# Heterozygosity and overdominance

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Consider \$m\$ loci with heterozygote advantage (overdominance) such that the fitness of the two homozygotes is \$1-frac{s}{2}\$ and the fitness of the heterozygotes is \$1+frac{s}{2}\$, where \$s>0\$. We'll assume that the fitness of an individual is given by the multiplication of the fitness component on each locus. In consequence, the fitness of the best possible genotype is given by \$left(1+frac{s}{2} ight)^m\$.

According to this book, an individual is heterozygote at \$j\$ of these \$m\$ loci with probability

\$\${mchoose j}left(frac{1}{2} ight)^m\$\$

and the equilibrium population mean fitness \$hat w\$ is

\$\$hat w = sum_{j=0}^m {mchoose j}left(frac{1}{2} ight)^m left(1+frac{s}{2} ight)^j left(1-frac{s}{2} ight)^{m-j} = 1\$\$

I don't understand any of these two equations! Can you help me to understand how they have been calculated?

For the moment I just enjoyed proving that \$hat w = 1\$. We can reformulate \$hat w\$ as

\$\$hat w = left(frac{1}{2} ight)^m sum_{j=0}^m {mchoose j} left(1+frac{s}{2} ight)^j left(1-frac{s}{2} ight)^{m-j} = 1\$\$

,then using the binomial identity

\$\$ hat w = left(frac{1}{2} ight)^m left(left(1+frac{s}{2} ight) + left(1-frac{s}{2} ight) ight)^{space m}\$\$ \$\$hat w = left(frac{1}{2} ight)^m 2^m\$\$ \$\$ hat w = 1 \$\$

If the fitness of a heterozygote is \$(1+s/2)\$ and of a homozygote is \$(1-s/2)\$ then why is the probability for a given state \$(1+s/2)^j(1-s/2)^{m-k}\$

\$\$inom mj (1/2)^j(1/2)^{m-j}= inom mj (1/2)^m~~ ?\$\$

As you pointed out earlier, in the general case it need not be true that \$p = q = 1/2\$ but that is what the form of the probability above implies. So the threshold question is why this particular heterozygous-dominant model implies equilibrium probabilities \$p = q = 1/2.\$ I think the ideas below begin to address this.

The simplest case is for one locus, two alleles, and there are many good derivations online. I think if you understand the situation for one locus you can generalize to higher numbers. (Hopefully I will supplement this answer, time allowing. I think a shortcut would be to assume \$p = q = 1/2\$ and use your fitness weights. That gives us \$ar{w}=1\$ as a denominator. Now for reasons of symmetry I think you can show that the relative frequencies of \$p\$ and \$q\$ are equal and so \$p' = p.\$ Then you have to show that this equilibrium solution is unique.)

For a single locus the derivations of 'heterozygote advantage' I have found,(1)(2), assign fitness weights as follows:

AA = (1 - s), Aa = 1, aa = (1 - t )

in which \$s,t > 0, s eq t\$ in general, from which they derive as a condition for equilibrium

\$\$Delta q = frac{pq(sp-tq)}{ar{W}} = 0\$\$

so

\$\$hat{p} = frac{t}{s+t} ~~~ ext{and}~~~ hat{q} = frac{s }{s+t}\$\$

in which \$hat{p}, hat{q}\$ are equilibrum frequencies for each allele, respectively, and s and t are rates of mutation. Nowhere did I see a model in which they had assigned the same fitness to both homozygous cases (AA, aa) but it's just a special case of the heterozygote advantage model.

fitness (AA) = fitness(aa) = (1 - s/2), fitness(Aa) = (1+ s/2).

So if we subtract s/2 from each fitness score (or normalize with respect to Aa to the same effect), we get:

fitness(AA) = fitness (aa) = (1- s ) and fitness (Aa) = 1 as in the two references above, except that now the two homozygous states have equal fitness.

But then we have \$\$Delta q = frac{pq(sp-sq)}{ar{W}} = frac{pqs(p - q)}{ar{W}} \$\$

and the only nontrivial solution is \$p = q = 1/2.\$

So what I am suggesting is that when you assign the same fitness to both \$AA\$ and \$aa\$ you no longer have the general case. As for the value of s being forced, the following seems relevant.

