Haplotype diversity is a measure of the uniqueness of a particular haplotype in a given population. The haplotype diversity (H) is computed as: [ 13 ]

 H
 =

 N

 N
 −
 1

 (
 1
 −

 ∑

 i

 x

 i

 2

 )

 {\displaystyle H={\frac {N}{N-1}}(1-\sum _{i}x_{i}^{2})}

where

 x

 i

 {\displaystyle x_{i}}

  is the (relative) haplotype frequency of each haplotype in the sample and

 N

 {\displaystyle N}

  is the sample size. Haplotype diversity is given for each sample.


The term "haplotype" was first introduced by  MHC  biologist  Ruggero Ceppellini  during the Third International Histocompatibility Workshop to substitute "pheno-group". [ 14 ] [ 15 ]


In  population genetics , the  Hardy–Weinberg principle , also known as the  Hardy–Weinberg equilibrium ,  model ,  theorem , or  law , states that  allele  and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences.  These influences include  genetic drift ,  mate choice ,  assortative mating ,  natural selection ,  sexual selection ,  mutation ,  gene flow ,  meiotic drive ,  genetic hitchhiking ,  population bottleneck ,  founder effect ,   inbreeding  and  outbreeding depression .


In the simplest case of a single locus with two  alleles  denoted  A  and  a  with frequencies  f (A) =  p  and  f (a) =  q , respectively, the expected genotype frequencies under random mating are  f (AA) =  p 2  for the AA  homozygotes ,  f (aa) =  q 2  for the aa homozygotes, and  f (Aa) = 2 pq  for the  heterozygotes . In the absence of selection, mutation, genetic drift, or other forces, allele frequencies  p  and  q  are constant between generations, so equilibrium is reached.


The principle is named after  G. H. Hardy  and  Wilhelm Weinberg , who first demonstrated it mathematically. Hardy's paper was focused on debunking the view that a  dominant  allele would automatically tend to increase in frequency (a view possibly based on a misinterpreted question at a lecture [ 1 ] ). Today, tests for Hardy–Weinberg genotype frequencies are used primarily to test for  population stratification  and other forms of non-random mating.


Consider a population of  monoecious   diploids , where each organism produces male and female gametes at equal frequency, and has two alleles at each gene locus. We assume that the population is so large that it can be treated as infinite. Organisms reproduce by random union of gametes (the "gene pool" population model). A locus in this population has two alleles, A and a, that occur with initial frequencies  f 0 (A) =  p  and  f 0 (a) =  q , respectively. [ note 1 ]  The allele frequencies at each generation are obtained by pooling together the alleles from each  genotype  of the same generation according to the expected contribution from the homozygote and heterozygote genotypes, which are 1 and 1/2, respectively:


The different ways to form genotypes for the next generation can be shown in a  Punnett square , where the proportion of each genotype is equal to the product of the row and column allele frequencies from the current generation.


The sum of the entries is  p 2  + 2 pq  +  q 2  = 1 , as the genotype frequencies must sum to one.


Note again that as  p  +  q  = 1 , the binomial expansion of  ( p  +  q ) 2  =  p 2  + 2 pq  +  q 2  = 1  gives the same relationships.


Summing the elements of the Punnett square or the binomial expansion, we obtain the expected genotype proportions among the offspring after a single generation:


These frequencies define the Hardy–Weinberg equilibrium. It should be mentioned that the genotype frequencies after the first generation need not equal the genotype frequencies from the initial generation, e.g.  f 1 (AA) ≠  f 0 (AA) .  However, the genotype frequencies for all  future  times will equal the Hardy–Weinberg frequencies, e.g.   f t (AA) =  f 1 (AA)  for  t  > 1 . This follows since the genotype frequencies of the next generation depend only on the allele frequencies of the current generation which, as calculated by equations ( 1 ) and ( 2 ), are preserved from the initial generation:


For the more general case of  dioecious   diploids  [organisms are either male or female] that reproduce by random mating of individuals, it is necessary to calculate the genotype frequencies from the nine possible matings between each parental genotype ( AA ,  Aa , and  aa ) in either sex, weighted by the expected genotype contributions of each such mating. [ 2 ]  Equivalently, one considers the six unique diploid-diploid combinations:


and constructs a Punnett square for each, so as to calculate its contribution to the next generation's genotypes. These contributions are weighted according to the probability of each diploid-diploid combination, which follows a  multinomial distribution  with  k  = 3 . For example, the probability of the mating combination  (AA,aa)  is   2  f t (AA) f t (aa)  and it can only result in the  Aa  genotype:  [0,1,0] . Overall, the resulting genotype frequencies are calculated as:


As before, one can show that the allele frequencies at time  t  + 1  equal those at time  t , and so, are constant in time.  Similarly, the genotype frequencies depend only on the allele frequencies, and so, after time  t  = 1  are also constant in time.


If in either  monoecious  or  dioecious  organisms, either the allele or genotype proportions are initially unequal in either sex, it can be shown that constant proportions are obtained after one generation of random mating. If  dioecious  organisms are  heterogametic  and the gene locus is located on the  X chromosome , it can be shown that if the allele frequencies are initially unequal in the two sexes [ e.g ., XX females and XY males, as in humans],  f ′(a)  in the  heterogametic  sex 'chases'  f (a)  in the  homogametic  sex of the previous generation, until an equilibrium is reached at the weighted average of the two initial frequencies.


The seven assumptions underlying Hardy–Weinberg equilibrium are as follows: [ 3 ]


Violations of the Hardy–Weinberg assumptions can cause deviations from expectation.  How this affects the population depends on the assumptions that are violated.


If a population violates one of the following four assumptions, the population may continue to have Hardy–Weinberg proportions each generation, but the allele frequencies will change over time.


In real world genotype data, deviations from Hardy–Weinberg Equilibrium may be a sign of genotyping error. [ 4 ] [ 5 ] [ 6 ]


Where the A gene is  sex linked , the heterogametic sex ( e.g. , mammalian males; avian females) have only one copy of the gene (and are termed hemizygous), while the homogametic sex ( e.g. ,  human  females) have two copies.  The genotype frequencies at equilibrium are  p  and  q  for the heterogametic sex but  p 2 , 2 pq  and  q 2  for the homogametic sex.


For example, in humans  red–green colorblindness  is an X-linked recessive trait.  In western European males, the trait affects about 1 in 12, ( q  = 0.083) whereas it affects about 1 in 200 females (0.005, compared to  q 2  = 0.007), very close to Hardy–Weinberg proportions.


If a population is brought together with males and females with a different allele frequency in each subpopulation (males or females), the allele frequency of the male population in the next generation will follow that of the female population because each son receives its X chromosome from its mother.  The population converges on equilibrium very quickly.


The simple derivation above can be generalized for more than two alleles and  polyploidy .


Consider an extra allele frequency,  r .  The two-allele case is the  binomial expansion  of ( p  +  q ) 2 , and thus the three-allele case is the trinomial expansion of ( p  +  q  +  r ) 2 .


More generally, consider the alleles A 1 , ..., A n  given by the allele frequencies  p 1  to  p n ;


giving for all  homozygotes :


and for all  heterozygotes :


The Hardy–Weinberg principle may also be generalized to  polyploid  systems, that is, for organisms that have more than two copies of each chromosome.  Consider again only two alleles.  The diploid case is the  binomial expansion  of:


and therefore the polyploid case is the binomial expansion of:


where  c  is the  ploidy , for example with tetraploid ( c  = 4):


Whether the organism is a 'true' tetraploid or an amphidiploid will determine how long it will take for the population to reach Hardy–Weinberg equilibrium.


For

 n

 {\displaystyle n}

  distinct alleles in

 c

 {\displaystyle c}

 -ploids, the genotype frequencies in the Hardy–Weinberg equilibrium are given by individual terms in the  multinomial expansion  of

 (

 p

 1

 +
 ⋯
 +

 p

 n

 )

 c

 {\displaystyle (p_{1}+\cdots +p_{n})^{c}}

 :


Testing deviation from the HWP is generally performed using  Pearson's chi-squared test , using the observed genotype frequencies obtained from the data and the expected genotype frequencies obtained using the HWP.  For systems where there are large numbers of alleles, this may result in data with many empty possible genotypes and low genotype counts, because there are often not enough individuals present in the sample to adequately represent all genotype classes.  If this is the case, then the  asymptotic  assumption of the  chi-squared distribution , will no longer hold, and it may be necessary to use a form of  Fisher's exact test , which requires a  computer  to solve.  More recently a number of  MCMC  methods of testing for deviations from HWP have been proposed (Guo & Thompson, 1992; Wigginton  et al.  2005)


This data is from  E. B. Ford  (1971) on the  scarlet tiger moth , for which the  phenotypes  of a sample of the population were recorded.  Genotype–phenotype distinction  is assumed to be negligibly small.  The  null hypothesis  is that the population is in Hardy–Weinberg proportions, and the  alternative hypothesis  is that the population is not in Hardy–Weinberg proportions.


From this, allele frequencies can be calculated:


So the Hardy–Weinberg  expectation  is:


Pearson's chi-squared test  states:


There is 1  degree of freedom  (degrees of freedom for test for Hardy–Weinberg proportions are # genotypes − # alleles).  The 5%  significance level  for 1 degree of freedom is 3.84, and since the χ 2  value is less than this, the  null hypothesis  that the population is in Hardy–Weinberg frequencies is  not  rejected.


Fisher's exact test  can be applied to testing for Hardy–Weinberg proportions. Since the test is conditional on the allele frequencies,  p  and  q , the problem can be viewed as testing for the proper number of heterozygotes.  In this way, the hypothesis of Hardy–Weinberg proportions is rejected if the number of heterozygotes is too large or too small.  The conditional probabilities for the heterozygote, given the allele frequencies are given in Emigh (1980) as


where  n 11 ,  n 12 ,  n 22  are the observed numbers of the three genotypes, AA, Aa, and aa, respectively, and  n 1  is the number of A alleles, where

 n

 1

 =
 2

 n

 11

 +

 n

 12

 {\displaystyle n_{1}=2n_{11}+n_{12}}

 .


An example
Using one of the examples from Emigh (1980), [ 7 ]  we can consider the case where  n  = 100, and  p  = 0.34.  The possible observed heterozygotes and their exact significance level is given in Table 4.


Using this table, one must look up the significance level of the test based on the observed number of heterozygotes.  For example, if one observed 20 heterozygotes, the significance level for the test is 0.007.  As is typical for Fisher's exact test for small samples, the gradation of significance levels is quite coarse.


However, a table like this has to be created for every experiment, since the tables are dependent on both  n  and  p .


