F-statistics
Encyclopedia
In population genetics
, F-statistics (also known as fixation indices) describe the level of heterozygosity
in a population; more specifically the degree of (usually) a reduction in heterozygosity when compared to Hardy–Weinberg expectation. F-statistics can also be thought of as a measure of the correlation between genes drawn at different levels of a (hierarchically) subdivided population. This correlation is influenced by several evolutionary processes, such as mutation, migration, inbreeding
, natural selection
, or the Wahlund effect
, but it was originally designed to measure the amount of allelic fixation owing to genetic drift
.
The concept of F-statistics was developed during the 1920s by the American
geneticist
Sewall Wright
, who was interested in inbreeding in cattle
. However, because complete dominance causes the phenotype
s of homozygote
dominants and heterozygotes to be the same, it was not until the advent of molecular genetics
from the 1960s onwards that heterozygosity in populations could be measured.
The value for F is found by solving the equation for F using heterozygotes in the above inbred population. This becomes one minus the observed
number of heterozygotes in a population divided by its expected
number of heterozygotes at Hardy–Weinberg equilibrium:
where the expected value at Hardy–Weinberg equilibrium is given by
where p and q are the allele frequencies of A and a, respectively. It is also the probability that at any locus, two alleles from a random individuum of the population are identical by descent
.
For example, consider the data from E.B. Ford
(1971) on a single population of the scarlet tiger moth
:
From this, the allele frequencies can be calculated, and the expectation of ƒ(AA) derived:
The different F-statistics look at different levels of population structure. FIT is the inbreeding coefficient of an individual (I) relative to the total (T) population, as above; FIS is the inbreeding coefficient of an individual (I) relative to the subpopulation (S), using the above for subpopulations and averaging them; and FST is the effect of subpopulations (S) compared to the total population (T), and is calculated by solving the equation:
as shown in the next section.
Consider a population that has a population structure
of two levels; one from the individual (I) to the subpopulation (S) and one from the subpopulation to the total (T). Then the total F, known here as FIT, can be partitioned into FIS (or f) and FST (or θ):
This may be further partitioned for population substructure, and it expands according to the rules of binomial expansion, so that for I partitions:
A reformulation of the definition of F would be the ratio of the average number of differences between pairs of chromosomes sampled within diploid individuals with the average number obtained when sampling chromosomes randomly from the population (excluding the grouping per individual).
One can modify this definition and consider a grouping per sub-population instead of per individual. Population geneticists have used that idea to measure the degree of structure in a population.
Unfortunately, there is a large number of definitions for Fst, causing some confusion in the scientific literature. A common definition is the following:
where the variance of p is computed across sub-populations and p(1 −p) is the expected frequency of heterozygotes.
Population genetics
Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
, F-statistics (also known as fixation indices) describe the level of heterozygosity
Zygosity
Zygosity refers to the similarity of alleles for a trait in an organism. If both alleles are the same, the organism is homozygous for the trait. If both alleles are different, the organism is heterozygous for that trait...
in a population; more specifically the degree of (usually) a reduction in heterozygosity when compared to Hardy–Weinberg expectation. F-statistics can also be thought of as a measure of the correlation between genes drawn at different levels of a (hierarchically) subdivided population. This correlation is influenced by several evolutionary processes, such as mutation, migration, inbreeding
Inbreeding
Inbreeding is the reproduction from the mating of two genetically related parents. Inbreeding results in increased homozygosity, which can increase the chances of offspring being affected by recessive or deleterious traits. This generally leads to a decreased fitness of a population, which is...
, natural selection
Natural selection
Natural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....
, or the Wahlund effect
Wahlund effect
In population genetics, the Wahlund effect refers to reduction of heterozygosity in a population caused by subpopulation structure. Namely, if two or more subpopulations have different allele frequencies then the overall heterozygosity is reduced, even if the subpopulations themselves are in a...
, but it was originally designed to measure the amount of allelic fixation owing to genetic drift
Genetic drift
Genetic drift or allelic drift is the change in the frequency of a gene variant in a population due to random sampling.The alleles in the offspring are a sample of those in the parents, and chance has a role in determining whether a given individual survives and reproduces...
.
