Stress concentration

A simulation showing the evolution of linkage disequilibrium in randomly generated haploid populations with positive assortative mating

In population genetics, linkage disequilibrium is the non-random association of alleles at two or more loci, that descend from single, ancestral chromosomes. Linkage disequilibrium is wholly a measurement of proximal genomic space. It is necessary to refer to this as gametic phase disequilibrium or simply gametic disequilibrium because it is described through DNA recombination. 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 a second order phenomenon derived from 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 allelic frequencies and those expected from a homogenous, randomly distributed model. 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 mutation, genetic drift, non-random mating, and population 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.

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:

Haplotype	Frequency
$A_{1}B_{1}$	$x_{11}$
$A_{1}B_{2}$	$x_{12}$
$A_{2}B_{1}$	$x_{21}$
$A_{2}B_{2}$	$x_{22}$

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 $A_{1}B_{1}$ 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 $A_{1}B_{1}$ 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 called the linkage disequilibrium and is commonly denoted by a capital D:

$D=x_{11}-p_{1}q_{1}$

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.

	$A_{1}$	$A_{2}$	Total
$B_{1}$	$x_{11}=p_{1}q_{1}+D$	$x_{21}=p_{2}q_{1}-D$	$q_{1}$
$B_{2}$	$x_{12}=p_{1}q_{2}-D$	$x_{22}=p_{2}q_{2}+D$	$q_{2}$
Total	$p_{1}$	$p_{2}$	$1$

$D$ 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 suggested normalising D by dividing it by the theoretical maximum for the observed allele frequencies.

Thus:

$D'$ = ${\tfrac {D}{D_{\max }}}$

where

$D_{\max }={\begin{cases}\min(p_{1}q_{1},\,p_{2}q_{2})&{\text{when }}D<0\\\min(p_{1}q_{2},\,p_{2}q_{1})&{\text{when }}D>0\end{cases}}$

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

$r={\frac {D}{\sqrt {p_{1}p_{2}q_{1}q_{2}}}}$.

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.

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 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.

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 referred to by Vogel and Motulsky (1997).

Table 1. Association of HLA-A1 and B8 in unrelated Danes

			Antigen j		Total
			$+$	$-$
			$B8^{+}$	$B8^{-}$
Antigen i	$+$	$A1^{+}$	$a=376$	$b=237$	$C$
	$-$	$A1^{-}$	$c=91$	$d=1265$	$D$
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.

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

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.

Vogel and Motulsky (1997) 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 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.

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.
• The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by $\delta$ value to exceed 0.

Table 3. Association of ankylosing spondylitis with HLA-B27 allele

		Ankylosing spondylitis		Total
		Patients	Healthy controls
HLA alleles	$B27^{+}$	$a=96$	$b=77$	$C$
	$B27^{-}$	$c=22$	$d=701$	$D$
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. 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 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 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

Disease	HLA allele	Relative risk (%)	FAD (%)	FAP (%)	$\delta$
Ankylosing spondylitis	B27	90	90	8	0.89
Reiter's syndrome	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.

(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.$\delta$ value is expressed by

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

where $FAD$ and $FAP$ are HLA allele frequencies among patients and healthy populations, respectively. 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 $N\leq 30$), one should utilize Yates's correction for continuity or Fisher's exact test.

Resources

A comparison of different measures of LD is provided by Devlin & Risch

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— 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

Simulation software

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

See also

• Haploview
• Hardy-Weinberg principle
• Genetic linkage
• Co-adaptation
• Genealogical DNA test
• Tag SNP
• Association Mapping
• Family based QTL mapping

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Bibliography: Linkage Disequilibrium Analysis : a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918.

support.borinquenhealth.org - http://support.borinquenhealth.org/entries/35126754-Hostgator-Exchange. Hostgator is a huge privately possessed hosting company operating from numerous state of the art facilities in Dallas, Texas. Hostgator was started in 2002, since then they have actually grown rapidly and presently host over 400,000 sites.

Hostgator offers many various web hosting plans, and cater to a broad variety of consumers. From the first time web designer who requires simple, stress free hosting for their personal website; all the means through to huge companies, who require specialist dedicated hosting services.

Hostgator's hosting packages can be split into 3 teams; conventional shared hosting strategies (suitable for the huge bulk people), reseller hosting strategies (these are mainly for people and companies that want to "resell" their account resources to consumers of their own), and lastly devoted server strategies (these accounts give customer their own server, so they do not have to share its resources with anybody else). Very few of us will every require a specialized server so this evaluation will focus on the shared hosting plans that Hostgator offer.

