We begin by examining the most general way to express the relationship between genotype and fitness (i.e., to describe the fitness landscape). A map between a quantitative trait (such as fitness), $y$, and the underlying genotype can be expressed as a sum of combinations of $\mathrm{\ell }$ biallelic loci $x_1,x_2,\mathrm{\dots },x_{\mathrm{\ell }}$ that take on values $x_i=\pm 1$ (Hordijk and Stadler, 1998; Neher and Shraiman, 2011; Weinberger, 1991; Szendro et al., 2013; Weinreich et al., 2013):

where $\overline{y}$ is a constant that sets the overall scale of fitness. The symmetric convention $x_i=\pm 1$ for the two allelic variants is less often used than $x_i=0,1$, but it is an equivalent formulation, which we employ here because it will prove more convenient for our purposes (see Poelwijk et al., 2016 for a discussion). The coefficients of terms linear in x_i represent the additive contribution of each locus to the fitness (i.e., its fitness effect averaged across genotypes at all other loci), the higher-order terms quantify epistatic interactions of all orders, and $\overline{y}$ is the average fitness across all possible genotypes. Importantly, Equation (2) makes apparent the idiosyncrasies induced by epistasis: a mutation at a locus with $\mathrm{\ell }$ interacting partners has an effect composed of $2^{\mathrm{\ell }-1}$ contributions.

To explicitly compute the fitness effect of a mutation at locus $i$ on a particular genetic background, we simply flip the sign of x_i, keeping all other x_j constant, and write down the difference in fitness that results. This fitness effect will generally involve a sum over a large number of terms involving the $f$’s in Equation (2). While this may suggest that an analysis of fitness effects via Equation (2) is intractable, the analysis in fact simplifies considerably if the locus has a significant number of independent interactions that contribute to the fitness (i.e., provided that the number of independent, nonzero epistatic terms associated to the locus is large). In this case, we show that the fitness effects of individual mutations decrease linearly with background fitness and the fluctuations around this linear trend are normally distributed. In other words, widespread independent idiosyncratic epistatic interactions lead to the observed patterns of diminishing-returns and increasing-costs epistasis.

We present a derivation of this result in the SI. Here, we explain the key intuition using a heuristic argument. The argument is based on a simple idea: for a well-adapted organism ($y>\overline{y}$) with complex epistatic interactions, a mutation is more likely to disrupt rather than enhance fitness. To be quantitative, consider a highly simplified scenario where some number $N$ of the $f$’s in Equation (2) are ±1 at random and the others are 0. In this case, the fitness of a given genotype is a sum of $N_{+}$ and $N_{-}$ interactions that contribute positively and negatively to the trait, respectively, each with unit magnitude, so that $y=\overline{y}+N_{+}-N_{-}$. When positive and negative interactions balance, the organism is in a ‘neutrally adapted’ state ($y\approx \overline{y}$). By selecting for positive interactions, adaptation generates a bias so that $N_{+}>N_{-}$ and $y>\overline{y}$. If locus $i$ involved in a fraction v_i of all of $N=N_{+}+N_{-}$ interactions is mutated, the effect of the mutation, on average, is to flip the sign of $N_{+}{v}_i$ positive interactions and $N_{-}{v}_i$ negative interactions. The new fitness is then $y_i=y-2N_{+}{v}_i+2N_{-}{v}_i=\overline{y}+(1-2v_i)(y-\overline{y})$ and thus $s_i=y_i-y=-2v_i(y-\overline{y})$. The negative linear relation between the background fitness, $y$, and the fitness effect of the mutation, s_i, is immediately apparent and emerges as a systematic trend simply due to a sampling bias towards positive interactions. Of course, while this relation is true on average, it is possible that locus $i$ affects more or less positive interactions due to sampling fluctuations. Provided only that $N$ is large and the interactions are independent, these fluctuations are approximately Gaussian with magnitude $\sqrt{N{v}_i(1-v_i)}$.

