Hale Telescope

From Wikipedia, the free encyclopedia

Hale Telescope

Named after	George Ellery Hale
Part of	Palomar Observatory
Location(s)	Palomar Mountain, California, US
Coordinates	33°21′23″N 116°51′54″WCoordinates: 33°21′23″N 116°51′54″W
Altitude	1,713 m (5,620 ft)
Built	1936 –1948
First light	January 26, 1949, 10:06 pm PST
Discovered	Caliban, Sycorax, Jupiter LI
Telescope style	optical telescope reflecting telescope
Diameter	200 in (5.1 m)
Collecting area	31,000 sq in (20 m2)
Focal length	16.76 m (55 ft 0 in)
Mounting	Equatorial mount
Website	www.astro.caltech.edu/palomar/about/telescopes/hale.html
Location of Hale Telescope

The Hale Telescope is a 200-inch (5.1 m), f/3.3 reflecting telescope at the Palomar Observatory in California, US, named after astronomer George Ellery Hale. With funding from the Rockefeller Foundation in 1928, he orchestrated the planning, design, and construction of the observatory, but with the project ending up taking 20 years he did not live to see its commissioning. The Hale was groundbreaking for its time, with double the diameter of the second-largest telescope, and pioneered many new technologies in telescope mount design and in the design and fabrication of its large aluminum coated "honeycomb" low thermal expansion Pyrex mirror. It was completed in 1949 and is still in active use.

The Hale Telescope represented the technological limit in building large optical telescopes for over 30 years. It was the largest telescope in the world from its construction in 1949 until the Russian BTA-6 was built in 1976, and the second largest until the construction of the Keck Observatory Keck 1 in 1993.

History

Base of the tube

Crab Nebula, 1959

Hale supervised the building of the telescopes at the Mount Wilson Observatory with grants from the Carnegie Institution of Washington: the 60-inch (1.5 m) telescope in 1908 and the 100-inch (2.5 m) telescope in 1917. These telescopes were very successful, leading to the rapid advance in understanding of the scale of the Universe through the 1920s, and demonstrating to visionaries like Hale the need for even larger collectors.

The chief optical designer for Hale's previous 100-inch telescope was George Willis Ritchey, who intended the new telescope to be of Ritchey–Chrétien design. Compared to the usual parabolic primary, this design would have provided sharper images over a larger usable field of view. However, Ritchey and Hale had a falling-out. With the project already late and over budget, Hale refused to adopt the new design, with its complex curvatures, and Ritchey left the project. The Mount Palomar Hale Telescope turned out to be the last world-leading telescope to have a parabolic primary mirror.

In 1928 Hale secured a grant of $6 million from the Rockefeller Foundation for "the construction of an observatory, including a 200-inch reflecting telescope" to be administered by the California Institute of Technology (Caltech), of which Hale was a founding member. In the early 1930s, Hale selected a site at 1,700 m (5,600 ft) on Palomar Mountain in San Diego County, California, US, as the best site, and less likely to be affected by the growing light pollution problem in urban centers like Los Angeles. The Corning Glass Works was assigned the task of making a 200-inch (5.1 m) primary mirror. Construction of the observatory facilities and dome started in 1936, but because of interruptions caused by World War II, the telescope was not completed until 1948 when it was dedicated. Due to slight distortions of images, corrections were made to the telescope throughout 1949. It became available for research in 1950.

The 200-inch (510 cm) telescope saw first light on January 26, 1949, at 10:06 pm PST under the direction of American astronomer Edwin Powell Hubble, targeting NGC 2261, an object also known as Hubble's Variable Nebula. The photographs made then were published in the astronomical literature and in the May 7, 1949 issue of Collier's Magazine. 

The telescope continues to be used every clear night for scientific research by astronomers from Caltech and their operating partners, Cornell University, the University of California, and the Jet Propulsion Laboratory. It is equipped with modern optical and infrared array imagers, spectrographs, and an adaptive optics system. It has also used lucky cam imaging, which in combination with adaptive optics pushed the mirror close to its theoretical resolution for certain types of viewing.

One of the Corning Labs' glass test blanks for the Hale was used for the C. Donald Shane telescope's 120-inch (300 cm) primary mirror.

The collecting area of the mirror is about 31,000 square inches (20 square meters).

