## Results: 3

1.

Figure 3. From: Epistasis from functional dependence of fitness on underlying traits.

Numerically computed epistasis distributions show a generic negative trend for all possible proportions of beneficial mutations. Each bell-shaped histogram corresponds to the distribution of epistasis at a given fraction of beneficial mutations (

*ρ*). For visual clarity, bars associated with negative*ɛ*are depicted in light grey, while bars for positive*ɛ*are depicted in dark grey. The front slice (*ρ*= 0) is the same distribution shown in figure 2*d*. (*a*) The concave shape for the negative*ɛ*bars across different values of*ρ*indicates that the bias towards negative*ɛ*increases as the portion of beneficial allele moves away from 0.5. (*b*) Negative epistasis is more likely to occur when the single mutants are dominated by mostly beneficial (*ρ*≫ 0.5) of mostly deleterious alleles (*ρ*≪ 0.5).2.

Figure 1. From: Epistasis from functional dependence of fitness on underlying traits.

Schematic depiction of how we quantify epistasis relative to a fitness function that depends on two quantitative traits, or phenotypes. (

*a*) Two alleles or genetic perturbations*i*and*j*are assumed to potentially affect multiple traits, here*X*and*Y*(‘low-level traits’). The phenomenon in which a genetic perturbation affects multiple traits is called pleiotropy. Here we assume that there is no epistasis at the level of the individual traits*X*and*Y*. A ‘high-level trait’ (e.g. fitness*f*) is defined as a function*F*of the two traits*X*and*Y*. These assumptions allow us to predict how the functional shape of*F*affects epistasis between the two perturbations. Without any knowledge of this internal structure (dashed box), the presence of epistasis could only be measured experimentally, but not inferred mathematically. (*b*) The same model as described above, in the absence of pleiotropy. In this case, perturbations*i*and*j*affect each a single trait, i.e.*X*and*Y*respectively, and can be thought of acting on different modules. Depending on the function*F*, this may still lead to epistasis.3.

Figure 2. From: Epistasis from functional dependence of fitness on underlying traits.

Estimating epistasis through a geometrical representation of perturbations in phenotype space. (

*a*) The (*λ*,*θ*) plane, a geometrical representation of possible mutant alleles in a benefit–cost model of fitness. Any allele (e.g.*i*) can be represented as a point with coordinates (*λ*,_{i}*θ*) corresponding to the multiplicative alterations of the benefit and cost, respectively. We assume that both_{i}*λ*and*θ*can have values between zero and*W*. Throughout the paper, we assume*W*= 2, so that beneficial and deleterious mutations have equal chance of being chosen when sampling uniformly. The (*λ*,*θ*) plane is divided into four regions by the neutrality line (corresponding to mutants with fitness equal to the wild-type) and the isochange line (corresponding to mutations such that*λ*=_{i}*θ*). The intersection between these two lines (i.e. the point (_{i}*λ*,*θ*) = (1,1)) corresponds to the wild-type strain.*B*_{a}is the area containing beneficial alleles above the isochange line;*B*_{u}is the area containing beneficial alleles under the isochange line.*D*_{a}and*D*_{u}are similarly defined for deleterious alleles. The combination of two alleles both lying above the isochange line will give rise to negative*ɛ*, as evident from equation (3.1). In general, the sign of*ɛ*depends on the chance of selecting alleles from different regions in the (*λ*,*θ*) plane. The maximum value of*B*_{u}= (*W*− 1)^{2}/2 occurs when the slope of the neutrality line is zero (*c*_{0}= 0). The corresponding*B*_{a}in this situation is*B*_{u}+ (*W*− 1). When we increase the slope,*B*_{u}decreases (while*B*_{a}increases) monotonically as*c*_{0}goes up, until*B*_{u}reaches its minimum value at zero when*c*_{0}=*b*_{0}(slope of neutrality line = 1). Thus, it is always*B*_{a}>*B*_{u}. (*b*) Without imposing any constraint on whether mutations are beneficial or deleterious the regions above and under the isochange line have equal chance to occur (inset), leading to an unbiased epistasis distribution. (*c*,*d*) Negative bias between strictly beneficial alleles (*c*, region*B*_{a}>*B*_{u}shaded in inset) and between strictly deleterious alleles (*d*, region*D*_{u}>*D*_{a}shaded in inset) can be demonstrated analytically, and is confirmed here by simulations (see the electronic supplementary material, §A).