This basic argument holds beyond the simple model with unit interactions. In the more general case, if the mutation is directed from $x_i=-1\to +1$, we show in the SI that its fitness effect, s_i, on a background of fitness $y$ can be written as

where,

and ${\widetilde{ϵ}}_i$ is a genotype and locus-dependent term that is distributed across genotypes with mean zero and variance expressed in terms of the $f$’s from Equation (2) (see SI for details). In the following equation and similar ones henceforth, a summation such as ${\sum }_{j>k\ne i}{f}_{ijk}^2$ is meant to denote a sum over pairs $j,k$, where each pair appears only once and no pair that includes index $i$ appears. Symmetry of the $f$’s w.r.t. interchanged indices is also assumed (e.g., $f_{ijk}=f_{jik}$). The numerator of ${\tilde{v}}_i$ in Equation (4) is proportional to the covariance of fitness effects and background fitness, and the denominator is the variance of background fitness across genotypes. A similar equation for the case $x_i=+1\to -1$ can be derived. The choice of $+1\to -1$ or $-1\to +1$ is simply a matter of convention. If the convention is reversed, the coefficients of odd-order in Equation (2), that is, $f_i,f_{ijk},\mathrm{\dots }$, should also switch signs. It can be easily checked that reversing the signs of these quantities in the expression for ${\tilde{v}}_i$ above leads to the expression for ${\tilde{v}}_i$ when $x_i=+1\to -1$.

Note that in general ${\tilde{v}}_i$ is not guaranteed to be positive and ${\widetilde{ϵ}}_i$ is arbitrary and determined by the genotype-fitness map. However, consistent patterns emerge when locus $i$ has a large number of independent, nonzero epistatic terms and the additive effects $f_1,f_2,\mathrm{\dots }$ of its interacting partners are not much larger than the epistatic terms (defined further below), which we call the WE limit. In the WE limit, ${\widetilde{ϵ}}_i$ is normally distributed across genotypes with variance proportional to ${\tilde{v}}_i(1-{\tilde{v}}_i)$. This follows from the same reasoning as in our heuristic argument with unit interactions above (see SI for details). In addition, ${\tilde{v}}_i$ is typically positive, giving rise to a negative linear trend (i.e., diminishing-returns and increasing-costs). We can see this by taking the third- and higher-order terms in Equation (4) to be zero, in which case ${\tilde{v}}_i$ is positive if ${\sum }_{j\ne i}{f}_{ij}^2>{\sum }_{j\ne i}{f}_j{f}_{ij}$. This will typically be true in the WE limit because we expect ${\sum }_{j\ne i}{f}_{ij}^2$ to scale with the number of interacting partners $\mathrm{\ell }$, while each term in ${\sum }_{j\ne i}{f}_j{f}_{ij}$ can be positive or negative and thus the sum scales as $\sqrt{\mathrm{\ell }}$ if the terms are independent. Thus when locus $i$ has a large number of interacting partners, ${\tilde{v}}_i$ is typically positive unless the magnitude of the additive terms ($a$) is much larger than the magnitude of the epistatic terms ($e$), $a\gg e\sqrt{\mathrm{\ell }}$. This argument is easily extended to the case when the third- and higher-order terms are nonzero (see SI); the upshot is that the bias towards ${\tilde{v}}_i$ positive gets stronger with increasing epistasis.

The conditions for the WE limit are more likely to hold when the number of loci, $\mathrm{\ell }$, that affect the trait is large. Therefore, we expect to generically observe patterns of diminishing-returns and increasing-costs epistasis for a complex trait involving many loci. Importantly, whether we observe a negative linear trend does not depend on the magnitude of a locus’ epistatic interactions relative to its own additive effect, but rather relative to the additive effects of its interacting partners. If we are not in the WE limit, and instead the additive effects dominate (i.e., $a\gg e\sqrt{l}$), then Equation (4) suggests that the slope of the linear trend can be either positive or negative. We will show further below that recent experimental data demonstrates that both scenarios can be relevant: some loci have $a\ll e\sqrt{l}$ while others have $a\gg e\sqrt{l}$, with the former creating a bias towards the observed negative linear trends that characterize diminishing-returns and increasing-costs epistasis.