The full expression from (2) for the equilibrium condition is

\$\$p' = frac{p^2 W_{AA}+ pq W_{Aa}}{ar{W}} hspace{10mm}(1)\$\$

in which\$ W_{AA} = W_{aa} = 1- s\$ and \$W_{Aa} = 1\$ and

\$ar{W} = p^2 W_{AA}+2pqW_{Aa}+q^2W_{aa}\$

If the homozygotes are assigned the same fitness and \$p = q = 1/2\$ , equation (1) above becomes:

\$\$p' = frac{frac{1}{4} + frac{1}{4}(1-s)}{frac{1}{2} + frac{1}{2}(1-s)} \$\$

The program on page 583 of (1) is helpful. Let h:= heterozygous and m:= homozygous. If \$p = q = 1/2\$ then as long as fitness(h) and fitness(m) are equal the system is in equilibrium immediately.

If \$p eq 1/2\$ then as long as f(h) = f(m) the system reaches equilibrium at \$p = 1/2\$ asymtotically. If \$p = 1/2\$ but f(h) \$ eq\$ f(m) the asymtotic limit has to be calculated.

## Associative overdominance: Evaluating the effects of inbreeding and linkage disequilibrium

Expressions are obtained for the expected phenotypic values of homozygous and heterozygous genotypes for a neutral marker locus linked to a locus segregating for a recessive deleterious gene. The phenotypic values are functions of the allele frequencies at the marker locus, the inbreeding coefficient and the degree of association of the deleterious gene with the marker alleles. The analysis is extended to more than two alleles at the marker locus. Either linkage disequilibrium or inbreeding alone can produce an apparent superiority of heterozygotes for the marker locus (unless specified otherwise, the terms ‘homozygote’ and ‘heterozygote’ will refer to the marker locus). The effect of linkage disequilibrium on the difference between the heterozygote and homozygote values can be positive (associative overdominance) or negative (associative underdominance), depending on the frequencies of the marker alleles and the degree of their association with the deleterious gene. Inbreeding has always a positive effect. In general, the expected value of a homozygote is a positive function of its allele frequency. When the various homozygous genotypes are combined into one class and the various heterozygous genotypes into another, the phenotypic difference of the two classes is a function of the evenness of the allelic frequency distribution. Inbreeding is a more likely explanation of associative overdominance if the frequency of the deleterious gene is low, but its effect on the character high. Conversely, linkage disequilibrium is more likely if the frequency is high and the effect low. The degrees of association between marker alleles and the deleterious gene can, in principle, be estimated from the observed phenotypic scores and used to calculate expected multi-locus genotype scores. This could provide the basis for statistical tests of the associative overdominance hypothesis as an explanation of observed correlations between multi-locus heterozygosity and phenotypic traits.

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## Relationship between heterozygosity and quantitative traits:intralocus interactions and multiple-locus averaging

The dependence of the expression of genotypic values (Y) on gene dosage (X) have been analyzed for four types of intralocus interactions (additivity, dominance, overdominance, and neutrality) using a linear model. Artificial numerical examples have been used to demonstrate that X and Y are positively associated with each other in the cases of additivity, dominance, and overdominance and are not associated in the case of neutrality. The averaging of single-locus genotypic values to obtain multiple-locus genotypes yields different results for different types of intralocus interactions. Genotypic values and individual heterozygosity are positively correlated with each other in the cases of dominance and overdominance and are negatively correlated in the case of additivity. In the case of neutrality, there is still no correlation after averaging. The results obtained and their interpretation suggest a new view on the experimental studies and generalizations on the relationship between heterozygosity and quantitative traits.

## Heterozygosity predicts clutch and egg size but not plasticity in a house sparrow population with no evidence of inbreeding

We investigated the link between heterozygosity and the reaction norm attributes of reproductive performance in female house sparrows (Passer domesticus). We collected data on clutch size, egg size, hatching success and nestling survival in 2816 nesting attempts made by 791 marked individuals over a 16-year period. Pedigree analysis revealed no evidence of inbreeding. Neither parent-offspring regression nor an animal model revealed significant heritability in clutch or egg size. We selected 42 females that laid at least seven clutches at our study site and used a survey of 21 autosomal microsatellite loci to estimate heterozygosity for each female. We controlled for phenotypic plasticity and found that both clutch and egg size showed significant positive correlations with heterozygosity. We found no evidence that heterozygosity influenced the slope of individual reaction norms. Further analysis suggested that clutch size was affected by heterozygosity across the genome, but egg size had more complex relationships, with evidence favouring the influence of multiple loci. Given the apparent lack of inbreeding and large population size, our results suggest associative overdominance as the likely mechanism for the impact of heterozygosity, but also created a puzzle about the process producing associations between neutral markers and the genes affecting clutch size or egg size. One possible explanation is a long-term residual effect of the historical bottleneck that occurred when house sparrows were introduced into North America. The existence of heterozygosity-fitness correlations in a population with considerable phenotypic plasticity and little inbreeding implies that the effects of heterozygosity may be more significant than previously thought.

