definition dominance hypothesis

Dominance Theory

Dominance theory is the idea that one hemisphere of the brain (usually the left) dominates over the other in determining handedness. Some psychologists argue against this theory because it oversimplifies the complex nature of handedness and fails to account for individual variations.

Related terms

Lateralization : Lateralization refers to how certain cognitive functions are more dominant in one hemisphere of the brain than the other.

Ambidexterity : Ambidexterity is when an individual is able to use both hands equally well for various tasks.

Hemispheric Specialization : Hemispheric specialization refers to how different functions or abilities are primarily controlled by either the left or right hemisphere of the brain.

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Why do some psychologists argue against the dominance theory in explaining handedness?

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Social Psychology

Intergroup Relations Social Psychology

Social dominance theory.

Social dominance theory (SDT; Sidanius & Pratto, 1999 ) is a multi-level, integrative theory of intergroup relations. Its central aim has been to understand the ubiquity and stubborn stability of group-based inequalities, though our research program has begun to explore how to introduce instability in considering group-based social hierarchies.

What’s New in Social Dominance Theory

Social dominance theory in the workplace.

Social dominance theory adds to understanding how power and hierarchies operate within workplaces. For example, there is an association between employees’ levels of social dominance orientation and what influence tactics they use and respond well to, and this association shows the asymmetric effect that social dominance theory hypothesizes between hierarchy-enhancing and hierarchy-attenuating work contexts.

Aiello, A., Pierro, A., & Pratto, F. (2013). Framing Social Dominance Orientation and Power in organizational context. Basic and Applied Social Psychology, 5, 487-495.

Aiello, A.,  Tesi, A. , Pierro, A., Pratto, F. (2017). Social dominance and interpersonal power: Asymmetrical relationships within Hierarchy-enhancing and Hierarchy-attenuating work environments.  Journal of Applied Social Psychology.   (  http://onlinelibrary.wiley.com/doi/10.1111/jasp.12488/full  ) 

What’s wrong with the “Clash of Civilizations” Thesis

Bernard Lewis’s thesis, promoted especially by Samuel Huntington, argues that the conflict between Arabs and the West is based on a “clash of civilizations.” Our data show instead that both social identity and Arabs from Lebanon and Syria hold negative attitudes towards the west because they oppose Western hegemony.

Sidanius, J., Kteily, N., Levin, S., Pratto, F., & Obaidi, M. (2015). Support for asymmetric violence among Arab populations: The clash of cultures, social identity, or counterdominance?  Group Processes & Intergroup Relations . DOI: 1368430215577224.

International Support for the Arab Uprisings

We present data from several nations about people’s willingness (or not) to engage in sympathetic collective action on behalf of Arabs, employing both social dominance theory and social identity theory.

Stewart. A. L., Pratto, F., Bou Zeineddine, F., Sweetman, J., Eicher, V., Licata, L., Morselle, D., Saab, R., Aiello, A., Chryssochoou, X., Cichocka, A., Cidam, A., Foels, R., Giguère, B., Li, L., Prati, F., & van Stekelenburg, J. (2015). International Support for the Arab Uprisings: Understanding Sympathetic Collective Action Using Theories of Social Dominance and Social Identity.  Group Processes and Intergroup Relations.  Published online before print January 19, 2015, doi: 10.1177/1368430214558310.

International Limits & Expansion of Social Dominance Theory

We offer a critique of Social Dominance Theory and its limits, as well as providing suggestions for developing the theory and needed research.

Pratto, F., Stewart, A. L., & Bou Zeineddine, F. (2013). When Inequality Fails: Power, Group Dominance, and Societal Change. Journal of Social and Political Psychology , 1, 132-160. doi:10.5964/jspp.v1i1.97

Revised Long Social Dominance Orientation Scale

We also revised the long version of the Social Dominance Scale, predicting and finding that the Dominance component corresponds to harsher prejudice measures and policy attitudes, and the Anti-Egalitarian component corresponds to subtler prejudice measures and policy attitudes.

Ho, A. K., Sidanius, J., Pratto, F., Levin, S., Thomsen, L., Kteily N. & Sheehy-Skeffington, J. (2012). Social Dominance Orientation: Revisiting the structure and function of a variable predicting social and political attitudes. Personality and Social Psychology Bulletin, 38, 583-606.

Ho, A. K., Sidanius, J. Kteily, N., Sheehy-Skeffington, J. Pratto, F. Henkel, K. E., Foels, R., Stewart, A. L. (2015). The Nature of Social Dominance Orientation: Theorizing and Measuring Preferences for Intergroup Inequality Using the New SDO 7 Scale. Journal of Personality and Social Psychology, 109 , 1003-1038.   http://dx.doi.org/10.1037/pspi0000033

The Salience of Group Distinctions Moderates the Relationship between People’s Social Dominance Orientation and Their Political Attitudes

Social Dominance Theory predicts that people who oppose group-based dominance in general will support social and political policies and practices that promote low-power groups. In addition, SDT predicts that this relationship will be stronger for group distinctions that are especially salient in people’s society. We tested both predictions in a cross-national study, using a new Short SDO scale, and attitudes about women, the poor, and ethnic or religious minorities. Using national indicators pertaining to salience of these groups, we found that the expected negative relationship between SDO and each attitude was stronger in countries where that group distinction was stronger. The salience of each of the target groups was not related.

Pratto, F., Çidam, A., Stewart, A.L., Bou Zeineddine, F., Aranda, M., Aiello, A., Chryssochoou, X., Cichocka, A., Cohrs, C., Durrheim, K., Eicher, V., Foels, R., Górska, P., Lee, I., Licata, L., Li, L., Liu, J., Morselli, D., Meyer, I., Muldoon, O., Muluk, H., Petrovic, N., Prati, F., Papastamou, S., Petrovic, I., Prodromitis, G., Rubini, M., Saab, R., van Stekelenburg, J., Sweetman, J., Zheng, W., Henkel, K.E. (2013). Social Dominance in Context and in Individuals: Contextual Moderation of Robust Effects of Social Dominance Orientation in 15 languages and 20 countries. Social Psychological and Personality Science, 4, 587-599 . DOI: 10.1177/1948550612473663

For the paper, http://spp.sagepub.com/content/early/2013/02/18/1948550612473663. Note: The Simplified Chinese version in the on-line article is not correct. For the SSDO scale in 16 languages, click here SSDO

U.S. Hegemony and Regional Politics in the Levant

How do people in nations subordinated by other nations think about hegemonic nations? What is the relationship between their support or opposition to hegemonic nations and to political factions in their country or region? This paper examined Syrian and Lebanese citizens’ attitudes towards their own governments and toward Hezbollah in 2010, considering what kind of relationship those factions had toward their own government. We found that people who are generally opposed to group-based dominance (Social Dominance Orientation) disliked American influence over Arabs, and this predicted whether they liked factions that oppose the U.S. (Syrian government, Hezbollah) or that were friendly towards the U.S. (Lebanese government at the time of the study).

Pratto, F., Sidanius, J., Bou Zeineddine, F., Kteily, N., & Levin, S. (2013). When domestic politics and international relations intermesh: Subordinated publics’ factional support within layered power structures Foreign Policy Analysis, 1-22. doi: 10.1111/fpa.12023 . Also (2014), 10, 127-148.

You can download the pre-publication draft of the paper here: Domestic & International relations

Race and Sexual Dominance

We asked men who have sex with men and who identified as Asian or Asian-American about their feelings about themselves and conditions under which they would risk unsafe sex. We found that sexual positions, race of partners, and social dominance orientation together imply recapitulation of intersectional hierarchies in an important domain.

Tan, J. Y., Pratto, F., Operario, D. & Dworkin, S. (April 2, 2013). Sexual Positioning And Race-Based Attraction By Preferences For Social Dominance Among Gay Asian/Pacific Islander Men in the United States. Archives of Sexual Behavior. DOI 10.1007/s10508-013-0088-y

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The integrative biology of genetic dominance

Sylvain billiard.

1 Univ. Lille, CNRS, UMR 8198 ‐ Evo‐Eco‐Paleo, F‐59000 Lille France

Vincent Castric

Violaine llaurens.

2 Institut de Systématique, Evolution et Biodiversité, CNRS/MNHN/Sorbonne Université/EPHE, Museum National d'Histoire Naturelle, CP50, 57 rue Cuvier, 75005 Paris France

Dominance is a basic property of inheritance systems describing the link between a diploid genotype at a single locus and the resulting phenotype. Models for the evolution of dominance have long been framed as an opposition between the irreconcilable views of Fisher in 1928 supporting the role of largely elusive dominance modifiers and Wright in 1929, who viewed dominance as an emerging property of the structure of enzymatic pathways. Recent theoretical and empirical advances however suggest that these opposing views can be reconciled, notably using models investigating the regulation of gene expression and developmental processes. In this more comprehensive framework, phenotypic dominance emerges from departures from linearity between any levels of integration in the genotype‐to‐phenotype map. Here, we review how these different models illuminate the emergence and evolution of dominance. We then detail recent empirical studies shedding new light on the diversity of molecular and physiological mechanisms underlying dominance and its evolution. By reconciling population genetics and functional biology, we hope our review will facilitate cross‐talk among research fields in the integrative study of dominance evolution.

I. INTRODUCTION

‘ It is obvious that no single theory can explain all dominance mechanisms ’

Genetic dominance describes the relationship between the phenotype and the genotype at a diploid locus in heterozygotes. An allelic variant may behave as dominant when a single copy is sufficient for full phenotypic expression, co‐dominant when the effects of the two alleles are equally apparent, or recessive when a single copy of the allele has no detectable phenotypic effect. These interactions depend on the partner allele in the heterozygous genotype: a dominant allele in one genotype may be recessive when paired with another allelic variant. The dominance relationship among alleles also depends on the focal character: for instance, a pleiotropic allele can cause a dominant effect for one trait but a recessive effect for another trait. Starting from Mendel's observations on peas, it has been repeatedly documented that strict additivity is the exception rather than the rule for many traits. However, we know shockingly little about the actual distribution of dominance coefficients for new mutations (Manna, Martin & Lenormand,  2011 ), and even less for segregating variants in natural populations (Eyre‐Walker & Keightley,  2007 ). This lack of empirical data prevents estimating the extent to which mutations are visible to natural selection and therefore severely limits the power of population genomic approaches to predict the adaptive potential of species.

Understanding the biological phenomenon of dominance has been a topic of intense and continued interest, both from the population genetics and the functional biology communities. Nevertheless, these two scientific communities have approached this question in different ways. The evolution of dominance is thus an excellent case study of the sometimes conflictual, yet fruitful interactions between these disciplines. The population genetics community has explored the conditions under which natural selection acts on dominance interactions at the level of organismal fitness. These generic models largely neglected the molecular processes underlying these interactions. By contrast, the functional biology community provided detailed mechanistic models focusing on the genotype‐to‐phenotype map, detailing the molecular processes involved. However, by focusing on the organismal level, they largely ignored the complexity of predicting the evolutionary fate of mutations within populations. In particular, intragenomic conflicts among regulatory elements can lead to counter‐intuitive evolutionary outcomes and were initially ignored. Recent theoretical models have moved towards more integrative approaches, and are now considering the diversity of molecular processes by which gene expression can be linked to fitness through explicit phenotypes. These models are bridging the gap between the two research fields, revealing that just as gene expression and integrated phenotypes are evolvable properties, so are dominance interactions.

To embrace the complexity of dominance relationships and of their evolution, here we provide definitions for the dominance‐related terms used in different fields (Table  1 ) and specifically review the range of mechanisms that have been proposed to cause variation of dominance at three different levels (Fig.  1 ). First, at the fitness level, i.e. when dominance relationships result from contrasted evolutionary fates of alleles with a dominance assumed to be fixed. This process was initially formulated by Haldane ( 1927 ), and we review recent empirical examples consistent with this phenomenon. Second, at the level of phenotypic integration, whereby differences between homozygotes and heterozygotes translate into different levels of biochemical activity being integrated over successive cellular and developmental processes, ultimately resulting in different organismal phenotypes. This aspect is akin to the ‘physiological’ model proposed by Wright ( 1934 ). We review recent studies providing detailed molecular explanations for how dominance could arise as the consequence of such processes, beyond the classical case of enzymes. Third, at the level of allelic expression, through the qualitative modification of gene regulatory networks. This aspect is akin to the controversial model of ‘dominance modifiers’ proposed by (Fisher,  1928 ), for which recent studies have now provided compelling evidence in specific biological situations.

Definitions

From genes to fitness: three levels of integration giving rise to dominance evolution. Genotypes in an environment produce individual organisms through multiple steps of developmental and ecological integration. (1) Gene regulatory networks and gene expression: dominance evolution arises as a consequence of the evolution of differential expression between alleles. (2) Genotype‐to‐phenotype map: dominance evolution arises as the consequence of non‐linear maps between developmental levels n and n  + 1. (3) Haldane's sieve: dominance evolution arises as a consequence of the higher fixation probability of dominant‐favourable alleles.

Altogether, in this review, we argue that recent advances in theoretical modelling of the evolution of gene expression, along with the uncovering of molecular mechanisms involved in dominance, now provide a more comprehensive, less‐polarized view of the evolutionary biology of genetic dominance.

II. NATURAL SELECTION AND THE DISTRIBUTION OF DOMINANCE COEFFICIENTS

(1) the distribution of dominance coefficients in natural populations.

Population genetics models generally describe dominance using a coefficient of dominance ( h ) modulating the effect of the selection coefficient s on the fitness of heterozygous individuals (1– hs ). Classically, they considered this parameter as a fixed quantity. Recently, models have provided predictions on the distribution of dominance coefficients expected within populations. For instance, by considering explicit adaptive landscapes assuming concavity near the fitness optimum and studying resident genotypes close to the optimum, deleterious mutations are expected to be recessive, while favourable mutations are generally expected to be dominant (Manna et al .,  2011 ). Such studies based on adaptive landscapes have received much attention for their role in adaptive walks and the distribution of fitness effects, specifically regarding how their dimensionality, level of epistasis and pleiotropy might affect evolution (Fragata et al .,  2019 ), but predictions on how these properties affect the evolution of dominance are still scarce. By explicitly considering the functions linking genotypes to phenotypes, and the functions linking phenotypes to fitness, Martin ( 2014 ) also pinpointed that the distribution of fitness effects of mutations vary depending on the levels of pleiotropy considered and on the essentiality of the genes (i.e. the level of lethality provoked by their deletion). Such variation in the fitness landscape can then strongly affect the distribution of dominance coefficients. The current scarcity of data on actual fitness landscapes (e.g. in their ruggedness) is currently limiting our ability to derive general predictions on the distribution of dominance coefficients.

The few published distributions of dominance coefficients have been mostly based on the characterization of laboratory mutants in a few model organisms such as Drosophila , Caenorhabditis and yeast, and point at partial recessivity as the mean of the distribution (as reviewed in Manna et al .,  2011 ). In a recent study, Huber et al . ( 2018 ) took an original approach and compared the site frequency spectra in natural plant populations between the outcrosser Arabidopsis lyrata and the predominant selfer Arabidopsis thaliana . They explored the link between the strength of selection (as estimated by the selection coefficient s ) and dominance (as estimated by the dominance coefficient h ). In outcrossers, most single nucleotide polymorphisms (SNPs) are in a heterozygous state, and their fitness will therefore depend on the product hs . In selfers however, heterozygotes are expected to be rare such that the frequency at which SNPs segregate should depend on the selection coefficient s only and be largely independent from dominance. Assuming that the distribution of fitness effects of new mutations is identical between the two species, the contrast between the two site frequency spectra can thus be used to reveal general features of the distribution of dominance coefficients h . They showed that, on average, segregating variants significantly depart from additivity: the mean of the distribution was h  = 0.46, i.e. most mutations were only slightly recessive. They also showed that assuming a negative correlation between h and s improved the fit to the data, confirming predictions from the fitness landscape models and observations from laboratory mutants that more deleterious alleles generally tend to be more recessive (Orr,  1991 ). Interestingly, they found that the rate of decay of the h – s relationship was significantly higher for catalytic genes (which only need to be expressed at low levels) than for any other category of genes (such as structural genes that need to be expressed at high levels). This suggests that the cost of gene expression may be an important determinant of dominance.