We note that Equation (3) immediately leads to testable quantitative predictions: in the WE limit, the distribution of the residuals, ${\widetilde{ϵ}}_i$, obtained from regressing s_i and $y$ is entirely determined by the slope of the regression, $-2{\tilde{v}}_i$. Specifically, we predict that these residuals (the deviations of individual genotype fitnesses from the overall diminishing-returns or increasing-costs trend) should be normally distributed with a variance proportional to ${\tilde{v}}_i(1-{\tilde{v}}_i)$. However, this condition only applies if diminishing-returns arises from the WE limit. It does not hold if epistasis is negligible, if locus $i$ interacts significantly with only a few other dominant loci, or if the epistatic terms are interrelated (e.g., when global epistasis arises from a nonlinearity applied to an unobserved additive trait; Starr and Thornton, 2016; Sailer and Harms, 2017; Otwinowski et al., 2018). The latter case may still lead to a negative linear trend, but the statistics of the residuals will differ from Equation (3) (see SI for a discussion).

It is convenient to subsequently work with the symmetric version of Equation (3), where the fitness effects of both $x_i=-1\to +1$ and its reversion $x_i=+1\to -1$ (whose fitness effect is negative of the former) are included in the regression against their respective background fitness. In this case, the additive term is averaged out, and we show (SI) that in the WE limit,

where ${\eta }_i$ depends on the genetic background and the locus, and is normally distributed with zero mean and variance $V$, and

Here, $V$ is the total genetic variance due to all loci (i.e., the variance in fitness across all possible genotypes) while $V_i$ is the contribution to the total variance by the $f$’s involving locus $i$. We therefore refer to v_i as the variance fraction (VF) of locus $i$. We show further below that for certain fitness landscapes v_i can also be interpreted as the fraction of pathways affected by a locus. For these reasons, we focus on v_i, which is half of the negative slope, rather than the slope. Note that the v_i’s do not sum to one unless there is no epistasis (with epistasis, ${\sum }_i{v}_i>1$, reflecting the fact that the variance contributed by different loci overlap). While the directed mutation case discussed previously is the relevant one when presenting experimental data (e.g., Figure 1c, d), it is conceptually simpler to work with the symmetric case. These two cases coincide and $v_i\approx {\tilde{v}}_i$ in the WE limit if the additive effect of a locus is small (i.e., $f_i^2\ll {\sum }_{j\ne i}{f}_{ij}^2+{\sum }_{j>k\ne i}{f}_{ijk}^2+\mathrm{\dots }$).

Our results show that the VF v_i plays an important role. It determines the slope of the negative relationship between the fitness effect and background fitness. At the same time, it determines the magnitude of the idiosyncratic fluctuations away from this trend. We also note that this slope can be used to experimentally probe the contribution of a locus to the trait (i.e., its VF) taking into account all orders of epistasis, which circumvents the estimation of the individual $f$’s in Equation (2). The theory additionally predicts that the slope obtained by regressing the sum of fitness effects of two mutations at loci $i,j$ against background fitness is proportional to $v_{ij}=v_i+v_j-2e_{ij}$, where $e_{ij}$ quantifies the magnitude of epistatic interactions of all orders between $i$ and $j$ (SI).

Importantly, while the fitness effects of individual mutations (and hence the DFEs) may change over the course of evolution due to epistasis, the distribution of variance fractions (DVF) across loci, $P(v)$, is an invariant measure of the range of effect sizes available to the organism during adaptation. As we will see, this means that the DVF plays an important role in determining long-term adaptability.