Components

The Hale was not just big, it was better, and not just but better but combined breakthrough technologies including a new lower expansion glass from Corning, a newly invented Serruier truss, and vapor deposited aluminum.

Mounting structures

The Hale Telescope uses a special type of equatorial mount called a "horseshoe mount", a modified yoke mount that replaces the polar bearing with an open "horseshoe" structure that gives the telescope full access to the entire sky, including Polaris and stars near it. The optical tube assembly (OTA) uses a Serrurier truss, then newly invented by Mark U. Serrurier of Caltech in Pasadena in 1935, designed to flex in such a way as to keep all of the optics in alignment. Theodore von Karman designed the lubrication system to avoid potential issues with turbulence during tracking.

Left: The 200-inch (508 cm) Hale Telescope inside on its equatorial mount.
Right: Principle of operation of a Serrurier truss similar to that of the Hale Telescope compared to a simple truss. For clarity, only the top and bottom structural elements are shown. Red and green lines denote elements under tension and compression, respectively.

200-inch mirror

The 5 meter (16 ft. 8 in.) mirror in December 1945 at the Caltech Optical Shop when grinding resumed following World War 2. The honeycomb support structure on the back of the mirror is visible through the surface.

Originally, the Hale Telescope was going to use a primary mirror of fused quartz manufactured by General Electric, but instead the primary mirror was cast in 1934 at Corning Glass Works in New York State using Corning's then new material called Pyrex (borosilicate glass). Pyrex was chosen for its low expansion qualities so the large mirror would not distort the images produced when it changed shape due to temperature variations (a problem that plagued earlier large telescopes).

Entrance door to 200 inch Hale telescope dome

The mirror was cast in a mold with 36 raised mold blocks (similar in shape to a waffle iron). This created a honeycomb mirror that cut the amount of Pyrex needed down from over 40 short tons (36 t) to just 20 short tons (18 t), making a mirror that would cool faster in use and have multiple "mounting points" on the back to evenly distribute its weight (note – see external links 1934 article for drawings). The shape of a central hole was also part of the mold so light could pass through the finished mirror when it was used in a Cassegrain configuration (a Pyrex plug for this hole was also made to be used during the grinding and polishing process). While the glass was being poured into the mold during the first attempt to cast the 200-inch mirror, the intense heat caused several of the molding blocks to break loose and float to the top, ruining the mirror. The defective mirror was used to test the annealing process. After the mold was re-engineered, a second mirror was successfully cast.

After cooling several months, the finished mirror blank was transported by rail to Pasadena, California. Once in Pasadena the mirror was transferred from the rail flat car to a specially designed semi-trailer for road transport to where it would be polished. In the optical shop in Pasadena (now the Synchrotron building at Caltech) standard telescope mirror making techniques were used to turn the flat blank into a precise concave parabolic shape, although they had to be executed on a grand scale. A special 240 in (6.1 m) 25,000 lb (11 t) mirror cell jig was constructed which could employ five different motions when the mirror was ground and polished. Over 13 years almost 10,000 lb (4.5 t) of glass was ground and polished away reducing the weight of the mirror to 14.5 short tons (13.2 t). The mirror was coated (and still is re-coated every 18–24 months) with a reflective aluminum surface using the same aluminum vacuum-deposition process invented in 1930 by Caltech physicist and astronomer John Strong.

The Hale's 200 in (510 cm) mirror was near the technological limit of a primary mirror made of a single rigid piece of glass. Using a monolithic mirror much larger than the 5-meter Hale or 6-meter BTA-6 is prohibitively expensive due to the cost of both the mirror, and the massive structure needed to support it. A mirror beyond that size would also sag slightly under its own weight as the telescope is rotated to different positions, changing the precision shape of the surface, which must be accurate to within 2 millionths of an inch (50 nm). Modern telescopes over 9 meters use a different mirror design to solve this problem, with either a single thin flexible mirror or a cluster of smaller segmented mirrors, whose shape is continuously adjusted by a computer-controlled active optics system using actuators built into the mirror support cell.

Dome

The moving weight of the upper dome is about 1000 US tons, and can rotate on wheels. The dome doors weigh 125 tons each.

The dome is made of welded steel plates about 10 mm thick.

Observations and research

Dome of the 200-inch aperture Hale telescope

The first observation of the Hale telescope was of NGC 2261 on January 26, 1949.