(2) Natural selection on the dominance of emerging alleles

A large body of theoretical literature has studied the extent to which dominance determines the fate of new alleles, and in return how selective processes have shaped the distribution of dominance coefficients in natural populations over the long term. Haldane ( 1927 ) noted that in contrast to the widespread recessivity of deleterious mutations, adaptive alleles are more likely to become fixed within populations when dominant. Indeed, when adaptive variations arise by mutation, they are initially at the heterozygous state with wild‐type alleles, so that the effect of positive selection depends on their expression in the heterozygous state. Hence, fully recessive adaptive variations need to occur in the homozygous state before they induce any beneficial phenotypic change, and therefore have a high risk of being lost by drift before ever being expressed. In the long run, this is expected to lead to a filtering on the dominance level of new adaptive variants (the so‐called Haldane's ‘sieve’) leading to a turnover of increasingly dominant alleles over the course of evolution. However, direct empirical evidence of Haldane's sieve has remained scarce because adaptive alleles usually replace their ancestral versions, preventing direct comparison of dominance between them. Three indirect lines of evidence nevertheless suggest that Haldane's sieve is indeed a potent force in natural populations.

The study of highly polymorphic loci, where several alleles jointly persist within populations over long evolutionary times (as expected, for example when balancing selection is acting) allow such a comparison and offer a first line of evidence. In these systems, the ancestral versus derived status of the different segregating alleles can be determined either by comparing allelic variations in sister species or, in cases of alleles that emerged from chromosomal inversions, by reconstructing the mutational scenario given synteny information from sister species. In many polymorphic loci, the derived alleles have been shown to be dominant over the ancestral ones (Llaurens, Whibley & Joron,  2017 ), consistent with the action of Haldane's sieve.

A second line of evidence for Haldane's sieve has been obtained by studying the migration of adaptive alleles through different populations. In the case of negative frequency‐dependent selection for instance, migration of dominant alleles across populations might indeed be more effective than that of recessive ones (Pannell, Dorken & Eppley,  2005 ), as theoretically predicted in the case of increased migration of dominant female‐sterility alleles in gynodioecious plant species (Pannell,  1997 ). The action of Haldane's sieve on the migration success of dominant alleles has been observed in transition zones between contrasting environments, where clines in the frequency of variants adapted to either environment are typically observed, as for instance between alleles controlling flower colours in connected populations of Anthirrhinum majus (Whibley et al .,  2006 ) or between alleles controlling mimetic coloration in Heliconius erato butterflies (Mallet & Barton,  1989 ). These clines are maintained by positive frequency‐dependent selection acting at the local scale, whereby locally abundant phenotypes benefit from increased attraction of pollinators or protection against predators in these respective examples. However, the position of the cline is predicted to vary among alleles depending on their dominance (Mallet,  1986 ). Dominant alleles, by being expressed more frequently, can gradually invade transition zones between populations, leading to a forward movement of the cline towards the population displaying phenotypes controlled by recessive alleles.

Finally, a third line of evidence stems from the emergence of adaptive variants through introgression from closely related species, allowing ancestral variants to be distinguished from their derived forms. For instance, melanism in the grey wolf Canis lupus has been shown to arise from a past hybridization event with domestic dogs (Anderson et al .,  2009 ). The introgressed melanic form is dominant over the ancestral coat colour and reaches high frequency in forest habitats, where it is under positive selection. After hybridization, genomic regions under positive selection might be more easily retained through generations of introgression in the accepting genome when they are dominant, in line with Haldane's hypothesis.

It is unclear how often adaptation proceeds from de novo mutations, migration or introgression, i.e. from alleles that initially arose at low frequency. Instead, it is possible that many adaptive variations emerge from standing genetic variation, i.e. were initially segregating neutrally and became advantageous only after a change in the ecological or genomic environment occurred. Orr & Betancourt ( 2001 ) showed that Haldane's sieve predictions no longer hold when adaptive variations emerge from standing genetic variation, because adaptive variants can then be promoted by natural selection even when recessive. Similarly, mechanisms favouring the rapid formation of homozygotes, for instance selfing reproductive mode, should favour the recruitment of recessive or partially recessive adaptive alleles, making Haldane's sieve a less‐prominent process. By studying quantitative trait loci (QTLs) associated with the domestication syndrome in crops, Ronfort & Glemin ( 2013 ) showed that adaptive alleles in selfing species are predominantly recessive or partially recessive, while in outcrossing species, adaptive variants are more frequently dominant. This highlights that the filtering of recessive adaptive variants might be counterbalanced when the reproduction regime favours the rapid formation of homozygotes (Hartfield, Bataillon & Glémin,  2017 ).

Overall, it is clear that dominance strongly affects the fate of genetic variants, from their initial emergence to their spread across spatially structured populations, modulating the effect of natural selection in a variety of ways.

(3) Evolution of dominance through modifiers

While the distribution of dominance can be shaped by direct selection on the mutants themselves, indirect selection through the evolution of dominance modifiers can also play a role. This question of the proximate and ultimate mechanisms underlying the emergence of variation in genetic dominance has been a subject of sustained interest in genetics, leading in particular to the development of a large theoretical literature since the first debate between R.A. Fisher and S. Wright. In brief, Fisher ( 1928 ) hypothesized that deleterious mutations are generally recessive because of modifiers of dominance. These genetic elements would be favoured by natural selection because they decrease the disadvantage of deleterious mutations in heterozygous individuals. This hypothesis has been extensively investigated since Fisher's seminal paper, which led to the development of a series of population genetics models collectively referred to as the ‘modifier theory’. The modifier theory relies on multilocus models to determine the conditions for the invasion of alleles at a given locus not selected by themselves (they are generally considered neutral) but because of their effects on other loci (Karlin & McGregor,  1974 ). Several modifier models have been analysed to study the evolution of dominance after the Fisher–Wright debate (Mayo & Burger,  1997 ). In particular, Feldman & Karlin ( 1971 ) and Bürger ( 1983 ) confirmed Wright's ( 1929 ) initial claim that the conditions for the invasion of modifiers of dominance are very restrictive: positive selection on such modifiers would be proportional to the frequency of heterozygous individuals in the population, i.e. of the same order as the frequency of deleterious mutations. Precisely because deleterious mutations are expected to be kept at low frequency by purifying selection, dominance modifiers would have little opportunity to play significant roles in natural populations. However, while the evolution of dominance modifiers was unlikely for deleterious mutations because they segregate at low frequencies, Wright ( 1929 ) nevertheless acknowledged that dominance modifiers could still evolve when heterozygotes are frequent.

Accordingly, the dominance modifier hypothesis was later investigated under various ecological and genetic mechanisms known to promote heterozygosity, at least transiently: in the case of a polymorphic trait such as in Batesian mimicry (Clarke & Sheppard,  1960 ; O'Donald & Barrett,  1973 ; Charlesworth & Charlesworth,  1975 ), during the sweep of a favourable mutant (Wagner & Bürger,  1985 ), in the case of a trait with heterozygote advantage or in an heterogeneous environment with dispersal (Otto & Bourguet,  1999 ), in genetic systems under frequency‐dependent selection because of competition for resources (Peischl & Burger,  2008 ; Peischl & Schneider,  2010 ), in sporophytic self‐incompatibility systems (Llaurens et al .,  2009 ; Schoen & Busch,  2009 ) or in the case of sexually antagonistic selection (Spencer & Priest,  2016 ). This series of models showed that in a wide variety of contexts, heterozygous genotypes can indeed be frequent enough for dominance modifiers to spread in a population. The fundamental mechanism promoting dominance evolution is the mitigation of the deleterious effects in heterozygotes in a specific genetic background, for instance in the emergence of sex‐specific dominance (Spencer & Priest,  2016 ). The strength of indirect selection on the dominance modifier depends on the fitness of the regulated alleles, as well as on the recombination rate with the locus under direct selection. The recombination rate sometimes dramatically changes the evolutionary outcome. For instance, only strongly linked dominance modifiers invade and go to fixation in sporophytic self‐incompatibility (Schoen & Busch,  2009 ), while in other cases recombination has no effect (e.g. in the case of sexually antagonistic selection; Spencer & Priest,  2016 ). The role of recombination can be hard to predict since it depends on the complex interaction between the frequency of the modifier in the population and the level of polymorphism at the locus at which balancing selection acts directly (Wagner & Bürger,  1985 ; Otto & Bourguet,  1999 ). Overall, dominance modifiers are expected to evolve in a large variety of genetic and ecological conditions, as long as heterozygotes are frequent, either transiently (in the case of recurrent selective sweeps) or at equilibrium (under various forms of balancing selection).

How often these genetic and ecological conditions are met in nature remains a matter of debate, but empirical case studies have provided evidence of selection acting on dominance, especially at polymorphic loci where several allelic variants are maintained within populations by balancing selection. In the polymorphic butterfly mimetic species Heliconius numata , Arias et al . ( 2016 ) showed that the predation risk was higher for intermediate heterozygotes as compared to mimetic homozygotes, highlighting the stringent selection against co‐dominance. Furthermore, when the polymorphic locus is controlling a trait composed of multiple developmental modules, natural selection may promote the coordination of dominance across the different features of the trait. In the butterfly Papilio dardanus for instance, several mimetic wing morphs also segregate, some of which can be shared across natural populations. Wing morphs show strong dominance when crosses are performed between individuals of the same populations: heterozygote phenotypes closely resemble the phenotype of one of the two parents. Strikingly however, crosses between the same wing morphs but between individuals collected from distant populations often result in mosaic phenotypes between parental morphs (Nijhout,  2003 ). This suggests that natural selection exerted on heterozygote phenotypes promotes the coordination of dominance throughout the whole wing colour pattern: such selection on dominance can occur only when alleles are in geographic contact and thus when heterozygous genotypes can be formed and submitted to natural selection. A similar result was found in the polymorphic mimetic species H. numata , where strict dominance between mimetic alleles is generally observed in sympatry, generating mimetic phenotypes in heterozygotes (Le Poul et al .,  2014 ). One notable exception was observed in some populations of this species, where one pair of mimetic alleles generates an intermediate phenotype in heterozygotes. This intermediate phenotype does not match the mimicry rings corresponding to the parental alleles, but is nevertheless selected positively because of its mimicry towards another mimetic community. This highlights that the fine‐tuning of dominance relationships between alleles of polymorphic loci depends on the associated fitness landscape.

Other cases of coordinated dominance are observed in loci controlling the male and female components of a trait. In Primula plants for instance, two genetically controlled morphs of flowers co‐exist: pin flowers with a short style and long anthers, and thrum flowers with conversely a long style but short anthers. The two morphs also differ in the size of the pollen grains they produce. For each of these three components of the complex trait (length of the style, position of anther and size of pollen grains), strong dominance is observed, each time in the same direction: the short‐style morph is dominant over the long‐style morph (Nowak et al .,  2015 ). A similar pattern is observed in the self‐incompatibility locus ( S ‐locus) of Brassicacea where a large number of genetically encoded recognition specificities stably segregate in natural populations. The S ‐locus contains the tightly linked genes SCR (encoding for a cysteine‐rich protein) and SRK (encoding for a receptor kinase). Male specificities are determined by SCR ligand proteins displayed in the pollen coat and female specificities are determined by SRK receptors in the pistil. The genes encoding the male and female recognition specificities segregate as a single genetic unit, but they are clearly distinct from one another. Yet, dominance between alleles of the SCR and SRK genes is coordinated, with an overall highly consistent pattern of dominance between pollen and pistil phenotypes and no case of opposite dominance between the two genes for any given pair of specificities (Llaurens et al .,  2008 ). The dominance relationships are however not fully identical between the two genes. While a strictly linear hierarchy is observed for the pollen phenotype, co‐dominance is often observed in the pistil phenotype (i.e. heterozygous genotypes jointly express their two S‐alleles), allowing rejection of a greater number of pollen phenotypes by the pistil. This difference may stem from the difference in reproductive investment between the male and female functions: seed production comes at a high metabolic cost, so avoiding the formation of inbred seeds is advantageous even in the face of rejection of a higher number of potential mates. By contrast, the male fitness through pollen increases with access to a greater number of mates, promoting the phenotypic expression of a single self‐incompatibility allele, i.e. strict dominance. Strikingly, pollen dominance seems to be the rule in such systems, with instances of co‐dominant expression being rare and restricted to pairs of alleles that are already high in the dominance hierarchy [S 13 S 20 in Arabidopsis halleri (Llaurens et al .,  2008 ) and class I S‐alleles in Brassica (Hatakeyama et al .,  1998 )]. More generally, sexually antagonistic selection acting within a locus has been shown to promote the evolution of dominance. For instance, in the Atlantic salmon Salmo salar , selection favours an earlier age of maturity in males than in females. The gene VGLL3 , an adiposity regulator, controls a large part of variations in size and age at maturity and is highly polymorphic within salmon populations. Sex‐dependent dominance is observed at this gene, whereby heterozygotes males have the precocious age at maturity, while heterozygous females display late maturity phenotypes (Barson et al .,  2015 ). This sex‐specific dominance may have evolved in response to the sexually antagonistic selection exerted on this trait.

Overall, recent theoretical frameworks have been developed that provide more detailed predictions for how the shape of fitness landscapes should translate into general patterns of dominance, but the distribution of dominance coefficients has been empirically documented in a systematic manner in only a very limited number of cases. Recently, theoretical and empirical studies have shown that natural selection can modify patterns of dominance, either through direct selection of allelic variants, or through indirect selection on dominance modifiers in a number of genetic and ecological situations. A central tenet of the predictions and observations above is that the mechanistic details of how dominance arises is of central importance to predict whether and how natural selection can act on it, either directly or indirectly. We thus review below the main features of the mechanistic models for dominance, and detail several empirical examples that demonstrate the variety of molecular mechanisms by which genetic dominance can arise.

III. DOMINANCE EMERGING FROM THE SHAPE OF GENOTYPE‐TO‐PHENOTYPE MAPS

(1) regulatory network models describing dominance as a by‐product of the genotype‐to‐phenotype map.

Wright ( 1934 ) was the first to argue that dominance is an inherent property of biological systems. He focused on enzymatic reactions, where biochemical fluxes typically show a saturating relationship with enzyme concentration. This saturating function is justified by biochemical reactions themselves, as fluxes are limited by the quantity of free enzymes. The biochemical theory of dominance then developed in many directions, sometimes with contradicting conclusions and interpretations. Kacser & Burns ( 1981 ) showed that biochemical reactions due to a sequence of enzymatic reactions should necessarily generate dominance because variations in the efficiency of one focal enzyme in the pathway only have a small effect on the general flux. Mutations decreasing enzymatic efficiency would then necessarily be recessive (assuming that the enzymatic flux generates a concave genotype‐to‐phenotype curve; Fig.  2 ). Kacser & Burns ( 1981 ) and followers (e.g. Orr,  1991 ; Keightley,  1996 ; Porteous,  1996 ) therefore concluded that dominance was due to metabolic properties, neglecting the effect of selective processes that could act directly on its evolution.

Non‐linear relationships between integration levels and consequences on dominance relationships. Here, additivity is observed at integration level n (for example RNA expression level or protein concentration), and depending on the genotype/phenotype map, different dominance levels can be observed at the integration level n  + 1. (1) Assuming a concave shape (blue line), Aa and AA genotypes display similar phenotypes, therefore A is dominant over a . (2) Assuming a convex shape (red line) Aa and aa genotypes display similar phenotypes, therefore a is dominant over A . (3) Assuming a sigmoid shape (purple line): if homozygotes are located on either side of the inflexion point (as shown on the plot), then the phenotype Aa will be intermediate (additivity or semi‐dominance). By contrast, if both homozygotes are located on the concave or convex part, then the allele A is dominant or recessive respectively, similarly to cases 1 and 2.