The equivalence tests are developed in order to establish sufficiently good agreement of the observed genotype frequencies and Hardy Weinberg equilibrium. Let

 M

 {\displaystyle {\mathcal {M}}}

  denote the family of the genotype distributions under the assumption of Hardy Weinberg equilibrium. The distance between a genotype distribution

 p

 {\displaystyle p}

  and Hardy Weinberg equilibrium is defined by

 d
 (
 p
 ,

 M

 )
 =

 min

 q
 ∈

 M

 d
 (
 p
 ,
 q
 )

 {\displaystyle d(p,{\mathcal {M}})=\min _{q\in {\mathcal {M}}}d(p,q)}

 , where

 d

 {\displaystyle d}

  is some distance. The equivalence test problem is given by

 H

 0

 =
 {
 d
 (
 p
 ,

 M

 )
 ≥
 ε
 }

 {\displaystyle H_{0}=\{d(p,{\mathcal {M}})\geq \varepsilon \}}

   and

 H

 1

 =
 {
 d
 (
 p
 ,

 M

 )
 <
 ε
 }

 {\displaystyle H_{1}=\{d(p,{\mathcal {M}})<\varepsilon \}}

 , where

 ε
 >
 0

 {\displaystyle \varepsilon >0}

  is a tolerance parameter. If the hypothesis

 H

 0

 {\displaystyle H_{0}}

   can be rejected then the population is close to Hardy Weinberg equilibrium with a high probability. The equivalence tests for the biallelic case are developed among others in Wellek (2004). [ 8 ]  The equivalence tests for the case of multiple alleles are proposed in Ostrovski (2020). [ 9 ]


The inbreeding coefficient,

 F

 {\displaystyle F}

  (see also  F -statistics ), is one minus the observed frequency of heterozygotes over that expected from Hardy–Weinberg equilibrium.


where the expected value from Hardy–Weinberg equilibrium is given by


For example, for Ford's data above:


For two alleles, the chi-squared goodness of fit test for Hardy–Weinberg proportions is equivalent to the test for inbreeding,

 F
 =
 0

 {\displaystyle F=0}

 .


The inbreeding coefficient is unstable as the expected value approaches zero, and thus not useful for rare and very common alleles.  For:

 F

 |

 E
 =
 0
 ,
 O
 =
 0

 =
 −
 ∞

 {\displaystyle F{\big |}_{E=0,O=0}=-\infty }

 ;

 F

 |

 E
 =
 0
 ,
 O
 >
 0

 {\displaystyle F{\big |}_{E=0,O>0}}

  is undefined.


Mendelian genetics  were rediscovered in 1900.  However, it remained somewhat controversial for several years as it was not then known how it could cause continuous characteristics.  Udny Yule  (1902) argued against Mendelism because he thought that dominant alleles would increase in the population. [ 10 ]  The  American   William E. Castle  (1903) showed that without  selection , the genotype frequencies would remain stable. [ 11 ]   Karl Pearson  (1903) found one equilibrium position with values of  p  =  q  = 0.5. [ 12 ]   Reginald Punnett , unable to counter Yule's point, introduced the problem to  G. H. Hardy , a  British   mathematician , with whom he played  cricket .  Hardy was a  pure mathematician  and held  applied mathematics  in some contempt; his view of biologists' use of mathematics comes across in his 1908 paper where he describes this as "very simple": [ 13 ]


The principle was thus known as  Hardy's law  in the  English-speaking world  until 1943, when  Curt Stern  pointed out that it had first been formulated independently in 1908 by the German physician  Wilhelm Weinberg . [ 14 ] [ 15 ]   William Castle  in 1903 also derived the ratios for the special case of equal allele frequencies, and it is sometimes (but rarely) called the Hardy–Weinberg–Castle Law.


Hardy's statement begins with a  recurrence relation  for the frequencies  p , 2 q , and  r . These recurrence relations follow from fundamental concepts in probability, specifically  independence , and  conditional probability . For example, consider the probability of an offspring from the generation

 t

 {\displaystyle \textstyle t}

  being homozygous dominant. Alleles are inherited independently from each parent. A dominant allele can be inherited from a homozygous dominant parent with probability 1, or from a heterozygous parent with probability 0.5. To represent this reasoning in an equation, let

 A

 t

 {\displaystyle \textstyle A_{t}}

  represent inheritance of a dominant allele from a parent. Furthermore, let

 A

 A

 t
 −
 1

 {\displaystyle \textstyle AA_{t-1}}

  and

 A

 a

 t
 −
 1

 {\displaystyle \textstyle Aa_{t-1}}

  represent potential parental genotypes in the preceding generation.


The same reasoning, applied to the other genotypes yields the two remaining recurrence relations. Equilibrium occurs when each proportion is constant between subsequent generations. More formally, a population is at equilibrium at generation

 t

 {\displaystyle \textstyle t}

  when


By solving these equations necessary and sufficient conditions for equilibrium to occur can be determined. Again, consider the frequency of homozygous dominant animals. Equilibrium implies


First consider the case, where

 p

 t
 −
 1

 =
 0

 {\displaystyle \textstyle p_{t-1}=0}

 , and note that it implies that

 q

 t
 −
 1

 =
 0

 {\displaystyle \textstyle q_{t-1}=0}

  and

 r

 t
 −
 1

 =
 1

 {\displaystyle \textstyle r_{t-1}=1}

 . Now consider the remaining case, where

 p

 t
 −
 1

 ≠

 0

 {\displaystyle \textstyle p_{t-1}\neq \textstyle 0}

 :


where the final equality holds because the allele proportions must sum to one. In both cases,

 q

 t
 −
 1

 2

 =

 p

 t
 −
 1

 r

 t
 −
 1

 {\displaystyle \textstyle q_{t-1}^{2}=p_{t-1}r_{t-1}}

 . It can be shown that the other two equilibrium conditions imply the same equation. Together, the solutions of the three equilibrium equations imply sufficiency of Hardy's condition for equilibrium. Since the condition always holds for the second generation, all succeeding generations have the same proportions.


An example computation of the genotype distribution given by Hardy's original equations is instructive. The phenotype distribution from Table 3 above will be used to compute Hardy's initial genotype distribution. Note that the  p  and  q  values used by Hardy are not the same as those used above.


As checks on the distribution, compute


For the next generation, Hardy's equations give


Again as checks on the distribution, compute


which are the expected values. The reader may demonstrate that subsequent use of the second-generation values for a third generation will yield identical results.


The Hardy–Weinberg principle can also be used to estimate the frequency of carriers of an  autosomal recessive condition  in a population based on the frequency of suffers.


Let us assume an estimated

 1
 2500

 {\displaystyle \textstyle {\frac {1}{2500}}}

  babies are born with  cystic fibrosis , this is about the frequency of homozygous individuals observed in Northern European populations. We can use the Hardy–Weinberg equations to estimate the carrier frequency, the frequency of heterozygous individuals,

 2
 p
 q

 {\displaystyle \textstyle 2pq}

 .


As

 1
 50

 {\displaystyle \textstyle {\frac {1}{50}}}

  is small we can take  p ,

 1
 −

 1
 50

 {\displaystyle \textstyle 1-{\frac {1}{50}}}

 , to be 1.


We therefore estimate the carrier rate to be

 1
 25

 {\displaystyle \textstyle {\frac {1}{25}}}

 , which is about the frequency observed in Northern European populations.


This can be simplified to the carrier frequency being about twice the square root of the birth frequency.


It is possible to represent the distribution of genotype frequencies for a bi-allelic locus within a population graphically using a  de Finetti diagram . This uses a triangular plot (also known as trilinear, triaxial or  ternary plot ) to represent the distribution of the three genotype frequencies in relation to each other. It differs from many other such plots in that the direction of one of the axes has been reversed. [ 16 ]  The curved line in the diagram is the Hardy–Weinberg  parabola  and represents the state where  alleles  are in Hardy–Weinberg equilibrium. It is possible to represent the effects of  natural selection  and its effect on allele frequency on such graphs. [ 17 ]  The de Finetti diagram was developed and used extensively by  A. W. F. Edwards  in his book  Foundations of Mathematical Genetics . [ 18 ]


Heterosis ,  hybrid vigor , or  outbreeding enhancement  is the improved or increased function of any biological quality in a  hybrid  offspring. An offspring is heterotic if its  traits  are enhanced as a result of mixing the genetic contributions of its parents. The heterotic offspring often has traits that are more than the simple addition of the parents' traits, and can be explained by  Mendelian  or  non-Mendelian inheritance . [ 1 ]  Typical heterotic/hybrid traits of interest in agriculture are higher yield, quicker maturity, stability, drought tolerance [ 2 ]  etc.


In proposing the term  heterosis  to replace the older term  heterozygosis ,  G.H. Shull  aimed to avoid limiting the term to the effects that can be explained by heterozygosity in Mendelian inheritance. [ 3 ]


The physiological vigor of an organism as manifested in its rapidity of growth, its height and general robustness, is positively correlated with the degree of dissimilarity in the gametes by whose union the organism was formed … The more numerous the differences between the uniting gametes — at least within certain limits — the greater on the whole is the amount of stimulation … These differences need not be Mendelian in their inheritance … To avoid the implication that all the genotypic differences which stimulate cell-division, growth and other physiological activities of an organism are Mendelian in their inheritance and also to gain brevity of expression I suggest … that the word 'heterosis' be adopted.


Heterosis is often discussed as the opposite of  inbreeding depression , although differences in these two concepts can be seen in evolutionary considerations such as the role of  genetic variation  or the effects of  genetic drift  in small populations on these concepts. Inbreeding depression occurs when related parents have children with  traits  that negatively influence their  fitness  largely due to  homozygosity .  In such instances,  outcrossing  should result in heterosis.


Not all outcrosses result in heterosis. For example, when a hybrid inherits traits from its parents that are not fully compatible, fitness can be reduced.  This is a form of  outbreeding depression , the effects of which are similar to inbreeding depression. [ 4 ]


Since the early 1900s, two competing genetic hypotheses, not necessarily mutually exclusive, have been developed to explain hybrid vigor. More recently, an  epigenetic  component of hybrid vigor has also been established. [ 5 ] [ 6 ]


When a population is small or inbred, it tends to lose genetic diversity.  Inbreeding depression  is the loss of fitness due to loss of genetic diversity. Inbred strains tend to be  homozygous  for  recessive   alleles  that are mildly harmful (or produce a trait that is undesirable from the standpoint of the breeder). Heterosis or hybrid vigor, on the other hand, is the tendency of outbred strains to exceed both inbred parents in fitness.