Halley's Comet (1P) upcoming 1986 approach to the Sun was first detected by astronomers David C. Jewitt and G. Edward Danielson on 16 October 1982 using the 200-inch Hale telescope equipped with a CCD camera.

Two moons of the planet Uranus were discovered in September 1997, in addition to the planet's 15 other known moons at that time. One was Caliban (S/1997 U 1), which was discovered on 6 September 1997 by Brett J. Gladman, Philip D. Nicholson, Joseph A. Burns, and John J. Kavelaars using the 200-inch Hale telescope. The other Uranian moon discovered then is Sycorax (initial designation S/1997 U 2) and was also discovered using the 200 inch Hale telescope.

In 1999, astronomers used a near-infrared camera and adaptive optics to take some of the best Earth-surface based images of planet Neptune up to that time. The images were sharp enough to identify clouds in the ice giant's atmosphere.

The Cornell Mid-Infrared Asteroid Spectroscopy (MIDAS) survey used the Hale Telescope with a spectrograph to study spectra from 29 asteroids. An example of a result from that study, is that the asteroid 3 Juno was determined to have average radius of 135.7±11 km using the infrared data.

In 2009, using a coronograph, the Hale telescope was used to discover the star Alcor B, which is a companion to Alcor in the famous Big Dipper constellation.

In 2010, a new satellite of planet Jupiter was discovered with the 200-inch Hale, called S/2010 J 1 and later named Jupiter LI.

In October 2017 the Hale telescope was able to record the spectrum of the first recognized interstellar object, 1I/2017 U1 ("ʻOumuamua"); while not specific mineral was identified it showed the visitor had a reddish surface color.

Direct imaging of exoplanets

Up until the year 2010, telescopes could only directly image exoplanets under exceptional circumstances. Specifically, it is easier to obtain images when the planet is especially large (considerably larger than Jupiter), widely separated from its parent star, and hot so that it emits intense infrared radiation. However, in 2010 a team from NASA's Jet Propulsion Laboratory demonstrated that a vortex coronagraph could enable small scopes to directly image planets. They did this by imaging the previously imaged HR 8799 planets using just a 1.5 m portion of the Hale Telescope.

Direct image of exoplanets around the star HR8799 using a vortex coronagraph on a 1.5m portion of the Hale Telescope

Comparison

Size comparison of the Hale Telescope (upper left, blue) to some modern and upcoming extremely large telescopes

The Hale had four times the light-collecting area of the second-largest scope when it was commissioned in 1949. Other contemporary telescopes were the Hooker Telescope at the Mount Wilson Observatory and the Otto Struve Telescope at the McDonald Observatory.

The three largest telescopes in 1949

#	Name / Observatory	Image	Aperture	Altitude	First Light	Special advocate(s)
1	Hale Telescope Palomar Obs.		200-inch 508 cm	1713 m (5620 ft)	1949	George Ellery Hale John D. Rockefeller Edwin Hubble
2	Hooker Telescope Mount Wilson Obs.		100-inch 254 cm	1742 m (5715 ft)	1917	George Ellery Hale Andrew Carnegie
3	Otto Struve Telescope McDonald Obs.		82-inch 210 cm	2070 m (6791 ft)	1939	Otto Struve

Linkage disequilibrium

From Wikipedia, the free encyclopedia

 

In population genetics, linkage disequilibrium is the non-random association of alleles at different loci in a given population. Loci are said to be in linkage disequilibrium when the frequency of association of their different alleles is higher or lower than what would be expected if the loci were independent and associated randomly.

Linkage disequilibrium is influenced by many factors, including selection, the rate of genetic recombination, mutation rate, 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 processes 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). Furthermore, linkage disequilibrium is sometimes referred to as gametic phase disequilibrium; 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 ${\displaystyle p_{A}}$ at one locus (i.e. ${\displaystyle p_{A}}$ is the proportion of gametes with A at that locus), while at a different locus allele B occurs with frequency ${\displaystyle p_{B}}$. Similarly, let ${\displaystyle p_{AB}}$ be the frequency with which both A and B occur together in the same gamete (i.e. ${\displaystyle p_{AB}}$ is the frequency of the AB haplotype). 