## Materials and Methods

### Populations

Genotypes for over 3 million SNPs are available from The HapMap project Web page for the following populations: 60 unrelated individuals from the Yoruba population in Ibadan, Nigeria (YRI), 45 unrelated individuals from Tokyo, Japan (JPT), 45 unrelated individuals from Han Chinese in Beijing, China (CHB), and 60 unrelated Utah residents with ancestry from northern and western Europe, from the Centre d'Etude de Polymorphisms Human (CEU). We grouped JPT and CHB data into one new group termed CHB + JPT.

### Databases

To estimate the ratio of observed heterozygosity to expected gene diversity for every autosomal coding nonsynonymous SNP, we used the HapMap SNP information (release 22, based on National Center for Biotechnology Information build 36 and dbSNP, the Single Nucleotide Polymorphism database, build 126). Additional related information was downloaded from the UCSC, the University of California Santa Cruz, public MySQL server (ftp://hgdownload.cse.ucsc.edu/mysql/hg18/).

Genome coordinates for copy number variations (CNVs build 36) were taken from http://projects.tcag.ca/variation/. Genome coordinates for segmental duplications (build 36) were obtained from http://projects.tcag.ca/humandup/, and pseudogenes genome coordinates (build 36) from http://www.pseudogene.org.

Genomic coordinates for olfactory receptor (OR) pseudogenes were obtained from HORDE (Human Olfactory Receptor Data Exploratorium), (Available at http://bioportal.weizmann.ac.il/HORDE/). We also used classifier for olfactory receptor pseudogenes (CORP), a probabilistic method for annotation of OR pseudogenes, in order to increase our chances to detect pseudogenized ORs (http://bioportal.weizmann.ac.il/HORDE/CORP/).

### Coalescent Simulations

For the neutral simulations, we used the coalescent-based program ms by Hudson (2002). To correct for demography, we used an out-of-Africa model of human evolution. Demographic parameters were initially set to those reported by Schaffner et al. (2006). Iteratively, these values were modified until the pairwise FST distributions simulated under these values showed a good agreement with pairwise FST distributions obtained from real genomic data from the 3 major geographical human groups from the HapMap project (see Izagirre et al. 2006) (P values for the Kolmogorov–Smirnov D statistic: 0.891 for Caucasians vs. Asians, 0.104 for Caucasians vs. Africans, and 0.683 for Asians vs. Africans, which indicate that both simulated and real FST distributions are not statistically different). Finally, demographic parameters assumed an ancestral population size of 24 000 for the African population (population 1) and 7 700 for both Asians and Caucasians (populations 2 and 3, respectively), with a migration rates matrix Mij = <0, 0.05, 0.4, 0.1, 0 3, 0.8, 2.5, 0>for i and j values going from 1 to 3. Going backward in time, we assumed 2 bottlenecks with instant population reduction followed each by a population fusion: one at approximately 40 000 years ago, in which the Chinese population reduced its size to approximately one-sixth. About 2 000 years after this episode, the Chinese population fusses with the European population. Assuming a generation time of 20 years, this represents an F value of 0.04 for this bottleneck. A second population bottleneck takes place about 90 000 years ago. In this episode, the Eurasian population suffers a reduction in size to one-sixth of its previous size. About 10 000 years after this bottleneck (F = 0.21), the Eurasian population fusses with the African population.

Subsequent simulations to estimate the probability of a given ratio value were run under these demographic parameters using again ms program of Hudson (2002). We fixed the number of segregating sites to 1 and sample sizes were equal to the corresponding HapMap sample sizes of each population group. For each of the 10 000 simulation and for each sample set of 2n alleles, we formed n genotypes from pairs of consecutive alleles (i.e., first allele plus second allele formed genotype number 1 and so on). For each set of so formed genotypes, we scored the number of observed heterozygotes and calculated the expected heterozygosity from the sample allele frequencies. Finally, we obtained a distribution of 10 000 ratio values from which we estimated the cumulative probability of a given ratio value.

