Linkage disequilibrium

In population genetics, linkage disequilibrium is the non-random association of alleles at different loci i.e. the presence of statistical associations between alleles at different loci that are different from what would be expected if alleles were independently, randomly sampled based on their individual allele frequencies.^[1] If there is no linkage disequilibrium between alleles at different loci they are said to be in linkage equilibrium.

Linkage disequilibrium is influenced by many factors, including selection, the rate of recombination, the rate of mutation, genetic drift, the system of mating, population structure, and genetic linkage. As a result, the pattern of linkage disequilibrium in a genome is a powerful signal of the population genetic forces that are structuring it.

In spite of its name, linkage disequilibrium may exist between alleles at different loci without any genetic linkage between them and independently of whether or not allele frequencies are in equilibrium (not changing with time).^[1] Furthermore, linkage disequilibrium is sometimes referred to as gametic phase disequilibrium;^[2] however, the concept also applies to asexual organisms and therefore does not depend on the presence of gametes.

Formal Definition

Suppose that among the gametes that are formed in a sexually reproducing population, allele A occurs with frequency $p_A$ at one locus (i.e. $p_A$ is the proportion of gametes with A at the first locus), while at a different locus allele B occurs with frequency $p_B$ . Similarly, let $p_{AB}$ be the frequency of A and B occurring together in the same gamete (i.e. $p_{AB}$ is the frequency of the AB haplotype).

The association between the alleles A and B can be regarded as completely random when the probability of A and B occurring together in a randomly selected gamete $p_{AB}$ is just the probability that A and B independently occur in the same gamete, which is given by $p_{A} p_{B}$ . If $p_{AB}$ differs from $p_{A} p_{B}$ for any reason, then there is non-random association between A and B and we say that there is linkage disequilibrium between these alleles.

The level of linkage disequilibrium between A and B can be quantified by the coefficient of linkage disequilibrium $D_{AB}$ , which is defined as

D_{AB} = p_{AB} - p_{A}p_{B}

Linkage disequilibrium corresponds to $D_{AB} \neq 0$ . In the case $D_{AB}=0$ we have $p_{AB} = p_{A} p_{B}$ and the alleles A and B are said to be in linkage equilibrium. The subscript AB on $D_{AB}$ emphasizes that it is a property of the alleles A and B. Different alleles at the same loci may have different coefficients of linkage disequilibrium.

Linkage disequilibrium can be defined in a similar way for asexual populations using population allele frequencies. Furthermore, it is also possible to define linkage disequilibrium between three or more alleles, however these higher order associations are not commonly used in practice.^[1]

Measures of linkage disequilibrium derived from $D$

The coefficient of linkage disequilibrium $D$ is not always a convenient measure of linkage disequilibrium because its range of possible values depends on the frequencies of the alleles it refers to. This makes it difficult to compare the level of linkage disequilibrium between different pairs of alleles.

Lewontin^[3] suggested normalising D by dividing it by the theoretical maximum for the observed allele frequencies as follows:

D' = D/D_\max

where

D_\max = \begin{cases} \min\{p_A p_B,\,(1-p_A)(1-p_B)\} & \text{when } D < 0\\ \min\{p_A (1-p_B),\,(1-p_A) p_B\} & \text{when } D > 0 \end{cases}

An alternative to $D'$ is the correlation coefficient between pairs of loci, expressed as

$r=\frac{D}{\sqrt{p_A(1-p_A)p_B (1-p_B)}}$ .

