# Nonadditive Variation Is Likely Insignificant At The Population Level

With the question of ‘missing heritability’ (Clark & Cooper 2010; Eichler et. al 2010Manolio et. al 2009; Nolte et. al 2017) haunting geneticists, epidemiologists and psychologists alike, some have turned to question the foundations of quantitative genetics itself (Murphy 1979; Nelson et. al 2013). The most prominent critique of the typical polygenic additive model (Visscher & Wray 2015) involves the invocation of nonadditive genetic variation contributing to phenotypes (Bubb & Queitsch 2015Zuk et. al 2012).

## Theory

### Evolution on Epistasis

Ávila et. al (2014) models traits under different types of selection and shows that the contribution of epistatic variants is expected to be low, especially under stabilizing selection. Hermani et. al (2013), on the other hand, proposes a model where linkage disequilibrium can easily inflate additive and deflate nonadditive components of variance. Finally, Hill (2017) suggests that drift can ‘convert’ epistatic contributions to additive variance over time.

### Epistasis on Evolution

The effect of epistatis between different loci has long been a controversy in evolutionary and quantitative genetics (Csillery et. al 2018).

Some models suggest little effect of epistasis on different types of selection (Barton 2017; Crow 2010; McCandlish et. al 2014), while others suggest that even subtle epistasis can influence population structure and subsequent evolution (Neher & Shraiman 2009), with higher-order interactions being even more substantial (Sailer & Harms 2017) and the idea that epistasis has little influence on selection is the result of conceptual confusion (Hansen 2013) or measurement issues (Hansen 2015). Other researchers solve the question of how selection can generate variation (Gerhart 2005) by showing how epistasis in combination with natural selection can alter the path and frequency of mutations (Jones et. al 2014), as well as the speed (Rouzic & Álvarez-Castro 2016). Finally, others suggest that the relationship between selection and epistasis depends on the fitness effects of the alleles in question, the number of chromosomes for segregation and the magnitude of the selection (Paixão & Barton 2016)

Some empirical results show that while epistasis does play a role in modulating and altering the additive components, including it can cause increased error in modeling (Forneris et. al 2017). In species with larger effective population sizes, epistasis may also substantially impact the trajectory and magnitude of evolution (Arnold et. al 2018). Additionally, epistasis seems to be particularly important in molecular evolution (Breen et. al 2012).

### Population Genetics

Zuk et. al (2012) reignited the question of nonadditive variation in phenotypic variance by suggesting the ‘missing heritability’ may be found in high-order interactions. Other models suggest that gene interactions may actually contribute to additive variance at the population level, with the nonadditive part is relatively small due to high heterozygosity, despite there being ubiquitous nonadditive action at the genetic level (Hill et. al 2008Mäki-Tanila & Hill 2014), meaning that while it may not contain the missing heritability, it limits individual prediction (Sackton & Hartl 2016).

## Family Data

We also have considerable data from family studies [1] on the contribution of nonadditive variation to population phenotypic variance. For instance, Kaplanis et. al (2018) exploited massive volumes of family tree data to estimate additive, dominance and epistatic components to variance, finding small (4%) dominance and negligible (~0%) epistatic contributions [2].

Several reviews of twin data show that the twin studies are consistent with very little nonadditive contribution to phenotypic variance (Hill et. al 2008Lakhani et. al 2019; Polderman et. al 2015). Hill et. al‘s rough summary of the data indicated most traits were consistent with a $2r_{DZ}-r_{MZ}$ figure near 0, while Lakhani et. al (2019) found ~52% of traits were consistent with an additive model compared to the 62.5% in Polderman et. al (2015). Some specific traits are likely to have nonadditive components, however, like antisocial behavior (Rhee & Waldman 2002), age at menarche (Treloar & Martin 1990), pedophilic behavior (Alanko et. al 2013), BMI and lipid traits (Herzig et. al 2018).

There is the possibility that dominance and other nonadditive components can be ‘masked’ by shared environmental variance (Chen et. al 2015; Hill et. al 2008), though for many traits it seems that this explanation doesn’t suffice (Zaitlen et. al 2013).

## Genomic Data

Some research has suggested that up to 60% of total heritability may be explained by additive common variants alone (Golan et. al 2012), suggesting that there is little room left for epistasis.