### Gene Ontology Analysis

For the Gene Ontology (GO) analysis, we used Onto-Express (http://vortex.cs.wayne.edu/Projects.html). The reference list of genes consisted of all those genes for which nonsynonymous SNPs had been genotyped in HapMap, excluding both possible pseudogenes and those genes contained within segmental duplications and CNV regions. Lists of genes containing nonsynonymous SNPs with ratios higher than the established cutoff point were the query for the overrepresentation analysis. These lists were further refined as indicated in the main text. Molecular function terms were explored by means of the hypergeometric function. Multiple test correction used the false discovery rate approach implemented in Onto-Express.

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. [1]

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.

Dominance versus overdominance is a scientific controversy in the field of genetics that has persisted for more than a century. [2] These two alternative hypotheses were first stated in 1908.

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: [3]

• Dominance hypothesis. The dominance hypothesis attributes the superiority of hybrids to the suppression of undesirable recessive alleles from one parent by dominant alleles from the other. It attributes the poor performance of inbred strains to loss of genetic diversity, with the strains becoming purely homozygous at many loci. The dominance hypothesis was first expressed in 1908 by the geneticist Charles Davenport. [4] Under the dominance hypothesis, deleterious alleles are expected to be maintained in a random-mating population at a selection–mutation balance that would depend on the rate of mutation, the effect of the alleles and the degree to which alleles are expressed in heterozygotes. [5]
• Overdominance hypothesis. Certain combinations of alleles that can be obtained by crossing two inbred strains are advantageous in the heterozygote. The overdominance hypothesis attributes the heterozygote advantage to the survival of many alleles that are recessive and harmful in homozygotes. It attributes the poor performance of inbred strains to a high percentage of these harmful recessives. The overdominance hypothesis was developed independently by Edward M. East (1908) [6] and George Shull (1908). [7] Genetic variation at an overdominant locus is expected to be maintained by balancing selection. The high fitness of heterozygous genotypes favours the persistence of an allelic polymorphism in the population. [5]

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.

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. [8] 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.

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. [8]

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 ] . However, there are two problems with this claim:

• First, according to an article published in the journal Genome Biology, "genetic superiority" is an ill-defined term and not generally accepted terminology within the scientific field of genetics. [9] A related term fitness is well defined, but it can rarely be directly measured. Instead, scientists use objective, measurable quantities, such as the number of seeds a plant produces, the germination rate of a seed, or the percentage of organisms that survive to reproductive age. [10] From this perspective, crossbred plants and animals exhibiting heterosis may have "superior" traits, but this does not necessarily equate to any evidence of outright "genetic superiority". Use of the term "superiority" is commonplace for example in crop breeding, where it is well understood to mean a better-yielding, more robust plant for agriculture. Such a plant may yield better on a farm, but would likely struggle to survive in the wild, making this use open to misinterpretation. In human genetics any question of "genetic superiority" is even more problematic due to the historical and political implications of any such claim. Some may even go as far as to describe it as a questionable value judgement in the realm of politics, not science. [9]
• Second, not all hybrids exhibit heterosis (see outbreeding depression).

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.

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. [11] [12]

The genetic dominance hypothesis attributes the superiority of hybrids to the masking of expression of undesirable (deleterious) recessive alleles from one parent by dominant (usually wild-type) alleles from the other (see Complementation (genetics)). It attributes the poor performance of inbred strains to the expression of homozygous deleterious recessive alleles. The genetic overdominance hypothesis states that some combinations of alleles (which can be obtained by crossing two inbred strains) are especially advantageous when paired in a heterozygous individual. This hypothesis is commonly invoked to explain the persistence of some alleles (most famously the Sickle cell trait allele) that are harmful in homozygotes. In normal circumstances, such harmful alleles would be removed from a population through the process of natural selection. Like the dominance hypothesis, it attributes the poor performance of inbred strains to expression of such harmful recessive alleles. 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.