The association between the alleles A and B can be regarded as completely random—which is known in statistics as independence—when the occurrence of one does not affect the occurrence of the other, in which case the probability that both A and B occur together is given by the product ${\displaystyle p_{A}p_{B}}$ of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever ${\displaystyle p_{AB}}$ differs from ${\displaystyle p_{A}p_{B}}$ for any reason. 

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

${\displaystyle D_{AB}=p_{AB}-p_{A}p_{B}}$,

provided that both ${\displaystyle p_{A}}$ and ${\displaystyle p_{B}}$ are greater than zero. Linkage disequilibrium corresponds to ${\displaystyle D_{AB}\neq 0}$. In the case ${\displaystyle D_{AB}=0}$ we have ${\displaystyle p_{AB}=p_{A}p_{B}}$ and the alleles A and B are said to be in linkage equilibrium. The subscript "AB" on ${\displaystyle D_{AB}}$ emphasizes that linkage disequilibrium is a property of the pair {A, B} of alleles and not of their respective loci. Other pairs of alleles at those same two loci may have different coefficients of linkage disequilibrium. 

Linkage disequilibrium in asexual populations can be defined in a similar way in terms of population allele frequencies. Furthermore, it is also possible to define linkage disequilibrium among three or more alleles, however these higher-order associations are not commonly used in practice.

Measures derived from ${\displaystyle D}$

The coefficient of linkage disequilibrium ${\displaystyle D}$ 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 suggested normalising D by dividing it by the theoretical maximum difference between the observed and expected haplotype frequencies as follows: 

${\displaystyle D'=D/D_{\max }}$

where 

${\displaystyle D_{\max }={\begin{cases}\max\{-p_{A}p_{B},\,-(1-p_{A})(1-p_{B})\}&{\text{when }}D<0 amp="" min="" p_="" text="" when="">0\end{cases}}}$

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

${\displaystyle r={\frac {D}{\sqrt {p_{A}(1-p_{A})p_{B}(1-p_{B})}}}}$.

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
${\displaystyle A_{1}B_{1}}$	${\displaystyle x_{11}}$
${\displaystyle A_{1}B_{2}}$	${\displaystyle x_{12}}$
${\displaystyle A_{2}B_{1}}$	${\displaystyle x_{21}}$
${\displaystyle A_{2}B_{2}}$	${\displaystyle x_{22}}$

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

Allele	Frequency
${\displaystyle A_{1}}$	${\displaystyle p_{1}=x_{11}+x_{12}}$
${\displaystyle A_{2}}$	${\displaystyle p_{2}=x_{21}+x_{22}}$
${\displaystyle B_{1}}$	${\displaystyle q_{1}=x_{11}+x_{21}}$
${\displaystyle B_{2}}$	${\displaystyle q_{2}=x_{12}+x_{22}}$

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

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: 

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

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

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

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 ${\displaystyle D}$ converges to zero along the time axis at a rate depending on the magnitude of the recombination rate ${\displaystyle c}$ between the two loci.

Using the notation above, ${\displaystyle D=x_{11}-p_{1}q_{1}}$, we can demonstrate this convergence to zero as follows. In the next generation, ${\displaystyle x_{11}'}$, the frequency of the haplotype ${\displaystyle A_{1}B_{1}}$, becomes

${\displaystyle x_{11}'=(1-c)\,x_{11}+c\,p_{1}q_{1}}$

This follows because a fraction ${\displaystyle (1-c)}$ of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction ${\displaystyle x_{11}}$ of those are ${\displaystyle A_{1}B_{1}}$. A fraction ${\displaystyle c}$ have recombined these two loci. If the parents result from random mating, the probability of the copy at locus ${\displaystyle A}$ having allele ${\displaystyle A_{1}}$ is ${\displaystyle p_{1}}$ and the probability of the copy at locus ${\displaystyle B}$ having allele ${\displaystyle B_{1}}$ is ${\displaystyle q_{1}}$, 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 

${\displaystyle x_{11}'-p_{1}q_{1}=(1-c)\,(x_{11}-p_{1}q_{1})}$

so that 

${\displaystyle D_{1}=(1-c)\;D_{0}}$

where ${\displaystyle D}$ at the ${\displaystyle n}$-th generation is designated as ${\displaystyle D_{n}}$. Thus we have 

${\displaystyle D_{n}=(1-c)^{n}\;D_{0}}$.