The concept of F-statistics was developed during the 1920s by the American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
geneticist
Geneticist
A geneticist is a biologist who studies genetics, the science of genes, heredity, and variation of organisms. A geneticist can be employed as a researcher or lecturer. Some geneticists perform experiments and analyze data to interpret the inheritance of skills. A geneticist is also a Consultant or...
Sewall Wright
Sewall Wright
Sewall Green Wright was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. With R. A. Fisher and J.B.S. Haldane, he was a founder of theoretical population genetics. He is the discoverer of the inbreeding coefficient and of...
, who was interested in inbreeding in cattle
Cattle
Cattle are the most common type of large domesticated ungulates. They are a prominent modern member of the subfamily Bovinae, are the most widespread species of the genus Bos, and are most commonly classified collectively as Bos primigenius...
. However, because complete dominance causes the phenotype
Phenotype
A phenotype is an organism's observable characteristics or traits: such as its morphology, development, biochemical or physiological properties, behavior, and products of behavior...
s of homozygote
Zygosity
Zygosity refers to the similarity of alleles for a trait in an organism. If both alleles are the same, the organism is homozygous for the trait. If both alleles are different, the organism is heterozygous for that trait...
dominants and heterozygotes to be the same, it was not until the advent of molecular genetics
Molecular genetics
Molecular genetics is the field of biology and genetics that studies the structure and function of genes at a molecular level. The field studies how the genes are transferred from generation to generation. Molecular genetics employs the methods of genetics and molecular biology...
from the 1960s onwards that heterozygosity in populations could be measured.
Definitions and equations
The measures FIS, Fst, and FIT are related to the amounts of heterozygosity at various levels of population structure. Together, they are called F-statistics, and are derived from F, the inbreeding coefficient. In a simple two-allele system with inbreeding, the genotypic frequencies are:The value for F is found by solving the equation for F using heterozygotes in the above inbred population. This becomes one minus the observed
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...
number of heterozygotes in a population divided by its expected
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
number of heterozygotes at Hardy–Weinberg equilibrium:
where the expected value at Hardy–Weinberg equilibrium is given by
where p and q are the allele frequencies of A and a, respectively. It is also the probability that at any locus, two alleles from a random individuum of the population are identical by descent
Identity by descent
Two or more alleles are identical by descent if they are identical copies of the same ancestral allele. This property is often used in genetic linkage to identify alleles which are potential candidates for harboring mutations causing a genetic disease....
.
For example, consider the data from E.B. Ford
E.B. Ford
Edmund Brisco "Henry" Ford FRS Hon. FRCP was a British ecological geneticist. He was a leader among those British biologists who investigated the role of natural selection in nature. As a schoolboy Ford became interested in lepidoptera, the group of insects which includes butterflies and moths...
(1971) on a single population of the scarlet tiger moth
Scarlet tiger moth
The Scarlet Tiger Moth is a colorful moth of Europe, Turkey, Transcaucasus, northern Iran. It belongs to the tiger moth family, Arctiidae....
:
| Genotype | White-spotted (AA) | Intermediate (Aa) | Little spotting (aa) | Total |
|---|---|---|---|---|
| Number | 1469 | 138 | 5 | 1612 |
From this, the allele frequencies can be calculated, and the expectation of ƒ(AA) derived:
The different F-statistics look at different levels of population structure. FIT is the inbreeding coefficient of an individual (I) relative to the total (T) population, as above; FIS is the inbreeding coefficient of an individual (I) relative to the subpopulation (S), using the above for subpopulations and averaging them; and FST is the effect of subpopulations (S) compared to the total population (T), and is calculated by solving the equation:
as shown in the next section.
Partition due to population structure
Population structure
Population structure may refer to many aspects of population ecology:* Population stratification* Population pyramid* Age class structure* F-statistics* Population density* Population distribution* Population dynamics and population growth...
of two levels; one from the individual (I) to the subpopulation (S) and one from the subpopulation to the total (T). Then the total F, known here as FIT, can be partitioned into FIS (or f) and FST (or θ):
This may be further partitioned for population substructure, and it expands according to the rules of binomial expansion, so that for I partitions:
One can modify this definition and consider a grouping per sub-population instead of per individual. Population geneticists have used that idea to measure the degree of structure in a population.
Unfortunately, there is a large number of definitions for Fst, causing some confusion in the scientific literature. A common definition is the following:
where the variance of p is computed across sub-populations and p(1 −p) is the expected frequency of heterozygotes.