An epigenetic contribution to heterosis has been established in plants, [12] and it has also been reported in animals. [13] 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. [14] In hybrid plants, most miRNAs have non-additive expression (it might be higher or lower than the levels in the parents). [12] This suggests that the small RNAs are involved in the growth, vigor and adaptation of hybrids. [12]

'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. [15] 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). [15] [16] It has been shown [11] 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 and/or methylation of specific amino acids in histone H3, a protein closely associated with DNA, which can either activate or repress associated genes.

### Major histocompatibility complex in animals Edit

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 will be more similar in smaller, more closely related populations. Breeding between more genetically distant individuals will decrease the chance of inheriting two alleles which are the same or similar, allowing a more diverse range of peptides to be presented. This therefore gives a decreased chance that any particular pathogen will not be recognised, and means that more antigenic proteins on any pathogen are likely to be recognised, giving a greater range of T-cell activation and therefore a greater response. This will also mean that the immunity acquired to the pathogen will be against a greater range of antigens, meaning that the pathogen must mutate more before immunity is lost. Thus hybrids will be less likely to be succumb to pathogenic disease and will be more capable of fighting off infection.

On the other hand, this may be the cause 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.

### Corn (maize) Edit

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 non stiff stalk. [ citation needed ]

### Rice (Oryza sativa) Edit

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 of

15 tons per hectare. In India also, several varieties have shown high vigor, including RH-10 and Suruchi 5401.

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 such as "beefalo" are also used for specialty markets.

### Poultry Edit

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 with a marketable carcass 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. [17]

John Scott and John L. Fuller performed a detailed study of purebred cocker spaniels, purebred basenjis, and hybrids between them. [18] 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. [19]

Sometimes people working on a dog breeding program find no useful heterosis. [20]

In 2014, a study undertaken by the Centre for Integrative Ecology at Deakin University in Geelong, Victoria concluded that intraspecific hybrids between the subspecies flaveolus and elegans of the Crimson rosella (Platycercus elegans) were more likely to fight off diseases than their pure counterparts. [21]

Human beings are all extremely genetically similar to one another. [22] [23] [24] 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 twentieth century.

## Overdominance maintaining polymorphism

One of the classic ways to maintain genetic variation with a population is "overdominance," in short, a state where heterozygotes exhibit greater fitness than the homozygote genotypes. Imagine for example a locus, A, with two alleles, A 1 & A 2 . Now, assume the fitness is distributed like so across the genotypes: A 1 A 1 = 0.75 A 1 A 2 = 1.00 A 2 A 2 = 0.75 In a random mating population the equilibrium genotypes given particular allele frequencies are described by the Hardy-Weinberg Equilibrium like so: p ^2 + 2pq + q ^2 In a diallelic scenario q is by definition 1 - p, resulting in some algebraic simplification (i.e., the above equation is equivalent to p ^2 + 2p <1 - p>+ <1 - p>^2 ). We can imagine that A 1 is equivalent to p, while A 2 is equivalent to q. From the fitness values above we then know that the fitness of the genotypes are: p ^2 = 0.75 2pq = 1.00 q ^2 = 0.75 Intuitively what would you expect? The frequencies of p & q need to equilibrate so that 2pq, the heterozygote, is maximized! Here is the formalism which describes the frequency of q at equilibirum: q = (selection against p)/(selection against p + selection against q) Now, fitness equals 1 minus the selection coefficient, that is, 1 - s . So, from the numbers above: 0.50 = (0.25)/(0.25 + 0.25) In other words, when homozygotes are less fit than the heterozygote in a diallelic model, and they are of equal fitness, they will be extant within the population at equal frequencies. By the nature of the Hardy-Weinberg Equilibrium the maximum heterozygosity attainable is 0.50 with a diallelic model and this is exactly what is produced by p & q frequencies of 0.5 within the population . Note that if the selection coefficients "favor" one allele over the the other in a homozygote state the ratio would differ, but if the heterozygote is modally fit then polymporphism will be maintained. Of course, this is all abstract. I just assigned "fitness" values without elaborating any scenario. Now, consider this from Introduction to Quantitative Genetics :

. Another example is the resistance of wild rates to the anti-coagulant poison warfarin. The gene conferring resistance is dominant, so the heterozygotes and homozygotes are resistant. Homozygotes, however, have a much increased requirement for vitamin K, which is not met by the normal diet. So in areas where the poison is being used, one homozygote is selected against by the poison and the other by the vitamin K deficiency, leading to an equilibrium frequency of the resistance gene, which was about 0.35 in the area studied.

