Calculating Tetraploid Genotype Frequencies in Solanum nigrum Using Hardy-Weinberg Principles

Polyploidy is common in plants and adds complexity to genetic calculations. In Solanum nigrum, a wild species with diploid, tetraploid, and hexaploid forms, understanding genotype frequencies in polyploid populations is essential for evolutionary and breeding studies. Here’s how to calculate expected genotype frequencies for a tetraploid population at a locus with two alleles (A and a), given known allele frequencies.

Problem Overview

• Alleles: A (frequency $p=0.8$), a (frequency $q=0.2$)

• Population: Tetraploid (each plant has 4 alleles per locus)

• Genotypes observed: AAAA, AAAa, AAaa, Aaaa, aaaa

• Population size: 1,000 plants

Step 1: Hardy-Weinberg for Tetraploids

For a tetraploid, the expansion is:

(p+q)4=p4+4p3q+6p2q2+4pq3+q4(p+q)4=p4+4p3q+6p2q2+4pq3+q4

This gives the genotype frequencies:

• AAAA: p4p4

• AAAa: 4p3q4p3q

• AAaa: 6p2q26p2q2

• Aaaa: 4pq34pq3

• aaaa: q4q4

Step 2: Calculate Each Genotype Frequency

Plug in $p=0.8$, $q=0.2$:

• AAAA: (0.8)4=0.4096(0.8)4=0.4096

• AAAa: 4×(0.8)3×0.2=4×0.512×0.2=0.40964×(0.8)3×0.2=4×0.512×0.2=0.4096

• AAaa: 6×(0.8)2×(0.2)2=6×0.64×0.04=0.15366×(0.8)2×(0.2)2=6×0.64×0.04=0.1536

• Aaaa: 4×0.8×(0.2)3=4×0.8×0.008=0.02564×0.8×(0.2)3=4×0.8×0.008=0.0256

• aaaa: (0.2)4=0.0016(0.2)4=0.0016

Step 3: Calculate Expected Numbers in 1,000 Plants

Multiply each frequency by 1,000:

• AAAA: $0.4096\times 1000=409.6\approx 409$

• AAAa: $0.4096\times 1000=409.6\approx 409$

• AAaa: $0.1536\times 1000=153.6\approx 154$

• Aaaa: $0.0256\times 1000=25.6\approx 26$

• aaaa: $0.0016\times 1000=1.6\approx 2$

Step 4: Match with Provided Options

• (1) 409 : 409 : 154 : 26 : 2

• (2) 420 : 420 : 140 : 18 : 2

• (3) 409 : 409 : 144 : 36 : 2

• (4) 409 : 420 : 144 : 25 : 2

The correct answer is (1) 409 : 409 : 154 : 26 : 2.

Why This Calculation Matters

• Polyploid genetics: Accurate genotype frequency estimates are crucial for plant breeding and evolutionary studies.

• Conservation: Helps track genetic variation in wild populations.

• Research: Provides a basis for understanding inheritance and selection in polyploid species.

Conclusion

For a tetraploid Solanum nigrum population with allele frequencies $p=0.8$ and $q=0.2$, the expected genotype numbers among 1,000 plants are 409 AAAA, 409 AAAa, 154 AAaa, 26 Aaaa, and 2 aaaa. This demonstrates the power of Hardy-Weinberg principles in polyploid genetics.