To illustrate our analytical results, we first demonstrate that the effects described above are reproduced in numerical simulations. To do so, we numerically generated a genotype-phenotype map of the form in Equation (2), with $\mathrm{\ell }=400$ loci and an exponential DVF, $P(v)={\overline{v}}^{-1}{e}^{-v/\overline{v}}$, where $\overline{v}=0.02$ (Materials and methods). This DVF is shown in Figure 2a. Note that $\overline{v}\mathrm{\ell }\gg 1$ corresponds to an epistatic landscape; $\overline{v}\mathrm{\ell }=8$ chosen here thus corresponds to a model within the WE limit (note that ${\tilde{v}}_i\approx v_i$ in this parameter range). Using this numerical landscape, we measured the fitness effect of mutations at 30 loci across 640 background genotypes with a range of fitnesses (Figure 2b). Our results recapitulate the predicted linear dependence on background fitness (Figure 1c, d), with a negative slope equal to twice the VF predicted from Equation (5). We further simulated the evolution of randomly generated genotypes similar to the experimental procedure used in Kryazhimskiy et al., 2014 (Figure 2c), finding that our results reproduce the patterns of declining adaptability observed in experiments (Figure 1b). Note that ∼10 mutations are fixed during this simulated evolution; declining adaptability here is not due to a finite-sites effect.

Figure 2

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As described previously, Equation (5) implies a proportional relationship between the magnitude of global epistasis (quantified by the slope of the relationship between the fitness effect of a mutation and the background fitness) and the magnitude of microscopic epistasis (quantified by the residual variance around this linear trend); see also Figure 3a. We verify this relationship in simulations (Figure 2d). We predict that the slope obtained by regressing the sum of fitness effects of two mutations at loci $i,j$ against background fitness is proportional to $v_{ij}=v_i+v_j-2e_{ij}$. We further assume that $e_{ij}=O({\overline{v}}^2)$ (specifically, $e_{ij}=v_i{v}_j$ for the genotype-phenotype map used for numerics). Since v_i and v_j are typically small for a complex trait, we expect near-additivity $v_{ij}\approx v_i+v_j$ and that any deviations are sub-additive, which is confirmed in simulations (Figure 2e, f).

Figure 3

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While testing the latter prediction on double mutants requires further experiments, we can immediately test the relationship between the slope and the distribution of residuals from existing experimental data. To do so, we reanalyzed the data from Johnson et al., 2019, which measured the fitness effect of 91 insertion mutants on about 145 backgrounds. These background strains were obtained by crossing two yeast strains that differed by $\approx 40,000$ single-nucleotide polymorphisms (SNPs). Of these 40,000 loci, $\mathrm{\ell }\approx 40$ have been identified as causal loci with currently available mapping resolution (Bloom et al., 2015). In Figure 3, we show the estimated ${\tilde{v}}_i$ (negative one-half of the slope of the best-fit line) and the VF v_i for each of the 91 mutations. These mutations were selected after screening for nonzero effect, and thus the DVF is biased upwards. The mean VF is $\overline{v}\approx 0.06$. The wide range of v_i observed in the data implies that the epistatic influence of loci varies greatly across loci, and we will show further below that this is crucial for maintaining a supply of beneficial mutations even when the organism is well-adapted to the environment.

Our theoretical results imply that we expect the linear relationship between background fitness and fitness effect to be negative if the additive effects of a locus’ interacting partners are not much larger than the epistatic terms. Specifically, we define the additivity of interacting loci (AoIL) for locus $i$ as

which we show can be estimated from data (Materials and methods and SI). If the AoIL is less than half, Equation (4) implies that the linear trend is guaranteed to be negative. If instead the AoIL is greater than 0.5, the trend can be either positive or negative. The data shows a range of AoIL between 0 and 1 across loci. As predicted by our theory, we find that the loci with AoIL lt_0.5 always show negative trends and the ones with AoIL gt_0.5 show both negative and positive trends (Figure 3c). Importantly, the sign of the trend is determined by the AoIL and not by the additivity of the mutated locus, which we define as

The additivity across loci also has a wide range. However, small additivity does not necessarily imply a negative trend (Figure 3d).