Example: Two-loci and two-alleles

Consider the haplotypes for two loci A and B with two alleles each—a two-locus, two-allele model. Then the following table defines the frequencies of each combination:

Haplotype	Frequency
$A_1B_1$	$x_{11}$
$A_1B_2$	$x_{12}$
$A_2B_1$	$x_{21}$
$A_2B_2$	$x_{22}$

Note that these are relative frequencies. One can use the above frequencies to determine the frequency of each of the alleles:

Allele	Frequency
$A_1$	$p_{1}=x_{11}+x_{12}$
$A_2$	$p_{2}=x_{21}+x_{22}$
$B_1$	$q_{1}=x_{11}+x_{21}$
$B_2$	$q_{2}=x_{12}+x_{22}$

If the two loci and the alleles are independent from each other, then one can express the observation $A_1B_1$ as " $A_1$ is found and $B_1$ is found". The table above lists the frequencies for $A_1$ , $p_1$ , and for $B_1$ , $q_1$ , hence the frequency of $A_1B_1$ is $x_{11}$ , and according to the rules of elementary statistics $x_{11} = p_{1} q_{1}$ .

The deviation of the observed frequency of a haplotype from the expected is a quantity^[4] called the linkage disequilibrium^[5] and is commonly denoted by a capital D:

D = x_{11} - p_1q_1

The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.

	$A_1$	$A_2$	Total
$B_1$	$x_{11}=p_1q_1+D$	$x_{21}=p_2q_1-D$	$q_1$
$B_2$	$x_{12}=p_1q_2-D$	$x_{22}=p_2q_2+D$	$q_2$
Total	$p_1$	$p_2$	$1$

Role of recombination

In the absence of evolutionary forces other than random mating, Mendelian segregation, random chromosomal assortment, and chromosomal crossover (i.e. in the absence of natural selection, inbreeding, and genetic drift), the linkage disequilibrium measure $D$ converges to zero along the time axis at a rate depending on the magnitude of the recombination rate $c$ between the two loci.

Using the notation above, $D= x_{11}-p_1 q_1$ , we can demonstrate this convergence to zero as follows. In the next generation, $x_{11}'$ , the frequency of the haplotype $A_1 B_1$ , becomes

x_{11}' = (1-c)\,x_{11} + c\,p_1 q_1

This follows because a fraction $(1-c)$ of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction $x_{11}$ of those are $A_1 B_1$ . A fraction $c$ have recombined these two loci. If the parents result from random mating, the probability of the copy at locus $A$ having allele $A_1$ is $p_1$ and the probability of the copy at locus $B$ having allele $B_1$ is $q_1$ , and as these copies are initially in the two different gametes that formed the diploid genotype, these are independent events so that the probabilities can be multiplied.

This formula can be rewritten as

x_{11}' - p_1 q_1 = (1-c)\,(x_{11} - p_1 q_1)

so that

D_1 = (1-c)\;D_0

where $D$ at the $n$ -th generation is designated as $D_n$ . Thus we have

D_n = (1-c)^n\; D_0

If $n \to \infty$ , then $(1-c)^n \to 0$ so that $D_n$ converges to zero.

If at some time we observe linkage disequilibrium, it will disappear in the future due to recombination. However, the smaller the distance between the two loci, the smaller will be the rate of convergence of $D$ to zero.

Example: Human Leukocyte Antigen (HLA) alleles

HLA constitutes a group of cell surface antigens as MHC of humans. Because HLA genes are located at adjacent loci on the particular region of a chromosome and presumed to exhibit epistasis with each other or with other genes, a sizable fraction of alleles are in linkage disequilibrium.

An example of such linkage disequilibrium is between HLA-A1 and B8 alleles in unrelated Danes^[6] referred to by Vogel and Motulsky (1997).^[7]

Table 1. Association of HLA-A1 and B8 in unrelated Danes^[6]
			Antigen j		Total
			$+$	$-$
			$B8^{+}$	$B8^{-}$
Antigen i	$+$	$A1^{+}$	$a=376$	$b=237$	$C$
Antigen i	$-$	$A1^{-}$	$c=91$	$d=1265$	$D$
Total			$A$	$B$	$N$
			No. of individuals

Because HLA is codominant and HLA expression is only tested locus by locus in surveys, LD measure is to be estimated from such a 2x2 table to the right.^[7]^[8]^[9]^[10]

expression ( $+$ ) frequency of antigen $i$ :

pf_i = C/N = 0.311\!