Selective breeding of plants and animals, including hybridization, began long before there was an understanding of underlying scientific principles. In the early 20th century, after  Mendel's laws  came to be understood and accepted, geneticists undertook to explain the superior vigor of many plant hybrids. Two competing hypotheses, which are not mutually exclusive, were developed: [ 7 ]


Dominance and overdominance have different consequences for the  gene expression profile  of the individuals. If overdominance is the main cause for the fitness advantages of heterosis, then there should be an over-expression of certain genes in the heterozygous offspring compared to the homozygous parents. On the other hand, if dominance is the cause, fewer genes should be under-expressed in the heterozygous offspring compared to the parents. Furthermore, for any given gene, the expression should be comparable to the one observed in the fitter of the two parents.  In any case, outcross matings provide the benefit of masking deleterious recessive alleles in progeny. This benefit has been proposed to be a major factor in the maintenance of sexual reproduction among eukaryotes, as summarized in the article  Evolution of sexual reproduction .


Which of the two mechanisms are the "main" reason for heterosis has been a  scientific controversy  in the field of  genetics . [ 12 ]  Population geneticist  James Crow  (1916–2012) believed, in his younger days, that overdominance was a major contributor to hybrid vigor. In 1998 he published a retrospective review of the developing science. [ 13 ]  According to Crow, the demonstration of several cases of  heterozygote advantage  in  Drosophila  and other organisms first caused great enthusiasm for the overdominance theory among scientists studying plant hybridization.  But overdominance implies that yields on an inbred strain should decrease as inbred strains are selected for the performance of their hybrid crosses, as the proportion of harmful recessives in the inbred population rises. Over the years, experimentation in plant genetics has proven that the reverse occurs, that yields increase in both the inbred strains and the hybrids, suggesting that dominance alone may be adequate to explain the superior yield of hybrids. Only a few conclusive cases of overdominance have been reported in all of genetics. Since the 1980s, as experimental evidence has mounted, the dominance theory has made a comeback.


Crow wrote:


The current view ... is that the dominance hypothesis is the major explanation of inbreeding decline and [of] the high yield of hybrids. There is little statistical evidence for contributions from overdominance and  epistasis . But whether the best hybrids are getting an extra boost from overdominance or favorable epistatic contributions remains an open question. [ 13 ]


An  epigenetic  contribution to heterosis has been established in plants, [ 6 ]  and it has also been reported in animals. [ 14 ]   MicroRNAs  (miRNAs), discovered in 1993, are a class of non-coding small RNAs which repress the translation of messenger RNAs (mRNAs) or cause degradation of mRNAs. [ 15 ]  In hybrid plants, most miRNAs have non-additive expression (it might be higher or lower than the levels in the parents). [ 6 ]  This suggests that the small RNAs are involved in the growth, vigor and adaptation of hybrids. [ 6 ]


'Heterosis without hybridity' effects on plant size have been demonstrated in genetically isogenic F1 triploid (autopolyploid) plants, where paternal genome excess F1 triploids display positive heterosis, whereas maternal genome excess F1s display negative heterosis effects. [ 16 ]  Such findings demonstrate that heterosis effects, with a genome dosage-dependent epigenetic basis, can be generated in F1 offspring that are genetically isogenic (i.e. harbour no heterozygosity). [ 16 ] [ 17 ]  It has been shown [ 5 ]  that hybrid vigor in an allopolyploid hybrid of two  Arabidopsis  species was due to epigenetic control in the upstream regions of two genes, which caused major downstream alteration in chlorophyll and starch accumulation. The mechanism involves acetylation or methylation of specific amino acids in  histone H3 , a protein closely associated with DNA, which can either activate or repress associated genes.


One example of where particular genes may be important in vertebrate animals for heterosis is the  major histocompatibility complex  (MHC). Vertebrates inherit several copies of both  MHC class I  and  MHC class II  from each parent, which are used in  antigen presentation  as part of the adaptive immune system. Each different copy of the genes is able to bind and present a different set of potential peptides to  T-lymphocytes . These genes are highly polymorphic throughout populations, but are more similar in smaller, more closely related populations. Breeding between more genetically distant individuals decreases the chance of inheriting two alleles that are the same or similar, allowing a more diverse range of peptides to be presented. This, therefore, increases the chance that any particular pathogen will be recognised, and means that more antigenic proteins on any pathogen are likely to be recognised, giving a greater range of T-cell activation, so a greater response. This also means that the immunity acquired to the pathogen is against a greater range of antigens, meaning that the pathogen must  mutate  more before immunity is lost. Thus, hybrids are less likely to succumb to pathogenic disease and are more capable of fighting off infection. This may be the cause, though, of autoimmune diseases. [ citation needed ]


Crosses between inbreds from different  heterotic groups  result in vigorous F1 hybrids with significantly more heterosis than F1 hybrids from inbreds within the same heterotic group or pattern. Heterotic groups are created by plant breeders to classify inbred lines, and can be progressively improved by reciprocal recurrent selection.


Heterosis is used to increase yields, uniformity, and vigor.  Hybrid breeding methods are used in  maize ,  sorghum ,   rice ,  sugar beet ,  onion ,  spinach ,  sunflowers ,  broccoli  and to create a more  psychoactive   cannabis .


Nearly all field corn ( maize ) grown in most  developed nations  exhibits heterosis. Modern corn hybrids substantially outyield conventional cultivars and respond better to  fertilizer .


Corn heterosis was famously demonstrated in the early 20th century by  George H. Shull  and  Edward M. East  after hybrid corn was invented by Dr.  William James Beal  of  Michigan State University  based on work begun in 1879 at the urging of  Charles Darwin . Dr. Beal's work led to the first published account of a  field experiment  demonstrating hybrid vigor in corn, by  Eugene Davenport  and  Perry Holden , 1881. These various pioneers of botany and related fields showed that crosses of inbred lines made from a Southern dent and a Northern flint, respectively, showed substantial heterosis and outyielded conventional cultivars of that era. However, at that time such hybrids could not be economically made on a large scale for use by farmers.  Donald F. Jones  at the  Connecticut   Agricultural Experiment Station ,  New Haven  invented the first practical method of producing a high-yielding hybrid maize in 1914–1917. Jones' method produced a double-cross hybrid, which requires two crossing steps working from four distinct original inbred lines. Later work by corn breeders produced inbred lines with sufficient vigor for practical production of a commercial hybrid in a single step, the single-cross hybrids. Single-cross hybrids are made from just two original parent inbreds. They are generally more vigorous and also more uniform than the earlier double-cross hybrids. The process of creating these hybrids often involves  detasseling .


Temperate maize hybrids are derived from two main heterotic groups: 'Iowa Stiff Stalk Synthetic', and nonstiff stalk. [ citation needed ]


Hybrid rice sees cultivation in many countries, including China, India, Vietnam, and the Philippines. [ 18 ]  Compared to inbred lines, hybrids produce approximately 20% greater yield, and comprise 45% of rice planting area in China. [ 19 ]  Rice production has seen enormous rise in China due to heavy uses of hybrid rice. In China, efforts have generated a super hybrid rice strain ('LYP9') with a production capability  around 15 tons per hectare. In India also, several varieties have shown high vigor, including 'RH-10' and 'Suruchi 5401'. [ citation needed ]


Since rice is a self-pollinating species, it requires the use of male-sterile lines to generate hybrids from separate lineages. The most common way of achieving this is using lines with genetic male-sterility, as manual emasculation is not optimal for large-scale hybridization. [ 20 ]  The first generation of hybrid rice was developed in the 1970s. It relies on three lines: a  cytoplasmic male sterile (CMS)  line, a maintainer line, and a restorer line. [ 19 ]  The second generation was widely adopted in the 1990s. [ 19 ]  Instead of a CMS line, it uses an environment-sensitive genic male sterile line (EGMS), which can have its sterility reversed based on light or temperature. [ 20 ]  This removes the need for a maintainer, making the hybridization and breeding process more efficient (albeit still high-maintenance). Second generation lines show a yield increase of 5-10% over first generation lines. [ 20 ]  The third and current generation uses a nuclear male sterile line (NMS). Third generation lines have a recessive sterility gene, and their cultivation is more lenient towards maintainer lines and environmental conditions. Additionally, transgenes are only present in the maintainer, so hybrid plants can benefit from hybrid vigor without requiring special oversight. [ 19 ]


The concept of heterosis is also applied in the production of commercial  livestock . In cattle, crosses between  Black Angus  and  Hereford  produce a cross known as a " Black Baldy ". In  swine , "blue butts" are produced by the cross of  Hampshire  and Yorkshire. Other, more exotic hybrids (two different species, so genetically more dissimilar), such as " beefalo " which are hybrids of cattle and bison, are also used for specialty markets.


Within  poultry ,  sex-linked  genes have been used to create hybrids in which males and females can be sorted at one day old by color. Specific genes used for this are genes for barring and wing feather growth. Crosses of this sort create what are sold as Black Sex-links, Red Sex-links, and various other crosses that are known by trade names.


Commercial  broilers  are produced by crossing different strains of White Rocks and White Cornish, the Cornish providing a large frame and the Rocks providing the fast rate of gain. The hybrid vigor produced allows the production of uniform birds at a marketable carcass weight at 6–9 weeks of age.


Likewise, hybrids between different strains of White Leghorn are used to produce laying flocks that provide the majority of white  eggs  for sale in the United States.


In 2013, a study found that mixed breeds live on average 1.2 years longer than pure breeds. [ 21 ]


John Scott and  John L. Fuller  performed a detailed study of purebred Cocker Spaniels, purebred Basenjis, and hybrids between them. [ 22 ]
They found that hybrids ran faster than either parent, perhaps due to heterosis. Other characteristics, such as basal heart rate, did not show any heterosis—the dog's basal heart rate was close to the average of its parents—perhaps due to the additive effects of multiple genes. [ 23 ]


Sometimes people working on a dog-breeding program find no useful heterosis. [ 24 ]


All this said, studies do not provide definitive proof of hybrid vigor in dogs. This is largely due to the unknown heritage of most mixed breed dogs used. Results vary wildly, with some studies showing benefit and others finding the mixed breed dogs to be more prone to genetic conditions. [ 25 ] [ 26 ] [ 27 ]


In 2014, a study undertaken by the Centre for Integrative Ecology at  Deakin University  in Geelong, Victoria, concluded that intrasubspecific hybrids between the subspecies  Platycercus elegans flaveolus  and  P. e. elegans  of the  crimson rosella  ( P. elegans ) were more likely to fight off diseases than their pure counterparts. [ 28 ]


Human beings are all extremely genetically similar to one another. [ 29 ] [ 30 ] [ 31 ]  Michael Mingroni has proposed heterosis, in the form of hybrid vigor associated with historical reductions of the levels of inbreeding, as an explanation of the  Flynn effect , the steady rise in IQ test scores around the world during the 20th century,  [ citation needed ]  though a review of nine studies found that there is no evidence to suggest inbreeding has an effect on IQ. [ 32 ]


The term heterosis often causes confusion and even controversy, particularly in selective breeding of  domestic animals , because it is sometimes (incorrectly) claimed that all  crossbred  plants and animals are "genetically superior" to their parents, due to heterosis, [ citation needed ] . but two problems exist with this claim:


An example of the ambiguous value judgements imposed on hybrids and hybrid vigor is the  mule . While mules are almost always infertile, they are valued for a combination of hardiness and temperament that is different from either of their horse or donkey parents. While these qualities may make them "superior" for particular uses by humans, the infertility issue implies that these animals would most likely become extinct without the intervention of humans through  animal husbandry , making them "inferior" in terms of  natural selection .


