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)

Genetic linkage

From Wikipedia, the free encyclopedia

 

Genetic linkage is the tendency of DNA sequences that are close together on a chromosome to be inherited together during the meiosis phase of sexual reproduction. Two genetic markers that are physically near to each other are unlikely to be separated onto different chromatids during chromosomal crossover, and are therefore said to be more linked than markers that are far apart. In other words, the nearer two genes are on a chromosome, the lower the chance of recombination between them, and the more likely they are to be inherited together. Markers on different chromosomes are perfectly unlinked.

Genetic linkage is the most prominent exception to Gregor Mendel's Law of Independent Assortment. The first experiment to demonstrate linkage was carried out in 1905. At the time, the reason why certain traits tend to be inherited together was unknown. Later work revealed that genes are physical structures related by physical distance.

The typical unit of genetic linkage is the centimorgan (cM). A distance of 1 cM between two markers means that the markers are separated to different chromosomes on average once per 100 meiotic product, thus once per 50 meioses...

Discovery

Gregor Mendel's Law of Independent Assortment states that every trait is inherited independently of every other trait. But shortly after Mendel's work was rediscovered, exceptions to this rule were found. In 1905, the British geneticists William Bateson, Edith Rebecca Saunders and Reginald Punnett cross-bred pea plants in experiments similar to Mendel's. They were interested in trait inheritance in the sweet pea and were studying two genes—the gene for flower colour (P, purple, and p, red) and the gene affecting the shape of pollen grains (L, long, and l, round). They crossed the pure lines PPLL and ppll and then self-crossed the resulting PpLl lines. 

According to Mendelian genetics, the expected phenotypes would occur in a 9:3:3:1 ratio of PL:Pl:pL:pl. To their surprise, they observed an increased frequency of PL and pl and a decreased frequency of Pl and pL:

Bateson, Saunders, and Punnett experiment

Phenotype and genotype	Observed	Expected from 9:3:3:1 ratio
Purple, long (P_L_)	284	216
Purple, round (P_ll)	21	72
Red, long (ppL_)	21	72
Red, round (ppll)	55	24

Their experiment revealed linkage between the P and L alleles and the p and l alleles. The frequency of P occurring together with L and p occurring together with l is greater than that of the recombinant Pl and pL. The recombination frequency is more difficult to compute in an F2 cross than a backcross, but the lack of fit between observed and expected numbers of progeny in the above table indicate it is less than 50%. This indicated that two factors interacted in some way to create this difference by masking the appearance of the other two phenotypes. This led to the conclusion that some traits are related to each other because of their near proximity to each other on a chromosome. This provided the grounds to determine the difference between independent and codependent alleles.

The understanding of linkage was expanded by the work of Thomas Hunt Morgan. Morgan's observation that the amount of crossing over between linked genes differs led to the idea that crossover frequency might indicate the distance separating genes on the chromosome. The centimorgan, which expresses the frequency of crossing over, is named in his honour.

Linkage map

Thomas Hunt Morgan's Drosophila melanogaster genetic linkage map. This was the first successful gene mapping work and provides important evidence for the chromosome theory of inheritance. The map shows the relative positions of alleles on the second Drosophila chromosome. The distances between the genes (centimorgans) are equal to the percentages of chromosomal crossover events that occur between different alleles.

 

A linkage map (also known as a genetic map) is a table for a species or experimental population that shows the position of its known genes or genetic markers relative to each other in terms of recombination frequency, rather than a specific physical distance along each chromosome. Linkage maps were first developed by Alfred Sturtevant, a student of Thomas Hunt Morgan. 

A linkage map is a map based on the frequencies of recombination between markers during crossover of homologous chromosomes. The greater the frequency of recombination (segregation) between two genetic markers, the further apart they are assumed to be. Conversely, the lower the frequency of recombination between the markers, the smaller the physical distance between them. Historically, the markers originally used were detectable phenotypes (enzyme production, eye colour) derived from coding DNA sequences; eventually, confirmed or assumed noncoding DNA sequences such as microsatellites or those generating restriction fragment length polymorphisms (RFLPs) have been used. 

Linkage maps help researchers to locate other markers, such as other genes by testing for genetic linkage of the already known markers. In the early stages of developing a linkage map, the data are used to assemble linkage groups, a set of genes which are known to be linked. As knowledge advances, more markers can be added to a group, until the group covers an entire chromosome. For well-studied organisms the linkage groups correspond one-to-one with the chromosomes. 

A linkage map is not a physical map (such as a radiation reduced hybrid map) or gene map.

