Reference:https://www.ndsu.edu/pubweb/~mcclean/plsc731/Linkage%20Disequilibrium%20-%20Association%20Mapping%20in%20Plants-lecture-overheads.pdf

Linkage disequilibrium (LD)

• Measures the degree to which alleles at two loci are associated

  • The non-random associations between alleles at two loci

    • Based on expectations relative to allele frequencies at

      two loci

Step1: Haplotype frequencies for two loci with two alleles

<img src="C:\Users\pro\AppData\Roaming\Typora\typora-user-images\image-20210118193845455.png" alt="image-20210118193845455" style="zoom:50%;" />

• The frequency of each allele at each locus can be calculated

  • Using traditional population genetic nomenclature

    • p and q for Allele frequencies at loci A and B.

Step2:allele frequencies based on haplotype frequencies:

<img src="C:\Users\pro\AppData\Roaming\Typora\typora-user-images\image-20210118194040407.png" alt="image-20210118194040407" style="zoom:50%;" />

To measure linkage disequilibrium (LD)

• Compare the observed and expected frequency of one haplotype

• The difference between these two values is considered the

  deviation or D

Relationships among haplotype and allelic frequencies relative to the deviation

<img src="C:\Users\pro\AppData\Roaming\Typora\typora-user-images\image-20210118194320889.png" alt="image-20210118194320889" style="zoom:50%;" />

Step3: Standard measure of LD

$$

D = x_{11} – p_{1}q_{1}

$$

OR

$$

D = (x_{11})(x_{22}) – (x_{12})(x_{21})

$$

• If two loci are in linkage equilibrium, then

  D = 0

• If the two loci are in linkage disequilibrium, then

  D ≠ 0

From the definition of D we can determine

• The relationship of haplotype frequencies (Table 1)

• D and allelic frequencies (Table 2).

D depends on allele frequencies

Value can range from -0.25 to 0.25

• Researchers suggested the value should be normalized

  • Based on the theoretical maximum and minimum relative to the value of D

• When D ≥ 0 ,

• When D<0,

Another LD measure(correlation)

• Correlation between a pair of loci is calculated using the following formula(r, )

Ranges from

• = 0

• Loci are in complete linkage equilibrium,different information

• = 1

• Loci are in complete linkage disequilibrium, same information

Calculation of

<img src="C:\Users\pro\AppData\Roaming\Typora\typora-user-images\image-20210118195322000.png" alt="image-20210118195322000" style="zoom:80%;" />

Haplotype frequency

| haplotype | frequency |

| :----: | :----: |

| AC | =0.6 |

| Ac | =0.4 |

|aC|=0|

|ac|=0|

Allele frequency

| haplotype | frequency |

| :--: | :--: |

| A | =1|

| a | =0 |

|C|=0.6|

|c|=0|

so the correlation =0, loci are in complete linkage equilibrium

Another example： calculating haplotype frequency using allele frequency