This view was challenged by several authors (e.g. Savageau, 1992 ; Omholt et al .,  2000 ; Gilchrist & Nijhout,  2001 ; Bagheri‐Chaichian et al .,  2003 ), who showed that the Kacser & Burns ( 1981 ) model, by focusing on enzymatic flux, does not necessarily apply to all regulatory networks (for an extended review, see Bagheri,  2006 ). For instance, Omholt et al . ( 2000 ) showed that biochemical reactions implying feedback loops and regulation in a three‐locus system can lead to a large variety of results, including additivity, dominance, recessivity and even overdominance. Gilchrist & Nijhout ( 2001 ) showed that a model of spatial diffusion of gene products can give rise to non‐linearity in the genotype‐to‐phenotype map, and therefore generates various dominance patterns. More recently, Veitia and collaborators (Veitia,  2003 ; Bost & Veitia,  2014 ; Bottani & Veitia,  2017 ) used biochemical reaction models to predict the level of expression of a gene as a function of the rate at which transcription factors bind to its promoters. Their model allows the possibility of multiple binding sites and assumes that the transcriptional response depends on the number of sites occupied. In these conditions, the genotype‐to‐phenotype map is a sigmoid curve, whose steepness depends on the number of binding sites. As a consequence, whether a mutant is recessive, dominant or has an additive effect depends on the arbitrary position of the homozygote on the map (see Fig.  2 ). Along the same lines, Porter, Johnson & Tulchinsky ( 2017 ) explicitly modelled biochemical reactions underlying the level of expression of a gene assuming two or three interacting genes in a sequential manner, with genes located upstream in the reaction chain regulating the expression of genes located downstream. In line with Omholt et al . ( 2000 ), they modelled gene promoters explicitly, so that alleles can vary in their coding region, in their promoters, or both, and the expression level of a gene depends on the binding affinity of its promoter with the protein produced by the gene located directly upstream in the cascade (effectively functioning as a transcription factor). In this model, gene products compete to bind to the promoters of the downstream genes, which introduces a non‐linearity between concentration of the transcription factors and expression of the genes they regulate. By explicitly modelling the effect of variations in either transcription factors, transcription factor promoter regions, or in the promoter region of the gene targeted by the transcription factor, Porter et al . ( 2017 ) predicted that the location at which mutations emerge should lead to contrasting levels of dominance. While variations in the transcription factor itself are likely to generate dominance because of competitive binding to their target sites, variations in the promoter regions of transcription factors are expected to be associated with simple changes in levels of allele‐specific expression, and should therefore more likely lead to additivity. The molecular nature of the mutations and their effects on the binding properties of the transcription factor are therefore important biophysical properties shaping dominance relationships.

Epistatic interactions among alleles at different genes thus play an important role in dominance variation. Nevertheless, the effect of natural selection on these changes is scarcely explored. Population genetics models evaluating the invasion propensity of mutants with given epistatic effects may allow this question to be tackled. Bagheri & Wagner ( 2004 ) analysed a population genetics model with an explicit genotype‐to‐phenotype‐to‐fitness map, where the evolution of dominance emerges from the selective pressure applied to an entire gene expression network. In their model, fitness was assumed to be proportional to the flux of a simple biochemical chain comprising two enzymes. The enzyme concentration and their catalytic activities can be modified by four different loci (two for each enzyme) at which mutation can occur. The modifier alleles affecting the concentration of enzymes can be interpreted as modifiers of gene expression. As expected, natural selection on the introduced mutations led to an increase of the biochemical flux, in particular through an increase in the concentration of enzymes. More importantly, they showed that dominance is expected to evolve as a side effect of the increase of the flux rate, because of the concave relationship between enzyme concentration and flux rate: as the flux rate increased, the heterozygotes became closer to the wild‐type homozygotes (Fig.  2 ). Even though it was not the primary aim of the authors, this is to our knowledge the first model explicitly to link selection on modifiers of gene expression and dominance evolution. Their result can likely be generalized to other cases as long as a concave relationship exists between genotype and phenotype: if a modifier making this relationship steeper invades and goes to fixation, a necessary side‐effect is the increase of dominance (Fig.  2 ). Further theory is clearly needed to explore in detail such models of dominance linking gene expression to fitness, but they are still crucially lacking from the literature (Bagheri & Wagner,  2004 ; Bürger et al .,  2008 ).

(2) Empirical evidence for non‐linear relationships between levels of gene expression and phenotypes

Overall, it is clear from theory that a comprehensive understanding of the patterns of dominance in a genetic system requires access to general properties of the genotype‐to‐phenotype map. Recent evidence shows that the genotype‐to‐phenotype map can indeed be non‐linear (Kemble, Nghe & Tenaillon,  2019 ). In the following section, we review a series of empirical studies illustrating the diversity of molecular processes by which non‐linearity can arise in genetic networks and along developmental processes, leading to different dominance relationships.

(a) Threshold effect in gene expression induces dominance

In the peppered moth Biston betularia , the melanic phenotype carbonaria was promoted by industrial activities darkening tree trunks (Haldane,  1956 ), and stems from a transposable element (TE) insertion in an intron of the ‘melanization’ gene cortex . The TE insertion is associated with an increase in cortex transcript numbers (van't Hof et al .,  2016 ), and the level of expression of cortex in heterozygote developing larvae is intermediate as compared to the respective homozygotes (i.e. the effect is additive at the transcriptional level). However, at the phenotypic level, a single copy of the TE is sufficient to trigger full expression of the melanic morph, illustrating that additive transcription of the causal genes can still result in strong phenotypic dominance. This example of a strongly selected variant has been used to illustrate Haldane's sieve effect, and demographic inference suggests that this variant arose from a single recent mutation that rapidly spread across natural populations (van't Hof et al .,  2016 ).

In another emblematic example of natural selection, lactase persistence allowing human adults to digest milk in pastoralist populations, cis ‐regulatory elements acting on the LCT gene encoding lactase have been described. Lactose tolerance evolved in parallel across pastoralist populations from Europe and Africa through distinct genetic variants causing different modifications of LCT expression. The different identified SNPs are associated with different levels of blood glucose after milk ingestion in homozygote and heterozygote individuals (Tishkoff et al .,  2007 ). However, intermediate efficiency in lactose digestion seems to be sufficient to permit full lactose tolerance. Dominance of lactose tolerance thus stems from the fact that intermediate levels of lactase confer full tolerance, pointing to a non‐linear relationship between lactase expression and the tolerance phenotype. Overall, similar threshold effects seem to be common.

(b) Non‐additive relationships between alleles: loss of function and protein heterodimers

Genetic variations leading to a loss of function at the biochemical level are often expected to be recessive because the presence of the alternative allele is generally sufficient for the gene function to be ensured at the organismal level. However, loss of function does not necessarily lead to recessivity, and a number of ‘dominant‐negative’ mutations have been described (Veitia, Caburet & Birchler,  2018 ). For instance, mutations in the PSEN1 gene are associated with elevated risk of Alzheimer's disease in humans. Loss‐of‐function mutations of this gene are dominant because they are able to suppress the wild‐type γ‐secretase activity (Heilig et al .,  2013 ). Zhou et al . ( 2017 ) further revealed that this loss of function is due to hetero‐oligomerization of the mutant with the wild‐type protein, forming a non‐functional duplex.

The formation of multimeric proteins is indeed frequently associated with non‐additive dominance relationships, and the effect of various protein interactions such as dimerization, often has been described to lead to ‘dominant‐negative’ mutations (Herskowitz,  1987 ). If the subunits of a dimeric protein are produced by two highly differentiated alleles in a heterozygote, then the phenotype will depend on the biochemical activity of the heterodimer composed of different subunits. Assuming equal protein dosage, the heterodimer is expected to represent half of the population of proteins in the cell, so any deviation from additivity in the activity of the heterodimer would result in dominance. For instance, a non‐functional protein might suffice to render the heterodimer non‐functional by ‘poisoning’ the complex, so that 75% of the population of dimers would be non‐functional (Veitia,  2006 ).

This situation may also apply to binding specificity of the polymorphic positive regulatory domain zinc finger protein 9 ( PRDM9 ) locus of mice, controlling the localization of recombination hotspots in mammals by binding to specific DNA sites, whose identity differs among PRDM9 alleles. Importantly, the PRDM9 proteins form dimers, and heterozygous individuals are expected to form 50% heterodimers, whose binding activity may be dominated by one of the two alleles. The majority of PRDM9 proteins (~75%) will thus be targeted towards DNA sites of the dominant allele, resulting in limited recombination on the sites targeted by the recessive PRDM9 allele. Moreover, protein variants compete for the recombination machinery, further altering the distribution of recombination hotspots in heterozygotes (Baker et al .,  2015 ).

A similar situation may occur at the SRK gene, controlling female self‐incompatibility specificity in plants of the Brassicacea family. Most individuals at this gene are heterozygotes, and in many cases both alleles are expressed at the phenotypic level. In a number of cases, however, heterozygotes express only one of their two alleles at the phenotypic level, resulting in dominance interactions (Llaurens et al .,  2008 ). This dominance phenomenon does not seem to involve transcriptional silencing for the female phenotype, since transcripts of both SRK alleles are typically detected in heterozygote genotypes (Kusaba et al .,  2002 ; Burghgraeve et al .,  2020 ). Interestingly, the SRK proteins form dimers, and Naithani et al . ( 2007 ) used yeast assays to suggest that affinity towards their cognate pollen proteins may depend on the binding properties of heterodimers. Direct measurement of the binding affinity of the heterodimers in a large number of combinations will now be necessary to evaluate the generality of this mechanism.

A final example has been described in disease‐resistance genes in wheat ( Triticum aestivum ), where the combination of multiple alleles of the nucleotide‐binding leucine‐rich repeat (LRR) resistance protein (Pm3) in the same plant by genetic transformation can result in suppression of Pm3‐based resistance. The suppression effect is post‐transcriptional, as both transcript and protein levels remain unchanged, and involve the LRR domain of the protein (Stirnweis et al .,  2014 ). This case provides evidence that alleles of a single gene (and even partial copies of them) can block the activity of other alleles of the same gene, providing molecular evidence for interallelic interactions that can underlie the mechanism of dominance. The generality of such post‐transcriptional mechanisms remains to be determined.

(c) Duplications leading to variations in dominance

Because of complex interactions between alleles or copies of the same genes, gene duplications may interfere with dominance relationships. Haldane ( 1933 ) and Fisher ( 1935 ) already proposed that gene duplications could act as dominance modifiers: fixation of a duplicated copy of the wild‐type allele of a dosage‐sensitive gene would buffer the difference between homozygous and heterozygous genotypes at the original copy. Accordingly, haplo‐insufficient genes have on average more paralogs that haplo‐sufficient genes, suggesting that gene dosage may be an important factor for the initial fixation of duplicated copies. The evolution of gene duplication might thus be linked to the dominance relationships at the duplicated loci. In turn, dominance at one locus might be tuned by the presence of additional gene copies. Similar processes are believed to play a crucial role in the retention of gene duplicates over time (Kaltenegger & Ober,  2015 ; Diss et al .,  2017 ). Our understanding of the evolution of dominance may thus benefit from the sharing of a common theoretical framework with that of genes with multiple copies (Kondrashov & Koonin,  2004 ).

A striking example of the link between selection on dominance and on duplication is illustrated by the insecticide‐resistance gene ace‐1 in mosquitoes, where heterozygotes carrying a susceptible ( S ) and a resistant ( R ) allele benefit from increased fitness. RR homozygotes have increased survival in treated areas, but also suffer from several fitness costs (e.g. decreased fertility and reproductive success) as compared to SS homozygotes. As the result of this trade‐off, RS heterozygotes have the greatest overall fitness. This overdominant selection has favoured the emergence of a duplicated copy of the gene, leading to an haplotype D combining the susceptible and resistant alleles (Milesi et al .,  2017 ). Over the long term, favourable combinations of alleles can thus become fixed on a single haplotype, avoiding the constant reshuffling implied by their initial allelic status.

(d) Non‐linear protein activity

Different mutations targeting the same gene may induce similar phenotypes, but with different dominance levels, depending on how the encoded protein is affected. For instance, independently derived mutations in the coding sequence of the gene MC1R involved in skin colour variation have emerged in three lizard species ( Sceloporus undulatus , Aspidoscelis inornata and Holbrookia maculata ), all leading to an adaptive light phenotype in a sandy environment (Rosenblum et al .,  2010 ). The parallel mutations in S. undulatus and A. inordata led to a reduction of pigmentation by distinct mechanisms (reduction in melanocyte membrane integration efficacy versus disruption of receptor signalling, respectively), with opposite consequences on dominance (the light allele is dominant in S. undulatus but is recessive in A. inordata ). This shows that inferring dominance mechanisms from genetic variation alone is not straightforward, even in genes for which genetic pathways towards the phenotype have been well studied, and requires specific functional studies.

In some cases, different alleles of a given gene may activate distinct downstream pathways, so predicting dominance in heterozygotes requires a detailed understanding of these pathways. Some of the best‐studied examples of this phenomenon are the human leukocyte antigen (HLA) alleles in humans, which are commonly associated with elevated risks of developing autoimmune diseases. Some HLA alleles are known to overshoot non‐self targets and recognize self‐epitopes, therefore triggering an inappropriate inflammatory response. In some cases, this risk can be masked in the heterozygote state. For instance, carriers of the HLA‐DR15 allele are at greater risk of developing Goodpasture Disease, and Ooi et al . ( 2017 ) showed that the risk of developing an autoimmune response in HLA‐DR15/HLA‐DR1 heterozygotes is as low as that of HLA‐DR1 homozygotes, suggesting a dominant protective effect of HLA‐DR1 . The HLA‐DR15 and HLA‐DR1 alleles exhibit distinct peptide repertoires and binding preferences, and they present the collagen epitope in different ways. This leads to the recruitment of a different population of T‐cells (‘conventional’ versus ‘regulatory’ T‐cells). In heterozygotes, the regulatory T‐cells recruited by HLA‐DR1 seem to be sufficient to prevent the inflammatory response by the conventional T‐cells recruited by HLA‐DR15, hence providing a molecular explanation for the dominant protective phenotype of HLA‐DR1 over the HLA‐DR15 allele. While important molecular details of this phenomenon and the broader related phenomenon of ‘immunodominance’ (Akram & Inman,  2012 ) remain to be determined, this example provides a compelling illustration that even equal production of different proteins in heterozygotes can lead to dominance at the phenotypic level, because of complex interactions with the downstream pathways.

(e) A significant role of protein interactions in dominance

Overall, various interactions involving the expressed proteins can lead to dominance relationships at the organismal level. Dosage sensitivity in different functional categories of genes may explain their contrasting levels of dominance/recessivity, with enzymes more commonly found among haplo‐sufficient genes, whereas transcription factors are more common among haplo‐insufficient genes (Jimenez‐Sanchez, Childs & Valle,  2001 ). Accordingly, multiprotein complexes tend to be encoded by haplo‐insufficient genes, which are generally sensitive to dosage (Papp, Pal & Hurst,  2003 ). The type of protein therefore strongly impacts the non‐linearity between levels of protein concentrations and heterozygote phenotype, and therefore the dominance relationships between alleles. The effect of natural selection on dominance might thus differ depending on the molecular properties of the targeted proteins.

IV. DOMINANCE EVOLUTION TRIGGERED BY CHANGES IN ALLELE‐SPECIFIC EXPRESSION

Accumulating theoretical and empirical evidence is now showing that dominance can arise from the effect of modifiers that cause changes in the relative abundance of the gene products of the two alleles in a diploid genotype.

(1) Specificity and linkage of the modifiers towards the targeted gene matter

In many metabolic models (Kacser & Burns,  1981 ; Gilchrist & Nijhout,  2001 ; Bagheri‐Chaichian et al .,  2003 ; Veitia,  2003 ; Bagheri & Wagner,  2004 ; Bottani & Veitia,  2017 ), biochemical reactions occurring in heterozygotes are hypothesized to rely on the simple additive effect of the two alleles, such that their genotypic values (or enzyme/transcription factor concentration) are intermediate between the two respective homozygotes. Empirical data on allele‐specific expression (see Section  IV .2), suggest that this assumption does not apply in a number of situations. Models by Omholt et al . ( 2000 ), Gjuvsland et al . ( 2007 ) and Porter et al . ( 2017 ) are the only ones where biochemical reactions are derived separately for homozygote and heterozygote genotypes. Among these, only Porter et al . ( 2017 ) considers the possibility of allele‐specific interactions. Metabolic models also generally assume that interaction rates between molecules are the product of their concentrations. However, genes are in limited number of copies within each cell and competition between transcription factors for binding sites may be high, such that stochastic effects can considerably impact allelic dominance at the cellular level [see the case of olfactory receptor (OR) genes in Section  IV .2].

