.

The same logic applies independently to the other two chromosome pairs (Pair 2 and Pair 3). For each pair i (i=1, 2, 3), there are 4 unique combinations of chromosomes possible in the zygote: $(M_i^A, F_i^A)$, $(M_i^A, F_i^B)$, $(M_i^B, F_i^A)$, and $(M_i^B, F_i^B)$.

Since the composition of each chromosome pair in the zygote is determined independently, the total number of unique zygotic combinations is the product of the number of unique combinations for each pair. Total unique zygotic combinations = (Combinations for Pair 1) $\times$ (Combinations for Pair 2) $\times$ (Combinations for Pair 3) Total unique zygotic combinations = $4 \times 4 \times 4 = 4^3$.

Given n=3, the number of unique zygotic combinations is $4^n$. $4^3 = 64$.

Alternatively, we can consider the total number of possible male gametes ($2^n$) and the total number of possible female gametes ($2^n$). Any male gamete can fuse with any female gamete. The total number of possible fusions is $2^n \times 2^n = 2^{2n}$. A zygote is defined by the specific male gamete and female gamete that fused. Since each of the $2^n$ male gametes is genetically unique (due to independent assortment of distinct chromosomes from the male's parents) and each of the $2^n$ female gametes is genetically unique (due to independent assortment of distinct chromosomes from the female's parents), the fusion of any unique male gamete with any unique female gamete will result in a unique zygotic combination, provided the contributing chromosomes are distinct as stated in the problem. Let $G_M = (C_{1M}, C_{2M}, C_{3M})$ and $G'_M = (C'_{1M}, C'_{2M}, C'_{3M})$ be two different male gametes. Let $G_F = (C_{1F}, C_{2F}, C_{3F})$ and $G'_F = (C'_{1F}, C'_{2F}, C'_{3F})$ be two different female gametes. The zygote from $(G_M, G_F)$ is $((C_{1M}, C_{1F}), (C_{2M}, C_{2F}), (C_{3M}, C_{3F}))$. The zygote from $(G'_M, G'_F)$ is $((C'_{1M}, C'_{1F}), (C'_{2M}, C'_{2F}), (C'_{3M}, C'_{3F}))$. If $(G_M, G_F) \neq (G'_M, G'_F)$, then either $G_M \neq G'_M$ or $G_F \neq G'_F$ (or both). If $G_M \neq G'_M$, then for at least one pair i, $C_{iM} \neq C'_{iM}$. Since $C_{iM}$ and $C'_{iM}$ are distinct chromosomes ($M_i^A$ or $M_i^B$), and are also distinct from female chromosomes ($F_i^A, F_i^B$), the resulting zygote composition for pair i, $(C_{iM}, C_{iF})$ will be different from $(C'_{iM}, C'_{iF})$ unless $C_{iF} = C'_{iF}$ and the order didn't matter, but here the pair is an ordered set of chromosomes inherited from male and female gametes. For example, $(M_1^A, F_1^A)$ is different from $(M_1^B, F_1^A)$. If $G_M = G'_M$ but $G_F \neq G'_F$, then for at least one pair i, $C_{iF} \neq C'_{iF}$. Since $C_{iF}$ and $C'_{iF}$ are distinct chromosomes ($F_i^A$ or $F_i^B$), and are also distinct from male chromosomes ($M_i^A, M_i^B$), the resulting zygote composition for pair i, $(C_{iM}, C_{iF})$ will be different from $(C_{iM}, C'_{iF})$. For example, $(M_1^A, F_1^A)$ is different from $(M_1^A, F_1^B)$.

Since all $2^n \times 2^n = 4^n$ combinations of male and female gametes result in a unique zygotic chromosomal composition under the assumption of distinct parental chromosomes and no crossing over, the total number of unique zygotic combinations is $4^n$. Given n=3, the number is $4^3 = 64$.