The strength of genetic interactions scales weakly with mutational effects
© Velenich et al.; licensee BioMed Central Ltd. 2013
Received: 11 February 2013
Accepted: 26 July 2013
Published: 26 July 2013
Genetic interactions pervade every aspect of biology, from evolutionary theory, where they determine the accessibility of evolutionary paths, to medicine, where they can contribute to complex genetic diseases. Until very recently, studies on epistatic interactions have been based on a handful of mutations, providing at best anecdotal evidence about the frequency and the typical strength of genetic interactions. In this study, we analyze a publicly available dataset that contains the growth rates of over five million double knockout mutants of the yeast Saccharomyces cerevisiae.
We discuss a geometric definition of epistasis that reveals a simple and surprisingly weak scaling law for the characteristic strength of genetic interactions as a function of the effects of the mutations being combined. We then utilized this scaling to quantify the roughness of naturally occurring fitness landscapes. Finally, we show how the observed roughness differs from what is predicted by Fisher's geometric model of epistasis, and discuss the consequences for evolutionary dynamics.
Although epistatic interactions between specific genes remain largely unpredictable, the statistical properties of an ensemble of interactions can display conspicuous regularities and be described by simple mathematical laws. By exploiting the amount of data produced by modern high-throughput techniques, it is now possible to thoroughly test the predictions of theoretical models of genetic interactions and to build informed computational models of evolution on realistic fitness landscapes.
KeywordsEpistasis Evolution Fitness landscapes Genetic interactions Yeast
Genetic interactions  have shaped the evolutionary history of life on earth. They have been found to limit the accessibility of evolutionary paths , to confine populations to suboptimal evolutionary states and, on larger time scales, to control the rate of speciation . Epistatic interactions can also be relevant to the development of complex human diseases such as diabetes . Complex traits and diseases are determined by a multiplicity of genomic loci , whose independent effects and interactions  are often necessary to understand the phenotype of interest. Despite the broad implications of epistatic interactions, a quantitative characterization of their typical strength is still lacking. In this study, we consider growth rate in yeast as an example of a complex trait modulated by genetic interactions.
Previous studies [7–10] on the relation between the growth effects of a mutation and its epistatic interactions have often been based on a handful of mutations, and only in recent years has anecdotal evidence started being replaced by robust statements based on large data sets. Perhaps the most impressive of these datasets is the one made publicly available with the publication of the article entitled 'The genetic landscape of a cell' by Costanzo et al. . The genome of the budding yeast Saccharomyce cerevisiae includes approximately 6,000 genes, about 1,000 of which are essential. Viable mutants can be constructed by knocking out any of the approximately 5,000 non-essential genes, by reducing the expression of the essential genes, or by partially compromising the functionality of the gene products. The dataset (see Additional file 1, Figure S1) has been compiled with the growth rates of about 5.4 million double knockout mutants, a sizable fraction of all possible double knockout mutants in yeast. Supported by the Costanzo et al. dataset, we consider the fundamental question of whether mutations with larger effects have stronger genetic interactions.
Results and discussion
An unbiased definition of genetic interactions
depends only on which pair of genes is considered, being a geometric measure for the 'curvature' of the fitness landscape (Figure 1b).
The definition of E has found some favor in the theoretical literature [7, 16], but it is not routinely used to analyze experimental data apart from rare exceptions [8, 17]. Its main drawback is that synthetic lethals have a log growth rate of -∞, and require a separate although simpler analysis in which lethal interactions can simply be counted. The definition of E proves instead to be extremely valuable when quantifying the strength of non-lethal genetic interactions.
Epistatic interactions scale weakly with mutational effects
With the appropriate definition of epistasis, a simple relation between the growth rate effects of two mutations and the expected strength of their interaction emerges.
that is, the variance of the random variable E relative to the bin labeled by growth rates G01 and G10. In the rest of the paper we will refer to such variance as var(G01, G10), emphasizing that the variance in the strength of epistatic interactions is, eventually, a function of G01 and G10 (Figure 2a). The square root of the variance, σ(G01, G10), then represents the expected strength of epistasis as a function of the independently varying effects of the two single knockouts. A natural expectation for the dependence of epistasis on the effect of the combined mutations comes from rescaling Figure 1a; if all the log growth effects of single and double knockouts increase by a factor of two, then the strength of epistasis should also increase by a factor of two. Unexpectedly, however, when combining deleterious mutations, the strength of epistatic interactions does grow with the effects of the mutations that are combined, but the dependence is much weaker; when the effect of both single knockouts is doubled, the strength of epistasis increases only by a factor of √2 (Figure 2).