In models of variation in gene expression triggered by modifiers (Bagheri & Wagner,  2004 ; Nuismer & Otto,  2005 ), specific relationships between modifiers and alleles of the regulated gene often have been ignored, so that the modifier affects gene expression of the two heterozygous alleles indiscriminately. More recently, the evolution of dominance has been studied in models where the mode of action of modifiers of gene expression was explicit. Fyon, Cailleau & Lenormand ( 2015 ) investigated the invasion probability of modifier alleles affecting the expression of a partially linked locus, assuming cis interactions. These expression modifiers change the affinity with the transcription factors and therefore modify the level of expression of the gene copy linked in cis . The partially linked regulated locus is subject to purifying selection, and is hit by recurrent deleterious mutations. In addition, a concave relationship between gene expression and fitness is assumed. The promoter regions compete for access to the transcription factors, so that modifiers increasing the expression of one allele automatically decrease expression of the alternative allele. They showed that modifier alleles increasing gene expression are favoured when they are tightly linked to the regulated locus under purifying selection. This outcome results from the balance between two consequences of expressing deleterious alleles. A modifier increasing the expression of a deleterious allele is expected to be lost more often because it has a greater chance of being hitch‐hiked during the purging process. However, because purging is more efficient when deleterious mutations are more expressed, the modifier allele can also be found more often in association with a purged background than with a deleterious allele. The model therefore predicts that expression enhancers tightly linked to regulated genes are recurrently promoted by a runaway process, in which binding affinity of the promotor regions of the regulated gene steadily increases over the course of evolution. Nevertheless, the dominance relationship between a pair of alleles remains identical after the fixation of the expression modifier since the modifier becomes associated with the different alleles at the gene. The overall level of gene expression in the heterozygotes thus remains the same as in the ancestral state, because both alleles attract equivalent amounts of the transcription factor. However, as soon as regulatory genetic regions are polymorphic, one allele at the selected locus can be expressed more than the other, translating into different fitnesses in the heterozygotes.

In addition to this general model of evolution of gene expression modifying the fitness of heterozygotes, other models have studied the effect of modifiers of expression on specific traits. For instance, Llaurens, Joron & Billiard ( 2015 ) investigated the evolution of expression at the locus controlling mimetic colour patterns in species where individuals have chemical defences deterring predators. In this case, the colour pattern polymorphism is maintained in structured populations connected by dispersal because predators exert different selective pressures among demes. The different colour patterns are assumed to be controlled by a single bi‐allelic locus, where homozygotes have optimal colour patterns. Fitness depends on the distance to the optimal colour pattern in a given deme. As a consequence, when heterozygotes show intermediate colour patterns, they are counter‐selected across the whole population. Llaurens et al . ( 2015 ) looked at the fate of a modifier of gene expression, partially linked to the colour pattern locus, with different modes of action: the modifier can repress or enhance expression, in cis ‐, in trans ‐ or through both types of interactions, and it can show allelic specificity or not (i.e. the modifier can modify only one allele of colour pattern, or it can affect both). They showed that the evolutionary fate of the modifier depends strongly on its mode of action. In particular, only a modifier enhancing gene expression with an allele‐specific mode of action can invade the population, with no significant effect of the recombination rate. This model thus predicts that if dominance indeed evolved because of selection on colour pattern gene expression, then it should be due to allele‐specific enhancing modifiers.

Overall, these models by Fyon et al . ( 2015 ) and Llaurens et al . ( 2015 ) suggest that the way genetic elements control gene expression can affect the outcome of the evolutionary process. It should be noted that these models have explored only a small fraction of all possible ecological and genetic contexts, and the analysis of other models will now be necessary in order to reach a broader understanding of the link between gene expression and dominance evolution. We believe that this research avenue holds great potential to provide novel testable predictions to guide experiments and empirical discoveries.

(2) Biased allele‐specific expression can be caused by trans ‐acting factors

The models presented above make specific assumptions about the way allelic expressions are controlled by molecular interactions that take place either between alleles, or within or among genes (Fig.  3 ). But how common are these phenomena, and what do we know about their molecular mechanisms? High‐throughput transcriptomic approaches are now allowing the routine comparison of allele‐specific transcript levels [‘the allelome’; e.g. Lappalainen et al . ( 2013 ) in humans; Crowley et al . ( 2015 ) in the mouse], and have revealed that differences in transcript levels of the two alleles in a diploid genotype are common. Identifying the general molecular mechanisms by which this bias is determined and established remains an active area of research (Gaur et al .,  2013 ). Below, we review a series of genetic phenomena in which genetic interactions result from trans ‐acting factors acting at the same locus, but interallelically (cases ④ and ⑤ in Fig.  3 ). The genetics literature does not have a specific name for this kind of interactions, but they are essential to understanding how the rewiring of gene regulatory networks can result in a dominance phenotype upon which natural selection can act.

Interactions between genes (A and B) and alleles (1 and 2) in heterozygotes in a diploid individual. Two genes (A and B) located on the same chromosome pair with their respective regulatory regions (crossed boxes) are exhibited here, all at a heterozygous state (haplotypes 1 and 2). Arrows describe the influence of a genomic region on another one: cis ‐ and trans ‐ acting factors are shown, acting either between genes (dotted lines) or between alleles (solid lines). Enclosed numbers involve different molecular mechanisms described in the text. Depending on the different enhancing and repressing effects and specificity of the target, various departures from additivity in expression levels can be observed in heterozygotes.

Some of these phenomena have been revealed most clearly in interspecific hybrids. An example of interallelic interaction in trans‐ is the Sc locus of rice. This gene occurs as a single copy in Oryza japonica (the Sc‐j allele), while O. indica contains three duplicated copies of Sc (the Sc‐i allele). In Sc‐i/Sc‐j hybrids, high expression of Sc‐i results in suppression of the Sc‐j allele in the (diploid) sporophytic cells. This suppression is then stably maintained throughout development, leading to selective abortion of the (haploid) pollen grains carrying the Sc‐j allele, due to the lack of expression of this essential gene. This results in drastic segregation distortion (Shen et al .,  2017 ). Overall, the Sc‐i / Sc‐j heterozygotes thus express only Sc proteins from their Sc‐i allele, which amounts to full dominance of Sc‐i alleles caused by a within‐gene trans‐ acting repressive effect (case ⑤ in Fig.  3 ).

Another example is nucleolar dominance, a widespread phenomenon initially documented in Crepis hybrids (Navashin,  1934 ). In diploids, individuals inherit ribosomal RNA (rRNA) genes from their two parents, but in a number of hybrids, only one of the two sets of rRNA genes is actively expressed. In Arabidopsis , nucleolar dominance is controlled by reversible, chromatin‐mediated alterations in gene expression (Pontes et al .,  2003 ). Preuss et al . ( 2008 ) suggest that small non‐coding RNAs may be the agents of the trans‐ modification of expression observed in the interspecific hybrid Arabidopis suecica (cases ④, ⑤, ⑥ or ⑦ in Fig.  3 ). However, Mohannath, Pontvianne & Pikaard ( 2016 ) suggested instead a role of chromosome positional effects in Arabidopis (see also Durica & Krider,  1978 ). In fact, two clusters of rRNA genes exist in A. thaliana , with typically only one being transcribed, and there is variation among A. thaliana accessions in the identity of the cluster that is expressed. Nucleolar dominance was shown to evolve rapidly, with only two generations being necessary for one of the two sets of rRNA genes to become silenced in synthetic A. suecica lines newly obtained by experimental crosses in the laboratory (Chen, Comai & Pikaard,  1998 ). However, Joly et al . ( 2004 ) did not observed such rapid evolution of nucleolar dominance in Glycine hybrids. Overall, it is clear that nucleolar dominance arises through both allelic (in cis , case ② or ③ in Fig.  3 ) and epistatic interactions (in trans , cases ④, ⑤, ⑥ or ⑦ in Fig.  3 ) (Rabanal et al .,  2017 ).

These biases in relative expression of alleles observed in hybrids reflect the evolution of regulatory networks that diverged among species or lineages. However, direct selection on dominance in hybrids is unlikely to be common in the wild, where hybrids are typically rare. Selection on dominance is more likely to take place within species, where heterozygotes can be formed more frequently. In fact, similar interallelic interactions have also been observed within species. In Arabidopsis , comparison of the methylome of F1 hybrids obtained by crossing different natural accessions with that of their parental lines revealed a large number of non‐additive methylation patterns brought about by the processes of trans‐ chromosomal methylation and demethylation, whereby the methylation state of one allele is altered to resemble that of the other allele. The majority of these loci are associated with transposable element‐derived sequences, and a proportion of these alterations of the cytosine methylation state are stably inherited to the F2 generation (Greaves et al .,  2014 ). In the majority of cases, these altered methylation states are associated with the production of 24‐nucleotide small interfering RNA (siRNA) molecules and involve the RNA‐directed DNA methylation pathway (Greaves et al .,  2016 ). In most cases, the ‘methylated’ state seems to be dominant over the ‘unmethylated’ state. This phenomenon represents hundreds of loci across the genome, although it remains unclear how often they lead to transcriptional changes in the nearby genes. In these situations, it is clear that the epigenetic state of an allele on one chromosome is not independent from the presence and epigenetic state of the allele on the other chromosome. These paramutation‐like phenomena remain incompletely understood, but are also common for natural epi‐alleles in the tomato Solanum lycopersicum (Gouil & Baulcombe,  2018 ) and also have been found in mice (Herman et al .,  2003 ), suggesting that they could play a role in some instances of dominance modification in plants and animals.

Transvection, or trans‐ sensing effects (Henikoff & Comai,  1998 ), has been observed in plants, fungi and mammals, and is another genetic phenomenon that can lead to trans‐ activation or trans‐ inhibition of alleles. In some cases, transvection seems to require direct physical interaction or at least proximity between the homologous chromosomes carrying the two alleles, and occurs when the two homologous chromosomes are paired during meiosis. A possible mechanism in this case is transcriptional activation of the promotor of a gene from one chromatid through recruitment of the transcriptional machinery by enhancers from the other homologous chromatid (hence implying that enhancers are acting not exclusively in cis ‐; see Liu et al .,  2008 ). Data in human, mouse and Drosophila seem to be consistent with this first model (but see Rodriguez et al .,  2017 ). For instance, expression of the human cyclin D1 gene is controlled by trans‐ allelic regulatory effects, whereby the presence of a translocated copy of the gene can alter the methylation and expression status of the other allele. The two interacting allelic sequences may be tethered at a peripheral region of the nucleolus, where transcriptional activity is low. The prevalence of transvection remains unclear, but insertion of a number of transgenes in different genomic locations in Drosophila revealed that about 30–52% of gene regulatory sequences may be associated with transvection (Mellert & Truman, 2012 ). Given the phylogenetic span over which epigenetic trans‐ repression was identified (including mammals, insects and plants), this may be an underappreciated phenomenon of gene regulation, with a substantial impact for our understanding of dominance.

Another fascinating example of trans ‐ interaction is provided by olfactory receptor (OR) genes in mammals. Monoallelic expression of OR genes is observed at the level of an entire gene family, enabling high olfactive discrimination by individual cells. In the mouse for instance, coordinated expression of the 1,075 OR intact paralogs, each binding to specific chemical compounds, contributes to a broad spectrum of smell perception. These genes are expressed in neuronal cells of the main olfactory epithelium located in the nose in a monogenic and monoallelic fashion, whereby individual neuronal cells express a single allele of a single OR gene (Monahan & Lomvardas,  2015 ), therefore involving regulation both within (cases ④ and ⑤ in Fig.  3 ) and among (cases ②, ③, ⑥ and ⑦ in Fig.  3 ) a large repertoire of OR genes. This is accomplished at the single‐cell level by a complex process starting with collective epigenetic silencing of all OR genes of the genome by repressive histone marks such as H3K9me3 and H4K20me3. Expression of the lysine‐specific demethylase 1 protein (LSD1) results in slow local demethylation, de‐condensing the chromatin structure, progressively making a set of specific sequence motifs in the promotor available for transcription factor binding and contact with long‐range DNA enhancers (Lyons et al .,  2013 ). The first randomly targeted OR sequence then becomes associated with the H3K4me3 histone mark typical of transcriptional initiation and ultimately becomes transcriptionally active. The produced OR protein then initiates a negative feedback loop that results in LSD1 downregulation, ensuring that other OR sequences can no longer become demethylated and access the active state. Ultimately, the silenced sequences become sequestered in repressive nuclear foci from which the long‐distance DNA enhancers are apparently excluded, further preventing transcription. Overall, the targeting of the expressed OR sequence corresponds to a ‘winner‐takes‐all’ process, whereby the first OR sequence to be transcribed and translated at a sufficient level induces inhibition of the other OR sequences in the genotype, a phenomenon known as ‘allelic exclusion’. The inhibition process not only concerns the other OR genes in the cluster but also the other allele of the chosen OR gene, hence in a ‘ cis–trans ’ manner akin to that defined herein. Any factor modifying the probability of expression of one given allele (and silencing of the others) would directly affect the dominance relationship in heterozygote individuals. At this point, however, how LSD1 is targeted to the chosen OR and whether LSD1 has different affinity for the different OR genes or different allelic variants and thus affects their propensity to become the chosen sequence is not known. Strikingly, inactivating components of the silencing feedback loop results in strong and reproducible bias towards expression of a small number of ORs (Lyons et al .,  2014 ), suggesting a co‐evolutionary process between the strength of the promotor of the OR genes and the strength of the silencing phenomenon, resulting in approximately equal opportunity for all OR genes to be expressed in wild‐type genotypes. Hence, the ability to activate the silencing feedback loop rather than the ability to become demethylated in the first place might be the key to biasing transcriptional activity in heterozygotes. The evolutionary significance of such phenomena remains to be investigated, but they are clearly of interest in the context of how dominance could be controlled at the transcriptional level.

Several of the best‐documented cases of monoallelic expression involve self‐recognition genes. An early example concerns the gene encoding Toll‐like receptor 4 in the mouse, a situation that resembles allelic exclusion (Pereira et al .,  2003 ). Mammalian T‐cells also show a deterministic developmental switch from monoallelic to biallelic transcription of the Gata3 (a transcription factor specifically recognizing GATA sequences) in about half of the developing T‐cell progenitors. This mono‐allelic versus bi‐allelic transcription is stably established in a parent‐of‐origin‐independent manner and the identity of the transcribed allele does not correlate with the classical repressive H3K4me3 and activating H3K27me3 epigenetic marks (Ku et al .,  2015 ). How the switch is controlled (how the gene on the silent chromosome transitions from a silent to an expressed state) is currently under study but is clearly of interest to understand dominance between alleles: any factor modifying the expression threshold would directly affect the dominance relationship in heterozygote individuals.

Finally, a recently uncovered mechanism of mono‐allelic expression stems from dominance at the gene controlling self‐incompatibility in pollen of Brassicaceae plants ( SCR ), in which the interallelic interaction is mediated by small RNAs (sRNAs) produced by the dominant SCR allele targeting specific sequence motifs in the promotor of the recessive allele, leading to transcriptional repression possibly through DNA methylation (Tarutani et al .,  2010 ; Durand et al ., 2014 ). Specific interactions between the regulatory regions of the different SCR alleles therefore generate a dominance hierarchy between alleles (case ④ in Fig.  3 ), preventing expression of co‐dominant phenotypes that would be characterized by limited mating success. The evolution of these sRNAs seems complex with several emergences and losses (Durand et al .,  2014 ), revealing a high potential for dominance evolution at the SCR gene, probably driven by strong positive selection on dominance at the SCR gene enhancing pollen mating success (Llaurens et al .,  2009 ; Schoen & Busch,  2009 ).

Overall, it is clear that a variety of mechanisms can cause allele‐specific expression, as described in Fig.  3 , such that transcription of the two alleles in heterozygote individuals does not need to be purely additive, as is assumed in many models of gene expression. Instead, the two alleles in a diploid genotype can interact to various degrees, potentially leading to differences in transcript levels ranging from subtle allele‐specific biases to massive mono‐allelic expression. In other words, there is a number of ways by which the relative doses of the two alleles in a heterozygote can depart from additivity, and these variations can result in dominance at the phenotypic level. While the molecular mechanisms underlying these phenomena are still not fully characterized, recent studies have highlighted the role of small RNAs and DNA methylation on modifications of allele expression, although the importance of such mechanisms at the genome scale is still unclear.

V. DISCUSSION

(1) dominance as an evolved property of the genotype‐to‐phenotype map.

The importance of dominance on the evolutionary fate of alleles, as well as the possibility for dominance to evolve have been highlighted by population geneticists since the early 20th century. The molecular mechanisms involved in dominance relationships and their modifications have been investigated by functional geneticists, sometimes using a drastically different vocabulary (Table  1 ). Results obtained in these research fields have rarely been combined, preventing drawing general conclusions on the evolutionary significance of dominance interactions between alleles. Dominance can be viewed as a specific genetic interaction occurring between alleles within a locus, and is as such sometimes covered by general models of gene expression based on molecular networks. The important theoretical literature on the evolution of gene regulatory networks might thus complement population genetics models focusing on dominance modifiers. Models of evolution of regulatory networks usually assumed an interaction graph between genes so that mutants at a given locus can affect the expression of another gene, depending on the property of the graph.