In other words, in regards to resistance to warfarin the mutant allele exhibits physiological dominance. But, in regards to vitamin K deficiency it does not (insofar as one copy of the wild type allele is enough to prevent the nutritional deficit). The combination of these two physiological processes result in a natural advantage for the heterozygote genotype within the population, which preserves genetic variation and extant frequencies of both the mutant resistance allele and the wild type. I thought this example would be apropos since Larry Moran has a post up on the biochemistry of warfarin . So go check it out! Nothing in biology makes sense except in light of evolution, but evolutionary biology without mechanism is lame.

## Introduction

Hybrid vigor or heterosis is one of the most fascinating phenomena in genetics, evolutionary biology and applied breeding. In general, hybrid vigor refers to the higher performance of an F1 hybrid over the mean of the two parents. In agricultural sense, the F1 should outperform the better parent to be useful. Heterosis has also been applied to adaptive traits like increased fecundity, viability and resistance to biotic and abiotic stress (Dobzhansky, 1950). In this report unless noted otherwise, we use heterosis in the agricultural sense for data analysis. Although breeders and farmers have long used hybrid varieties to produce high-yield and quality agricultural products, nonetheless the genetic basis for heterosis still remains unclear (Coors and Pandey, 1999). Therefore, understanding the underlying mechanisms is not only a great intellectual challenge but also has a promising applied side to it.

Two popular hypotheses, namely, dominance and overdominance, have stimulated interesting debates as to whether heterosis is caused by dominant complementation of slightly deleterious recessive alleles (Bruce, 1910 Jones, 1917 Xiao et al, 1995 Cockerham and Zeng, 1996) or by overdominant gene action in which genes have greater expression when they are heterozygous (Shull, 1908 East, 1936 Crow, 1948 Stuber et al, 1992 Mitchell-Olds, 1995). According to the former, highest performance should follow the maximum accumulation of dominant favorable genes from both parents in homozygous conditions. If the latter holds true, heterosis should reach its peak at the maximum levels of heterozygosity and dissipate when approaching homozygosity. In many cases, however, overdominance is accompanied by nonallelic interactions. Removing significant nonallelic interactions also results in reduction or disappearance of apparent overdominant effects (Jinks, 1955). The third model suggests that heterosis results from epistatic interactions among alleles at different loci. Indeed, epistasis is involved in most quantitative trait loci (QTLs) associated with inbreeding depression and heterosis in corn (Stuber et al, 1992) and rice (Li et al, 2001 Luo et al, 2001).

The relationship between heterozygosity and F1 hybrid vigor remains unclear. For example, the association of genetic distance with heterosis in elite inbred lines of corn may be very strong (Lee et al, 1989 Smith et al, 1990) or weak (Godshalk et al, 1990 Dudley et al, 1991), because the correlation between marker distance and F1 performance depends on the origin of lines studied (Melchinger et al, 1990 Boppenmaier et al, 1993). Genetic background plays an important role in heterosis. Doebley et al (1995) showed that one of the two QTLs controlling differences in plant morphology and inflorescence architecture between maize and its ancestor (teosinte) has strong phenotypic effects in the teosinte background but reduced effects in the maize genetic background (Doebley et al, 1995). However, when the two QTLs are combined into one genotype, both plant morphology and inflorescence architecture are changed.

To investigate the genetic architecture of various phenotypic traits and mapping QTLs, we used the collection of recombinant inbred lines (RILs) (Lister and Dean, 1993) in North Carolina Design III (NCIII) (Comstock and Robinson, 1952). We observed superior F1 performance of many RILs when crossed to its parents (Col and Ler). Since this population is extensively mapped using a large number of (>1000) molecular markers, genotype/heterozygosity of each test cross (homozygous RIL crossed to Col or Ler) relative to the parents may be easily determined. The precise genotypes of all crosses based on published marker intervals are determined and the hybrid performance or heterosis among test cross families is investigated by comparing them with RILs and the two parents (Col and Ler) and their reciprocal hybrids. Significant QTLs are mapped for various morphological and developmental traits to determine their effects on homozygous and heterozygous states including different genetic backgrounds. Using linear and multiple linear regression and QTL analyses, we show that the varying degrees of heterozygosity combined with epistasis, genetic background and origin of plant materials are associated with high F1 performance. Chromosomes bearing QTLs for most traits have elevated levels of heterozygosity for high-performing hybrids showing that various QTLs in heterozygous state have positive effects on heterosis. These crosses with known genotypes combined with the excellent genomic resources of Arabidopsis provide unprecedented data and materials effectiveness for studying the relationships among heterozygosity, locus interactions and hybrid vigor.