;

expression ( $+$ ) frequency of antigen $j$ :

pf_j = A/N = 0.237\!

;

frequency of gene $i$ :

gf_i = 1 - \sqrt{1 - pf_i} = 0.170\!

and

hf_{ij} = \text{estimated frequency of haplotype } ij = gf_i \; gf_j = 0.0215\!

Denoting the '―' alleles at antigen i to be 'x,' and at antigen j to be 'y,' the observed frequency of haplotype xy is

o[hf_{xy}]=\sqrt{d/N}

and the estimated frequency of haplotype xy is

e[hf_{xy}]=\sqrt{(D/N)(B/N)}

Then LD measure $\Delta_{ij}$ is expressed as

\Delta_{ij}=o[hf_{xy}]-e[hf_{xy}]=\frac{\sqrt{Nd}-\sqrt{BD}}{N}=0.0769

Standard errors $SEs$ are obtained as follows:

SE\text{ of }gf_i=\sqrt{C}/(2N)=0.00628

SE\text{ of }hf_{ij}=\sqrt{\frac{(1-\sqrt{d/B})(1-\sqrt{d/D})-hf_{ij}-hf_{ij}^2/2}{2N}}=0.00514

SE\text{ of }\Delta_{ij}=\frac{1}{2N}\sqrt{a-4N\Delta_{ij}\left (\frac{B+D}{2\sqrt{BD}}-\frac{\sqrt{BD}}{N}\right )}=0.00367

Then, if

t=\Delta_{ij}/(SE\text{ of }\Delta_{ij})

exceeds 2 in its absolute value, the magnitude of $\Delta_{ij}$ is statistically significantly large. For data in Table 1 it is 20.9, thus existence of statistically significant LD between A1 and B8 in the population is admitted.

Table 2. Linkage disequilibrium among HLA alleles in Pan-europeans^[10]
HLA-A alleles i	HLA-B alleles j	$\Delta_{ij}$	$t$
A1	B8	0.065	16.0
A3	B7	0.039	10.3
A2	Bw40	0.013	4.4
A2	Bw15	0.01	3.4
A1	Bw17	0.014	5.4
A2	B18	0.006	2.2
A2	Bw35	-0.009	-2.3
A29	B12	0.013	6.0
A10	Bw16	0.013	5.9

Table 2 shows some of the combinations of HLA-A and B alleles where significant LD was observed among pan-europeans.^[10]

Vogel and Motulsky (1997)^[7] argued how long would it take that linkage disequilibrium between loci of HLA-A and B disappeared. Recombination between loci of HLA-A and B was considered to be of the order of magnitude 0.008. We will argue similarly to Vogel and Motulsky below. In case LD measure was observed to be 0.003 in Pan-europeans in the list of Mittal^[10] it is mostly non-significant. If $\Delta_0$ had reduced from 0.07 to 0.003 under recombination effect as shown by $\Delta_n=(1-c)^n \Delta_0$ , then $n\approx 400$ . Suppose a generation took 25 years, this means 10,000 years. The time span seems rather short in the history of humans. Thus observed linkage disequilibrium between HLA-A and B loci might indicate some sort of interactive selection.^[7]

Further information: HLA A1-B8 haplotype

The presence of linkage disequilibrium between an HLA locus and a presumed major gene of disease susceptibility corresponds to any of the following phenomena:

Relative risk for the person having a specific HLA allele to become suffered from a particular disease is greater than 1.^[11]
The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by $\delta$ value^[12] to exceed 0.

Table 3. Association of ankylosing spondylitis with HLA-B27 allele^[13]
		Patients	Healthy controls	Total
		Ankylosing spondylitis		Total
HLA alleles	$B27^+$	$a=96$	$b=77$	$C$
HLA alleles	$B27^-$	$c=22$	$d=701$	$D$
Total		$A$	$B$	$N$

2x2 association table of patients and healthy controls with HLA alleles shows a significant deviation from the equilibrium state deduced from the marginal frequencies.