The scaling described above is seen only for deleterious knockouts. When combining the beneficial knockouts in the dataset instead, the strength of epistasis is close to zero (Figure 2c, inset). This might be because the slightly beneficial knockouts are not adaptive mutations, but simply remove genes that are not needed in the conditions chosen for the experiment, so that their interactions are likely to be negligible. However, in apparent contrast to this observation, recent studies [8, 18] on adaptive mutations in Escherichia coli suggest that genetic interactions between adaptive mutations are mostly negative. In fact, during adaptation, the prevalence of negative interactions is likely to be caused by biased sampling, because the mutations that fix in the population are likely to be the ones that solve environmental or biological challenges for an organism. Diminishing returns arise because the appearance of multiple 'solutions' to the same challenge is not necessarily preferable over the presence of a single solution. Rather than focusing on mutations that fix during a bout of adaptation, the Costanzo et al. dataset includes a large fraction of all possible pairs of genes in the yeast genome. Because for most pairs the two genes are involved in unrelated biological processes, interactions are often vanishingly small. We did observe, however, that the distribution of epistatic interactions is asymmetric, with a heavy tail of deleterious interactions (Figure 2d).
Experimental uncertainty generates spurious epistatic interactions
When inferring genetic interactions from experimental data, it is important to take into account that each measured growth rate is affected by some uncertainty, and that measurement errors in the growth rates could erroneously be interpreted as genetic interactions. Importantly, for each single and double mutant, the Costanzo et al. dataset provides the mean growth rate together with its estimated experimental uncertainty (the growth rate of each mutant being measured at least four times).
Comparison between theory and experiment
The scaling of epistasis observed in the Costanzo et al. dataset (Figure 2) is in sharp contrast to the predictions of Fisher's geometric model , a popular model of epistasis in which genetic interactions emerge from geometry. As we saw, when the effects of the two knockouts are similar (G01 = G10 = G), the variance of epistasis is approximately proportional to G. By contrast, in the Fisher's model, the variance var(G, G) would grow faster than G2 (Figure 2c; see Additional file 1, Supplementary text 2), a much stronger dependence than the linear dependence observed experimentally.
Notice that it is not unlikely that epistasis will cancel the effect of the second mutation, so that the growth rate of the double knockout mutant is greater than 0.95, that is, greater than the growth rate of either of the single knockout mutants.
In this case, a deviation from the null model that is greater than three standard deviations would be needed for the double knockout mutant to have a growth rate greater than that of the single knockout (0.60), making the event extremely unlikely.
Epistasis constrains the evolutionary dynamics
The previous section provided two examples of reciprocal sign epistasis, realized when two deleterious mutations produce a double mutant that is fitter than either of the two single mutants (Figure 4a). In those cases, a fitness valley limits the evolutionary accessibility of the fitter double mutant, and only on longer time scales may the simultaneous appearance of two mutations [20, 21] drive a population to the new local fitness maximum. In this context, the scaling behavior of epistasis is of great importance, because it determines the number and the topology of the evolutionarily accessible paths [2, 22, 23], ultimately affecting the possible outcomes of the evolutionary process.
In particular, if σ(G, G) is proportional to G, then the probability of observing sign epistasis is independent of G. The Fisher's model implies a super-linear dependence of σ(G, G) on G, thus predicting a greater probability of observing sign epistasis among mutations with strong effects. Instead, if the scaling of σ(G, G) is proportional to √G (Figure 2), then sign epistasis is more likely to occur among mutations with small effects (Figure 3b). When the relative growth rate effects of the single knockouts are small (<2 to 3%), experimental uncertainty prevents us from pinpointing which pairs of genes are epistatic. This does not mean, however, that mutations with small effects do not interact. Assuming that the scaling of epistasis we measured directly for mutations with intermediate and large effects extends to mutations with small effects, a consequence of the observed scaling of epistasis is the roughening of the local fitness landscape in the proximity of an evolutionary optimum; when the fitness effects of available mutations become small , epistatic interactions become increasingly relevant [25, 26], reducing the accessibility of evolutionary paths and further slowing down the rate of adaptation [27, 28]. The evolutionary dynamics on correlated fitness landscapes [10, 29] with the realistic correlations described here certainly deserves further experimental and theoretical investigation.