## General Model

To determine the precise conditions under which population heterozygote advantage will be observed, we consider a general model that predicts the expected outcome of a comparison between heterozygotes and homozygotes in an epidemiological study as a function of (i) the frequencies of resistant and susceptible alleles at a particular locus and (ii) the relationship between genotype at that locus and phenotype. Note that this is not a model for the evolution of genotype frequencies or for the maintenance of MHC heterozygosity, but simply an algebraic framework for predicting the outcome of an epidemiological study of the type cited above, given current allele frequencies and genotype-phenotype mappings.

The model is summarized in Table 1. Suppose that individual alleles of a particular locus confer either susceptibility or resistance to a given disease, and that there are m resistance alleles, R 1,R 2. R mwith frequencies p 1,p 2. p min the population, and n susceptibility alleles S 1,S 2. S nwith frequencies q 1,q 2. q n. Let be the total frequency of resistant alleles, and be the total frequency of susceptible alleles, with p + q = 1. Further, define and are the sums of squared frequencies of the resistant and susceptible alleles, respectively. We assume Hardy-Weinberg genotype frequencies [23] throughout. Thus, Π is the frequency of RR homozygotes and is an inverse measure of the diversity of resistant alleles, while Θ, the frequency of SS homozygotes, is an inverse measure of the diversity of susceptible alleles.

We assume that SS homozygotes have a probability x of a favorable disease course, and that SS heterozygotes (carrying two different susceptible alleles), SR heterozygotes, RR heterozygotes, and RR homozygotes have probabilities ax, bx, cx, and dx respectively (see Table 1). To give meaning to the notions of resistant and susceptible alleles, we assume d > 1 (RR homozygotes do better than SS homozygotes) and abc (given that one is heterozygous, more R alleles are better). This model can accommodate dominance (a = 1 b = c = d), additivity (a = 1c = d b = (1 + d)/2), or recessiveness (a = b = 1 and c = d) of the resistant alleles. It can also accommodate overdominance of the resistant alleles (a > 1 b >d c >d), in which each heterozygote does better than either corresponding homozygote, and underdominance, in which each heterozygote does worse than either corresponding homozygote (a < 1 b < 1 c <d). In this model, susceptibility and resistance are relative and simply refer respectively to lower and higher probabilities of favorable disease course, given infection. For simplicity, we assume that all S alleles are equivalent and all R alleles are equivalent our conclusions could obviously be generalized to cases where there is a whole range of effects for different alleles.

Under these assumptions, we can calculate f homand f het, the probability of a favorable disease course for homozygotes (the first and fifth classes in Table 1) and for heterozygotes (the 2 nd , 3 rd , and 4 th classes in Table 1).

The relative risk (RR) of a favorable outcome for a heterozygote compared to a homozygote is defined as:

Population heterozygote advantage corresponds to RR > 1. Various formulations for relative risk (or odds ratio, used in case-control studies, such as [6]) are used in studies of HLA heterozygosity and disease outcome, but in all cases the cutoff is one and a value larger or smaller than one (depending on the precise definition used) corresponds to population heterozygote advantage. Using the relative risk equation above, it can be shown by taking partial derivatives with respect to the parameters p, Π and Θ, that (under the assumptions stated above) the relative risk increases with p and Θ and decreases with Π. Thus, population heterozygote advantage is most likely to be observed when resistant alleles are common (p large) but highly diverse (Π is small), and when susceptible alleles are not diverse (large Θ). These trends can be understood intuitively. Homozygotes will be predominantly resistant if R alleles are common and have little diversity and sensitive when S alleles are common and have little diversity. Heterozygotes will be predominantly resistant if R alleles are highly diverse and sensitive if S alleles are highly diverse.

Figure 1 shows the parameter regions in which population heterozygote advantage is expected and those in which the contrary is expected: homozygotes on average are more likely to have a favorable disease course. Each panel reflects a different assumption about the true genetic basis of resistance – assuming that resistant alleles are overdominant dominant additive recessive or underdominant. In each case, population heterozygote advantage, shown as the black region and corresponding to RR > 1, is most likely when resistant alleles are highly diverse and susceptible alleles have low diversity (bottom right of each panel).