Dominance has been considered as one of the simplest form of robustness, i.e. as an evolved property of the genotype‐to‐phenotype map, buffering the effect of deleterious mutations during development (Bagheri & Wagner,  2004 ; Bagheri,  2006 ). This idea has assumed several names following the debate between Wright, Haldane and Fisher about the evolution of dominance. Haldane ( 1930 ) introduced the concept of ‘factor of safety’ and considered the plateau of the enzymatic metabolic pathway as a ‘buffer’ to mutations. In the literature dealing with the evolution of genotype‐to‐phenotype maps, for instance of genes implied in development (Wagner, Booth & Bagheri‐Chaichian,  1997 ; Rice,  2002 ; Siegal & Bergman,  2002 ), dominance was rather referred to as a form of ‘canalization’ (Rendel,  1967 ) or ‘robustness’ (Bagheri & Wagner,  2004 ). Strikingly, however, with the exception of Bagheri & Wagner ( 2004 ), most of the recent studies interested in the evolution of canalization or robustness ignored dominance as a possible evolved by‐product. Indeed, most models either considered haploid individuals or additive effects of alleles at a given locus (e.g. Siegal & Bergman,  2002 ; Rünneburger & Le Rouzic, 2016 ). Most importantly, in such models, phenotype and fitness landscapes do not emerge from an explicit mechanistic or physiological model, which make them arbitrary to some extent, and precludes the evolution of dominance. As acknowledged by Bagheri ( 2006 ), synthesizing the physiological and evolutionary mechanisms underlying dominance is a long‐standing issue. Such a synthesis has not been achieved yet, possibly because of the still limited cross‐talk between functional biologists and evolutionary biologists (Plutynski,  2008 ; Billiard & Castric,  2011 ).

The relative independence between the modifier theory for the evolution of dominance or gene expression on the one hand, and the evolution of regulatory networks, canalization and robustness on the other hand is surprising for several reasons. First, because population genetics models dealing with the evolution of gene regulatory networks are essentially modifier models, with several interacting modifier loci. This is the approach taken by Wagner et al . ( 1997 ), Bagheri & Wagner ( 2004 ) and Fyon et al . ( 2015 ). Second, because G.P. Wagner was originally involved in both the development of the modifier theory, in particular to study the evolution of dominance (Wagner,  1981 ; Wagner & Bürger,  1985 ; Bagheri & Wagner,  2004 ) and of the theory dealing with the evolution of canalization and robustness (Wagner et al .,  1997 ). Actually, Wagner explicitly links the evolution of dominance with the evolution of robustness (Bagheri & Wagner,  2004 ). Finally, one can see no fundamental reason why the evolutionary and population genetics of robustness, canalization, dominance and gene expression should follow separate paths. Specific models assuming competition of alleles for the transcription machinery or cooperation between alleles enhancing attraction of transcription factors could shed light on dominance at the expression level. We thus believe that a synthesis between the two theoretical frameworks is crucially needed, for instance to clarify the link between the concepts of robustness and dominance. This synthesis would aim at bringing a general answer to the question ‘how is dominance changed by the fixation of a mutant at any locus in the gene network, via the evolution of the genotype‐to‐phenotype map?’. This would require determining the conditions under which mutants affecting dominance (among other things) can invade a population in ( i ) a given gene network, ( ii ) a physiological model linking genotype to phenotype, and ( iii ) an ecological model linking phenotype to fitness (Fig.  1 ). Such a formulation would embrace all models dealing with the evolution of dominance, and would help clarify the relationship between dominance and epistasis.

(2) From models to empirical data: a necessary cross‐talk between evolutionary and functional biology

The limited cross‐talk between evolutionary and functional biology has prevented questioning the relevance of the assumptions used in the models of dominance evolution via modifiers. Three categories of assumptions can be questioned (Fig.  1 ): ( i ) how likely is the genetic architecture assumed to underlie the dominance modification; ( ii ) is the effect of the dominance modifier on the phenotype ecologically relevant; and ( iii ) how does the phenotype translate into fitness variation? Following these criteria, three main classes of model can be identified. ( i ) Most modifier models assume that dominance modifiers directly modify the fitness of the heterozygotes, without providing any hypothesis on the molecular mechanisms involved (e.g. Fisher,  1928 ; Otto & Bourguet,  1999 ; see Section  I ). These models are conceptual and only prove that dominance can evolve, but are difficult to apply to actually documented genes. ( ii ) Some models explicitly consider a gene network (e.g. Omholt et al .,  2000 ; Gilchrist & Nijhout,  2001 ) but they do not define a fitness function, and as such, they cannot be used to analyse the invasion of dominance modifiers. Hence, while their assumptions about the processes underlying the genotype‐to‐phenotype map can be verified in experiments, their predictions for the evolution of dominance in natural populations are not straightforward. The model by Bagheri & Wagner ( 2004 ) makes specific assumptions about the links between the gene network, the phenotype and the effect of a modifier, but this model also assumes a linear relationship between the enzymatic flux and fitness. Under which ecological context this assumption is relevant remains undefined. ( iii ) Finally, models combining relatively precise genetic mechanisms for dominance, the effect of a modifier on the phenotype and on fitness have been developed more recently (e.g. Nuismer & Otto,  2005 ; Llaurens et al .,  2009 , 2015 ; Schoen & Busch,  2009 ; Fyon et al .,  2015 ). In these models, the genotype–phenotype‐fitness map is clearly defined in a specific ecological context. Their specific predictions allow experimental validations of the evolution of dominance. The predictions of models on the evolution of dominance in the self‐incompatibility system in flowering plants (Llaurens et al .,  2009 ; Schoen & Busch,  2009 ) have already partially been verified: the modifier genes have been detected (for a review see Billiard & Castric,  2011 ), their effects on the genotype‐to‐phenotype map and their tight association with the self‐incompatibility locus validated (Durand et al .,  2014 ), as well as the asymmetry between female and male phenotypes (Llaurens et al .,  2008 ). Hence, we hope that our review will stimulate the development of precise models about the evolution of dominance in ecologically relevant contexts, providing predictions that could be experimentally tested.

(3) Documenting the molecular basis of dominance to shed light on its evolution

In this review, we noted that dominance modification through different pathways (involving cis or trans mechanisms for instance, see Fig.  3 ) lead to different evolutionary outcomes, so that population genetics models investigating the evolution of dominance or of regulatory networks may be more relevant when considering known molecular mechanisms involved in changes in allele expression. Conversely, understanding the observed dominance patterns may require an evolutionary perspective: for instance, changes in allele expression often involve methylation mechanisms, allowing rapid changes in expression levels triggered by a limited number of genetic changes. Because selection on dominance acts in heterozygotes only and because polymorphism is transient for most loci, changes in expression triggered by simple variations might be favoured. Contrasting selection regimes acting on the different alleles might therefore favour different molecular mechanisms of dominance.

Furthermore, our review of molecular mechanisms highlights that dominance can arise at different levels of integration (Fig.  1 ), so that equal allelic expression may still lead to dominance at the protein or organismal levels through a variety of mechanisms: understanding dominance therefore requires precise dissection of the developmental mechanisms involved in trait variations. Finally, investigating the consequences of dominance variations on individual fitness is needed to predict evolutionary outcomes that might differ among environments. We hope that this review will motivate the combination of research efforts from different fields for a better understanding of dominance mechanisms and evolution.

VI. CONCLUSIONS

• The literature on the evolution of dominance has been very active in recent years, and overall it is now clear that dominance may arise at different levels of integration: from biases in allele‐specific expression to organismal traits, involving a diverse array of molecular interactions and physiological and developmental properties, with contrasting consequences on dominance evolution.
• Accordingly, mounting empirical evidence also shows that dominance can and has evolved within and across species. This can be an indirect consequence of the evolution of regulatory networks within or between species, but also of natural selection acting directly on the dominance of emerging or persistent alleles within populations.
• In this review, we argue that it is now possible to provide an integrative view of dominance as resulting from the combination of general processes of gene regulation, and being an emerging property of these more general processes rather a property in itself.
• Overall, our review highlights the diversity of processes by which dominance/recessivity can arise and evolve. We argue that these different processes can be seen as different layers of the same genetic, phenotypic and ecological integration framework (Fig.  1 ) rather than as mutually exclusive explanations, as the literature in this field has too often suggested.

VII. ACKNOWLEDGEMENTS

The authors would like to thank Louis Bernatchez for inspiring this literature review and Dominique de Vienne and Vincent Colot for insightful comments. We acknowledge funding by the European Research Council (NOVEL project, grant number 648321).

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• Published: 01 June 2021

Molecular basis of heterosis and related breeding strategies reveal its importance in vegetable breeding

• Daoliang Yu 1 ,
• Xingfang Gu 1 ,
• Shengping Zhang 1 ,
• Shaoyun Dong 1 ,
• Han Miao 1 ,
• Kiros Gebretsadik   ORCID: orcid.org/0000-0002-0540-0180 1 , 2 &
• Kailiang Bo   ORCID: orcid.org/0000-0001-8841-5195 1  

Horticulture Research volume  8 , Article number:  120 ( 2021 ) Cite this article

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• Gene regulation
• Plant breeding

Heterosis has historically been exploited in plants; however, its underlying genetic mechanisms and molecular basis remain elusive. In recent years, due to advances in molecular biotechnology at the genome, transcriptome, proteome, and epigenome levels, the study of heterosis in vegetables has made significant progress. Here, we present an extensive literature review on the genetic and epigenetic regulation of heterosis in vegetables. We summarize six hypotheses to explain the mechanism by which genes regulate heterosis, improve upon a possible model of heterosis that is triggered by epigenetics, and analyze previous studies on quantitative trait locus effects and gene actions related to heterosis based on analyses of differential gene expression in vegetables. We also discuss the contributions of yield-related traits, including flower, fruit, and plant architecture traits, during heterosis development in vegetables (e.g., cabbage, cucumber, and tomato). More importantly, we propose a comprehensive breeding strategy based on heterosis studies in vegetables and crop plants. The description of the strategy details how to obtain F 1 hybrids that exhibit heterosis based on heterosis prediction, how to obtain elite lines based on molecular biotechnology, and how to maintain heterosis by diploid seed breeding and the selection of hybrid simulation lines that are suitable for heterosis research and utilization in vegetables. Finally, we briefly provide suggestions and perspectives on the role of heterosis in the future of vegetable breeding.

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Introduction.

Heterosis occurs in a variety of species and has been observed and recorded in China since ancient times. For example, Jia Sixie described in “The Manual of Important Arts for the People” that interbreeding between horses and donkeys produced stronger mules, and the famous agricultural work “Tian Gong Kai Wu” also recorded crossbreeding techniques for silkworms. Heterosis has also been extensively studied in other countries. In 1763, the German scholar Koelreuter 1 was the first to present concrete evidence that the growth of hybrid tobacco is superior to that of its parents. By comparing the height of hybrid and self-crossing offspring in maize, Darwin 2 found that the average height of hybrid offspring was higher than that of self-crossing offspring. Beal 3 found that the yield of maize hybrid offspring was greater than that of both parents. Shull 4 , 5 observed heterosis in maize hybrid offspring and first proposed the concept of heterosis; he then formally named this phenomenon “heterosis.” Heterosis was first applied to genetic breeding in maize, and many excellent maize hybrids have been produced since the 1930s. Since 2011, the yield of maize increased by at least eightfold in America, due mostly to the cultivation of hybrids 6 .

As heterosis has been applied in cereal crop production, crossbreeding in vegetables has also rapidly progressed. Under natural planting conditions, 40–80% of seeds produced are usually hybrids due to fertilization competition between self-pollination and pollen from other plants 7 . Although the traits of randomly generated hybrid seeds are not organized at first, F 1 hybrids exhibit higher yield, better adaptability, and higher stress resistance than pure line seeds under optimum production and fertilization protection management conditions. Therefore, farmers have paid much attention to the cultivation of hybrid seeds 8 . The first hybrid of eggplant ( Solanum melongena ) was released in 1924 9 . Subsequently, hybrids of other vegetables, such as watermelon ( Citrullus lanatus L.), cucumber ( Cucumis sativus L.), radish ( Raphanus sativus L.), tomato ( Solanum lycopersicum L.), and cabbage ( Brassica oleracea L.), were developed over the next 20 years 7 . The number of hybrid vegetable varieties is rapidly increasing, at a rate of 8–10% each year, while nonhybrid vegetable varieties are gradually being eliminated 10 .

The application of heterosis to vegetable cultivation was first proposed by Hayes and Jones 11 using cucumbers. However, because of the high cost of producing hybrid seeds, hybrid cucumber seeds were not used until the 1930s 7 . Similarly, self-pollination and the occasional presence of indehiscent anthers in eggplant 12 and styles that are shorter than anthers in tomato 13 have resulted in a high degree of self-pollination, which in turn has limited hybrid utilization. Pearson (1933) and Jones and Clarke (1943) used the mechanisms of self-incompatibility in cabbage and cytoplasmic male sterility in onion, respectively, to produce pure line and hybrid seeds on a large scale 8 . To avoid undesirable selfing, various genetic and nongenetic mechanisms, including genic male sterility, cytoplasmic male sterility, self-incompatibility, gynoecious lines, auxotrophy, and the use of sex regulators and chemical hybridizing agents, have been applied to facilitate hybrid seed production in vegetables 8 , 14 . The various traits that exhibit remarkable heterosis in F 1 hybrids, including yield, earliness, growth vigor, and stress tolerance 15 , 16 , 17 , 18 , have become a major area of research on vegetables. In an experiment with hybrid eggplant conducted by Balwani et al. 19 and Makani et al. 20 heterosis in the optimal F 1 hybrid resulted in yield increases of 125.78% and 88.88%, respectively. A more productive eggplant hybrid will effectively decrease the time to first harvest 18 . Transgressive phenotypes have also been observed in other Solanaceae 21 , 22 , Cruciferae 23 , 24 , and Cucurbitaceae vegetables 25 , 26 .

Although heterosis in vegetables has historically been used in research and crossbreeding experiments, its genetic mechanism remains elusive. Different genetic models for heterosis have been described in various reviews 27 , 28 , 29 , 30 , 31 . However, it is apparent that the classical genetic hypothesis of heterosis cannot explain all mechanisms of heterosis. Therefore, genetic models of heterosis have been included in this review. In addition to genetic models, we also present a schematic diagram depicting the involvement of epigenetics in heterosis. Simultaneously, we discuss studies on heterosis at the molecular level based on QTL effects and differential gene expression analyses. We also describe the effects of QTL on heterosis in crop plants based on Shang et al. 32 to guide future research studies on the genetic mechanisms of heterosis. We summarize recent findings on the interactions of QTL sites with regard to heterosis and discuss the contribution of various QTL effects to heterosis. Differential expression analysis of genes related to heterosis can also provide a different perspective on heterosis 31 . In addition, we present morphological improvement as another measure to increase yield and an important component of breeding 7 and describe how to combine heterosis utilization and morphological improvement.

To date, studies on heterosis in vegetables mainly involve obtaining F 1 hybrids through crossbreeding. The utilization of cucumber hybrids proposed by Hayes and Jones 11 was likely the first instance of effective vegetable breeding that exploits heterosis. Kumar et al. 30 introduced methods of predicting heterosis in eggplant hybrids, such as genetic distance prediction and combining ability tests, and proposed the application of a sterile line system as well as transgenic and gene editing techniques in eggplant breeding. Herath et al. 33 summarized the QTL mapping of yield-related traits in chili, introduced the use of heterosis breeding to improve the economic and agronomic traits of chili, and suggested the use of genomic technology and sterile line materials in chili breeding. Mallikarjunarao et al. 34 reviewed the progress of various balsam pear (bitter gourd) hybridization tests and indicated that heterosis does occur in the yield of balsam pear hybrids. However, studies on the genetic mechanisms of heterosis in vegetables are limited, which hinders the application of heterosis in vegetable breeding. Therefore, in this review, we describe the progress of research on the genetic mechanisms of heterosis, analyze the use of hybrid production systems and molecular biology technology in vegetable production, and propose a breeding strategy that can predict, obtain, and maintain heterosis. This review will provide a reference for the utilization of heterosis in vegetable breeding.

Study on the genetic mechanisms of heterosis

Genetic regulation of heterosis.

Heterosis is a complex biogenetic phenomenon caused by the combination of many factors that is manifested in the performance of hybrid offspring. The classical hypotheses for the genetic mechanisms of heterosis include the dominance and overdominance hypotheses, which are based on allelic interactions, and epistasis, which is based on nonallelic interactions.