### Modeling

Detailed models and statistical methods are under development for the detection of epistatic influences on phenotypes in statistical genetics (Crawford et. al 2017; Goudey et. al 2013Lewinger et. al 2013; Liu et. al 2012McKinney & Pajewski 2012; Prabhu & Pe’er 2012; Stephan et. al 2015Wei et. al 2014). The search for epistasis is extremely computationally intensive and will likely require the incorporation of biological, genetic and chemical knowledge (Ritchie 2014) as well as extremely large sample sizes (Gauderman 2002). However, many tests for epistasis have extremely high rates of false positives (Stephan et. al 2015; Wei et. al 2014Wood et. al 2014) due to statistical artifacts (Fish et. al 2016), and recent work has shown that the presence of linkage disequilibrium can generate phantom epistasis (de los Campos et. al 2019) as well as mask it (Joiret et. al 2019). Finally, it may be that extreme computational power and complex modeling will be necessary to distinguish epistasis from randomness and chance [3] (Sailer & Harms 2018).

### Humans

In humans, both gene-environment interactions and gene-gene interactions have been purportedly associated with phenotypic traits, with gene-gene interactions explain ~4.3% of variance, and in some cases interaction explaining more than the additive part (Brown et. al 2014). Some argue that epistasis is crucial to understanding the development of complex disease (Mackay & Moore 2014), while others believe their contributions are negligible (Hall & Ebert 2013). Some associations are even gene-gene-environment (GxGxE) interactions (Sullivan et. al 2013), meaning that that polygenic scores could have substantially limited predictive validity (Clayton 2009).

There have been numerous associations reported in the literature, such as colorectal cancer (Jiao et. al 2012) and autoimmune diseases (Rose & Bell 2012), although their replicability is still in question (Wei et. al 2014), with some arguing their contribution has been overestimated (Hall & Ebert 2013).

Other forms of statistical modeling have concluded that dominance variation contributes little to the missing heritability (Nolte et. al 2017; Sanjak et. al 2016Zhu et. al 2015)

### Model Organisms

The contribution of high-order epistatic interactions has received recent interest in the genetics community (Taylor & Ehrenreich 2015), with research in C. elegans (Gaertner et. al 2012), Drosophilia [4] (Corbett-Detig et. al 2013; Huang et. al 2012) and other prototypical model organisms.

#### Plants and Fungi

In plants, there is some accumulating evidence that epistasis and other higher-order interactions may be involved in varying processes.

In apples, both dominance and epistasis have significant contributions to phenotypic variance (Kumar et. al 2015). In yeast, a classic model organism, there have been conflicting reports. Young & Durbin (2014) report substantial two-way interaction and higher-order interactions, while Edwards et. al (2014) have found that idiosyncratic interactions between different genomic elements can explain the missing heritability, but Bloom et. al (2015) reports that the contribution of nonadditive variance is typically much smaller than that of additive variance, though a later paper by the same group shows that nonadditive variance is necessary in predicting extreme phenotypes (Forsberg et. al 2016) and may actually increase prediction in some species (Dudley & Johnson 2013).

For instance, Arabidopsis thaliana has been shown to have some significant gene-by-gene-by environment interactions (Kerwin et. al 2017), and the same has been reported in cotton (Jia et. al 2014).

Despite somewhat widespread findings of nonadditive contribution at individual loci (Kelly & Mojica 2011), Monnahan & Kelly (2015) report that most variance at the population level is likely to be additive.

#### Mammals

Research in other mammals is less frequent and harder to find as well as replicate, but there have been some studies showing the presence of epistasis in mice (Casellas et. al 2014), pigs (Crooks & Guo 2017; Reiner et. al 1999), and cattle (Khatib et. al 2009).

#### Footnotes

[1] Though in the face of universal violation of family study assumptions, we should remain a bit skeptical. For instance, as mentioned below, common environment, nonadditive and assortative mating components can all mask each other.

[2] They only tested two-way and three-way epistatic models, so in theory we should only be able to rule out those, and 4+ interaction epistasis could still contribute. However, these models are evolutionarily unlikely and computationally intractable.

[3] This may be a more general consequence of the unpredictability of evolution (Sailer & Harms 2017).

[4] Note, however, that Huang et. al (2012)‘s results are susceptible to subtle biases (Campbell et. al 2018).