A  hypostatic gene  is one whose  phenotype  is altered by the expression of an  allele  at a separate locus, in an  epistasis  event. [ 1 ]


Example:  In  labrador retrievers , the chocolate coat colour is a result of  homozygosity  for a gene that is  epistatic  to the "black vs. brown" gene.  The alleles determining whether the dog is black or brown, are that of the  hypostatic gene  in this event.


Hartwell, Leland; L. Hood; M. Goldberg; A. Reynolds; L. Silver; R. Veres (2004).  Genetics: From Genes to Genomes . New York, NY: McGraw-Hill. pp.  56– 57.  ISBN   0-07-246248-5 .


A  DNA  segment is  identical by state  (IBS) in two or more individuals if they have identical  nucleotide sequences  in this segment. An IBS segment is  identical by descent  ( IBD ) in two or more individuals if they have inherited it from a common ancestor without  recombination , that is, the segment has the same ancestral origin in these individuals. DNA segments that are IBD are IBS per definition, but segments that are not IBD can still be IBS due to the same  mutations  in different individuals or recombinations that do not alter the segment. [ citation needed ]


All individuals in a finite population are related if traced back long enough and will, therefore, share segments of their  genomes  IBD. During  meiosis  segments of IBD are broken up by recombination. Therefore, the expected length of an IBD segment depends on the number of generations since the  most recent common ancestor  at the locus of the segment. The length of IBD segments that result from a common ancestor  n  generations in the past (therefore involving 2 n  meiosis) is exponentially distributed with mean 1/(2 n )  Morgans  (M). [ 1 ]  The expected number of IBD segments decreases with the number of generations since the common ancestor at this locus. For a specific DNA segment, the probability of being IBD decreases as 2 −2 n  since in each meiosis the probability of transmitting this segment is 1/2. [ 2 ]


Identified IBD segments can be used for a wide range of purposes. As noted above the amount (length and number) of IBD sharing depends on the familial relationships between the tested individuals. Therefore, one application of IBD segment detection is to quantify relatedness. [ 3 ] [ 4 ] [ 5 ] [ 6 ]  Measurement of relatedness can be used in  forensic genetics , [ 7 ]  but can also increase information in  genetic linkage  mapping [ 3 ] [ 8 ]  and help to decrease  bias  by undocumented relationships in standard  association studies . [ 6 ] [ 9 ]
Another application of IBD is  genotype imputation  and  haplotype   phase  inference. [ 10 ] [ 11 ] [ 12 ]  Long shared segments of IBD, which are broken up by short regions may be indicative for phasing errors. [ 5 ] [ 13 ] : SI


IBD mapping [ 3 ]  is similar to linkage analysis, but can be performed without a known pedigree on a cohort of unrelated individuals. IBD mapping can be seen as a new form of association analysis that increases the  power  to map genes or genomic regions containing multiple rare disease susceptibility variants. [ 6 ] [ 14 ]


Using simulated data, Browning and  Thompson  showed that IBD mapping has higher power than association testing when multiple rare variants within a gene contribute to disease susceptibility. [ 14 ]  Via IBD mapping, genome-wide  significant  regions in isolated populations as well as outbred populations were found while standard association tests failed. [ 11 ] [ 15 ]  Houwen et al. used IBD sharing to identify the chromosomal location of a gene responsible for benign recurrent intrahepatic  cholestasis  in an isolated fishing population. [ 16 ]  Kenny et al. also used an isolated population to fine-map a signal found by a  genome-wide association study  (GWAS) of plasma  plant sterol  (PPS) levels, a surrogate measure of cholesterol absorption from the intestine. [ 17 ]  Francks et al. was able to identify a potential susceptibility locus for  schizophrenia  and  bipolar disorder  with genotype data of case-control samples. [ 18 ]  Lin et al. found a genome-wide significant linkage signal in a dataset of  multiple sclerosis  patients. [ 19 ]  Letouzé et al. used IBD mapping to look for  founder mutations  in  cancer  samples. [ 20 ]


Detection of  natural selection  in the human genome is also possible via detected IBD segments. Selection will usually tend to increase the number of IBD segments among individuals in a population. By scanning for regions with excess IBD sharing, regions in the human genome that have been under strong, very recent selection can be identified. [ 21 ] [ 22 ]


In addition to that, IBD segments can be useful for measuring and identifying other influences on population structure. [ 6 ] [ 23 ] [ 24 ] [ 25 ] [ 26 ]  Gusev et al. showed that IBD segments can be used with additional modeling to estimate demographic history including  bottlenecks  and  admixture . [ 24 ]  Using similar models Palamara et al. and Carmi et al. reconstructed the  demographic history  of  Ashkenazi Jewish  and Kenyan  Maasai  individuals. [ 25 ] [ 26 ] [ 27 ]  Botigué et al. investigated differences in African ancestry among European populations. [ 28 ]  Ralph and Coop used IBD detection to quantify the common ancestry of different European populations [ 29 ]  and Gravel et al. similarly tried to draw conclusions of the genetic history of populations in the Americas. [ 30 ]  Ringbauer et al. utilized geographic structure of IBD segments to estimate dispersal within Eastern Europe during the last centuries. [ 31 ]  Using the  1000 Genomes  data Hochreiter found differences in IBD sharing between African, Asian and European populations as well as IBD segments that are shared with ancient genomes like the  Neanderthal  or  Denisova . [ 13 ]


Programs for the detection of IBD segments in unrelated individuals:


Introgression , also known as  introgressive hybridization , in  genetics  is the transfer of genetic material from one species into the  gene pool  of another by the repeated  backcrossing  of an  interspecific hybrid  with one of its parent species. Introgression is a long-term process, even when artificial; it may take many hybrid generations before significant backcrossing occurs. This process is distinct from most forms of  gene flow  in that it occurs between two populations of different species, rather than two populations of the same species.


Introgression also differs from simple  hybridization . Simple hybridization results in a relatively even mixture; gene and allele frequencies in the first generation will be a uniform mix of two parental species, such as that observed in  mules . Introgression, on the other hand, results in a complex, highly variable mixture of genes, and may only involve a minimal percentage of the donor genome.


Introgression or introgressive  hybridization  is the incorporation (usually via hybridization and backcrossing) of novel genes or  alleles  from one taxon into the  gene pool  of a second, distinct taxon. [ 1 ] [ 2 ] [ 3 ] [ 4 ]  This introgression is considered 'adaptive' if the genetic transfer results in an overall increase in the recipient taxon's fitness. [ 5 ]


Ancient introgression events can leave traces of extinct species in present-day genomes, a phenomenon known as  ghost introgression . [ 6 ]


Introgression is an important source of genetic variation in natural populations and may contribute to adaptation and even  adaptive radiation . [ 7 ]  It can occur across  hybrid zones  due to chance, selection or hybrid zone movement. [ 8 ]  There is evidence that introgression is a ubiquitous phenomenon in plants and animals, [ 9 ] [ 10 ]  including humans, [ 11 ]  in which it may have introduced the  microcephalin  D allele. [ 12 ]


It has been proposed that, historically, introgression with wild animals is a large contributor to the wide range of diversity found in domestic animals, rather than multiple independent domestication events. [ 13 ]


Introgressive hybridization has also been shown to be important in the evolution of domesticated crop species, possibly providing genes that help in their expansion into different environments. A genomic study from New York University Abu Dhabi Center for Genomics and Systems Biology showed that domesticated date palm varieties from North Africa show introgressive hybridization of between 5–18% of its genome from the wild Cretan palm  Phoenix theophrasti  into Middle East date palms  P. dactylifera . This process is also similar to the evolution of apples by hybridization of Central Asian apples with the European crabapple. [ 14 ]  It has also been shown that indica rice arose when Chinese japonica rice arrived in India about ~4,500 years ago and hybridized with an undomesticated proto-indica or wild  O. nivara , and transferred key domestication genes from japonica to indica. [ 15 ]


There is strong evidence for the introgression of  Neanderthal  genes [ 16 ]  and  Denisovan  genes [ 17 ]  into parts of the modern human gene pool.


The  Mallard  duck is possibly the world's most capable bird to hybridise with other duck species, often to the point of the loss of genetic identity of these species. For example, feral mallard populations have significantly reduced wild populations of the  Pacific black duck  in New Zealand and Australia through cross-breeding. [ citation needed ]


One important example of introgression has been observed in studies of mimicry in the butterfly genus  Heliconius . [ 18 ]  This genus includes 43 species and many races with different color patterns. Congeners exhibiting overlapping distributions show similar color patterns. The subspecies  H. melpomene amaryllis  and  H. melpomene timareta  ssp. nov. overlap in distribution.


Using the  ABBA/BABA test , some researchers have observed that there is about 2% to 5% introgression between the pair of subspecies. Importantly, the introgression is not random. The researchers saw significant introgression in chromosomes 15 and 18, where important mimicry loci are found (loci B/D and N/Yb). They compared both subspecies with  H. melpomene agalope , which is a subspecies near  H. melpomene amaryllis  in entire genome trees. The result of the analysis was that there is no relation between those two species and  H. melpomene agalope  in the loci B/D and N/Yb. Moreover, they performed the same analysis with two other species with overlapping distributions,  H. timareta florencia  and  H. melpomene agalope . They demonstrated introgression between the two taxa, especially in the loci B/D and N/Yb.


Finally, they concluded their experiments with sliding-window phylogenetic analyses, estimating different phylogenetic trees depending on the different regions of the loci. When a locus is important in the color pattern expression, there is a close phylogenetic relationship between the species. When the locus is not important in the color pattern expression, the two species are phylogenetically distant because there is no introgression at such loci.


