Mutation load under additive fitness effects

Summary Under the traditional mutation load model based on multiplicative fitness effects, the load in a population is 1−e−U, where U is the genomic deleterious mutation rate. Because this load becomes high under large U, synergistic epistasis has been proposed as one possible means of reducing the load. However, experiments on model organisms attempting to detect synergistic epistasis have often focused on a quadratic fitness model, with the resulting general conclusion being that epistasis is neither common nor strong. Here, I present a model of additive fitness effects and show that, unlike multiplicative effects, the equilibrium frequency of an allele under additivity is dependent on the average absolute fitness of the population. The additive model then results in a load of U/(U +1), which is much lower than 1−e−U for large U. Numerical iterations demonstrate that this analytic derivation holds as a good approximation under biologically relevant values of selection coefficients and U. Additionally, regressions onto Drosophila mutation accumulation data suggest that the common method of inferring epistasis by detecting large quadratic terms from regressions is not always necessary, as the additive model fits the data well and results in synergistic epistasis. Furthermore, the additive model gives a much larger reduction in load than the quadratic model when predicted from the same data, indicating that it is important to consider this additive model in addition to the quadratic model when inferring epistasis from mutation accumulation data.


Measure of skewness created by the numerical iterations
Skewness of deleterious mutation number was measured as a deviation from that predicted under a Poisson distribution (which is the distribution when there is independence among sites). This was accomplished in the numerical iterations by calculating the average number of deleterious mutations per genome ( ̅ ) for a given and . In this case, where is the number of deleterious mutations in an individual and ( ) is the frequency of individuals in the iteration at equilibrium with mutations.
If a distribution is Poisson (as is the case with independence among sites) the skewness should be equal to The actual skew in the distribution at equilibrium was calculated by If there is independence among sites, the ratio � � = = 1. If the actual skewness of the mutations in the distribution deviates from predicted, the ratio will deviate from 1. Figure S2 shows this ration, � �, for the values of analyzed in the iterations. Table 1 Numerical iterations were performed using the fitness functions in Table 1 to predict �. Iterations were run using the quadratic fitness model

Numerical iterations using regression coefficients from
as in Charlesworth (1990), with = ℎ and = 2ℎ 2 , where and are the linear and quadratic regression coefficients, respectively, and ℎ is the dominance coefficient. Here I use and from the quadratic regressions in Table 1, and ℎ = 0.2 as in Charlesworth (1990). Similarly, numerical iterations were also run using the additive fitness model ( ) = 1 − (17) Here, = ℎ, where is the linear regression coefficient from the additive regressions in Table  1. Lastly, numerical iterations were run using the multiplicative fitness model ( ) = − (18) where again = ℎ and is the linear regression coefficient from the multiplicative regressions.
These fitness models in Eqs (16) -(18) were then used to modify − in Eq (15). Specifically, for the quadratic model and Pr( ) and Pr( ) are the same as described above for the numerical iterations. Similarly, for the multiplicative case we have where Eq (19) and (20) were used for the value of ( − ) in Eq (15) and iterations were run with a of 2.2. For the additive model, was obtained as described for Eq (17) and used in the additive iterations as described above. Because Eqs (19) and (20) are fitness functions that never reach a fitness of zero, and therefore the in the iterations will approach infinity, a truncation point of 650 mutations per genome was selected where fitness would equal zero. In the iterations the frequency of individuals with 649 mutations was on the order of 10 −38 or lower, demonstrating that this truncation point does not impact the accuracy of the iteration.    Table S1. P-values from comparisons of the regressions for the three models (multiplicative, additive, and quadratic) for each of the three experiments (CH, PQ, and RT) from Mukai et al. (1972). Regressions were compared using var.test in R 3.0.3.

File S2
The following alternative method of deriving load equations under additive fitness is based on derivations provided to me by Brian Charlesworth (personal communication). This method takes into account the departure in the variance of mutation number from Poisson that occurs in 1 generation of selection.
Take a trait , where p( ) is the probability distribution function and follows a standardized Gaussian distribution, with a mean [ ] = 0 anda variance [ ] = 1 as in Shnol and Kondrashov (1993). From Eq (2) in Shnol and Kondrashov (1993), the variance of after selection, [ ], is where = ∫ ( ) ( ) . With additive fitness effects, the fitness function can be defined as ( ) = + . The average fitness ( �) is given by 0 and will therefore be 0 = � = .
Additionally, 1 = ∫ ( ) ( ) will give 1 = and 2 = ∫ 2 ( ) ( ) will give 2 = . Therefore, The change in variance due to selection is equal to (1 − ̅ ) 2 Assuming that the distribution of before selection is Poisson, the variance before selection will be equal to ̅ . Eqs (9) and (10) in Charlesworth (1990) indicate that ∆ represents the departure of the variance from Poisson. That is, ̅ − = ∆, so that Again assuming a normal distribution, we know that the average change in fitness due to selection is 9SI in accordance with Fisher's fundamental theorem of natural selection. Because = 2 and as above � = (1 − ̅ ) so that ∆ � = (∆ ̅ ). Eq (24) can then can be written as which represents the decrease in average number of deleterious mutations per individual due to selection. At equilibrium, the decrease in deleterious mutations due to selection will be equal to the increase due to new mutations, so that Substituting (26) into (23) gives ̅ − = 2 . Therefore, replacing in (26) with ̅ − 2 and solving (26) for ̅ gives and therefore � is Eqs (27) and (28) are similar to (12) and (13) in the main text except with the addition of + and -, respectively. Eq (28) helps capture the decrease in fitness that occurs as the value of increases. This derivation assumes that the distribution of mutation number is Guassian, even after selection. However, as the numerical iterations indicate ( Figure S2), additive effects create skewness in the distribution. Also, similar to the derivations for (12) and (13), these derivations also ignore the effect of the truncation point where 1 − ≤ 0 gives a fitness of zero. As can be seen in Figure S4, though fitness decreases with increasing s using (28), like Eq (13), it also quickly loses accuracy under larger .