When several genes are considered together, the proportion of homozygotes for all the genes in a backcross generation in self-pollinated crops is given by the formula :
Show Hint
- $[(2^m + 1)/2^m]^n$
- $[(2^m - 1)/2^n]^m$
- $[(2^m + 1)/2^n]^m$
- $[(2^m - 1)/2^m]^n$
The Correct Option is D
Solution and Explanation
The backcross method in plant breeding is used to transfer a specific desirable gene (such as disease resistance) from a donor parent to an otherwise high-performing recurrent parent.
As backcrossing proceeds, the proportion of the recurrent parent's genome increases, and heterozygous loci progressively segregate into homozygosity.
Key Formula or Approach:
For a single gene, the probability of obtaining a homozygous genotype in a segregating or backcross generation after $m$ generations of breeding is:
\[ \text{Proportion of homozygosity} = \frac{2^m - 1}{2^m} \]
Step 2: Detailed Explanation:
Let us extend this mathematical relationship to multiple genes:
If we consider $n$ independent, unlinked genes segregating simultaneously, the probability of obtaining a plant that is homozygous for all $n$ genes is calculated using the product rule of probability.
Since each gene segregates independently, the total probability is the product of the individual probabilities for each of the $n$ genes.
- For 1 gene, the proportion of homozygotes is:
\[ \frac{2^m - 1}{2^m} \]
- For $n$ independent genes, the proportion of plants homozygous for all $n$ loci is:
\[ \left[ \frac{2^m - 1}{2^m} \right] \times \left[ \frac{2^m - 1}{2^m} \right] \times \dots \text{ ($n$ times)} = \left[ \frac{2^m - 1}{2^m} \right]^n \]
This mathematical formula allows breeders to calculate the minimum population size needed to secure a desired homozygous recombinant individual in backcross generations.
Step 3: Final Answer:
The proportion of homozygotes for all the genes is given by the formula $[(2^m - 1)/2^m]^n$, which corresponds to option (D).
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