(1) Relative risk

Relative risk of an HLA allele for a disease is approximated by the odds ratio in the 2x2 association table of the allele with the disease. Table 3 shows association of HLA-B27 with ankylosing spondylitis among a Dutch population.^[13] Relative risk $x$ of this allele is approximated by

x=\frac{a/b}{c/d}=\frac{ad}{bc}\;(=39.7,\text{ in Table 3 })

Woolf's method^[14] is applied to see if there is statistical significance. Let

y=\ln (x)\;(=3.68)

and

\frac{1}{w}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\;(=0.0703)

Then

\chi^2=wy^2\;\left [=193>\chi^2(p=0.001,\; df=1)=10.8 \right ]

follows the chi-square distribution with $df=1$ . In the data of Table 3, the significant association exists at the 0.1% level. Haldane's^[15] modification applies to the case when either of $a,\; b,\;c,\text{ and }d$ is zero, where replace $x$ and $1/w$ with

x=\frac{(a+1/2)(d+1/2)}{(b+1/2)(c+1/2)}

and

\frac{1}{w}=\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}

respectively.

Table 4. Association of HLA alleles with rheumatic and autoimmune diseases among white populations^[11]
Disease	HLA allele	Relative risk (%)	FAD (%)	FAP (%)	$\delta$
Ankylosing spondylitis	B27	90	90	8	0.89
Reactive arthritis	B27	40	70	8	0.67
Spondylitis in inflammatory bowel disease	B27	10	50	8	0.46
Rheumatoid arthritis	DR4	6	70	30	0.57
Systemic lupus erythematosus	DR3	3	45	20	0.31
Multiple sclerosis	DR2	4	60	20	0.5
Diabetes mellitus type 1	DR4	6	75	30	0.64

In Table 4, some examples of association between HLA alleles and diseases are presented.^[11]

(1a) Allele frequency excess among patients over controls

Even high relative risks between HLA alleles and the diseases were observed, only the magnitude of relative risk would not be able to determine the strength of association.^[12] $\delta$ value is expressed by

\delta=\frac{FAD-FAP}{1-FAP},\;\;0\le \delta \le 1

where $FAD$ and $FAP$ are HLA allele frequencies among patients and healthy populations, respectively.^[12] In Table 4, $\delta$ column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high $\delta$ values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk $=6$ .

(2) Discrepancies from expected values from marginal frequencies in 2x2 association table of HLA alleles and disease

This can be confirmed by $\chi^2$ test calculating

\chi^2=\frac{(ad-bc)^2 N}{ABCD}\;(=336,\text{ for data in Table 3; }P<0.001)

where $df=1$ . For data with small sample size, such as no marginal total is greater than 15 (and consequently $N \le 30$ ), one should utilize Yates's correction for continuity or Fisher's exact test.^[16]

Resources

A comparison of different measures of LD is provided by Devlin & Risch ^[17]

The International HapMap Project enables the study of LD in human populations online. The Ensembl project integrates HapMap data with other genetic information from dbSNP.

Analysis software

PLINK - whole genome association analysis toolset, which can calculate LD among other things
LDHat
Haploview
LdCompare^[18]— open-source software for calculating LD.
SNP and Variation Suite- commercial software with interactive LD plot.
GOLD - Graphical Overview of Linkage Disequilibrium
TASSEL -software to evaluate linkage disequilibrium, traits associations, and evolutionary patterns
rAggr - finds proxy markers (SNPs and indels) that are in linkage disequilibrium with a set of queried markers, using the 1000 Genomes Project and HapMap genotype databases.