Introgression can have a significant impact between wild and domestic populations of animals. This includes household pets, as seen  in cats [ 19 ]  or  in dogs . [ 20 ]


Introgression has been observed in several plant species. For instance, a species of  iris  from southern  Louisiana  has been studied by Arnold and Bennett (1993) regarding the increased fitness of hybrid variants. [ 21 ] [ 22 ]


Espinasa et al. found that introgression between a surface-dwelling members of  Astroblepus  and a  troglomorphic  species,  Astroblepus pholeter,  resulted in the development of previously lost traits in offspring, such as distinct eyes and optic nerves. [ 23 ]


An  introgression line  ( IL ) is a  crop  species that contains genetic material artificially derived from a wild relative population through repeated backcrossing. An example of a collection of ILs (called an  IL-Library ) is the use of  chromosome  segments from  Solanum pennellii  (a wild species of  tomato ) that was introgressed into  Solanum lycopersicum  (the cultivated tomato). The lines of an IL-library usually cover the complete  genome  of the donor. Introgression lines allow the study of  quantitative trait loci , but also the creation of new varieties by introducing exotic  traits . [ 24 ]


Lineage fusion  is an extreme variant of introgression that results from the merging of two distinct species or populations. This eventually results in a single population that displaces or replaces the parental species in the region. [ 25 ]  Some lineage fusion occurs soon after two taxa diverge or speciate, especially if there are few reproductive barriers between lineages. [ 26 ]  It is not strictly necessary for the two lineages to be closely related, but rather have the ability to produce viable offspring.


Lethal alleles  (also referred to as  lethal  or  lethals ) are  alleles  that cause the death of the organism that carries them. They are usually a result of mutations in genes that are essential for growth or development. [ 1 ]  Lethal alleles can be recessive, dominant, conditional, perinatal, or postnatal after an extended period of apparently normal development depending on the gene or genes involved.


Lethal alleles may specifically refer to embryonically lethal alleles, in which the fetus will never survive to term. Such alleles are a cause of non- Mendelian  patterns of inheritance, such as the observation of traits in a 2:1 ratio.


Lethal alleles were first discovered by  Lucien Cuénot  in 1905 while studying the inheritance of coat colour in mice. The  agouti  gene  in mice is largely responsible for determining coat colour. The wild-type allele produces a blend of yellow and black pigmentation in each hair of the mouse. This yellow and black blend may be referred to as 'agouti' in colour. [ 3 ]  One of the mutant alleles of the  agouti  gene results in mice with a much lighter, yellowish colour. When these yellow mice were crossed with  homozygous   wild-type  mice, a 1:1 ratio of yellow and dark grey offspring were obtained. This indicated that the yellow mutation is dominant, and all the parental yellow mice were  heterozygotes  for the mutant allele.


By mating two yellow mice, Cuénot expected to observe a usual 1:2:1 Mendelian ratio of homozygous agouti to heterozygous yellow to homozygous yellow. Instead, he always observed a 1:2 ratio of agouti to yellow mice. He was unable to produce any mice that were homozygous for the yellow agouti allele.


It was not until 1910 that  W. E. Castle  and  C. C. Little  confirmed Cuénot's work, further demonstrating that one quarter of the offspring were dying during embryonic development. This was the first documented example of a recessive embryonic lethal allele.


Lethal alleles can also refer to any allele that can result in a terminal condition.


A pair of identical alleles that are both present in an organism that ultimately results in death of that organism are referred to as recessive lethal alleles.  Though recessive lethals may code for  dominant  or  recessive  traits, they are only fatal in the homozygous condition. Heterozygotes will sometimes display a form of diseased phenotype in addition to an apparently dominant phenotype, as yellow mice are particularly susceptible to diabetes and obesity.  [ 4 ]


An example of a lethal allele in humans are the BRCA mutations; inheriting one defective BRCA allele results in a greatly increased risk of  breast cancer  and  ovarian cancer , while inheriting both defective alleles is embryonically lethal in almost all cases.  [ 5 ]  For live cases, inheriting both mutations lead to a grave prognosis where survival almost never extends beyond childhood. [ 6 ]  This is because the BRCA mutations also result in a severe subtype of  Fanconi anemia  (FA-S for BRCA1, FA-D1 for BRCA2), itself an extremely rare medical condition.


Another example of a recessive lethal allele occurs in the  Manx cat . Manx cats possess a heterozygous mutation resulting in a shortened or missing tail. Crosses of two heterozygous Manx cats result in two-thirds of surviving offspring displaying the heterozygous shortened tail phenotype, and one-third of surviving offspring of normal tail length that is homozygous for a normal allele. Homozygous offspring for the mutant allele cannot survive birth and are therefore not seen in these crosses. [ 7 ]


A lethal allele may refer to any allele encoding the disease that results in a terminal condition only in the homozygous or biallelic state. The heterozygous and homozygous phenotype is still expressed in most cases if two different disease-causing alleles are present.  Achondroplasia  is a skeletal system disorder caused by a recessive allele that can still result in a live birth in the homozygous state.  [ 8 ]  One mutant allele for achondroplasia can be tolerated, but having two results in death. In the case of homozygous achondroplasia, death almost invariably occurs before birth or in the perinatal period. Not all heterozygotes for recessive lethal alleles will show a mutant  phenotype , as is the case for  cystic fibrosis  carriers. If two cystic fibrosis carriers have children, they have a 25 percent chance of producing offspring having two copies of the allele, eventually resulting in the death of the child without intensive treatment. [ 9 ]


Alleles that need only be present in one copy in an organism to be fatal are referred to as dominant lethal alleles. These alleles are not commonly found in populations because they usually result in the death of an organism before it can transmit its lethal allele on to its offspring. As a result, few dominant embryonically lethal alleles are documented as they would never show up in the population.


[ 8 ]  An example in humans of a dominant lethal allele is Huntington's disease, a rare neurodegenerative disorder that ultimately results in premature death. However, because of its late-onset (i.e., often after reproduction has already occurred), it is able to be maintained in populations. A person exhibits Huntington's disease when they carry a single copy of a  repeat-expanded   Huntington  allele on chromosome 4. [ 10 ]


Alleles that will only be fatal in response to some environmental factor are referred to as conditional lethals. One example of a conditional lethal is  favism , a sex-linked inherited condition that causes the carrier to develop  hemolytic anemia  when they eat  fava beans . [ 8 ]


An infection of an  E. coli  host cell by a  bacteriophage (phage) T4  temperature sensitive (ts) conditionally  lethal mutant  at a high restrictive temperature leads to lack of viable phage production. However growth of such mutants can still occur at a lower temperature.  Such conditionally lethal ts mutants have been used to identify and characterize the function of many of the phage's genes. [ 11 ]  Thus genes employed in the  repair of DNA damages  were identified using ts mutants, [ 12 ] [ 13 ]  as well as genes affecting  genetic recombination . [ 14 ] [ 15 ]   For example, growing a ts DNA repair mutant at an intermediate temperature will allow some progeny phage to be produced.  However, if that ts mutant is irradiated with UV light, its survival will be more strongly reduced compared to the reduction of survival of irradiated wild-type phage T4.  In addition, cold sensitive conditional lethal mutants able to grow at high temperatures, but unable to grow at low temperatures, were also isolated in phage T4. [ 16 ]   These cold sensitive conditional lethal mutants also defined a set of phage genes.  Another class of conditional lethal phage T4 mutants, called  amber mutants , are able to grow on some strains of  E. coli  but not on others. [ 11 ] [ 17 ] [ 18 ]   These mutants were also used to initially identify and characterize many of the phage T4  genes , including genes whose encoded proteins function in  DNA repair ,  genetic recombination ,  DNA replication  and molecular  morphogenesis .  In addition, it was found that an amber mutation produces a "nonsense codon" within a gene that causes polypeptide chain termination during  translation . This finding provided insight into a significant aspect of the  genetic code .


Malecot's coancestry coefficient ,

 f

 {\displaystyle f}

 , refers to an indirect  measure  of genetic similarity of two individuals which was initially devised by the  French  mathematician  Gustave Malécot .


f

 {\displaystyle f}

  is defined as the probability that any two  alleles ,  sampled  at random (one from each individual), are identical copies of an ancestral allele.  In species with well-known lineages (such as domesticated crops),

 f

 {\displaystyle f}

  can be calculated by examining detailed pedigree records.  Modernly,

 f

 {\displaystyle f}

  can be estimated using  genetic marker  data.


In a finite size  population , after some generations, all individuals will have a  common ancestor  :

 f
 →
 1

 {\displaystyle f\rightarrow 1}

 .
Consider a non-sexual population of fixed size

 N

 {\displaystyle N}

 , and  call

 f

 i

 {\displaystyle f_{i}}

  the inbreeding coefficient of generation

 i

 {\displaystyle i}

 . Here,

 f

 {\displaystyle f}

  means the probability that two individuals picked at random will have a common ancestor. At each generation, each individual produces a large number

 k
 ≫
 1

 {\displaystyle k\gg 1}

  of descendants, from the pool of which

 N

 {\displaystyle N}

  individual will be chosen at random to form the new generation.


At generation

 n

 {\displaystyle n}

 , the probability that two individuals have a common ancestor is "they have a common parent" OR "they descend from two distinct individuals which have a common ancestor" :


What is the source of the above formula? Is it in a later paper than the 1948 Reference.


This is a  recurrence relation  easily solved. Considering the worst case where at generation zero, no two individuals have a common ancestor,


The scale of the fixation time (average number of generation it takes to homogenize the population) is therefore


This computation trivially extends to the inbreeding coefficients of alleles in a sexual population by changing

 N

 {\displaystyle N}

  to

 2
 N

 {\displaystyle 2N}

  (the number of  gametes ).