Davenport 35 first proposed the dominance hypothesis (Fig. 1A ), and Bruce 36 and Jones 37 developed it further. In the dominant hypothesis, favorable genes controlling growth and development are dominant, and unfavorable genes are recessive. In the hybrid generation, the alleles from the two parents are complementary, and the unfavorable recessive genes are suppressed by the favorable dominant genes; therefore, the hybrid generation exhibits heterobeltiosis.

Suppose that the biomass is the sum of the genetic effects (A, B, C) and that the biomass of an organism is represented by the circular area. A Dominance effect: the dominant allele ( A ) inhibits the recessive allele (a); ( B ) overdominance effect: a single heterozygous allele (B/B − ) promotes the development of heterosis; ( C ) Epistasis effect: nonallelic (A 1 /B 1 ) interactions in the parents promote the development of heterosis; ( D ) active gene effect: genes from parents ( C ) promote heterosis when heterozygous and produce genome imprinting when homozygous, which inhibits the occurrence of heterosis; ( E ) gene network system: genes from parents (A, B, C) are combined into a coordinated gene network system that enables F 1 to develop heterosis; ( F ) single-cross hybrids P 1 (AB) and P 2 (CD) produced from four homozygous inbred tetraploids (with genotypes A, B, C, and D) are crossed to produce F 1 (ABCD), a double-cross tetraploid hybrid

The overdominance hypothesis (Fig. 1B ) was originally proposed by Shull 4 and East 38 as the opposite of the dominance hypothesis. This hypothesis denies that there is dominant-recessive relationship between alleles and suggests that the main cause of heterosis is the interaction of heterogeneous alleles from parents. Heterozygous alleles interact more strongly than homozygous alleles; thus, the hybrids exhibit heterobeltiosis. Using the isozyme technique, Dranginis 39 found that the enzymes in heterozygotes exhibit many unique conformations of hybrid enzymes. For example, the regulatory proteins of heterozygotes often present as polymers that regulate genes, and different heterozygous and homozygous proteins consistently show different activity characteristics. In addition, the anthocyanin content heterobeltiosis that occurs due to the heterozygosity of a single locus ( pl ) in maize 40 and the yield heterosis induced by the heterozygosity of a single locus ( sft ) in tomato 15 also provide experimental evidence for the overdominance hypothesis. However, the interaction of closely linked alleles can also result in an overdominance effect that is known as pseudo-overdominance 41 .

The dominance and overdominance hypotheses for the heterosis phenomenon both suggest that heterosis is caused by individual allele loci. However, several reports have shown that plant traits such as yield and growth vigor are complex quantitative traits 42 . Wright 43 visualized the network structure of population genotypes, i.e., multiple loci control the variations in most traits; in such networks, the replacement of anu gene may affect multiple traits. Based on this perspective, Sheridan 44 proposed the concept of epistasis. He believed that heterosis may arise from interactions between nonalleles. In genetics, the phenomenon in which the genetic effect of a nonallele deviates from its additive effect is called epistasis (Fig. 1C ). The significant special combining ability (SCA) effects in the hybridization experiment of Sao and Mehta indicated that epistasis plays a predominant role in the genetic control of eggplant heterosis 45 . Using a genetic map that covered the whole rice ( Oryza sativa ) genome, QTL mapping for yield-related traits was conducted in 250 F 2:3 lines. The results showed that the correlation between marker heterozygosity and yield-related traits was low and that the interaction between most genes could not be detected on the basis of single-gene loci; the interactions were classified as dominance by dominance, additive by dominance, and additive by additive 46 . Therefore, Yu et al. 46 also believed that epistasis is an important genetic basis for the development of heterosis.

Other ideas in addition to the classical hypotheses have been proposed. Zhong 47 proposed the active gene effect hypothesis (Fig. 1D ) by comparing the relationship between genomic imprinting and heterosis; this hypothesis suggests that heterosis is caused by additive effects between the active genes. When alleles are homozygous, only one of them is active. When genes are heterozygous, genomic imprinting does not occur, and all genes are active, showing all effects. The interaction between active genes increases the overall effect of gene expression; as a result, the hybrid exhibits heterosis. For example, in maize, the red1 ( r1 ) gene, when inherited from both parents, causes different colors in corn kernels 48 . Genomic imprinting affects the differential expression of genes by affecting DNA methylation and histone modification 49 . Bao 50 suggested that individuals have a specific set of genetic information that controls their growth. Genetic information is expressed as different coding genes in organisms; these genes form an orderly network of expression, and the activities of each gene are related to each other. An alteration in a single gene may cause changes in the entire network. The network of F 1 hybrids is a new gene network system that is formed from the two different gene networks of the parents. If the interactions between alleles bring the whole genetic network system to an optimal state, the F 1 hybrid exhibits heterosis; otherwise, it remains typical (Fig. 1E ). In addition, the effects caused by genomic imprinting or active gene effects may be components of genomic dosage effects 51 ; the other part of genomic dosage effects usually caused by polyploidy, which is a specific phenomenon in polyploid plants called progressive heterosis (Fig. 1F ) 52 , 53 . The genomic dosage effects produced by allopolyploids are usually stronger than those produced by homologous polyploids 38 , 51 , 54 , 55 . The formation of polyploids is accompanied by extensive genetic and epigenetic changes 56 , which may provide the molecular basis for the development of heterosis.

Epigenetics is involved in the development of heterosis

Although many hypotheses have been proposed to explain the mechanisms of plant heterosis at the genetic level, studies have shown that the genetic mechanisms of heterosis cannot be fully explained by one or even several hypotheses at the genetic level. Through the intensive study of epigenetics, epigenetic factors such as DNA methylation, small RNAs, and histone modifications have been found to be involved in the development of heterosis in plants 57 , 58 , 59 , 60 , 61 , 62 .

Epigenetic modifications play an important role in the formation of plant phenotypes by regulating gene transcription and gene expression 63 , 64 , 65 . Alleles of known phenotypes have been studied more extensively in the context of DNA methylation than in the context of other epigenetic modifications 63 . RNA-directed de novo methylation (RdDM) is one of the pathways that triggers DNA methylation by 24 nt-siRNA, which is regulated by two key genes, namely, NRPD1 and NRPE1 66 (Fig. 2A, B ). A silent epigenetic variant caused by differentially methylated regions (DMRs) in the promoter, sulfurea ( sulf/+ ), can result in homozygous lethal tomato plants that exhibit only chlorotic leaf sectors 64 , 65 . This may occur due to the random combination of genetic information from the parents of the F 1 hybrids because their genotypes are more prone to heterozygosity at the DNA methylation level; this is in line with the findings of Shen et al. 59 . The gene effect caused by such heterozygosity may enable F 1 hybrids to avoid producing common phenotypes or hybrid weakness, thus achieving heterobeltiosis. Using experiments involving heterograft eggplants, Cerruti et al. 62 found that scion vigor is related to DNA methylation and that the reduction in methylation in the CHH context promotes scion vigor. Tomato grafting experiments revealed that RdDM can cause a heritable enhancement-through-grafting phenotype 67 , 68 .

A DNA methylation: De novo methylation was catalyzed by DRM2, a homologous enzyme of DNMT3. In maintenance methylation, CG is catalyzed by MET1, a homologous enzyme of DNMT1; CHG is catalyzed by CMT3; and CHH is still catalyzed by DRM2. B Small RNA: Includes the miRNA produced by premiRNA and the siRNA produced by dsRNA. In general, 24 nt-siRNA mediates de novo DNA methylation catalyzed by the AGO4 protein. C Histone modifications: The modifications of histone amino acid residue includes acetylation, phosphorylation, methylation, and ubiquitination processes. Epigenetic modifications are produced by the parents. New epigenetic modifications may occur in F 1 hybrids. D Epigenetic modification status of the parents and F 1 hybrid: the increase and decrease in or recombination of epigenetic modifications induces the F 1 hybrid to exhibit heterosis

Because de novo DNA methylation is mediated by siRNAs (Fig. 2B ), siRNAs may also be involved in the regulation of heterosis. The level of siRNAs decreased in different genome regions between parents and hybrids, but this phenomenon was limited to 24 nt-siRNAs; in contrast, the levels of siRNAs of other sizes did not decrease 67 . Noncoding small RNAs can be used as signaling molecules in plants 67 . Shivaprasad et al. 61 observed that miR395 is differentially expressed, mediates transgressive phenotypes in the hybrid progeny of tomato and is associated with suppression of the corresponding target genes, which indicates that the combination of parental genetic information can cause differences in miR395 abundance in the progeny. Simultaneously, 21–24 nt small RNAs can move through the intercellular filaments and phloem of the graft site 69 , and 24 nt sRNAs can guide genomic DNA methylation in recipient cells 70 ; this information provides a theoretical basis for guiding grafting. In addition, sRNAs in plants usually play a major role in inducing gene expression silencing and gene posttranscriptional silencing 71 , 72 . This may be due to the downregulation of sRNA levels in hybrids, which lifts the silencing of some favorable genes and thus allows hybrids to exhibit heterobeltiosis 71 , 72 .

Different modifications, such as acetylation, phosphorylation, methylation, and ubiquitination, occur at the amino terminus of histones (Fig. 2C ). These histone modifications can affect the binding of related proteins to chromatin and thereby affect the transcriptional activity of genes. At the same time, the combination of modifications of the amino terminus of histones expands the genetic information for and changes the phenotype of an individual 73 . Histone modifications are related to the stability of heterosis. Studies have shown that histone deacetylases cause the nonadditive expression of some genes in hybrids 58 . In addition, histone acetylation and methylation are related to the activation of regulatory (circadian-regulated) genes in F 1 hybrids 73 . The biological clock controls the physiological activities of plants, including the synthesis of physiological and biochemical substances. Therefore, histone modifications can influence plant biomass heterosis.

The recombination of genetic information from parents may lead to new combinations of epigenetic modifications in the F 1 generation (Fig. 2D ). Epigenetic modifications essentially affect the expression of genes, causing them to be overexpressed or silenced. Therefore, epigenetic modifications may indirectly influence the development of heterosis in F 1 by affecting the expression pattern of genes.

Study on heterosis at the molecular level

Progress in heterosis research based on qtl analysis.

The genome contains all the genetic information of a species and determines whether an individual gene is expressed as well as its degree of expression. Heterosis is usually indicated if the hybrid generation is superior to the parents in terms of quantitative traits. Thus, it is essential to conduct a genetic analysis of heterosis from the perspective of the whole genome. With the rapid development of genome sequencing technology, it has become possible to identify gene loci related to heterosis by genome-wide association studies 74 , which lay a foundation for the study of individual phenotypic differences. This review summarizes the QTL effects on heterosis based on 35 studies that mainly addressed 6 crops and vegetables, i.e., rice ( Oryza sativa ), maize ( Zea mays ), cotton ( Gossypium hirsutum ), oilseed rape ( Brassica campestris ), sorghum ( Sorghum vulgare ), and tomato ( Solanum lycopersicum ) (Table S1 ). Among the six types of QTL effects, dominance and epistasis had equal proportions (19%, 23%, Fig. 3 ). Interestingly, the overdominance effect accounted for the largest proportion of all the effects (42%, Fig. 3 ). This means that although there are many gene loci in the plant genome, these interacted to produce different, complex, hard-to-imitate effects and resulted in heterosis; among these effects, overdominance effects occurred consistently and contributed significantly to heterosis. In addition, the overdominance effect can be conveniently used for artificial breeding, which has been well demonstrated in tomato 15 . However, efficiently and accurately locating the gene loci that impart the overdominance effect is necessary to make use of this effect. Heterosis may be the result of many traits. In addition, the results of QTL mapping differ among species and even within different groups of the same species 75 , 76 , 77 . Therefore, it is necessary to select a suitable genetic population based on the genetic background of the plants exhibiting heterosis.

Statistical analysis of the effect of quantitative trait loci on crop heterosis. A In the statistical analysis of the effect of quantitative trait loci on crop heterosis, the species and frequency of each species were studied; ( B ) in the statistical analysis of the effect of quantitative trait loci on crop heterosis, the quantitative trait locus effect on each species and the proportion of each type of effect were analyzed

Advances in gene action related to heterosis based on differential expression analysis of genes

The genome controls the formation of a biological phenotype by regulating the differential expression of genes 78 , 79 . Molecular-based expression analyses, such as allele-specific expression, DNA microarray, expression quantitative trait loci, RNA-seq, quantitative SNP-based Sequenom technology, and allele-specific RT-PCR, have made it possible to detect differential gene expression.

Yield and biomass heterosis in F 1 hybrids may occur due to the altered expression patterns of genes that control biological functions such as carbon fixation, glucose metabolism, and circadian rhythm 80 . Gene Ontology (GO) analysis of pakchoi line parents and hybrids indicated that most of the differentially expressed genes between parents and hybrids enriched the photosynthetic pathway and that the enhancement of the photosynthetic capacity of the hybrids was related mainly to an increase in the number of thylakoids 17 . In addition, the increase in the number of thylakoids also promoted the enhancement of the carbon fixation capacity in the hybrids 17 ; this is similar to the finding that differentially expressed genes that significantly enrich the optical signaling pathway occur between F 1 and their parents in broccoli 24 . The same results were also found in other plants 79 , 81 . Transcriptome and differential gene expression analyses revealed that the modes of action of heterosis genes were mainly additive (F 1  = MPV), overdominance (F 1  > HPV), and underdominance (F 1  < LPV) 82 (Fig. 4 ). When the expression value of a differentially expressed gene in the hybrid line was higher or lower than that of the parent, the gene action patterns were classified as high-parent dominance (F 1  ≈ HPV) and low-parent dominance (F 1  ≈ LPV), respectively 82 (Fig. 4 ). Li et al. 24 reported that most genes exhibited additive expression patterns in hybrid broccoli and that nonadditive action was involved mainly in light and hormone signal pathways related to heterosis; a similar finding was reported in Chinese cabbage ( Brassica campestris ssp. pekinensis cv. “ spring flavor ”) 23 . These gene expression patterns may have occurred due to selective inhibition or activation by the epigenetic modification of hybrid F 1 genes 83 , 84 ; the genes from inactive inbred lines can be activated by genes or regulatory factors of active inbred lines 85 , 86 . Epigenetic modifications and the interactions of heterogeneous factors occur in only a few genes, and the genome that produces differential expression in F 1 hybrids and parents accounts for only a small part of the total genome 87 . Moreover, Springer and Stupar 88 have shown that additive gene expression accounts for the majority of gene expression, while nonadditive gene expression is responsible for a small proportion of gene expression. These findings suggest that nonadditive expression of this fraction facilitates the development of heterosis.

Midparent value [MPV = (HPV + LPV)/2]; High-parent value (HPV); low-parent value (LPV)

Traits contributing to yield heterosis in vegetables

Traits related to yield heterosis.

Hybrids that exhibit heterosis show significant heterobeltiosis in yield, which is a complex trait that is usually measured by weight. To clearly study the mechanisms of yield increase in hybrids, it is essential to divide yield into other, simpler traits. This review describes the traits that contribute to vegetable yields. Fruits are the source of the yield of most plants; the yield contributing traits related to fruits usually include the fruit number, fruit size and fruit weight; earliness is usually also taken into account. Cabbage is a typical leafy, head-forming vegetable in Cruciferae, so its main yield contributing traits are head weight and head size (Fig. 5A, C ). Similar to that of cabbage, the yield of radish is determined by its taproot. For leafy vegetables that do not form heads, the main yield heading traits are the number and size of the leaves. Unlike cruciferous vegetables, Cucurbitaceae and Solanaceae vegetables are produce multiple harvests and multiple fruits per plant (Fig. 5B, D ), so the average single fruit weight and fruit yield per plant should be taken into account. In addition, Solanaceae vegetable flowers consist mostly of compound inflorescences 89 , so the numbers of flowers per cluster and fruits per cluster contribute greatly to production. Cucurbitaceae are single-inflorescence vegetables; only the fruits on the main vine are harvested in production, and the first nodal position of female flowers and sex ratio (M/F) affect the days to first harvest and the number of fruits per plant, respectively. Regardless of the trait considered, the total yield can be affected only by changes in yield-related traits. Therefore, it is necessary to analyze the mechanisms that regulate yield-related traits.

