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Linkage disequilibrium  

2013-01-31 23:38:41|  分类: 默认分类 |  标签: |举报 |字号 订阅

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In population geneticslinkage disequilibrium is the non-random association of alleles at two or more loci, that may or may not be on the same chromosome. It is also referred to as gametic phase disequilibrium,[1] or simply gametic disequilibrium. In other words, linkage disequilibrium is the occurrence of some combinations of alleles or genetic markers in a population more often or less often than would be expected from a random formation of haplotypes from alleles based on their frequencies. It is not the same as linkage, which is the presence of two or more loci on a chromosome with limited recombination between them. The amount of linkage disequilibrium depends on the difference between observed and expected (assuming random distributions) allelic frequencies. Populations where combinations of alleles or genotypes can be found in the expected proportions are said to be in linkage equilibrium.

The level of linkage disequilibrium is influenced by a number of factors, including genetic linkage, selection, the rate of recombination, the rate of mutationgenetic driftnon-random mating, andpopulation structure. A limiting example of the effect of rate of recombination may be seen in some organisms (such as bacteria) that reproduce asexually and hence exhibit no recombination to break down the linkage disequilibrium. An example of the effect of population structure is the phenomenon of Finnish disease heritage, which is attributed to a population bottleneck.

Contents

  [hide

[edit]Definition

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:

HaplotypeFrequency
A_1B_1x_{11}
A_1B_2x_{12}
A_2B_1x_{21}
A_2B_2x_{22}

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

AlleleFrequency
A_1p_{1}=x_{11}+x_{12}
A_2p_{2}=x_{21}+x_{22}
B_1q_{1}=x_{11}+x_{21}
B_2q_{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_1p_1, and forB_1q_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[2] called the linkage disequilibrium[3] and is commonly denoted by a capital D:

D = x_{11} - p_1q_1

In the genetic literature the phrase "two alleles are in LD" usually means that D ≠ 0. Contrariwise, "linkage equilibrium" means D = 0.

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

A_1A_2Total
B_1x_{11}=p_1q_1+D    x_{21}=p_2q_1-D   q_1
B_2x_{12}=p_1q_2-Dx_{22}=p_2q_2+Dq_2
Total   p_1p_21

D is easy to calculate with, but has the disadvantage of depending on the frequencies of the alleles. This is evident since frequencies are between 0 and 1. If any locus has an allele frequency 0 or 1 no disequilibrium D can be observed. When the allelic frequencies are 0.5, the disequilibrium D is maximal. Lewontin[4] suggested normalising D by dividing it by the theoretical maximum for the observed allele frequencies.

Thus:

D' = \tfrac{D}{D_\max}

where

D_\max = \begin{cases} \max(-p_1q_1,\,-p_2q_2) & \text{when } D < 0\\ \min(p_1q_2,\,p_2q_1) & \text{when } D > 0 \end{cases}

Another measure of LD which is an alternative to D' is the correlation coefficient between pairs of loci, expressed as

r=\frac{D}{\sqrt{p_1p_2q_1q_2}}.

This is also adjusted to the loci having different allele frequencies.

In summary, linkage disequilibrium reflects the difference between the expected haplotype frequencies under the assumption of independence, and observed haplotype frequencies. A value of 0 for D' indicates that the examined loci are in fact independent of one another, while a value of 1 demonstrates complete dependency.

[edit]Role of recombination

In the absence of evolutionary forces other than random mating and Mendelian segregation, the linkage disequilibrium measure D converges to zero along the time axis at a rate depending on the magnitude of the recombination rate cbetween 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 on different loci, 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.

[edit]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[5] referred to by Vogel and Motulsky (1997).[6]

Table 1. Association of HLA-A1 and B8 in unrelated Danes[5]
Antigen jTotal
+-
B8^{+}B8^{-}
Antigen i+A1^{+}a=376b=237C
-A1^{-}c=91d=1265D
TotalABN
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.[6][7][8][9]

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 large statistically significantly. 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[9]
HLA-A alleles iHLA-B alleles j\Delta_{ij}t
A1B80.06516.0
A3B70.03910.3
A2Bw400.0134.4
A2Bw150.013.4
A1Bw170.0145.4
A2B180.0062.2
A2Bw35-0.009-2.3
A29B120.0136.0
A10Bw160.0135.9

Table 2 shows some of the combinations of HLA-A and B alleles where significant LD was observed among pan-europeans.[9]

Vogel and Motulsky (1997)[6] 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[9]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.[6]

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.[10]
  • The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by \delta value[11] to exceed 0.
Table 3. Association of ankylosing spondylitis with HLA-B27 allele[12]
Ankylosing spondylitisTotal
PatientsHealthy controls
HLA allelesB27^+a=96b=77C
B27^-c=22d=701D
TotalABN
  • 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.[12] Relative risk xof this allele is approximated by

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

Woolf's method[13] 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[14] modification applies to the case when either ofa,\; b,\;c,\text{ and }d is zero, where replace x and 1/wwith

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[10]
DiseaseHLA alleleRelative risk (%)FAD (%)FAP (%)\delta
Ankylosing spondylitisB27909080.89
Reiter's syndromeB27407080.67
Spondylitis in inflammatory bowel diseaseB27105080.46
Rheumatoid arthritisDR4670300.57
Systemic lupus erythematosusDR3345200.31
Multiple sclerosisDR2460200.5
Diabetes mellitus type 1DR4675300.64

In Table 4, some examples of association between HLA alleles and diseases are presented.[10]

(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.[11]\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.[11] In Table 4, \deltacolumn 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.[15]

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