Mendelian inheritance  (also known as  Mendelism ) is a type of  biological   inheritance  following the principles originally proposed by  Gregor Mendel  in 1865 and 1866, re-discovered in 1900 by  Hugo de Vries  and  Carl Correns , and later popularized by  William Bateson . [ 1 ]  These principles were initially controversial. When Mendel's theories were integrated with the  Boveri–Sutton chromosome theory  of inheritance by  Thomas Hunt Morgan  in 1915, they became the core of  classical genetics .  Ronald Fisher  combined these ideas with the theory of  natural selection  in his 1930 book  The Genetical Theory of Natural Selection , putting  evolution  onto a  mathematical  footing and forming the basis for  population genetics  within the  modern evolutionary synthesis . [ 2 ]


The principles of Mendelian inheritance were named for and first derived by  Gregor Johann Mendel , [ 3 ]  a nineteenth-century  Moravian   monk  who formulated his ideas after conducting simple hybridization experiments with pea plants  ( Pisum sativum )  he had planted in the garden of his monastery. [ 4 ]  Between 1856 and 1863, Mendel cultivated and tested some 5,000 pea plants. From these experiments, he induced two generalizations which later became known as  Mendel's Principles of Heredity  or  Mendelian inheritance . He described his experiments in a two-part paper,  Versuche über Pflanzen-Hybriden  ( Experiments on Plant Hybridization ), [ 5 ]  that he presented to the Natural History Society of  Brno  on 8 February and 8 March 1865, and which was published in 1866. [ 3 ] [ 6 ] [ 7 ] [ 8 ]


Mendel's results were at first largely ignored. Although they were not completely unknown to biologists of the time, they were not seen as generally applicable, even by Mendel himself, who thought they only applied to certain categories of species or traits. A major roadblock to understanding their significance was the importance attached by 19th-century biologists to the  apparent blending  of  many inherited traits  in the overall appearance of the progeny, [ citation needed ]  now known to be due to  multi-gene interactions , in contrast to the organ-specific binary characters studied by Mendel. [ 4 ]  In 1900, however, his work was "re-discovered" by three European scientists,  Hugo de Vries ,  Carl Correns , and  Erich von Tschermak . The exact nature of the "re-discovery" has been debated: De Vries published first on the subject, mentioning Mendel in a footnote, while Correns pointed out Mendel's priority after having read De Vries' paper and realizing that he himself did not have priority. De Vries may not have acknowledged truthfully how much of his knowledge of the laws came from his own work and how much came only after reading Mendel's paper. Later scholars have accused Von Tschermak of not truly understanding the results at all. [ 9 ] [ 10 ]


Regardless, the "re-discovery" made Mendelism an important but controversial theory. Its most vigorous promoter in Europe was  William Bateson , who coined the terms " genetics " and " allele " to describe many of its tenets. [ 11 ]  The model of  heredity  was contested by other biologists because it implied that heredity was discontinuous, in opposition to the apparently continuous variation observable for many traits. [ 12 ]  Many biologists also dismissed the theory because they were not sure it would apply to all species. However, later work by biologists and statisticians such as  Ronald Fisher  showed that if multiple Mendelian factors were involved in the expression of an individual trait, they could produce the diverse results observed, thus demonstrating that Mendelian genetics is compatible with  natural selection . [ 13 ] [ 14 ]   Thomas Hunt Morgan  and his assistants later integrated Mendel's theoretical model with the  chromosome  theory of inheritance, in which the chromosomes of  cells  were thought to hold the actual hereditary material, and created what is now known as  classical genetics , a highly successful foundation which eventually cemented Mendel's place in history. [ 3 ] [ 11 ]


Mendel's findings allowed scientists such as Fisher and  J.B.S. Haldane  to predict the expression of traits on the basis of mathematical probabilities. An important aspect of Mendel's success can be traced to his decision to start his crosses only with plants he demonstrated were  true-breeding . [ 4 ] [ 13 ]  He only measured discrete (binary) characteristics, such as color, shape, and position of the seeds, rather than quantitatively variable characteristics. He expressed his results numerically and subjected them to  statistical analysis . His method of data analysis and his large  sample size  gave credibility to his data. He had the foresight to follow several successive generations (P, F 1 , F 2 , F 3 ) of pea plants and record their variations. Finally, he performed "test crosses" ( backcrossing  descendants of the initial  hybridization  to the initial true-breeding lines) to reveal the presence and proportions of  recessive  characters. [ 15 ]


Punnett Squares are a well known genetics tool that was created by an English geneticist, Reginald Punnett, which can visually demonstrate all the possible genotypes that an offspring can receive, given the genotypes of their parents. [ 16 ] [ 17 ] [ 18 ]  Each parent carries two alleles, which can be shown on the top and the side of the chart, and each contribute one of them towards reproduction at a time. Each of the squares in the middle demonstrates the number of times each pairing of parental alleles could combine to make potential offspring. Using probabilities, one can then determine which genotypes the parents can create, and at what frequencies they can be created. [ 16 ] [ 18 ]


For example, if two parents both have a heterozygous genotype, then there would be a 50% chance for their offspring to have the same genotype, and a 50% chance they would have a homozygous genotype. Since they could possibly contribute two identical alleles, the 50% would be halved to 25% to account for each type of homozygote, whether this was a homozygous dominant genotype, or a homozygous recessive genotype. [ 16 ] [ 17 ] [ 18 ]


Pedigrees are visual tree like representations that demonstrate exactly how alleles are being passed from past generations to future ones. [ 19 ]  They also provide a diagram displaying each individual that carries a desired allele, and exactly which side of inheritance it was received from, whether it was from their mother's side or their father's side. [ 19 ]  Pedigrees can also be used to aid researchers in determining the inheritance pattern for the desired allele, because they share information such as the gender of all individuals, the phenotype, a predicted genotype, the potential sources for the alleles, and also based its history, how it could continue to spread in the future generations to come. By using pedigrees, scientists have been able to find ways to control the flow of alleles over time, so that alleles that act problematic can be resolved upon discovery. [ 20 ]


Five parts of Mendel's discoveries were an important divergence from the common theories at the time and were the prerequisite for the establishment of his rules.


According to customary terminology, the principles of inheritance discovered by Gregor Mendel are here referred to as Mendelian laws, although today's geneticists also speak of  Mendelian rules  or  Mendelian principles , [ 21 ] [ 22 ]  as there are many exceptions summarized under the collective term  Non-Mendelian inheritance . The laws were initially formulated by the geneticist  Thomas Hunt Morgan  in 1916. [ 23 ]


Mendel selected for the experiment the following characters of pea plants:


When he crossed purebred white flower and purple flower pea plants (the parental or P generation) by  artificial  pollination, the resulting flower colour was not a blend. Rather than being a mix of the two, the offspring in the first generation ( F 1 -generation ) were all purple-flowered. Therefore, he called this  biological trait  dominant. When he allowed  self-fertilization  in the uniform looking F 1 -generation, he obtained both colours in the F 2  generation with a purple flower to white flower ratio of 3 : 1. In some of the other characters also one of the traits was dominant.


He then conceived the idea of heredity units, which he called hereditary "factors". Mendel found that there are alternative forms of factors—now called  genes —that account for variations in inherited characteristics. For example, the gene for flower color in pea plants exists in two forms, one for purple and the other for white. The alternative "forms" are now called  alleles . For each trait, an organism inherits two alleles, one from each parent. These alleles may be the same or different. An organism that has two identical alleles for a gene is said to be  homozygous  for that gene (and is called a homozygote). An organism that has two different alleles for a gene is said to be  heterozygous  for that gene (and is called a heterozygote).


Mendel hypothesized that allele pairs separate randomly, or segregate, from each other during the production of the  gametes  in the seed plant ( egg cell ) and the pollen plant ( sperm ). Because allele pairs separate during gamete production, a  sperm  or  egg  carries only one allele for each inherited trait. When sperm and egg unite at  fertilization , each contributes its allele, restoring the paired condition in the offspring. Mendel also found that each pair of alleles segregates independently of the other pairs of alleles during gamete formation.


The  genotype  of an individual is made up of the many alleles it possesses. The  phenotype  is the result of the  expression  of all characteristics that are genetically determined by its alleles as well as by its environment. The presence of an allele does not mean that the trait will be expressed in the individual that possesses it. If the two alleles of an inherited pair differ (the heterozygous condition), then one determines the organism's appearance and is called the  dominant allele ; the other has no noticeable effect on the organism's appearance and is called the  recessive allele .


If two parents are mated with each other who differ in one  genetic characteristic  for which they are both  homozygous  (each pure-bred), all offspring in the first generation (F 1 ) are equal to the examined characteristic in  genotype  and  phenotype  showing the dominant trait. This  uniformity rule  or  reciprocity rule  applies to all individuals of the F 1 -generation. [ 30 ]


The principle of dominant inheritance discovered by Mendel states that in a heterozygote the dominant allele will cause the recessive allele to be "masked": that is, not expressed in the phenotype. Only if an individual is homozygous with respect to the recessive allele will the recessive trait be expressed. Therefore, a cross between a homozygous dominant and a homozygous recessive organism yields a heterozygous organism whose phenotype displays only the dominant trait.


The F 1  offspring of Mendel's pea crosses always looked like one of the two parental varieties. In this situation of "complete dominance", the dominant allele had the same phenotypic effect whether present in one or two copies.


But for some characteristics, the F 1  hybrids have an appearance  in between  the phenotypes of the two parental varieties. A cross between two four o'clock ( Mirabilis jalapa ) plants shows an exception to Mendel's principle, called  incomplete dominance . Flowers of heterozygous plants have a phenotype somewhere between the two homozygous genotypes. In cases of intermediate inheritance (incomplete dominance) in the F 1 -generation Mendel's principle of uniformity in genotype and phenotype applies as well. Research about intermediate inheritance was done by other scientists. The first was  Carl Correns  with his studies about Mirabilis jalapa. [ 28 ] [ 31 ] [ 32 ] [ 33 ] [ 34 ]


The Law of Segregation of genes applies when two individuals, both heterozygous for a certain trait are crossed, for example, hybrids of the F 1 -generation. The offspring in the F 2 -generation differ in genotype and phenotype so that the characteristics of the grandparents (P-generation) regularly occur again. In a dominant-recessive inheritance, an average of 25% are homozygous with the dominant trait, 50% are heterozygous showing the dominant trait in the phenotype ( genetic carriers ), 25% are homozygous with the recessive trait and therefore  express  the recessive trait in the phenotype. The genotypic ratio is 1: 2 : 1, and the phenotypic ratio is 3: 1.


In the pea plant example, the capital "B" represents the dominant allele for purple blossom and lowercase "b" represents the recessive allele for white blossom. The  pistil  plant and the  pollen  plant are both F 1 -hybrids with genotype "B b". Each has one allele for purple and one allele for white. In the offspring, in the F 2 -plants in the Punnett-square, three combinations are possible. The genotypic ratio is 1  BB  : 2  Bb  : 1  bb . But the phenotypic ratio of plants with purple blossoms to those with white blossoms is 3 : 1 due to the dominance of the allele for purple. Plants with homozygous "b b" are white flowered like one of the grandparents in the P-generation.


In cases of  incomplete dominance  the same segregation of alleles takes place in the F 2 -generation, but here also the phenotypes show a ratio of 1 : 2 : 1, as the heterozygous are different in phenotype from the homozygous because the  genetic expression  of one allele compensates the missing expression of the other allele only partially. This results in an intermediate inheritance which was later described by other scientists.


