Linkage disequilibrium

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Linkage_disequilibrium

In population genetics, linkage disequilibrium (LD) 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 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 ${\displaystyle D_{AB}\ne 0}$. 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 ${\displaystyle \{A,B\}}$ of alleles and not of their respective loci. Other pairs of alleles at those same two loci may have different coefficients of linkage disequilibrium.

For two biallelic loci, where a and b are the other alleles at these two loci, the restrictions are so strong that only one value of D is sufficient to represent all linkage disequilibrium relationships between these alleles. In this case, ${\displaystyle D_{AB}=-D_{Ab}=-D_{aB}=D_{ab}}$. Their relationships can be characterized as follows.

${\displaystyle D=P_{AB}-P_A{P}_B}$

${\displaystyle -D=P_{Ab}-P_A{P}_b}$

${\displaystyle -D=P_{aB}-P_a{P}_B}$

${\displaystyle D=P_{ab}-P_a{P}_b}$

The sign of D in this case is chosen arbitrarily. The magnitude of D is more important than the sign of D because the magnitude of D is representative of the degree of linkage disequilibrium. However, positive D value means that the gamete is more frequent than expected while negative means that the combination of these two alleles are less frequent than expected.

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 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:

${\displaystyle D^{\prime }=\frac{D}{D_{max}}}$

where

${\displaystyle D_{max}=\{\begin{array}{ll}max\{-p_A{p}_B,-(1-p_A)(1-p_B)\}& \text{when }D<0\\ min\{p_A(1-p_B),(1-p_A)p_B\}& \text{when }D>0\end{array}}$

An alternative to ${\displaystyle D^{\prime }}$ is the correlation coefficient between pairs of loci, usually expressed as its square, ${\displaystyle r^2}$

Limits for the ranges of linkage disequilibrium measures

The measures ${\displaystyle r^2}$ and ${\displaystyle D^{\prime }}$ have limits to their ranges and do not range over all values of zero to one for all pairs of loci. The maximum of ${\displaystyle r^2}$ depends on the allele frequencies at the two loci being compared and can only range fully from zero to one where either the allele frequencies at both loci are equal, ${\displaystyle P_A=P_B}$ where ${\displaystyle D>0}$, or when the allele frequencies have the relationship ${\displaystyle P_A=1-P_B}$ when ${\displaystyle D<0}$. While ${\displaystyle D^{\prime }}$ can always take a maximum value of 1, its minimum value for two loci is equal to ${\displaystyle |r|}$ for those loci.

Example: Two-loci and two-alleles

Consider the haplotypes for two loci A and B with two alleles each—a two-loci, 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, ${\displaystyle x_{11}^{\prime }}$, the frequency of the haplotype ${\displaystyle A_1{B}_1}$, becomes

${\displaystyle x_{11}^{\prime }=(1-c)x_{11}+c{p}_1{q}_1}$

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

${\displaystyle x_{11}^{\prime }-p_1{q}_1=(1-c)(x_{11}-p_1{q}_1)}$

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 ${\displaystyle n\to \mathrm{\infty }}$, 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.

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
• 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.
• Bcftools – utilities for variant calling and manipulating VCFs and BCFs.

Simulation software

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

Evolutionary tradeoff

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Evolutionary_tradeoff 

An evolutionary tradeoff is a situation in which evolution cannot advance one part of a biological system without distressing another part of it. In biology, and more specifically in evolutionary biology, tradeoffs refer to the process through which a trait increases in fitness at the expense of decreased fitness in another trait. A much agreed on theory on what causes evolutionary tradeoffs is that due to resources limitations (e.g. energy, habitat/space, time) the simultaneous optimization of two traits cannot be achieved. Another commonly accepted cause of evolutionary tradeoffs is that the characteristics of increasing the fitness in one trait negatively affects the fitness of another trait. This negative relationship is found in traits that are antagonistically pleiotropic (one gene responsible for multiple traits that are not all beneficial to the organism) or when linkage disequilibrium is present (non-random association of alleles at different loci during the gametic phase).

Background and theory

The general concept behind evolutionary tradeoffs is that in order to increase fitness (or function) in one trait it must come at the expense of the decrease in fitness/function of another trait. The 'Y-model' states that, within an individual, any two traits are determined by resources from a common pool. Although a useful tool that has provided valuable insight, the 'Y-model' has been oversimplified in much of the literature. Researchers have made different mathematical expansions to the 'Y model' in order to gain insights about evolutionary tradeoffs.

An important point that many authors make when discussing the concept of how tradeoffs affect evolutionary change is the ambiguous use of the word 'constraint'. The term 'constraint' has two meanings: hindering (slowing), but not stopping evolution in particular directions, or that there are certain evolutionary trajectories that are not available to selection. The distinction between the two senses of the word is important because according to the first definition all character states, or forms, are possible, where as according to the later definition some character states are unattainable. When discussing evolutionary tradeoffs it is important to make clear which sense of the word is being used.

Life history examples

Evolutionary tradeoffs can be present in a form called life history tradeoffs, which can be defined as the decrease in fitness (essentially, lifetime reproductive success) caused by one life history trait as a result of the increase in fitness caused by a different life history trait. Life history traits are traits closely linked to fitness, such as traits associated with growth rate, body size, stress response, timing of reproduction, offspring quantity/quality, longevity and dispersal.

A classic example of life history tradeoffs is a negative relationship between the age and the size of maturity. Growth rates are negatively correlated with maximal size so that the fastest growing individuals produce the smallest adults and slowly growing individuals produce large adults. Another classic example is the tradeoff between energy investment in reproduction versus survival. If an organism has a set amount of energy that must be allocated among all the functions that individual performs, then the more energy is allocated to reproduction (increased sexual activity/size of reproductive organs), the less is available for survival (longevity/weapon size). For example, through experimental manipulation in the lab researchers were able to see that an increase in reproductive activity is correlated with a decrease in longevity in the male fruit fly (Drosophila melanogaster). More evidence of the tradeoff between reproduction and survival comes from a study done on pinnipeds, where both genital length and testes mass are negatively associated with investment in precopulatory weaponry.

Life history tradeoffs can also be thought of in the context of adaptation to a specific environment. The general theory is that increased fitness within a selected environment will cause a loss of fitness in other nonelected environments. Researchers have used experimental evolution to test this theory in Escherichia coli evolved in a 20 °C environment. They were able to see that, although not universal (meaning all individuals showed it), generally there was a decrease in fitness of the evolved E. coli when grown in a 40 °C.

Human/clinical examples

Examples of tradeoffs can also be found in studies involving human subjects. A tradeoff can be seen between growth and immune function in human populations in which energy is a limiting factor. A study conducted on rural Bolivia found that children experiencing an elevated immune response had smaller gains in height than those with a normal level of immune response. This trend was stronger in children under 5 five years old, the ages when children experience rapid growth, as well as in children with less fat reserves. A tradeoff has also been observed between growth and reproduction. In a study of pregnant adolescents, researchers observed that less energy was allocated to fetuses of women still growing than those who had completed their growth. Tradeoffs have also been observed in clinical medicine. For example, hormone replacement therapy for post-menopausal women may reduce the risk of ovarian cancer and osteoporosis, but can increase the risk of breast cancer. This can be linked back to the fact that ovarian steroids act as both bone trophic hormones and mitotic stimulants in breast tissue.