Linkage analysis

Linkage analysis is a genetic method that searches for chromosomal segments that cosegregate with the ailment phenotype through families and is the analysis technique that has been used to determine the bulk of lipodystrophy genes. It can be used to map genes for both binary and quantitative traits. Linkage analysis may be either parametric (if we know the relationship between phenotypic and genetic similarity) or non-parametric. Parametric linkage analysis is the traditional approach, whereby the probability that a gene important for a disease is linked to a genetic marker is studied through the LOD score, which assesses the probability that a given pedigree, where the disease and the marker are cosegregating, is due to the existence of linkage (with a given linkage value) or to chance. Non-parametric linkage analysis, in turn, studies the probability of an allele being identical by descent with itself.

Parametric linkage analysis

The LOD score (logarithm (base 10) of odds), developed by Newton Morton, is a statistical test often used for linkage analysis in human, animal, and plant populations. The LOD score compares the likelihood of obtaining the test data if the two loci are indeed linked, to the likelihood of observing the same data purely by chance. Positive LOD scores favour the presence of linkage, whereas negative LOD scores indicate that linkage is less likely. Computerised LOD score analysis is a simple way to analyse complex family pedigrees in order to determine the linkage between Mendelian traits (or between a trait and a marker, or two markers). 

The method is described in greater detail by Strachan and Read. Briefly, it works as follows:

1. Establish a pedigree
2. Make a number of estimates of recombination frequency
3. Calculate a LOD score for each estimate
4. The estimate with the highest LOD score will be considered the best estimate

The LOD score is calculated as follows:

${\displaystyle {\text{LOD}}=Z=\log _{10}{\frac {\text{probability of birth sequence with a given linkage value}}{\text{probability of birth sequence with no linkage}}}=\log _{10}{\frac {(1-\theta )^{NR}\times \theta ^{R}}{0.5^{NR+R}}}}$

NR denotes the number of non-recombinant offspring, and R denotes the number of recombinant offspring. The reason 0.5 is used in the denominator is that any alleles that are completely unlinked (e.g. alleles on separate chromosomes) have a 50% chance of recombination, due to independent assortment. θ is the recombinant fraction, i.e. the fraction of births in which recombination has happened between the studied genetic marker and the putative gene associated with the disease. Thus, it is equal to R / (NR + R). 

By convention, a LOD score greater than 3.0 is considered evidence for linkage, as it indicates 1000 to 1 odds that the linkage being observed did not occur by chance. On the other hand, a LOD score less than −2.0 is considered evidence to exclude linkage. Although it is very unlikely that a LOD score of 3 would be obtained from a single pedigree, the mathematical properties of the test allow data from a number of pedigrees to be combined by summing their LOD scores. A LOD score of 3 translates to a p-value of approximately 0.05, and no multiple testing correction (e.g. Bonferroni correction) is required.

Limitations

Linkage analysis has a number of methodological and theoretical limitations that can significantly increase the type-1 error rate and reduce the power to map human quantitative trait loci (QTL). While linkage analysis was successfully used to identify genetic variants that contribute to rare disorders such as Huntington disease, it did not perform that well when applied to more common disorders such as heart disease or different forms of cancer. An explanation for this is that the genetic mechanisms affecting common disorders are different from those causing rare disorders.

Recombination frequency

Recombination frequency is a measure of genetic linkage and is used in the creation of a genetic linkage map. Recombination frequency (θ) is the frequency with which a single chromosomal crossover will take place between two genes during meiosis. A centimorgan (cM) is a unit that describes a recombination frequency of 1%. In this way we can measure the genetic distance between two loci, based upon their recombination frequency. This is a good estimate of the real distance. Double crossovers would turn into no recombination. In this case we cannot tell if crossovers took place. If the loci we're analysing are very close (less than 7 cM) a double crossover is very unlikely. When distances become higher, the likelihood of a double crossover increases. As the likelihood of a double crossover increases we systematically underestimate the genetic distance between two loci.

During meiosis, chromosomes assort randomly into gametes, such that the segregation of alleles of one gene is independent of alleles of another gene. This is stated in Mendel's Second Law and is known as the law of independent assortment. The law of independent assortment always holds true for genes that are located on different chromosomes, but for genes that are on the same chromosome, it does not always hold true.