In some literature sources, the principle of segregation is cited as the "first law". Nevertheless, Mendel did his crossing experiments with heterozygous plants after obtaining these hybrids by crossing two purebred plants, discovering the principle of dominance and uniformity first. [ 35 ] [ 27 ]


Molecular proof of segregation of genes was subsequently found through observation of  meiosis  by two scientists independently, the German botanist  Oscar Hertwig  in 1876, and the Belgian zoologist  Edouard Van Beneden  in 1883. Most alleles are located in  chromosomes  in the  cell nucleus . Paternal and maternal chromosomes get separated in meiosis because during  spermatogenesis  the chromosomes are segregated on the four sperm cells that arise from one mother sperm cell, and during  oogenesis  the chromosomes are distributed between the  polar bodies  and the  egg cell . Every individual organism contains two alleles for each trait. They segregate (separate) during meiosis such that each  gamete  contains only one of the alleles. [ 36 ]  When the gametes unite in the  zygote  the alleles—one from the mother one from the father—get passed on to the offspring. An offspring thus receives a pair of alleles for a trait by inheriting  homologous chromosomes  from the parent organisms: one allele for each trait from each parent. [ 36 ]  Heterozygous individuals with the dominant trait in the phenotype are  genetic carriers  of the recessive trait.


The Law of Independent Assortment proposes alleles for separate traits are passed independently of one another. [ 38 ] [ 35 ]  That is, the biological selection of an allele for one trait has nothing to do with the selection of an allele for any other trait. Mendel found support for this law in his dihybrid cross experiments. In his monohybrid crosses, an idealized 3:1 ratio between dominant and recessive phenotypes resulted. In dihybrid crosses, however, he found a 9:3:3:1 ratios. This shows that each of the two alleles is inherited independently from the other, with a 3:1 phenotypic ratio for each.


Independent assortment occurs in  eukaryotic  organisms during meiotic metaphase I, and produces a gamete with a mixture of the organism's chromosomes. The physical basis of the independent assortment of chromosomes is the random orientation of each bivalent chromosome along the metaphase plate with respect to the other bivalent chromosomes. Along with  crossing over , independent assortment increases genetic diversity by producing novel genetic combinations.


There are many deviations from the principle of independent assortment due to  genetic linkage .


Of the 46 chromosomes in a normal  diploid  human cell, half are maternally derived (from the mother's  egg ) and half are paternally derived (from the father's  sperm ). This occurs as  sexual reproduction  involves the fusion of two  haploid  gametes (the egg and sperm) to produce a zygote and a new organism, in which every cell has two sets of chromosomes (diploid). During  gametogenesis  the normal complement of 46 chromosomes needs to be halved to 23 to ensure that the resulting haploid gamete can join with another haploid gamete to produce a diploid organism.


In independent assortment, the chromosomes that result are randomly sorted from all possible maternal and paternal chromosomes. Because zygotes end up with a mix instead of a pre-defined "set" from either parent, chromosomes are therefore considered assorted independently. As such, the zygote can end up with any combination of paternal or maternal chromosomes. For human gametes, with 23 chromosomes, the number of possibilities is 2 23  or 8,388,608 possible combinations. [ 39 ]  This contributes to the genetic variability of progeny. Generally, the recombination of genes has important implications for many evolutionary processes. [ 40 ] [ 41 ] [ 42 ]


A Mendelian trait is one whose inheritance follows Mendel's principles—namely, the trait depends only on a single  locus , whose  alleles  are either  dominant  or recessive.


Many traits are inherited in a  non-Mendelian  fashion. [ 43 ]


Mendel himself warned that care was needed in extrapolating his patterns to other organisms or traits. Indeed, many organisms have traits whose inheritance works differently from the principles he described; these traits are called non-Mendelian. [ 44 ] [ 45 ]


For example, Mendel focused on traits whose genes have only two alleles, such as "A" and "a". However, many genes have  more than two alleles.  He also focused on traits determined by a single gene. But some traits, such as height, depend on many genes rather than just one. Traits dependent on multiple genes are called  polygenic traits .


A  merodiploid  is a partially diploid bacterium, which has its own  chromosome  complement and a chromosome fragment introduced by  conjugation , transformation or  transduction . [ 1 ]  It can also be defined as an essentially haploid  organism  that carries a second copy of a part of its genome. The term is derived from the Greek, meros = part, and was originally used to describe both unstable partial diploidy, such as that which occurs briefly in recipients after mating with an Hfr strain (1), and the stable state, exemplified by F-prime strains (see Hfr'S And F-Primes). Over time the usage has tended to confine the term to descriptions of stable genetic states.  Merodiploidy  refers to the partial duplication of chromosomes in a haploid organism. [ 2 ]


Microfluidic  whole genome haplotyping  is a technique for the physical separation of individual chromosomes from a metaphase cell followed by direct resolution of the  haplotype  for each allele.


Whole genome haplotyping is the process of resolving personal haplotypes on a whole  genome  basis. [ 1 ]   Current methods of  next generation sequencing  are capable of identifying heterozygous loci, but they are not well suited to identify which polymorphisms exist on the same (in cis) or allelic (in trans) strand of DNA. Haplotype information contributes to the understanding of the potential functional effects of variants in cis or in trans.  Haplotypes are more frequently resolved by inference through comparison with parental genotypes, or from population samples using statistical computational methods to determine linkage disequilibrium between markers.
Direct haplotyping is possible through isolation of  chromosomes  or chromosome segments. Most  molecular biology  techniques for haplotyping can accurately determine haplotypes of only a limited region of the genome.
Whole genome direct haplotyping involves the resolution of  haplotype  at the whole genome level, usually through the isolation of individual chromosomes.


A  haplotype  (haplo: from  Ancient Greek  ἁπλόος (haplóos, “single, simple”) is a contiguous section of closely linked segments of  DNA  within the larger  genome  that tend to be inherited together as a unit on a single  chromosome .  Haplotypes have no defined size and can refer to anything from a few closely linked loci up to an entire  chromosome .  The term is also used to describe groups of  single-nucleotide polymorphisms  (SNPs) that are statistically associated.
Most of the knowledge of SNP association comes from the effort of the  International HapMap Project , which has proved itself a powerful resource in the development of a publicly accessible database of human genetic variation.


Phasing  is the process of identifying the individual complement of  homologous chromosomes . Methods for phasing include pedigree analysis, allele-specific  PCR , linkage emulsion PCR haplotype analysis, [ 2 ]   polony  PCR, [ 3 ]  sperm typing,  bacterial artificial chromosome  cloning, construction of somatic cell hybrids, atomic force microscopy, among others. Haplotype phasing can also be achieved through computational inference methods.


Microfluidics  refers to the use of micro-sized channels on a micro-electro-mechanical system ( MEMS ). [ 4 ]   Microfluidic channels have a diameter of 10-100μm, making it possible to manipulate and analyze minute volumes. This technology combines engineering, physics, chemistry, biology, and optics. Over the past decades it has revolutionized micro and nanoscale biology, genetics and proteomics. Microfluidic devices can combine several analytical steps into one device. This technology has been coined by some as the "lab on a chip" technology. Most current molecular biology methods use some form of MEMS, including  microarray  technology and  next generation sequencing  instruments.


Direct deterministic phasing of individual chromosomes can be achieved by isolating single chromosomes for genetic analysis through the use of a  microfluidic  device. [ 5 ]


A single  metaphase  cell is isolated from solution. The chromosomes are then released from the nucleus, and the cytoplasm is digested enzymatically.  Next, the chromosome suspension is directed towards multiple partitioning channels. The chromosomes are physically directed into the partitioning channels using a series of valves. In the first description of this technique, Fan et al. designed a custom-made program (MatLab) to control this process.  Once separated, the chromosomes are prepared for amplification by sequential addition and washout of trypsin, denaturation buffer and neutralization solution. The DNA is then ready for further processing. Because of the small amount of DNA, amplification needs to be performed using kits specialized for very small initial DNA quantities. The amplified DNA is flushed out of the microfluidic device and solubilized by the addition of a buffer. The amplified DNA can now be analyzed by various methods.


Once the chromosomes have been isolated and amplified any molecular haplotyping can be applied as long as the chromosomes remain distinct.  This could be accomplished by keeping them physically separated, or identifying each sample by genotyping.  Once each chromosome has been identified each pair of homologs can be assorted into one of two haploid genomes.


Microfluidic direct deterministic phasing allows all the chromosomes to be isolated in the same experiment. This unique feature suggests possible applications within clinical, research and personal genomics realms. Some of the possible clinical applications for this technique include phasing of multiple mutations when parental samples are unavailable,  preimplantation genetic diagnosis ,  prenatal diagnosis  and in the characterization of cancer cells.


Whole genome haplotyping through microfluidics will increase the rate of discovery within the HapMap project, and provides an opportunity for corroboration and error detection within the existing database. It will further inform genetic association studies.


As methods for amplification of small amounts of DNA improve, single chromosome sequencing is possible using microfluidics to separate each individual chromosome. A cost-effective approach may be to barcode each individual chromosome and perform parallel resequencing of the entire individual genome. The amplification of each chromosome separately also provides a mechanism to potentially fill in some of the gaps that remain in the human  reference genome . Single chromosome sequencing will allow for unmapped sequences to be associated with a single chromosome. Additionally, single chromosome sequencing will be more accurate in the identification of copy number variants and repetitive sequences.


As of January 2011, only one publication has described use of this technique. [ 5 ]  The scientific commons awaits further validation of this method and its efficacy in isolating and amplifying analyzable amounts of DNA. While this method does streamline the process of chromosome isolation, certain parts in the process – such as the initial isolation of a metaphase cell – remain difficult and labour intensive. Other automated techniques for metaphase cell separation would improve throughput. In addition, this method is only applicable to cells in metaphase, which inherently limits the technique to cell types and tissues that undergo  mitosis . Single cell analysis does not account for the possibility of  mosaicism ; therefore, applications in cancer diagnosis and research would necessarily require processing of multiple cells.  Finally, since this entire process is based on amplification from a single cell, the accuracy of any genetic analysis is limited to the ability of commercially available platforms to produce sufficient amounts of unbiased and error free amplicon.


Chromosome microdissection  is another process for isolating single chromosomes for genetic analysis. As with the above technique microdissection begins with metaphase cells. The nucleus is lysed mechanically on a glass slide and part of the genetic material is partitioned under microscope. The actual microdissection of genetic material was initially accomplished through the careful use of a fine needle. Today computer-directed lasers are available. The genomic area isolated can range from part of a single chromosome, up to several chromosomes. To accomplish whole genome haplotyping the microdissected genomic section is amplified and genotyped or sequenced. Like with the microfluidic technique, specialized amplification platforms are necessary to address the problem of a small initial DNA sample. [ 6 ] [ 7 ] [ 8 ]


Randomly partitioning a complete diploid  fosmid  library into various pools of equal size presents an alternative method for haplotype phasing. In the proof of principle description of this technique [ 9 ]  115 pools were created containing ~5000 unique clones from the original fosmid library. Each of these pools contained roughly 3% of the genome. Between the 3% in each pool and the fact that each clone is a random sampling of the diploid genome, 99.1% of the time each pool contains DNA from a single homolog. Amplification and analysis of each pool provide haplotype resolution limited only by the size of the fosmid insert.


