Question

Consider the following three loci in humans, each of which is considered to be a complex locus with multiple alleles: \(\mathrm{ABO}\) locus (6 alleles), Rh locus (8 alleles) and MN locus ( 4 alleles). Assuming that each locus is independent of the other two, how many different genotypic combinations can there exist in a given population?

Short answer

There are 7560 different genotypic combinations that can exist in a given population considering the three loci (ABO, Rh, and MN), as each locus is independent. This is calculated by finding the possible genotypic combinations for each locus (21 for ABO, 36 for Rh, and 10 for MN) and then multiplying them together (21 x 36 x 10 = 7560).

Step-by-step solution

Determine possible genotypes at each locus

At each locus, an individual has two alleles (one from each parent). So the possible number of genotypes at a locus is equal to the number of ways two alleles can be chosen, with replacement, from the total number of alleles. This can be calculated using the formula for combinations with replacement: \[\binom{n+k-1}{k} = \frac{(n+k-1)!}{k!(n-1)!}\] Here, n is the number of different objects, and k is the number of objects to be chosen.

Calculate genotypes for the ABO locus

For the ABO locus, there are 6 alleles, and we are choosing 2 (n=6, k=2). Using the formula for combinations with replacement, we get: \[\binom{6+2-1}{2} = \frac{7!}{2!5!} = \frac{7\times6}{2} = 21\] So there are 21 possible genotypic combinations at the ABO locus.

Calculate genotypes for the Rh locus

For the Rh locus, there are 8 alleles, and we are choosing 2 (n=8, k=2). Using the formula, we get: \[\binom{8+2-1}{2} = \frac{9!}{2!7!} = \frac{9\times8}{2} = 36\] So there are 36 possible genotypic combinations at the Rh locus.

Calculate genotypes for the MN locus

For the MN locus, there are 4 alleles, and we are choosing 2 (n=4, k=2). Using the formula, we get: \[\binom{4+2-1}{2} = \frac{5!}{2!3!} = \frac{5\times4}{2} = 10\] So there are 10 possible genotypic combinations at the MN locus.

Calculate the total number of genotypic combinations

Since each of the three loci is independent, we can multiply the number of genotypic combinations at each locus to get the total possible combinations: \[21 \times 36 \times 10 = 7560\] Therefore, there are 7560 different genotypic combinations that can exist in a given population considering the three loci.

Key concepts

Combinations with Replacement

When dealing with genetic combinations where alleles can be paired in various forms, we often use a mathematical concept known as combinations with replacement. This concept allows us to determine how many ways we can choose a set of items from a larger group, where each item can be selected more than once.

In genetic terms, this means considering how alleles from each parent can combine. It is calculated using the formula:

• \( \binom{n+k-1}{k} = \frac{(n+k-1)!}{k!(n-1)!} \)

Here, \(n\) is the number of different alleles available, and \(k\) is the number of alleles we are choosing (usually two for diploid organisms like humans).

The reason we use combinations 'with' replacement is that each locus's alleles can be the same, meaning an allele can be chosen more than once, like having AA at a locus. Understanding this set-up is crucial to solving genetics problems that involve calculating diverse genotypic outcomes across multiple loci.

Loci in Genetics

In genetics, a locus (plural: loci) refers to the specific physical location of a gene or a significant sequence on a chromosome. Different alleles at a locus are variations of a gene that can influence an organism's traits or phenotype.

In this exercise, we are considering three loci: ABO, Rh, and MN. Each has a different number of alleles:

• **ABO locus**: has 6 alleles, influencing blood type.
• **Rh locus**: involves 8 alleles, affecting the Rh factor in blood.
• **MN locus**: contains 4 alleles, related to another blood group system.

Each of these loci is considered independently, which means the alleles at one locus do not affect the choices at another locus. This independence allows us to apply combinatorial principles separately to each locus. Understanding loci interaction and independence is key to mastering complexity in genetic problems.

Genotypic Combinations

Genotypic combinations refer to the various pairings of alleles that can occur at a genetic locus, resulting in different genotypes. In a diploid organism like humans, each individual inherits two alleles per locus, one from each parent.

To calculate the total number of genotypic combinations for multiple independent loci, we find how many combinations exist at each locus and then multiply these numbers. In our example:

• **ABO locus has**: 21 genotypic combinations.
• **Rh locus has**: 36 genotypic combinations.
• **MN locus has**: 10 genotypic combinations.

Thus, the total number of different genotypic combinations across all three loci is calculated as follows:

• \(21 \times 36 \times 10 = 7560\)

This multiplicative approach stems from the assumption of independence between loci, which means each locus's genotype is determined independently of the others. This process helps us grasp not only the diversity within a population but also enhances our understanding of genetic inheritance and variability.