If ${\displaystyle n\to \infty }$, then ${\displaystyle (1-c)^{n}\to 0}$ so that ${\displaystyle D_{n}}$ 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 ${\displaystyle D}$ to zero.

Example: Human leukocyte antigen (HLA) alleles

HLA constitutes a group of cell surface antigens also known as the 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
			${\displaystyle +}$	${\displaystyle -}$
			${\displaystyle B8^{+}}$	${\displaystyle B8^{-}}$
Antigen i	${\displaystyle +}$	${\displaystyle A1^{+}}$	${\displaystyle a=376}$	${\displaystyle b=237}$	${\displaystyle C}$
	${\displaystyle -}$	${\displaystyle A1^{-}}$	${\displaystyle c=91}$	${\displaystyle d=1265}$	${\displaystyle D}$
Total			${\displaystyle A}$	${\displaystyle B}$	${\displaystyle N}$
			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 (${\displaystyle +}$) frequency of antigen ${\displaystyle i}$ :

${\displaystyle pf_{i}=C/N=0.311\!}$ ;

expression (${\displaystyle +}$) frequency of antigen ${\displaystyle j}$ :

${\displaystyle pf_{j}=A/N=0.237\!}$ ;

frequency of gene ${\displaystyle i}$, given that individuals with '+/-', '+/+', and '-/+' genotypes are all positive for antigen ${\displaystyle i}$:

${\displaystyle gf_{i}=1-{\sqrt {1-pf_{i}}}=0.170\!}$ ,

and

${\displaystyle hf_{ij}={\text{estimated frequency of haplotype }}ij=gf_{i}\;gf_{j}=0.0215\!}$ .

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

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

and the estimated frequency of haplotype xy is

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

Then LD measure ${\displaystyle \Delta _{ij}}$ is expressed as

${\displaystyle \Delta _{ij}=o[hf_{xy}]-e[hf_{xy}]={\frac {{\sqrt {Nd}}-{\sqrt {BD}}}{N}}=0.0769}$.

Standard errors ${\displaystyle SEs}$ are obtained as follows:

${\displaystyle SE{\text{ of }}gf_{i}={\sqrt {C}}/(2N)=0.00628}$,

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

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

Then, if

${\displaystyle t=\Delta _{ij}/(SE{\text{ of }}\Delta _{ij})}$

exceeds 2 in its absolute value, the magnitude of ${\displaystyle \Delta _{ij}}$ 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

HLA-A alleles i	HLA-B alleles j	${\displaystyle \Delta _{ij}}$	${\displaystyle t}$
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 ${\displaystyle \Delta _{0}}$ had reduced from 0.07 to 0.003 under recombination effect as shown by ${\displaystyle \Delta _{n}=(1-c)^{n}\Delta _{0}}$, then ${\displaystyle n\approx 400}$. 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 ${\displaystyle \delta }$ value to exceed 0.

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

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

• 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 ${\displaystyle x}$of this allele is approximated by

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

Woolf's method is applied to see if there is statistical significance. Let

${\displaystyle y=\ln(x)\;(=3.68)}$

and

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

Then

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

follows the chi-square distribution with ${\displaystyle df=1}$. 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${\displaystyle a,\;b,\;c,{\text{ and }}d}$ is zero, where replace ${\displaystyle x}$ and ${\displaystyle 1/w}$with

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

and

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

respectively.

Table 4. Association of HLA alleles with rheumatic and autoimmune diseases among white populations

Disease	HLA allele	Relative risk (%)	FAD (%)	FAP (%)	${\displaystyle \delta }$
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.

(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.${\displaystyle \delta }$ value is expressed by

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

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

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

 

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

${\displaystyle \chi ^{2}={\frac {(ad-bc)^{2}N}{ABCD}}\;(=336,{\text{ for data in Table 3; }}P<0 .001="" annotation="">$.

where ${\displaystyle df=1}$. For data with small sample size, such as no marginal total is greater than 15 (and consequently ${\displaystyle N\leq 30}$), 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^[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.
• SNeP - Fast computation of LD and Ne for large genotype datasets in PLINK format.
• LDlink - A suite of web-based applications to easily and efficiently explore linkage disequilibrium in population subgroups. All population genotype data originates from Phase 3 of the 1000 Genomes Project and variant RS numbers are indexed based on dbSNP build 151.

Simulation software

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