As an example of independent assortment, consider the crossing of the pure-bred homozygote parental strain with genotype AABB with a different pure-bred strain with genotype aabb. A and a and B and b represent the alleles of genes A and B. Crossing these homozygous parental strains will result in F1 generation offspring that are double heterozygotes with genotype AaBb. The F1 offspring AaBb produces gametes that are AB, Ab, aB, and ab with equal frequencies (25%) because the alleles of gene A assort independently of the alleles for gene B during meiosis. Note that 2 of the 4 gametes (50%)—Ab and aB—were not present in the parental generation. These gametes represent recombinant gametes. Recombinant gametes are those gametes that differ from both of the haploid gametes that made up the original diploid cell. In this example, the recombination frequency is 50% since 2 of the 4 gametes were recombinant gametes.

The recombination frequency will be 50% when two genes are located on different chromosomes or when they are widely separated on the same chromosome. This is a consequence of independent assortment. 

When two genes are close together on the same chromosome, they do not assort independently and are said to be linked. Whereas genes located on different chromosomes assort independently and have a recombination frequency of 50%, linked genes have a recombination frequency that is less than 50%.

As an example of linkage, consider the classic experiment by William Bateson and Reginald Punnett. They were interested in trait inheritance in the sweet pea and were studying two genes—the gene for flower colour (P, purple, and p, red) and the gene affecting the shape of pollen grains (L, long, and l, round). They crossed the pure lines PPLL and ppll and then self-crossed the resulting PpLl lines. According to Mendelian genetics, the expected phenotypes would occur in a 9:3:3:1 ratio of PL:Pl:pL:pl. To their surprise, they observed an increased frequency of PL and pl and a decreased frequency of Pl and pL (see table below).

Bateson and Punnett experiment

Phenotype and genotype	Observed	Expected from 9:3:3:1 ratio
Purple, long (P_L_)	284	216
Purple, round (P_ll)	21	72
Red, long (ppL_)	21	72
Red, round (ppll)	55	24

Their experiment revealed linkage between the P and L alleles and the p and l alleles. The frequency of P occurring together with L and with p occurring together with l is greater than that of the recombinant Pl and pL. The recombination frequency is more difficult to compute in an F2 cross than a backcross, but the lack of fit between observed and expected numbers of progeny in the above table indicate it is less than 50%. 

The progeny in this case received two dominant alleles linked on one chromosome (referred to as coupling or cis arrangement). However, after crossover, some progeny could have received one parental chromosome with a dominant allele for one trait (e.g. Purple) linked to a recessive allele for a second trait (e.g. round) with the opposite being true for the other parental chromosome (e.g. red and Long). This is referred to as repulsion or a trans arrangement. The phenotype here would still be purple and long but a test cross of this individual with the recessive parent would produce progeny with much greater proportion of the two crossover phenotypes. While such a problem may not seem likely from this example, unfavourable repulsion linkages do appear when breeding for disease resistance in some crops. 

The two possible arrangements, cis and trans, of alleles in a double heterozygote are referred to as gametic phases, and phasing is the process of determining which of the two is present in a given individual. 

When two genes are located on the same chromosome, the chance of a crossover producing recombination between the genes is related to the distance between the two genes. Thus, the use of recombination frequencies has been used to develop linkage maps or genetic maps. 

However, it is important to note that recombination frequency tends to underestimate the distance between two linked genes. This is because as the two genes are located farther apart, the chance of double or even number of crossovers between them also increases. Double or even number of crossovers between the two genes results in them being cosegregated to the same gamete, yielding a parental progeny instead of the expected recombinant progeny. As mentioned above, the Kosambi and Haldane transformations attempt to correct for multiple crossovers.

Variation of recombination frequency

While recombination of chromosomes is an essential process during meiosis, there is a large range of frequency of cross overs across organisms and within species. Sexually dimorphic rates of recombination are termed heterochiasmy, and are observed more often than a common rate between male and females. In mammals, females often have a higher rate of recombination compared to males. It is theorised that there are unique selections acting or meiotic drivers which influence the difference in rates. The difference in rates may also reflect the vastly different environments and conditions of meiosis in oogenesis and spermatogenesis.

Meiosis indicators

With very large pedigrees or with very dense genetic marker data, such as from whole-genome sequencing, it is possible to precisely locate recombinations. With this type of genetic analysis, a meiosis indicator is assigned to each position of the genome for each meiosis in a pedigree. The indicator indicates which copy of the parental chromosome contributes to the transmitted gamete at that position. For example, if the allele from the 'first' copy of the parental chromosome is transmitted, a '0' might be assigned to that meiosis. If the allele from the 'second' copy of the parental chromosome is transmitted, a '1' would be assigned to that meiosis. The two alleles in the parent came, one each, from two grandparents. These indicators are then used to determine identical-by-descent (IBD) states or inheritance states, which are in turn used to identify genes responsible for diseases.