We next used the data from Johnson et al., 2019 to analyze the relationship between the slope of the linear trend and the residual variance around this trend. We find that the experimental data confirms our theoretical prediction that the residual variance is proportional to ${\tilde{v}}_i(1-{\tilde{v}}_i$) if the AoIL is small (Figure 3e, $R^2=0.5$ for loci with AoIL ¡ 0.5 and $R^2=0.42$ for all loci). The Gaussian-distributed term in Equation (3) also predicts the shape of the distribution of the residuals given the VFs, which aligns well with the empirical distribution of the residuals (Figure 3f).

Together, these theoretical results and our reanalysis of experimental data show that linear patterns of global diminishing-returns and increasing-costs epistasis are a simple consequence of widespread epistatic interactions. The DVFs observed in data (Figure 3b) further imply that the epistatic influence of different loci on fitness can vary across a wide range. In what follows, we show that these two observations can be put together to make general predictions about the DFEes, and consequently the long-term dynamics of adaptation. The key ingredient that enables this analysis (including Equation (5)) is that in the WE limit fitness and fitness effects are jointly normal (with respect to a uniform distribution over all possible genotypes), which allows us to quantify complex dependencies between these variables in terms of pairwise covariances.

Long-term adaptation is determined by the DFEs of possible mutations and the stochastic dynamical processes that lead to fixation. While Equation (5) represents the distribution of the fitness effects of a specific mutation at locus $i$ over all genotypes in the population that have fitness $y$, we are instead interested in the DFE, where fitness effects are measured for all the mutations arising in the background of a particular genotype that has fitness $y$. For now we ignore the influence of evolutionary history on the DFE; we expand on that complication in the following section.

Examining the DFE over $\mathrm{\ell }$ loci for a randomly chosen genotype of fitness $y$ can be thought of as sampling the fitness effects $s_1,s_2,\mathrm{\dots },s_{\mathrm{\ell }}$ from the conditional joint distribution $P(s_1,s_2,\mathrm{\dots },s_{\mathrm{\ell }}|y)$, which generally depends on epistasis. If the number of independent, nonzero epistatic terms is large, then $P(s_1,s_2,\mathrm{\dots },s_{\mathrm{\ell }}|y)$ is a multivariate normal distribution defined by the means and covariances of the $\mathrm{\ell }+1$ variables $y,s_1,s_2,\mathrm{\dots },s_{\mathrm{\ell }}$, which in turn can be computed in terms of the $f$’s from Equation (2). In particular, the conditional means and covariances are ${\displaystyle {\text{Mean}}_y(s_i)=-2v_i(y-\overline{y})}$, ${\displaystyle {\text{Cov}}_y(s_i,s_j)=4V(e_{ij}-v_i{v}_j)}$, where $e_{ij}$ is the epistatic VF between loci $i$ and $j$ and $e_{ii}=v_i$. This implies that the conditional correlation between fitness effects is $(e_{ij}-v_i{v}_j)/\sqrt{v_i{v}_j(1-v_i)(1-v_j)}$.

