Calculate dominant (p) and recessive (q) gene frequencies from observed genotype counts (AA, Aa, aa), disease phenotype prevalence, X-linked traits, and 3-allele ABO blood groups.
| Genotype | Observed Count | Observed % | Expected % (HWE) | Expected Count |
|---|---|---|---|---|
| Homozygous AA | 298 | 29.8% | 29.4% (p²) | 294.3 |
| Heterozygous Aa | 489 | 48.9% | 49.6% (2pq) | 496.4 |
| Homozygous aa | 213 | 21.3% | 20.9% (q²) | 209.3 |
In population genetics, an allele frequency (or gene frequency) represents the relative abundance of a specific allele among all gene copies at a designated chromosomal locus in an interbreeding population. While an individual diploid organism carries exactly two alleles per locus, a population's gene pool is characterized by the continuous proportions of all segregating alleles ($p$ and $q$ for biallelic loci):
Direct calculation from homozygous ($N_{AA}, N_{aa}$) and heterozygous ($N_{Aa}$) individuals. Because each homozygote carries 2 copies and each heterozygote carries 1 copy: $p = (2N_{AA} + N_{Aa}) / 2N$ and $q = (2N_{aa} + N_{Aa}) / 2N$.
When complete dominance prevents distinguishing $AA$ from $Aa$, Hardy-Weinberg equilibrium is assumed: $q = \sqrt{q^2} = \sqrt{\text{Recessive Prevalence}}$, and dominant allele frequency is $p = 1 - q$.
Formulated independently by British mathematician G. H. Hardy and German physician Wilhelm Weinberg in 1908, the Hardy-Weinberg principle demonstrates that in the absence of evolutionary disturbances, allele and genotype frequencies reach an equilibrium state described by the binomial theorem:
Differential reproductive fitness altering allele frequencies over time (e.g., sickle cell heterozygote advantage against severe malaria).
Stochastic sampling error causing random allele fixation or loss in small populations (Founder Effect and Population Bottlenecks).
Immigration/emigration of alleles between distinct populations, or positive assortative mating altering genotype proportions without changing $p$ and $q$.
In medical genetics and genetic counseling, the Hardy-Weinberg square root method enables exact carrier frequency estimation for autosomal recessive disorders:
Because males carry only one X chromosome (hemizygous $XY$), male phenotypic frequency directly equals the underlying allele frequency ($q$). Females carry two X chromosomes ($XX$), requiring two recessive alleles ($q^2$) for phenotypic expression:
For tri-allelic systems such as human ABO blood types ($I^A$ with frequency $p$, $I^B$ with frequency $q$, and $i$ with frequency $r$), trinomial expansion dictates $(p + q + r)^2 = p^2 + 2pr + q^2 + 2qr + 2pq + r^2 = 1$. Due to sampling error in field datasets, raw frequencies rarely sum to 1.0. German mathematician Felix Bernstein (1925) formulated the standard normalization algorithm:
To determine whether an observed sample deviates significantly from Hardy-Weinberg equilibrium, geneticists employ Pearson's Chi-Square ($\chi^2$) goodness-of-fit test:
Allele frequency (or gene frequency) represents the relative proportion of a specific allele among all gene copies at a given locus within a biological population. In a diploid population of N individuals (2N total alleles) with genotype counts N_AA, N_Aa, and N_aa: The dominant allele frequency is p = (2N_AA + N_Aa) / (2N), and the recessive allele frequency is q = (2N_aa + N_Aa) / (2N), where p + q = 1.0 (100%).
When complete dominance obscures heterozygotes (both AA and Aa exhibit the dominant phenotype), allele frequencies are estimated under the assumption of Hardy-Weinberg equilibrium using the square root method: (1) Calculate the observed frequency of the homozygous recessive phenotype: q² = N_recessive / N_total; (2) Solve for the recessive allele frequency: q = √(q²); (3) Solve for the dominant allele frequency: p = 1 - q; (4) Calculate the carrier frequency: 2pq = 2 × p × q.
Allele frequency measures the abundance of an individual allele (such as A or a) within the total haploid gametic gene pool (p and q). In contrast, genotype frequency measures the proportion of diploid individuals possessing a specific combination of two alleles (such as AA, Aa, or aa). Under Hardy-Weinberg equilibrium, genotype frequencies are the binomial expansion of allele frequencies: p² (AA) + 2pq (Aa) + q² (aa) = 1.
In species with chromosomal sex determination (females XX, males XY), females carry 2 copies of an X-linked gene while males carry only 1 copy (hemizygous). In a population of N_f females and N_m males (total X alleles = 2N_f + N_m): The dominant allele frequency is p = (2N_XAXA + N_XAXa + N_XAY) / (2N_f + N_m), and the recessive allele frequency is q = (2N_XaXa + N_XAXa + N_XaY) / (2N_f + N_m). In males, the phenotypic disease frequency directly equals the allele frequency (q), whereas in females, it equals q².
The Chi-Square goodness-of-fit test calculates χ² = Σ [(Observed - Expected)² / Expected] across all genotypes, where Expected_AA = N·p², Expected_Aa = 2N·pq, and Expected_aa = N·q². For a 2-allele system, there is 1 degree of freedom (df = genotypes - alleles = 3 - 2 = 1). If the calculated χ² is less than the critical value of 3.841 (p > 0.05), the population does not deviate significantly from Hardy-Weinberg equilibrium. If χ² ≥ 3.841 (p ≤ 0.05), the population violates HWE due to evolutionary forces (such as non-random mating, natural selection, genetic drift, or gene flow).
The human ABO blood group system features three alleles (I^A with frequency p, I^B with frequency q, and i with frequency r, where p + q + r = 1). Under equilibrium: (1) Recessive allele i frequency is r = √(Freq(O)); (2) Dominant allele I^A frequency is p = 1 - √(Freq(B) + Freq(O)); (3) Dominant allele I^B frequency is q = 1 - √(Freq(A) + Freq(O)). Bernstein's standard correction adjustment is then applied: D = 1 - (p + q + r), giving corrected frequencies p* = p(1 + D/2), q* = q(1 + D/2), and r* = (r + D/2)(1 + D/2) ensuring p* + q* + r* = 1.0 exactly.