Contributing traits of yield heterosis in cucumber, cabbage and tomato. A Traits contributing to yield heterosis in cucumber, cabbage, and tomato: cucumber yield contributing traits include the number of fruits, days to first female flowering, days to first harvest, first nodal position of female flower, sex ratio (M/F), fruit length, fruit diameter, and fruit weight; cabbage yield contributing traits include fruit length, fruit diameter, and fruit weight; tomato yield contributing traits include number of fruits, days to first female flowering, days to first harvest, number of flowers/fruits per cluster, fruit length, fruit diameter, and fruit weight. B Cucumber: cucumber model in production, gynoecious line with a small number of branches. C Cabbage: an aerial and cross-sectional model of cabbage consisting of leaves and heads. D Tomato: a tomato with single inflorescences and indeterminate growth is crossbred with a tomato with compound inflorescences and determinate growth to produce the hybrid F 1 with earlier fruiting, more compound inflorescences, and determinate growth

Relationship between yield heterosis and plant architecture

Since the “green revolution”, interest in breeding for specific plant architecture has significantly increased, and the idea of combining heterosis breeding with plant architecture breeding has been proposed 90 . Donald 91 conducted research on half-dwarf plant architecture, which gradually turned into the concept of the ideotype. Donald introduced the ideotype concept, which refers to the plant architecture form that results in the minimum competitive intensity in population breeding. Although this definition is no longer used, the concept of an ideal plant architecture has played a major role in promoting plant breeding for high yields. Research on ideotypes first made progress in rice. It is worth mentioning that a key gene regulating ideotype, IPA1 , was proven by Huang et al. 75 to influence genes that are important in heterosis by using the indica-japonica hybrid rice group. Studies of heterozygosity and ideotype were also combined effectively in tomato. The self-pruning ( sp ) gene promotes indeterminate growth in tomato, while the sft gene changes indeterminate growth into determinate growth by inhibiting the sp gene 92 . The sft gene results in the development of heterosis in tomatoes through the heterozygosity of a single gene 15 and induces changes in plant architecture on the ground, causing tomato to produce compound inflorescences rather than single inflorescences 93 . The earliness of F 1 was also higher than that of its parent (Fig. 5D ), which increased tomato yield. Other vegetables in addition to tomato may also have ideotypes, and the key genes controlling plant architecture may also be important genes that are involved in the development of heterosis. Therefore, it is particularly important to study the genetic mechanisms of heterosis. By identifying the important genes involved in heterosis, the key genes that control plant ideotypes can be characterized.

Advances in heterosis utilization and biotechnology in vegetables

Breeding for heterosis has been extensively studied in plants, and research on the heterobeltiosis of hybrid offspring in vegetables has focused mainly on yield 94 and disease resistance 29 . Wellington 95 and Tschermak 96 showed that tomato hybrids exhibit heterosis in early maturity and during yield production. Krieger et al. 15 cloned the single-gene sft that affects the female flower fertility rate in tomato by infiltrating the IL and TC populations. When the sft gene exhibited heterozygosity, the tomato yield exhibited heterosis. According to this study, tomato plants that showed yield heterosis also showed resistance to both biological and abiotic stresses. The heterozygous state of the Tm and Tm22 genes contributes to tobacco mosaic virus resistance 97 , 98 and high-temperature stress tolerance 99 , 100 . Naresh et al. 101 suggested that heterosis is the result of nonadditive gene effects and that it also plays an important role in improving Cercospora leaf spot resistance in eggplant in the field. Similar to studies on other vegetables, studies on heterosis in Cucurbitaceae vegetables have also focused mainly on yield and disease resistance. Pandey et al. 102 used 77 cucumber hybrid generations and their parents to study the yield heterosis and contributing traits of different cucumber hybrid varieties and found that DC–1 × B–159 and VRC–11–2 × Bihar–10 were the best hybrid combinations for yield and prematurity. Using 48 F 1 hybrids and their parents, the gene effects caused by diseases and insect pests under natural conditions 29 were investigated. The results indicated that nonadditive gene effects had a significant regulatory effect on other traits in cucumber (except morbidity caused by Drosophila), demonstrating the importance of heterosis in cucumber breeding for disease resistance.

Different molecular markers, such as simple sequence repeats (SSRs), inter-simple sequence repeats (ISSRs), amplified fragment length polymorphisms (AFLPs), random amplified polymorphic DNAs (RAPDs), and sequence-related amplified polymorphisms (SRAPS), have provided the molecular basis for the construction of genetic maps and the mapping of important trait genes (Table 1 ). Whole-genome sequencing has been conducted for a variety of vegetables (Table 1 ), which has provided a basis for whole-genome strategies. Whole-genome approaches can help obtain complete sequences of germplasm resources, increase the coverage of molecular markers, and increase the accuracy of genetic maps 103 . Molecular markers are often used for the determination of genetic distance and the classification of heterotic groups. To elucidate the breeding processes and to improve the efficiency of breeding techniques in cabbage, heterotic cabbages are usually divided into two groups: The round head type and the flat head type. Xing et al. 104 further divided 21 flat cabbage inbred lines into three heterotic groups and divided 42 round cabbage inbred lines into five heterotic groups in order to provide a more definite direction for the preparation of hybrid combinations of cabbage. The method of dividing heterotic groups by molecular markers and genetic distance is widely used in vegetable breeding (Table 1 ).

Chen 83 proposed that determining how to obtain hybrid seeds is the key to the utilization of heterosis. The purpose of obtaining hybrid seeds is to make heterosis in the offspring permanent. The sporophyte of cruciferous vegetables is a self-incompatible system 105 that can prevent self-pollination and produce normal seeds through cross-pollination. Hence, this system is convenient for the generation of hybrid seeds. In cabbage 106 , 107 and Chinese cabbage 108 , hybrids are usually obtained using self-incompatible and male-sterile lines. To produce hybrid tomato seeds, pollen-abortive type and functionally sterile lines are often used 109 , 110 , 111 . Cytoplasmic male sterility occurs in eggplant 112 , 113 and pepper 114 , 115 . Gynoecious lines tend to exist in Cucurbitaceae 116 . A new male-sterile system in tomato was developed by Du et al. 117 . Plant growth regulators such as ethylene, auxins, and brassinosteroids 118 , 119 can increase the number of female flowers in Cucurbitaceae; this effect and male sterility are both convenient for hybrid seed production.

Strategies for heterosis breeding in vegetables (with tomato as an example)

Obtaining f 1 hybrids that exhibit heterosis based on heterosis prediction.

It is not advisable to conduct extensive hybridization tests to obtain hybrid F 1 lines that exhibit heterosis, as this approach requires considerable resources and time and produces unreliable results 13 . Melchinger and Gumber 120 proposed that heterotic groups should be used as the basis for crossbreeding. The heterotic group is the population that is classified according to breeding requirements, with abundant genetic variation and high combining ability. Chen et al. 121 carried out a genome-wide association study (GWAS) on the yield traits, general combining ability (GCA), and SCA of rice. The study provided strong evidence for the use of combining ability to classify heterotic groups and provided a reference for studies on combining ability in vegetables (Fig. 6 ). Other studies have also shown that combining ability, genetic distance, and molecular markers can provide the basis for evaluating parental inbred lines and predicting F 1 hybrid heterosis in vegetables 122 , 123 , 124 , 125 .

There are two strategies for obtaining heterotic lines in crop breeding. The first is the use of crossbreeding or molecular biotechnology. Genealogical analysis, molecular markers, combining ability, and genetic distance can usually predict heterosis development, so they are often used to classify heterotic groups. The inbred lines from different heterotic groups can be crossed with each other to obtain elite lines that exhibit heterosis. The second strategy is to use modern molecular biotechnology. Elite lines were obtained based on GWAS and linkage analysis, mapping and cloning genes related to heterosis, gene editing, and gene transformation

The GCA characterizes the average performance of a set of hybrid combinations and is mainly the consequence of additive gene effects and additive × additive interactions; SCA evaluates the average performance of certain hybrid combinations compared to the parental lines and is the result of dominance, epistatic deviation and genotype × environmental interactions 126 . Parents with a high GCA effect have higher adaptability and fewer environmental effects 127 . Parents with superior traits do not always pass on their traits to offspring 126 ; hence, the evaluation of combining ability is more reliable than the performance of the lines per se. Many types of combining ability tests can be used to identify superior parental lines for developing heterotic hybrids, including line × tester analysis, topcross tests, single-cross tests, poly-cross tests, and diallel mating 128 . Singh et al. 129 conducted a complete diallel cross test on seven diverse bitter gourd lines and found that combinations with high × high GCA usually produced high SCA effects and could therefore be considered for use in developing superior variants through the pedigree method. High/low × low GCA combinations can also achieve high but unstable SCA effects that are suitable for heterosis breeding and are in line with the results of Kenga et al. 130 in sweet sorghum and Franco et al. 131 in common bean.

In addition to combining ability, heterotic groups are often classified by genealogical information 132 . For parents with known genealogical relationships, heterosis in hybrids can usually be predicted according to these genealogical relationships. Genetic distance is a quantitative description of the genetic differences that provide the genetic basis for the development of heterosis in offspring 133 , 134 . Parental lines with a longer genetic distance are more likely to produce hybrids with strong predominance 135 , 136 . Molecular markers can also be used to directly or indirectly classify heterotic groups by assessing their genetic distance 125 , 137 , 138 . RAPD and AFLP have been successfully used to detect the genetic distance between tested lines, and the yield of carrots was found to be significantly correlated with genetic distance 125 . Genetic distance has also been applied to predict hybrid pepper fruit diameter 139 and hybrid melon ( Cucumis melo L.) fruit shape diameter 140 . The scientific classification of heterotic groups improves the efficiency of selecting hybrid combinations of superior parents and utilizing heterosis (Fig. 6 ).

In addition, some omics approaches, such as genomics, transcriptomics, and metabolomics, have become tools for predicting hybrid yield in rice 141 . Xu et al. 141 analyzed metabolomic and genomic data from 21,445 hybrids developed by 210 recombinant inbred lines and found that metabolomic data were more effective than genomic data in predicting hybrid yield. Research on the prediction of heterosis in vegetables with omics data has not been published. However, the genome or epigenome is the most fundamental source of the plant phenotype, and the transcriptome, proteome, and metabolism are the direct sources of plant phenotypes. Therefore, omics data could represent a more accurate way to predict vegetable hybrid heterosis, and studies of crop hybrid yields can provide a reference for predicting heterosis in vegetables.

Obtaining elite lines based on molecular biotechnology

GWAS is a method used to identify the gene loci that control certain traits in a population by combining phenotypes with genotypes. GWAS is often used to identify certain traits, such as green flesh color or thermotolerance, in cucumber 142 , 143 but can also be used to analyze complex traits, such as yield and biomass 144 , 145 , 146 , 147 , 148 , 149 , 150 , 151 , 152 , 153 , 154 , 155 , 156 . In addition, whole-genome sequencing of various vegetables provides a basis for GWAS (Table 1 ). Due to the unique phenotype of heterosis and its genetic background sources, a genetic population can be composed of different populations or ecotype hybrid populations. A segregated F 2 population that was produced by a strongly predominant F 1 population is regarded as the best population for studying heterosis 27 . Such an F 2 population not only has a reasonable proportion of lines with heterozygous genotypes and homozygous genotypes but also has allele combinations that are distributed evenly at each site 27 .

DeVicente and Tanksley 157 randomly paired an RIL population obtained by strong F 1 self-crossing to produce a new population. This population not only preserves the genotype of the RIL population but also reproduces the F 2 population; thus, it is called an IF 2 population. At present, IF 2 populations have been established in rice 158 , 159 , 160 , 161 , maize 150 , 162 , 163 , 164 , 165 , 166 , 167 , 168 , 169 , cotton 170 , and other crops. In addition, there are also diverse F 1 156 , IL 171 , 172 , 173 , 174 , 175 , BILF 1 176 , 177 , and SSSL 178 populations that can be used to study heterosis. Except for two studies on tomato, there are few relevant studies on heterosis in vegetables using such populations that would provide a reference for conducting heterosis-related studies in other vegetables.

Using genome editing techniques to knockout adverse genes or overexpress favorable genes can transform ordinary lines into strong predominance lines. For example, biomass, plant height, and leaf photosynthetic pigment contents increased in rice expressing maize GLK genes compared with those in wild-type rice; 179 such results may cause researchers to think about studying mutual heterosis promotion among different vegetables. Dominance and overdominance effects account for a large proportion of the effects that produce heterosis and are easy to mimic (Fig. 3B ). Understanding the mechanisms of heterosis helps breeders to improve current varieties and generate novel cultivars 27 (Fig. 6 ).

Maintaining heterosis

The hybridization of the selfing line of two heterotic groups can generate hybrid offspring that exhibit heterosis. Through hybrid seed production, self-incompatibility and male-sterile line technology can be used to maintain the hybrid vigor of the hybrid F 1 line. Some of the characteristics of the vegetables themselves, such as the gynoecious characteristic of Cucurbitaceae 116 and asexual reproduction in potato ( Solanum tuberosum L.) 180 , are convenient for hybrid seed production or heterosis maintenance. In addition, some plant hormones or chemical reagents can also be used for plant sex regulation 14 . However, exogenous regulation is often not completely effective 14 , which may affect the purity of hybrid seeds. Therefore, it is necessary to study hybrid systems of vegetables for hybrid seed production.

Du et al. 117 used gene editing technology (Cas9) to knock out the male-specific gene SlSTR1 in tomato to obtain a sterile line and generated a maintainer line by transferring a fertility-restoration gene to the sterile line; it was easy to distinguish whether offspring of crosses between the maintainer and male-sterile lines were male-fertile maintainer plants because a seedling-color gene was linked to the fertility-restoration gene. This system combined tomato sterile lines and gene editing technology and represents a highly practical potential approach to hybrid seed production in tomatoes. Moreover, it may serve as an important reference for the use of gene editing technology for hybrid seed production in other vegetables.

Khanday et al. 181 and Wang et al. 182 found that genome editing can cause mitosis to replace meiosis in rice such that diploid clonal seeds have the original F 1 gene heterozygosity and maintain F 1 traits (Fig. 6 ). Unlike with knocking out the infertility gene using gene editing technology, with this method, fertilization and cell division are necessary for hybridization. Some vegetables do not have sterile line material. Therefore, this method, in which plant fertilization involves only mitosis and not meiosis, will be more widely applicable.

In addition, by repeatedly screening the F 2 lines that were close to the F 1 phenotype, Wang et al. 85 obtained pure F 5 /F 6 lines that were close to the F 1 phenotype; these were called hybrid simulation lines, indicating that the phenotype of the F 1 hybrids was fixed in this line. This method has also been used to maintain F 1 heterosis in other vegetables, such as tomatoes 183 and peas ( Pisum sativum L.) 184 . Therefore, the heterosis of hybrid F 1 vegetables produced by hybridization or molecular biotechnology can be maintained by diploid seed breeding and selection for hybrid simulation lines in the future (Fig. 6 ).

Conclusions and future perspectives

Research on vegetable heterosis has focused mainly on its applications in heterosis breeding. Studies on its genetic mechanism are limited, which hinders its utilization. Extensive progress has been made in the study of heterosis in cereal crops such as rice and maize. In vegetables, both hybrid production systems (male sterility lines, self-incompatibility lines, and gynoecious lines) and molecular biological techniques (gene editing, transgenosis, and asexual reproduction) have been used. Therefore, the methods and strategies proposed by this paper for studying the genetic mechanisms of heterosis can be applied to vegetable breeding. In the near future, we will identify certain heterosis-related gene loci in vegetables to understand the molecular genetics and mechanism of heterosis formation in vegetables and to make new breakthroughs in improving the yield, quality, and safety of vegetables. This review emphasizes the following points: (1) The application of heterosis in vegetable crops allows improvements in yield and quality and enhances plant resistance to biological and environmental stresses. (2) In the future, more attention should be paid to the study of the genetic mechanisms of vegetable heterosis to identify the important genes involved in the development of heterosis and to understand the regulation and activity modes of the key genes affecting vegetable heterosis. (3) By fully referencing and adapting the strategies used in cereal crop heterosis studies, exogenous genes can be applied to produce the same function in different species 179 . Therefore, transgenic and genomic editing technologies can significantly improve the efficiency of research on heterosis gene identification in vegetables. (4) Although a certain basic molecular knowledge of vegetable heterosis has been obtained, applying the knowledge acquired from cereal crops to vegetables will improve vegetable production and quality. It will also be useful to compare sterile line seed production with optimized transgenic systems to achieve more breakthroughs in vegetable production. (5) The study of heterosis can promote the study of ideal plant architecture in vegetable breeding. A breeding strategy that combines heterosis with the ideal plant architecture can achieve substantial gains in vegetable yield and quality. (6) Maintaining heterosis is the core factor of the extensive use of heterosis and has been reflected mainly in F 1 hybrid seed production. With the development of gene editing technology, sterile line gene editing systems, MiMe (Cas9) systems and even new biotechnology approaches will have opportunities to be widely applied; this will be of great significance for hybrid seed production. (7) Progressive heterosis caused by the dosage effect in polyploid hybrids is also an important component of the genetic mechanisms of heterosis, and these phenomena have been observed in different plants 55 , 185 . Polyploid systems allow experiments to be performed that are impossible in diploid systems; hence, polyploid crossbreeding may lead to different plant performance results than diploid breeding. However, polyploids have highly heterozygous genomes and complex genetic structures, and we may not be able to evaluate their phenotypes and genetic structures using diploid criteria. This topic deserves future investigation.