The DFE simplifies considerably if we make certain additional assumptions on the magnitude of epistatic interactions. If we assume the typical VF $\overline{v}$ is small (i.e., $\overline{v}\ll 1$) and also that $e_{ij}$ is $O({\overline{v}}^2)$, then correlations are $O(\overline{v})$ and thus negligible. Then, in a particular sample $s_1,s_2,\mathrm{\dots },s_{\mathrm{\ell }}$, we can think of each s_i as being drawn independently with mean $-2v_i(y-\overline{y})$ and variance $4v_{i}V$. To compute the DFE, $\rho (s|y)$, we first sample the VF from the DVF, $P(v)$, and then sample a Gaussian random variable with the aforementioned mean and variance. This leads to the DFE

where $\phi$ is the standard normal pdf. Curiously, the correlations between s_i’s vanish when $e_{ij}=v_i{v}_j$, in which case the above equation is exact and the DFE is determined entirely by the DVF. Further below, we introduce a specific fitness landscape model for which this relation does hold. Diminishing-returns is naturally incorporated in Equation (9): the mean of $s$ is $-2\overline{v}(y-\overline{y})$, that is, the DFE shifts progressively towards deleterious values with increasing fitness.

A key unresolved question is the extent to which evolutionary history influences the DFE and the dynamics of adaptation (Agarwala and Fisher, 2019). That is, what does our theory say about historical contingency?

Suppose a clonal population of fitness y₀ accumulates $k$ successive mutations resulting in fitnesses $y_1,y_2,\mathrm{\dots },y_k$. By virtue of arising on the same ancestral background, the fitness gain of a new mutation, $s_{k+1}$, is in general correlated with the full sequence of past fitnesses and the identity of the $k$ mutations through its epistatic interactions with them. Based on these correlations, we use well-known properties of conditional normal distributions (Eaton, 1983) to write

where the weights $w_{k+1,i}$ depend on the VF ($v_{k+1}$) of the new mutation and its epistatic interactions with past mutations. Here, $ϵ$ is the normally distributed residual that depends on the initial genotype and the weights (SI). Equation (10) is a generalization to a sequence of mutations of Equation (5), which we can think of as the special case where $k=0$.

To gain intuition, it is useful to first analyze Equation (10) when $k=1$ (i.e., to compute the effect of a second mutation conditional on the first). In this case, we show in the SI that

where $s_1=y_1-y_0$ is the fitness effect due to mutation 1. The first term on the right-hand side is the dependence on the fitness of the immediate ancestor, similar to the corresponding term in Equation (5). The second term quantifies the influence of epistasis between loci 1 and 2 on s₂. When $e_{12}=v_1{v}_2$, dependence on s₁ vanishes entirely and s₂ depends only on y₁. In contrast, if loci 1 and 2 do not interact, $e_{12}=0$, and s₂ is, on average, larger if the mutation at 1 is beneficial compared to when it is deleterious. This has an intuitive interpretation: diminishing-returns applies to the overall fitness and the mechanism through which it acts is epistasis. However, if mutations 1 and 2 do not interact, then the increase in fitness corresponding to mutation 1 does not actually reduce the effect of mutation 2 (as expected by diminishing-returns) so the expected effect of mutation 2 is larger. This analysis suggests that during adaptation, since selection favors mutations with stronger fitness effects on the current background, a mutation that interacts less with previous mutations is more likely to be selected.

To identify the conditions under which history plays a minimal role, we would like to examine when $s_{k+1}$ depends only on the current fitness, y_k, and is independent of both the past fitnesses and idiosyncratic epistasis. If this were true, then Equation (5) would apply for new mutations that arise through the course of a single evolutionary path (i.e., the fitness effect of a new mutation is ‘memoryless’ and depends only on its VF and the current fitness). Surprisingly, such a condition does exist. We show that this occurs when the magnitude of epistatic interactions between the new mutation and the $k$ previous mutations, $e_{k+1,1:k}$, satisfies a specific relation: $e_{k+1,1:k}=v_{k+1}{v}_{1:k}$, where $v_{1:k}$ is the combined VF of the $k$ previous mutations (SI). In general, this condition is not satisfied, implying that there will be historical contingency that can be analyzed using the framework above. Remarkably, it turns out that a fitness landscape model for which the condition is satisfied does exist and arises from certain intuitive assumptions on the organization of biological pathways and cellular processes. This fitness landscape model additionally serves as an example of a landscape where global epistasis can vary substantially across loci. We describe this model below.

