Explain Linkage Disequilibrium (LD) and its primary measures, including the D statistic and r2.

Answer:
The D statistic (coefficient of linkage disequilibrium) quantifies the non-random association between alleles at two different loci. It is calculated as the observed frequency of a specific haplotype minus the frequency expected if the alleles were in linkage equilibrium (i.e., assorting independently). For alleles A1, A2 at locus A and B1, B2 at locus B, $D = P(A1B1) - P(A1)P(B1)$.

The value of $D$ can range from $-0.25$ to $+0.25$ (though its exact maximum and minimum depend on allele frequencies).

• If $D = 0$, the loci are in linkage equilibrium, meaning alleles at the two loci are randomly associated.
• If $D > 0$ (positive sign), there is an excess of coupling haplotypes (A1B1 and A2B2) compared to what is expected by chance. This indicates that these specific combinations of alleles are found together more frequently.
• If $D < 0$ (negative sign), there is an excess of recombinant haplotypes (A1B2 and A2B1) compared to expectation. This indicates that these combinations are found together more frequently.

Answer:
The D' statistic is a normalized measure of linkage disequilibrium, designed to overcome the limitation that the raw $D$ value is heavily influenced by allele frequencies, making it difficult to compare across different genetic regions or populations. $D'$ normalizes $D$ by dividing it by its theoretical maximum ($D_{max}$) or minimum ($D_{min}$) possible value, given the observed allele frequencies.

• If $D \ge 0$, $D' = \frac{D}{D_{max}}$, where $D_{max} = \min(P(A1)P(B2), P(A2)P(B1))$.
• If $D < 0$, $D' = \frac{D}{D_{min}}$, where $D_{min} = \max(-P(A1)P(B1), -P(A2)P(B2))$.

$D'$ ranges from $-1$ to $+1$. A value of 1 or $-1$ indicates complete LD, meaning that only two of the four possible haplotypes exist in the population. This measure is particularly useful for identifying historical recombination events. A commonly accepted cutoff for considering a pair of alleles to be in significant linkage disequilibrium is when $D'$ is greater than 0.5.

Answer:
The r^2 statistic is another widely used measure of linkage disequilibrium, representing the squared correlation coefficient between alleles at two loci. It is calculated using the formula:

where $D$ is the coefficient of linkage disequilibrium, and $P(A1), P(A2), P(B1), P(B2)$ are the individual allele frequencies at loci A and B, respectively.

The value of $r^2$ ranges from 0 to 1, and its interpretation is straightforward:

• If $r^2 = 0$, the loci are in complete linkage equilibrium, meaning there is no statistical association between the alleles at the two loci.
• If $r^2 = 1$, the loci are in complete linkage disequilibrium, indicating a perfect correlation between the alleles. This means that knowing the allele at one locus perfectly predicts the allele at the other locus.
• Increasing values of $r^2$ between 0 and 1 denote progressively stronger linkage disequilibrium. $r^2$ is particularly useful in association studies because it directly reflects the statistical power to detect an association between a marker and a disease-causing variant.

• Observed haplotype data:
  • A1B1: 0.6
  • A1B2: 0.1
  • A2B1: 0.2
  • A2B2: 0.1
• Calculated allele data:
  • A1 ($p_1$): 0.7
  • A2 ($p_2$): 0.3
  • B1 ($q_1$): 0.8
  • B2 ($q_2$): 0.2 Calculate the D statistic and the r^2 statistic for these two loci.