Features

Hostgator's three main shared hosting strategies are named: "Hatchling" (the entry level strategy priced at $6.95 / month), "Baby" (this is the most popular plan, and it is likely to satisfy the requirements of a really broad range of customers), and "Swamp" (much the same as the "Infant" plan, but with boosts in bandwidth and disk space, priced at $14.95 / month).

For a complete list of functions and a side by side comparison of all hosting plans you ought to visit Hostgator's site below. Below is a review of the most essential functions of the "Infant" strategy, this is most likely the most suitable bundle for a lot of users, and it is our favored plan.

Disk area 100GB - This quantity has been just recently upgraded by Hostgator from 5GB to a huge 100GB. All users are likely to find it difficult to exhaust this amount of disk space.

Bandwidth 1000GB/month - Also enhanced is the bandwidth allotment, from 75GB to a seemingly extreme 1000GB. Again virtually absolutely no possibility of utilizing all that, but it's good to know that you certainly will not be facing extra costs for reviewing your limitation, even if you have a really busy website(s).

Website Studio website home builder - This is a wonderful free of cost program that allows you to build your site from scratch. You have over 500 design templates and color schemes to pick from. You require no HTML experience, or code writing understanding. It offers the simplest possible means of creating a professional looking site in a really brief space of time. However, there is no should take my word for it, as Hostgator offers us with a trial version of the software on their internet site!

Unrestricted add-on domains - This actually is the stand out function of this hosting strategy (the "Hatchling" plan just permits 1 domain), and enables you to host as many internet sites as you like on a single account, at no extra cost. This permits you to make complete use of your huge bandwidth and disk space allowances, and host numerous websites at a fraction of the regular expense.

99.9 % Uptime guarantee - This basically informs us that Hostgator is a serious host offering a trustworthy service. If uptime drops below this figure in any offered month then you don't spend for that months hosting, it's as simple as that! We would never ever consider using a host that did not provide a strong uptime assurance; this is simply because there is only one excellent reason why a host will not offer an uptime assurance - unreliable uptime!

30 Day refund guarantee - This has become a pretty conventional attribute in the hosting area, though it's nice to have for included assurance.

Instantaneous setup - Many hosting providers take 24-48 hours to setup your account but Hostgator guarantees to have you up and running in under 15 minutes (they do not charge a setup charge either)!

Unrestricted MySQL data sources - This is really useful since each Fantastico (see below) script needs its own MySQL data source.

Fantastico DeLuxe - This remarkable program enables you to instantaneously set up over 50 scripts via your control panel. Scripts consist of blogs, online forums, galleries, shopping carts, and more.

Endless e-mail accounts - Allows you to have as many, or as couple of, different e-mail addresses as you such as.

cPanel control panel - At the heart of any web hosting experience is the control panel, thankfully Hostgator makes use of among the best systems around, a cPanel. It's well laid out and easy to run. It contains numerous functions and offers great performance. Most importantly, there is a complete working demo on the Hostgator website, so you can check it out yourself!

Customer care and technical support

Hostgator gives us 24/7 phone support, and live online talk. The truth that you are provided two choices to receive instantaneous technical support at any time of the day is great. Our experience has actually constantly been very excellent when contacting Hostgator, their operatives are very respectful and most importantly they seem to know their stuff when dealing with technical concerns.

Efficiency

The performance from Hostgator's servers is outstanding! Hostgator place much tighter limits on the variety of websites sharing the exact same server compared to most various other shared hosting suppliers. This gives higher reliability due to the fact that less strain is put on the servers; and it also greatly improves the speed at which your websites run.

Server efficiency is an additional among the essential locations where Hostgator distinguish themselves from the crowd of various other webhosting.

Our judgment

Overall there is a lot to such as about the method Hostgator does business, they actually do appear to have a great grasp on what the typical consumer needs from an internet hosting carrier. Hardly ever do you discover reports of unhappy Hostgator customers, and after hosting with them ourselves we now know why! Dependability, performance, and uptime from the servers is first rate, also client service is means ahead of many other webhosting out there. Nevertheless, the location where Hostgator really excels is value for money. At just $9.95 / month for the "Infant" strategy (which consists of limitless domains); anybody looking to host even more than one internet site has a rather simple decision to make. The other fantastic aspect of this bundle is that you do not need to sign up to an agreement. You pay on a month-by-month basis, which leaves you free to cancel your account at any time without losing money. To sum up, we don't really have any bad things about Hostgator; at the rates they charge, we genuinely feel that they offer a phenomenal service. Sustain the great Hostgator!