The scaling of genetic interactions may be generic
A comparison between different definitions of epistasis
Finally, it is important to emphasize that the traditional definition of epistasis remains slightly more successful at discovering the functional relations between genes, as cataloged in the GO database (see Additional file 1, Figure S6). Part of the reason for this could be that some of those functional characterizations were suggested by the traditional definition of epistasis in the first place. It is certainly true, however, that many of the top-ranking interactions according to the geometric definition of epistasis involve single and double mutants with small growth rates; for those mutants, experimental noise is relatively large, and this may cause a few weakly interacting pairs to be incorrectly ranked as strongly interacting. It is likely that the experimental protocols could be easily adjusted to reduce the relative uncertainty on the growth rate of especially slow-growing mutants to avoid this issue (for example, by allowing for a much longer time for growth or by measuring the growth rates of additional replicates).
We analyzed the growth rates of about five million double mutants in the dataset associated with the work by Costanzo et al. We characterized how the strength of genetic interactions depends on the growth effects of the mutations being combined, and found a weaker dependence than that predicted by current theoretical models. Although the results were obtained mainly from entire gene knockouts, there is some evidence that the observed scaling might extend to the interactions between single point mutations. The scaling of epistasis might or might not be generic [35, 36]; important drivers could be the harshness of the environment , details about the evolutionary history [38–40], sexual versus asexual reproduction  and, perhaps most importantly, metabolic [42–45] and genetic complexity [46, 47]. In general, the experimentally observed scaling suggests a previously unexplored class of correlated fitness landscapes with tunable roughness, in which epistasis depends explicitly on the effects of the mutations being combined.
A clear limitation of our discussion is that only pair interactions were considered. Although high-throughput experiments will provide data on higher-order interactions, a solid understanding of pair interactions remains necessary before addressing n-mutation interactions. A genuine three-mutation interaction, for instance, should be defined as the unexplained deviation from what can be computed by combining the effects of all relevant mutations and their pair interactions [10, 48], perhaps using linear fits within the additive null model for log growth rates.
The results we present here were based on a geometric definition of epistasis. We compared this definition with a more standard definition, highlighting the desirable mathematical properties of the geometric definition and the simple phenomenological relations it produces.
In conclusion, although each epistatic interaction between specific genes depends on biological details and remains largely unpredictable from first principles, we have shown that the statistical properties of an ensemble of interactions can display conspicuous regularities, and can be described by simple mathematical laws.
Materials and methods
The Costanzo et al. dataset is publicly available . The file sgadata_costanzo2009_rawdata_101120.txt.gz was downloaded on August 17, 2010 and analyzed with Mathematica (code available at the Gore laboratory website ). We restricted our analysis to double knockout mutants whose growth rates were positive numerical values and for which the growth rates of both single mutants were numerical values (see Additional file 1, Figure S1). Some genes appear in the dataset both as query and array genes; care was taken to avoid double counting.
The exponentially growing intervals used for the binning of the log growth rate effects were defined as [-2n, -2n-1] for an appropriate range of integer n's. Owing to the rarity of extremely deleterious mutations, bins for positive n's contained only a few data points, while bins with large negative n's were extremely small. In the figures we reported only bins for n = -7 to 0, containing log growth rate effects ranging from -20 = -1 to -2-8 = -0.0039 or, alternatively, relative growth rate effects ranging from 2-1 = 0.5 to 2-0.0039 = 0.997. Different choices for the binning sizes and positions did not significantly alter the results of the analysis.
In order to quantify the contribution of experimental uncertainty to epistasis, we generated nine randomized mock datasets. The mean level of noise-generated epistasis in these nine datasets is reported in Figure 4 (dashed lines), and we provide an extensive discussion of the choice of Student's t-distributions to generate the mock datasets from the original dataset (see Additional file 1, Supplementary text 3).
The GO database go_201207-assocdb-tables.tar.gz was downloaded from the GO site  on July 19, 2012. The MySQL database was queried with Python and analyzed Mathematica (code available upon request).
Conflict of interest
The authors declare that they have no conflict of interest.
Decreased abundance by mRNA perturbation
We are grateful to Mingjie Dai for collaboration during the early stages of the study. We thank Kirill Korolev, Pankaj Mehta, and the members of the Gore laboratory for providing comments and advice on the manuscript. This research was funded by an NIH Pathways to Independence Award, NSF CAREER Award, Pew Biomedical Scholars Program, and Alfred P. Sloan Foundation Fellowship.
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