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Acknowledgements

This work was supported by the National Key Research and Development Program of China [2016YFD0101705], the Earmarked Fund for Modern Agro–industry Technology Research System [CARS–25] and the Key Laboratory of Biology and Genetic Improvement of Horticultural Crops, Ministry of Agriculture, China.

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Daoliang Yu, Xingfang Gu, Shengping Zhang, Shaoyun Dong, Han Miao, Kiros Gebretsadik & Kailiang Bo

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D.Y. and K.B. conducted the literature review and wrote the paper. X.G., S.Z., S.D., H.M. and K.G. helped review the crop heterosis data. K.B. conceived of and supervised the study. All authors reviewed and approved the final submission.

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Yu, D., Gu, X., Zhang, S. et al. Molecular basis of heterosis and related breeding strategies reveal its importance in vegetable breeding. Hortic Res 8 , 120 (2021). https://doi.org/10.1038/s41438-021-00552-9

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8.4: Dominance

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• Joseph Fields
• Southern Connecticut State University

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We’ve said a lot about the equivalence relation determined by Cantor’s definition of set equivalence. We’ve also, occasionally, written things like \(|A| < |B|\), without being particularly clear about what that means. It’s now time to come clean. There is actually a (perhaps) more fundamental notion used for comparing set sizes than equivalence — dominance. Dominance is an ordering relation on the class of all sets. One should probably really define dominance first and then define set equivalence in terms of it. We haven’t followed that plan for (at least) two reasons. First, many people may want to skip this section — the results of this section depend on the difficult Cantor-Bernstein-Schröder theorem 1 . Second, we will later take the view that dominance should really be considered to be an ordering relation on the set of all cardinal numbers – i.e. the equivalence classes of the set equivalence relation – not on the collection of all sets. From that perspective, set equivalence really needs to be defined before dominance.

One set is said to dominate another if there is a function from the latter into the former. More formally, we have the following

Definition: Dominance

If A and B are sets, we say “\(A\) dominates \(B\)” and write \(|A| > |B|\) iff there is an injective function \(f\) with domain \(B\) and codomain \(A\).

It is easy to see that this relation is reflexive and transitive. The Cantor-Bernstein-Schröder theorem proves that it is also anti-symmetric — which means dominance is an ordering relation. Be advised that there is an abuse of terminology here that one must be careful about — what are the domain and range of the “dominance” relation? The definition would lead us to think that sets are the things that go on either side of the “dominance” relation, but the notation is a bit more honest, “\(|A| > |B|\)” indicates that the things really being compared are the cardinal numbers of sets (not the sets themselves). Thus anti-symmetry for this relation is

\[(|A| > |B|) ∧ (|B| > |A|) \implies (|A| = |B|).\]

In other words, if \(A\) dominates \(B\) and vice versa, then \(A\) and \(B\) are equivalent sets — a strict interpretation of anti-symmetry for this relation might lead to the conclusion that \(A\) and \(B\) are actually the same set, which is clearly an absurdity.

Naturally, we want to prove the Cantor-Bernstein-Schröder theorem (which we’re going to start calling the C-B-S theorem for brevity), but first it’ll be instructive to look at some of its consequences. Once we have the C-B-S theorem we get a very useful shortcut for proving set equivalences. Given sets \(A\) and \(B\), if we can find injective functions going between them in both directions, we’ll know that they’re equivalent. So, for example, we can use C-B-S to prove that the set of all infinite binary strings and the set of reals in \((0, 1)\) really are equinumerous. (In case you had some remaining doubt. . . )

It is easy to dream up an injective function from \((0, 1)\) to \(\mathbb{F}^∞_2\): just send a real number to its binary expansion, and if there are two, make a consistent choice — let’s say we’ll take the non-terminating expansion.

There is a cute thought-experiment called Hilbert’s Hotel that will lead us to a technique for developing an injective function in the other direction. Hilbert’s Hotel has \(ℵ_0\) rooms. If any countable collection of guests show up there will be enough rooms for everyone. Suppose you arrive at Hilbert’s hotel one dark and stormy evening and the “No Vacancy” light is on — there are already a denumerable number of guests there — every room is full. The clerk sees you dejectedly considering your options, trying to think of another hotel that might still have rooms when, clearly, a very large convention is in town. He rushes out and says “My friend, have no fear! Even though we have no vacancies, there is always room for one more at our establishment.” He goes into the office and makes the following announcement on the PA system. “Ladies and Gentlemen, in order to accommodate an incoming guest, please vacate your room and move to the room numbered one higher. Thank you.” There is an infinite amount of grumbling, but shortly you find yourself occupying room number \(1\).

To develop an injection from \(\mathbb{F}^∞_2\) to \((0, 1)\) we’ll use “room number \(1\)” to separate the binary expansions that represent the same real number. Move all the digits of a binary expansion down by one, and make the first digit \(0\) for (say) the terminating expansions and \(1\) for the non-terminating ones. Now consider these expansions as real numbers — all the expansions that previously coincided are now separated into the intervals \(\left(0, \dfrac{1}{2} \right)\) and \(\left(\dfrac{1}{2}, 1\right)\). Notice how funny this map is, there are now many, many, (infinitely-many) real numbers with no preimages. For instance, only a subset of the rational numbers in \(\left(0, \dfrac{1}{2} \right)\) have preimages. Nevertheless, the map is injective, so C-B-S tells us that \(\mathbb{F}^∞_2\) and \((0, 1)\) are equivalent. There are quite a few different proofs of the C-B-S theorem. The one Cantor himself wrote relies on the axiom of choice. The axiom of choice was somewhat controversial when it was introduced, but these days most mathematicians will use it without qualms. What it says (essentially) is that it is possible to make an infinite number of choices. More precisely, it says that if we have an infinite set consisting of non-empty sets, it is possible to select an element out of each set. If there is a definable rule for picking such an element (as is the case, for example, when we selected the nonterminating decimal expansion whenever there was a choice in defining the injection from \((0, 1)\) to \(\mathbb{F}^∞_2\)) the axiom of choice isn’t needed. The usual axioms for set theory were developed by Zermelo and Frankel, so you may hear people speak of the ZF axioms. If, in addition, we want to specifically allow the axiom of choice, we are in the ZFC axiom system. If it’s possible to construct a proof for a given theorem without using the axiom of choice, almost everyone would agree that that is preferable. On the other hand, a proof of the C-B-S theorem, which necessarily must be able to deal with uncountably infinite sets, will have to depend on some sort of notion that will allow us to deal with huge infinities.

The proof we will present here 2 is attributed to Julius König. König was a contemporary of Cantor’s who was (initially) very much respected by him. Cantor came to dislike König after the latter presented a well-publicized (and ultimately wrong) lecture claiming the continuum hypothesis was false. Apparently, the continuum hypothesis was one of Cantor’s favorite ideas, because he seems to have construed König’s lecture as a personal attack. Anyway. . .

König’s proof of C-B-S doesn’t use the axiom of choice, but it does have its own strangeness: a function that is not necessarily computable — that is, a function for which (for certain inputs) it may not be possible to compute an output in a finite amount of time! Except for this oddity, König’s proof is probably the easiest to understand of all the proofs of C-B-S. Before we get too far into the proof it is essential that we understand the basic setup. The Cantor-Bernstein-Schröder theorem states that whenever \(A\) and \(B\) are sets and there are injective functions \(f : A \implies B\) and \(g : B \implies A\), then it follows that \(A\) and \(B\) are equivalent. Saying \(A\) and \(B\) are equivalent means that we can find a bijective function between them. So, to prove C-B-S, we hypothesize the two injections and somehow we must construct the bijection.

Figure \(8.4.1.\) has a presumption in it — that \(A\) and \(B\) are countable — which need not be the case. Nevertheless, it gives us a good picture to work from. The basic hypotheses, that \(A\) and \(B\) are sets and we have two functions, one from \(A\) into \(B\) and another from \(B\) into \(A\), are shown. We will have to build our bijective function in a piecewise manner. If there is a non-empty intersection between \(A\) and \(B\), we can use the identity function for that part of the domain of our bijection. So, without loss of generality, we can presume that \(A\) and \(B\) are disjoint. We can use the functions \(f\) and \(g\) to create infinite sequences, which alternate back and forth between \(A\) and \(B\), containing any particular element. Suppose \(a ∈ A\) is an arbitrary element. Since \(f\) is defined on all of \(A\), we can compute \(f(a)\). Now since \(f(a)\) is an element of \(B\), and \(g\) is defined on all of \(B\), we can compute \(g(f(a))\), and so on. Thus, we get the infinite sequence

\(a, f(a), g(f(a)), f(g(f(a))), . . .\)

If the element \(a\) also happens to be the image of something under \(g\) (this may or may not be so — since \(g\) isn’t necessarily onto) then we can also extend this sequence to the left. Indeed, it may be possible to extend the sequence infinitely far to the left, or, this process may stop when one of \(f^{−1}\) or \(g^{−1}\) fails to be defined.

\(. . . g^{−1} (f^{−1} (g^{−1} (a))), f^{−1} (g^{−1} (a)), g^{−1} (a), a, f(a), g(f(a)), f(g(f(a))), . . .\)

Now, every element of the disjoint union of \(A\) and \(B\) is in one of these sequences. Also, it is easy to see that these sequences are either disjoint or identical. Taking these two facts together it follows that these sequences form a partition of \(A ∪ B\). We’ll define a bijection \(\phi : A \implies B\) by deciding what it must do on these sequences. There are four possibilities for how the sequences we’ve just defined can play out. In extending them to the left, we may run into a place where one of the inverse functions needed isn’t defined — or not. We say a sequence is an \(A\)-stopper, if, in extending to the left, we end up on an element of \(A\) that has no preimage under \(g\) (see Figure \(8.4.2\)). Similarly, we can define a \(B\)-stopper. If the inverse functions are always defined within a given sequence there are also two possibilities; the sequence may be finite (and so it must be cyclic in nature) or the sequence may be truly infinite.

Finally, here is a definition for \(\phi\).

\( \phi(x)=\left\{ \begin{array}{ll} g^{−1}(x) \;\;\;\;\text{ if } x \text{ is in a } B\text{-stopper}\\ f(x) \;\;\;\;\;\;\;\text{ otherwise} \end{array} \right.\)

Notice that if a sequence is either cyclic or infinite it doesn’t matter whether we use \(f\) or \(g^{−1}\) since both will be defined for all elements of such sequences. Also, certainly \(f\) will work if we are in an \(A\)-stopper. The function we’ve just created is perfectly well-defined, but it may take arbitrarily long to determine whether we have an element of a \(B\)-stopper, as opposed to an element of an infinite sequence. We cannot determine whether we’re in an infinite versus a finite sequence in a prescribed finite number of steps.

Exercise \(\PageIndex{1}\)

How could the clerk at the Hilbert Hotel accommodate a countable number of new guests?

Exercise \(\PageIndex{2}\)

Let \(F\) be the collection of all real-valued functions defined on the real line. Find an injection from \(\mathbb{R}\) to \(F\). Do you think it is possible to find an injection going the other way? In other words, do you think that \(F\) and \(\mathbb{R}\) are equivalent? Explain.

Exercise \(\PageIndex{3}\)

Fill in the details of the proof that dominance is an ordering relation. (You may simply cite the C-B-S theorem in proving anti-symmetry.)

Exercise \(\PageIndex{4}\)

We can inject \(\mathbb{Q}\) into \(\mathbb{Z}\) by sending \(± \dfrac{a}{b}\) to \(±2^a 3^b\). Use this and another obvious injection to (in light of the C-B-S theorem) reaffirm the equivalence of these sets.

1. This theorem has been known for many years as the Schröder-Bernstein theorem, but, lately, has had Cantor’s name added as well. Since Cantor proved the result before the other gentlemen this is fitting. It is also known as the Cantor-Bernstein theorem (leaving out Schröder) which doesn’t seem very nice.

2. We first encountered this proof in a Wikipedia article [3] .

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AP®︎/College Biology

Course: ap®︎/college biology   >   unit 5, variations on mendel's laws (overview).

• Thomas Hunt Morgan and fruit flies
• The chromosomal basis of inheritance
• X-linked inheritance
• Genetic linkage & mapping
• Pedigree for determining probability of exhibiting sex linked recessive trait
• Pedigrees review
• Extranuclear inheritance 1
• Inheritance of mitochondrial and chloroplast DNA
• Non-Mendelian genetics

Review: Mendel's basic model

• Heritable traits are determined by heritable factors, now called genes . Genes come in pairs (that is, are present in two copies in an organism).
• Genes come in different versions, now called alleles . When an organism has two different alleles of a gene, one (the dominant allele) will hide the presence of the other (the recessive allele) and determine appearance.
• During gamete production, each egg or sperm cell receives just one of the two gene copies present in the organism, and the copy allocated to each gamete is random ( law of segregation ).
• Genes for different traits are inherited independently of one another ( law of independent assortment ).

Variations involving single genes

• Multiple alleles. Mendel studied just two alleles of his pea genes, but real populations often have multiple alleles of a given gene.
• Incomplete dominance. Two alleles may produce an intermediate phenotype when both are present, rather than one fully determining the phenotype.
• Codominance . Two alleles may be simultaneously expressed when both are present, rather than one fully determining the phenotype.
• Pleiotropy. Some genes affect many different characteristics, not just a single characteristic.
• Lethal alleles. Some genes have alleles that prevent survival when homozygous or heterozygous.
• Sex linkage. Genes carried on sex chromosomes, such as the X chromosome of humans, show different inheritance patterns than genes on autosomal (non-sex) chromosomes.
• Multiple alleles, incomplete dominance, and codominance (article)
• Pleiotropy and lethal alleles (article)
• Sex-linked traits (video)

Variations involving multiple genes

• Complementary genes. Recessive alleles of two different genes may give the same phenotype.
• Epistasis. The alleles of one gene may mask or conceal the alleles of another gene.

Polygenic inheritance and environmental effects

• Polygenic inheritance. Some characteristics are polygenic , meaning that they’re controlled by a number of different genes. In polygenic inheritance, traits often form a phenotypic spectrum rather than falling into clear-cut categories.
• Environmental effects. Most real-world characteristics are determined not just by genotype, but also by environmental factors that influence how genotype is translated into phenotype.
• Polygenic inheritance and environmental effects (article)

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Social Dominance Orientation (SDO)

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• First Online: 01 January 2021
• pp 7676–7684
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• Thomas Haarklau Kleppestø 3 ,
• Nikolai Haahjem Eftedal 3 &
• Lotte Thomsen 3 , 4  

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Between-group hierarchy ; Dominance ; Evolutionary political psychology ; Political attitudes ; Prejudice ; Sex differences ; Warfare

Social dominance orientation is a measurement of “the general desire to establish and maintain hierarchically structured intergroup relations regardless of the position of one’s own group(s) within this hierarchy” (Sidanius et al. 2016 , p. 152).

Introduction

Human societies tend to structure themselves as group-based social hierarchies such that some groups enjoy greater access to fitness-relevant resources such as prestige, wealth, social status, healthcare, food, homes, mates, and so on. Social dominance theory (SDT, Sidanius and Pratto 1999 ) asks the questions why and how group-based hierarchies are continuously reproduced, at least among surplus-producing societies. The theoretical framework spans macrostructural, institutional, ideological, social role, individual, and behavioral genetic levels of analysis to address this question and...

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Department of Psychology, University of Oslo, Oslo, Norway

Thomas Haarklau Kleppestø, Nikolai Haahjem Eftedal & Lotte Thomsen

Department of Political Science, Aarhus University, Aarhus, Denmark

Lotte Thomsen

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Kleppestø, T.H., Eftedal, N.H., Thomsen, L. (2021). Social Dominance Orientation (SDO). In: Shackelford, T.K., Weekes-Shackelford, V.A. (eds) Encyclopedia of Evolutionary Psychological Science. Springer, Cham. https://doi.org/10.1007/978-3-319-19650-3_2602

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DOI : https://doi.org/10.1007/978-3-319-19650-3_2602

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