Population heterozygote advantage as a function of allele-specific effects and allele frequencies. Parameter regions in which heterozygotes will on average have a higher probability of a favorable disease outcome than homozygotes (regions of population heterozygote advantage) are shown in black. Population heterozygote advantage occurs when diversity of resistant alleles is sufficiently high and diversity of susceptible alleles is sufficiently low i.e., toward the bottom right of the parameter space in each panel of the figure. Different panels indicate various assumptions about the genotype-specific relative risks a-d (defined in Table 1). Parameters: Overdominant (a = 1.1, b = 1.6, c = 2, d = 1.5) dominant (a = 1, b = c = d) additive (a = 1, b = (1 + c)/2, c = d) recessive (a = b = 1, c = d) underdominant (a = 0.5, b = 0.9, c = 1.4, d = 1.5). These curves are drawn for p = 0.5. Dominant, additive and recessive curves are valid for all possible values of the free parameters, while underdominant and overdominant curves are examples whose positions depend on the particular values of the parameters a, b, c and d.

Figure 1 shows that allele-specific overdominance (the biological phenomenon of interest) and population heterozygote advantage (the finding of epidemiological studies such as those cited above) are two different things. Population heterozygote advantage (black) may be observed even when resistance is not overdominant, but only dominant, additive, recessive or only underdominant, as long as allele frequencies are sufficiently far toward the lower right of the parameter space (high diversity of resistant alleles and low diversity of susceptible alleles). Figure 1 shows that this is possible under a fairly broad range of parameter combinations (i.e., there are substantial black areas in the dominant, additive and recessive panels).

The converse is also true, though only in what seem to be very special circumstances. That is, even when allele-specific overdominance holds, it is possible that heterozygotes on average will do worse than homozygotes, so population heterozygote advantage will not be observed. This occurs with genotype frequencies sufficiently far toward the top left of Figure 1, with a high diversity of susceptible alleles and low diversity of resistant ones. This occurs only rarely for the parameter values we have chosen (most of the leftmost panel is black), and this seems to be the case for a broad range of parameter values.

Although neither population heterozygote advantage nor allele-specific overdominance implies the other, the two phenomena are of course related. Specifically, the conditions to observe population heterozygote advantage are broadest when allele-specific overdominance holds and become narrower as the underlying genetics becomes more "different" from overdominance of resistance (dominance -> additivity -> recessiveness -> underdominance).

## Heterozygosity and overdominance - Biology

Although heterozygosity-fitness correlations (HFCs) are widely reported in the literature, most studies use too few markers to allow the proximate mechanisms to be convincingly resolved. Two competing hypotheses have been proposed: the general effect hypothesis, in which marker heterozygosity correlates with genome-wide heterozygosity and hence the inbreeding coefficient f, and the local effect hypothesis, in which one or more of the markers by chance exhibit associative overdominance. To explore the relative contributions of general and local effects in a free-ranging marine mammal population, we revisited a strong HFC found using 9 microsatellite loci for canine tooth size in 84 male Antarctic fur seals Arctocephalus gazella (Hoffman JI, Hanson N, Forcada J, Trathan PN, Amos W. 2010. Getting long in the tooth: a strong positive correlation between canine size and heterozygosity in the Antarctic fur seal Arctocephalus gazella. J Hered.). Increasing the number of markers to 76, we find that heterozygosity is uncorrelated across loci, indicating that inbred individuals are rare or absent. Similarly, while the HFC based on overall heterozygosity is lost, stochastic simulations indicate that when an HFC is due to inbreeding depression, increasing marker number invariably strengthens the HFC. Together these observations argue strongly that the original HFC was not due to inbreeding depression. In contrast, a subset of markers show individually significant effects, and these are nonrandomly distributed across the marker panel, being preferentially associated with markers cloned from other species. Using basic alignment search tool searches, we were able to locate 94% of loci to unique locations in the dog genome, but the local genes are functionally diverse, and the majority cannot be linked directly to growth. Our results suggest that inbreeding depression contributes little if at all to the relationship between heterozygosity and tooth size but that instead the primary mechanism involves associative overdominance. These findings contribute to a growing body of evidence suggesting that general effects are likely to be uncommon in natural populations