Detecting High-Order Epistasis in Nonlinear Genotype-Phenotype Maps

Article excerpt

(ProQuest: ... denotes formulae omitted.)

RECENT analyses of genotype-phenotype maps have revealed "high-order" epistasis-that is, interactions between three, four, and even more mutations (Ritchie et al. 2001; Segrè et al 2005; Xu et al 2005; Tsai et al. 2007; Imielinski and Belta 2008; Matsuura et al. 2009; da Silva et al. 2010; Pettersson et al 2011; Wang et al. 2012; Hu et al. 2013; Weinreich et al. 2013; Sun et al. 2014; Anderson et al. 2015; Yokoyama et al. 2015). The importance of these interactions for understanding biological systems and their evolution is the subject of current debate (Weinreich et al. 2013; Poelwijk et al. 2016). Can they be interpreted as specific, biological interactions between loci? Or are they misleading statistical correlations?

We set out to tackle one potential source of spurious epistasis: a mismatch between the "scale" of the map and the scale of the model used to dissect epistasis (Fisher 1918; Rothman et al. 1980; Frankel and Schork 1996; Cordell 2002; Phillips 2008; Szendro et al. 2013). The scale defines how to combine mutational effects. On a linear scale, the effects of individual mutations are added. On a multiplicative scale, the effects of mutations are multiplied. Other, arbitrarily complex scales, are also possible (Rokyta etal 2011; Schenk etal. 2013; Blanquart 2014).

Application of a linear model to a nonlinear map will lead to apparent epistasis (Fisher 1918; Rothman etal. 1980; Frankel and Schork 1996; Cordell 2002; Phillips 2008; Szendro et al. 2013). Consider a map with independent, multiplicative mutations. Analysis with a multiplicative model will give no epistasis. In contrast, analysis with a linear model will give epistatic coefficients to account for the multiplicative nonlinearity (Cordell 2002; Phillips 2008). Epistasis arising from a mismatch in scale is mathematically valid, but obscures a key feature of the map: its scale. It is also not parsimonious, as it uses many coefficients to describe a potentially simple, nonlinear function. Finally, it can be misleading because these epistatic coefficients partition global information about the nonlinear scale into (apparently) specific interactions between mutations.

Most high-order epistasis models assume a linear scale (or a multiplicative scale transformed onto a linear scale) (Heckendorn and Whitley 1999; Szendro et al. 2013; Weinreich et al. 2013; Poelwijk et al. 2016). These models sum the independent effects of mutations to predict multimutation phenotypes. Epistatic coefficients account for the difference between the observed phenotypes and the phenotypes predicted by summing mutational effects. The epistatic coefficients that result are, by construction, on the same linear scale (Heckendorn and Whitley 1999; Weinreich et al. 2013; Poelwijk et al. 2016).

Because the underlying scale ofgenotype-phenotype maps is not known a priori, the interpretation of high-order epistasis extracted on a linear scale is unclear. If a nonlinear scale can be found that removes high-order epistasis, it would suggest that high-order epistasis is spurious: a highly complex description of a simple, nonlinear system. In contrast, if no such scale can be found, high-order epistasis provides a window into the profound complexity of genotype-phenotype maps.

In this article, we set out to estimate the nonlinear scales of experimental genotype-phenotype maps. We then account for these scales in the analysis of high-order epistasis. We took our inspiration from the treatment of multiplicative maps, which can be transformed into linear maps using a log transform. Along these same lines, we set out to transform genotypephenotype maps with arbitrary, nonlinear scales onto a linear scale for analysis of high-order epistasis. We develop our methodology using simulations and then apply it to experimentally measured genotype-phenotype maps.

Methods

Experimental data sets

We collected a set of published genotype-phenotype maps for which high-order epistasis had been reported previously. …

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