Simulation software

Haploid — a C library for population genetic simulation (GPL)

References

1 2 3 Slatkin, Montgomery (June 2008). "Linkage disequilibrium — understanding the evolutionary past and mapping the medical future". Nature Reviews Genetics 9 (6): 477–485. doi:10.1038/nrg2361.
↑ Falconer, DS; Mackay, TFC (1996). Introduction to Quantitative Genetics (4th ed.). Harlow, Essex, UK: Addison Wesley Longman. ISBN 0-582-24302-5.
↑ Lewontin, R. C. (1964). "The interaction of selection and linkage. I. General considerations; heterotic models". Genetics 49 (1): 49–67. PMC 1210557. PMID 17248194.
↑ Robbins, R.B. (1 July 1918). "Some applications of mathematics to breeding problems III". Genetics 3 (4): 375–389. PMC 1200443. PMID 17245911.
↑ R.C. Lewontin and K. Kojima (1960). "The evolutionary dynamics of complex polymorphisms". Evolution 14 (4): 458–472. doi:10.2307/2405995. ISSN 0014-3820. JSTOR 2405995.
1 2 Svejgaard A, Hauge M, Jersild C, Plaz P, Ryder LP, Staub Nielsen L, Thomsen M (1979). The HLA System: An Introductory Survey, 2nd ed. Basel; London; Chichester: Karger; Distributed by Wiley, ISBN 3805530498(pbk).
1 2 3 4 Vogel F, Motulsky AG (1997). Human Genetics : Problems and Approaches, 3rd ed.Berlin; London: Springer, ISBN 3-540-60290-9.
↑ Mittal KK, Hasegawa T, Ting A, Mickey MR, Terasaki PI (1973). "Genetic variation in the HL-A system between Ainus, Japanese, and Caucasians," In Dausset J, Colombani J, eds. Histocompatibility Testing, 1972, pp. 187-195, Copenhagen: Munksgaard, ISBN 87-16-01101-5.
↑ Yasuda N, Tsuji K (1975). "A counting method of maximum likelihood for estimating haplotype frequency in the HL-A system." Jinrui Idengaku Zasshi 20(1): 1-15, PMID 1237691.
1 2 3 4 Mittal KK (1976). "The HLA polymorphism and susceptibility to disease." Vox Sang 31: 161-173, PMID 969389.
1 2 3 Gregersen PK (2009). "Genetics of rheumatic diseases," InFirestein GS, Budd RC, Harris ED Jr, McInnes IB, Ruddy S, Sergent JS, eds. (2009). Kelley's Textbook of Rheumatology, pp. 305-321, Philadelphia, PA: Saunders/Elsevier, ISBN 978-1-4160-3285-4.
1 2 3 Bengtsson BO, Thomson G (1981). "Measuring the strength of associations between HLA antigens and diseases." Tissue Antigens18(5): 356-363, PMID 7344182.
1 2 Nijenhuis LE (1977). "Genetic considerations on association between HLA and disease." Hum Genet38(2): 175-182, PMID 908564.
↑ Woolf B (1955). "On estimating the relation between blood group and disease." Ann Hum Genet 19(4): 251-253, PMID 14388528.
↑ Haldane JB (1956). "The estimation and significance of the logarithm of a ratio of frequencies." Ann Hum Genet20(4): 309-311, PMID 13314400.
↑ Sokal RR, Rohlf FJ (1981). Biometry: The Principles and Practice of Statistics in Biological Research. Oxford: W.H. Freeman, ISBN 0-7167-1254-7.
↑ Devlin B., Risch N. (1995). "A Comparison of Linkage Disequilibrium Measures for Fine-Scale Mapping" (PDF). Genomics 29 (2): 311–322. doi:10.1006/geno.1995.9003. PMID 8666377.
↑ Hao K., Di X., Cawley S. (2007). "LdCompare: rapid computation of single- and multiple-marker r2 and genetic coverage". Bioinformatics 23 (2): 252–254. doi:10.1093/bioinformatics/btl574. PMID 17148510.