Midphalangeal hair , or the presence/absence of hair on the middle  phalanx  of the  ring finger , is one of the most widely studied markers in classical genetics of human populations. Although this  polymorphism  was observed at other fingers as well, for this kind of research, the fourth finger of the hand has been conventionally selected.


In humans, hair is commonly present on all the basal segments of the  digits  and invariably absent from all the terminal ones. On the middle segments, there is wide fluctuation with apparent familial and  racial  tendencies. Hair is present on the middle segment of the fingers more frequently than on the middle segment of the toes. Hair is most often found on the middle segment of the fourth finger.


Willier (1974), [ 1 ]  related citations quoted Danforth as stating that 'the hair  follicle  is a kind of  biological  microcosm in which almost any problem relating to growth, differentiation, decline and rejuvenescence of  tissue  can be studied to advantage....' While riding on a streetcar in  Wilkes-Barre  one summer, Danforth observed, in his own words, that 'a man in front of me draped his arm over the back of the seat and I noticed that while his  arm  was very hairy the middle segments of his fingers were free of hair and so, I observed, were my own; but I knew this was not generally true.' So far as he was aware, no one before had recognized this variation as possibly  hereditary .


The  genetic  determination of presence or absence of hair on the dorsal aspect of the middle phalanx was first suggested by Danforth (1921). From a study of 80 families with a total of 178 children, he suggested that 'a  phylogenetically  progressive loss of hair is brought about through the action of one or more  recessive genes , or of one primary recessive gene with several modifying factors that regulate the distribution of hair when it is present.' Stated conversely, 'despite the fact that in evolutionary progress hair is disappearing from the mid-digital region, its presence...may be regarded as the manifestation of a dominant trait.'


Bernstein and Burks (1942)  [ 2 ]
suggested that 5  allelic  genes, A-0 to A-4, 'control the inheritance and distribution of middigital hair involving but a  single gene  substitution (the subscript denoting the number of fingers affected with middigital hair),' and that the genes for the presence of hair are  dominant  over the genes for its absence. [ 3 ]


From a  literature  review and their own study in Brazil, Saldanha and Guinsburg (1961) suggested that lack of middle phalangeal hair may be determined by a pair of  recessive  genes, but noted that the occurrence of  sex ,  age , and possibly  environmental  differences make genetic analysis of the  trait  difficult.


Egesi and Rashid  (2010) reviewed the subject of middigital hair and its clinical significance. [ 4 ]


Danforth (1921) reported that middigital hair was present in men more often than in women. Caucasians were found to have a higher incidence of middle phalangeal hair than other ethnic groups, including  Afro-Americans ,  American Indians , and  Japanese .


Saldanha and Guinsburg (1961) studied the presence or absence of middigital hair in a white population of  São Paulo , Brazil, including 131 males and 158 females, and compared their findings with those of previous reports. The frequencies of individuals with midphalangeal hair showed striking population differences. The range among northern  Europeans  varied from 70 to 80%, and among  Mediterraneans , from 50 to 70%. Among Japanese,  American Indians , and  Afro-Americans , the figures varied between about 10% and 40%. The trait was virtually absent among the  Inuit . [ 5 ]


[ 6 ] [ page needed ] [ 7 ] [ page needed ]


A  monohybrid cross  is a cross between two organisms with different variations at one genetic locus of interest. [ 1 ] [ 2 ]   The character(s) being studied in a monohybrid cross are governed by two or multiple variations for a single location of a gene.
Then carry out such a cross, each parent is chosen to be  homozygous  or true breeding for a given  trait  (locus). When a cross satisfies the conditions for a monohybrid cross, it is usually detected by a characteristic distribution of second-generation (F 2 ) offspring that is sometimes called the monohybrid ratio.


Generally, the monohybrid cross is used to determine the  dominance  relationship between two  alleles . The cross begins with the parental  generation. One parent is  homozygous  for one allele, and the other parent is homozygous for the other allele. The offspring make up the first filial ( F1 ) generation. Every member of the F1 generation is  heterozygous  and the phenotype of the F1 generation expresses the dominant trait. [ 3 ]  Crossing two members of the F1 generation produces the second filial (F2) generation. Probability theory predicts that three quarters of the F2 generation will have the dominant allele's phenotype. And the remaining quarter of the F2s will have the recessive allele's phenotype. This predicted 3:1 phenotypic ratio assumes  Mendelian inheritance .


Gregor Mendel  (1822–1884) was an Austrian monk who theorized basic rules of inheritance. [ 4 ]  From 1858 to 1866, he bred garden peas ( Pisum sativum ) in his monastery garden and analyzed the offspring of these matings. The garden pea was chosen as an experimental organism because
many varieties were available that bred true for qualitative traits and their pollination could be manipulated. The seven variable characteristics Mendel investigated in pea plants were.  [ 5 ]


. [ 6 ]
Peas are normally self-pollinated because the stamens and carpels are enclosed within the petals. By removing the stamens from unripe flowers, Mendel could brush pollen from another variety on the carpels when they ripened. [ 7 ]


All the peas produced in the second or hybrid generation were round.


All the peas of this F1 generation have an Rr genotype. All the haploid sperm and eggs produced by meiosis received one chromosome. All the zygotes received one R allele (from the round seed parent) and one r allele (from the wrinkled seed parent). Because the R allele is dominant to the r allele, the phenotype of all the seeds was round. The phenotypic ratio in this case of Monohybrid cross is 1.


Mendel  then allowed his hybrid peas to self-pollinate. The wrinkled trait—which did not appear in his hybrid generation—reappeared in 25% of the new crop of peas.


Random union of equal numbers of R and r gametes produced an F2 generation with 25% RR and 50% Rr—both with the round phenotype—and 25% rr with the wrinkled phenotype.


Mendel then allowed some of each phenotype in the F2 generation to self-pollinate. His results:


One-third of the round seeds and all of the wrinkled seeds in the F2 generation were homozygous and produced only seeds of the same phenotype.


But two thirds of the round seeds in the F2 were heterozygous and their self-pollination produced both phenotypes in the ratio of a typical F1 cross.


Phenotype ratios are approximate. [ 8 ]
The union of sperm and eggs is random. As the size of the sample gets larger, however, chance deviations become minimized and the ratios approach the theoretical predictions more closely. The table shows the actual seed production by ten of Mendel's F1 plants. While his individual plants deviated widely from the expected 3:1 ratio, the group as a whole approached it quite closely.


To explain his results, Mendel formulated a hypothesis that included the following:
In the organism, there is a pair of factors that controls the appearance of a given characteristic. (They are called genes.)
The organism inherits these factors from its parents, one from each.
A factor is transmitted from generation to generation as a discrete, unchanging unit. (The r factor in the F2 generation passed through the round-seeded F1 generation. In spite of this, the rr seeds in the F2 generation were no less wrinkled than those in the P generation.) When the gametes are formed, the factors separate and are distributed as units to each gamete. This statement is often called Mendel's rule of segregation.
If an organism has two unlike factors (called alleles) for a characteristic, one may be expressed to the total exclusion of the other (dominant vs recessive).


A good hypothesis meets several standards.


In order to test his hypothesis, Mendel predicted the outcome of a breeding experiment that he had not carried out yet. He crossed heterozygous round peas (Rr) with wrinkled (homozygous, rr) ones. He predicted that in this case one-half of the seeds produced would be round (Rr) and one-half wrinkled (rr).


To a casual observer in the monastery garden, the cross appeared no different from the P cross described above: round-seeded peas being crossed with wrinkled-seeded ones. But Mendel predicted that this time he would produce both round and wrinkled seeds and in a 50:50 ratio. He performed the cross and harvested 106 round peas and 101 wrinkled peas.


Mendel tested his hypothesis with a type of  backcross  called a  testcross . An organism has an unknown genotype which is one of two genotypes (like RR and Rr) that produce the same phenotype. The result of the test identifies the unknown genotype.


Mendel did not stop there. He went on to cross pea varieties that differed in six other qualitative traits. In every case, the results supported his hypothesis. He crossed peas that differed in two traits. He found that the inheritance of one trait was independent of that of the other and so framed his second rule: the rule of independent assortment. Today, it is known that this rule does not apply to some genes, due to  genetic linkage . [ 9 ]


A  multilocus genotype  is the combination of  alleles  found at two or more  loci  in a single individual.


For example, in a  diploid  species, if there are two  SNP  loci and the first locus has alleles A and G, while the second locus has alleles T and C, the multilocus genotype can be represented as {A/G,T/C}. If the genome is not haploid then the multilocus genotype does not necessarily determine which alleles co-occur on chromosomes. In the example, if the two loci are located on the same chromosome the possibilities are either {A-T,G-C} or {A-C,G-T}. Where A-T represents a  haplotype  with alleles A and T together on one chromosome and G and C together on the other. If the  haplotypes  are determined the multilocus genotype is referred to as a phased genotype, otherwise it is referred to as unphased. Some authors [ 1 ] [ 2 ]  suggest that the term multilocus genotype should only be applied to phased multilocus data while others [ 3 ]  apply it to unphased multilocus data as well. The combination of alleles at two or more loci on a single chromosome make up a haplotype and the two haplotypes in a diploid individual make up the diplotype (a synonym for a phased multilocus genotype).


Non-Mendelian inheritance  is any pattern  in which traits do not segregate in accordance with  Mendel's laws . These laws describe the inheritance of traits linked to single  genes  on  chromosomes  in the nucleus. In  Mendelian inheritance , each parent contributes one of two possible  alleles  for a trait. If the  genotypes  of both parents in a genetic cross are known, Mendel's laws can be used to determine the distribution of  phenotypes  expected for the population of offspring.  There are several situations in which the proportions of phenotypes observed in the progeny do not match the predicted values.


Certain inherited diseases and their presentation display non-Mendelian patterns, complicating making predictions from family history. [ 1 ]


Incomplete dominance, codominance, multiple alleles, and polygenic traits follow Mendel's laws, display Mendelian inheritance, and are explained as extensions of Mendel's laws. [ 2 ]


In cases of intermediate inheritance due to  incomplete dominance , the principle of dominance discovered by Mendel does not apply. Nevertheless, the principle of uniformity works, as all offspring in the F 1 -generation have the same genotype and same phenotype. Mendel's principle of segregation of genes applies too, as in the F 2 -generation homozygous individuals with the phenotypes of the P-generation [ clarification needed ]  appear. Intermediate inheritance was first examined by  Carl Correns  in flower colour of  Mirabilis jalapa . [ 3 ]   Antirrhinum majus  also shows intermediate inheritance of the pigmentation of the blossoms. [ 4 ]