We introduce the ‘connectedness’ model (CN model, for short). In this model, each locus $i$ is involved in a fraction ${\mu }_i$ of independent ‘pathways,’ where each pathway has epistatic interactions between all loci involved in that pathway (Figure 4a). The probability of an epistatic interaction between three loci ($i,j,k$) is then proportional to ${\mu }_i{\mu }_j{\mu }_k$ since this is the probability that these loci are involved in the same pathway. When the number of loci $\mathrm{\ell }$ is large, we show that in this model $v_i={\mu }_i/(1+{\mu }_i)$, and when $\mathrm{\ell }$ is small, $v_i={\mu }_i/\overline{\mu }\mathrm{\ell }$, where $\overline{\mu }$ is the average over all loci (SI). The CN model therefore has a specific interpretation: the outsized contribution to the fitness from certain loci (large v_i) is due to their involvement in many different complex pathways (large ${\mu }_i$) and not from an unusually large perturbative effect on a few pathways. The distribution, $P(\mu )$, across loci determines the DVF.

Figure 4

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Statistical fitness landscapes such as the NK model and the Rough Mt. Fuji model (Neidhart et al., 2013; Agarwala and Fisher, 2019; Kauffman and Weinberger, 1989; Stadler and Happel, 1999; Aita et al., 2000; Altenberg, 1997) are related to the CN model. Specifically, the CN model is a subclass of the broader class of generalized NK models (see Hwang et al., 2018 for a review). However, often-studied fitness landscape models have one important difference that distinguishes them and gives qualitatively different dynamics of adaptation (shown further below): in contrast to the CN model, classical fitness landscapes are typically ‘regular.’ That is, the VF of every locus is assumed to be the same (except the star neighborhood model, which has a bimodal DVF; Hwang et al., 2018).

The CN model is equivalent to a Gaussian fitness landscape with exponentially decaying correlations (SI). The CN model has tunable ruggedness, where the landscape transitions from additivity to maximal epistasis with increasing $\overline{\mu }$. Maximal epistasis corresponds to ${\mu }_i=1$ (and hence $v_i=1/2$) for all $i$. From Equation (5), this implies that the new fitness after a mutation occurs is independent of the previous fitness, consistent with the expectation from a House-of-Cards (HoC) model (Kauffman and Levin, 1987) (where genotypes have uncorrelated fitness). Regular fitness landscape models with exponentially decaying correlations have memoryless fitness effects under the restrictive assumption that every locus is equivalent (Agarwala and Fisher, 2019). We show that the dynamics of adaptation of the more general CN model are also memoryless, that is, the condition detailed in the previous section holds true (SI). Yet, as we show below, the predicted dynamics for the CN model are very different to those from a regular fitness landscape model.

We emphasize that the well-connectedness assumed for the CN model is not a requirement for Equation (5) to hold. However, how diminishing-returns influences the long-term dynamics of adaptation depends on the specific genetic architecture and the corresponding fitness landscape. Consider, for example, an alternative model of genetic networks organized in a modular structure (Figure 4b). In this model, each locus is part of a single module and interacts epistatically with other loci in that module to determine the fitness of that module; overall fitness is then determined as a function of the module fitnesses. In this case, the variance contributed by a locus is due to its additive contribution and from epistasis between loci restricted to its module. While the argument for diminishing-returns still applies to the fitness as a whole, it follows from the same argument that diminishing-returns should also apply to each module separately. Consequently, the dynamics of adaptation for the modular model are different from the CN model. For simplicity, we analyze the dynamics of adaptation for the CN model and postpone a discussion of how the dynamics differ for different models to subsequent work.

We now examine the DFE that follows from Equation (5) and what that implies for long-term adaptation under the conditions for memoryless fitness effects. We henceforth assume a large number of loci with sparse epistasis (though the total number of nonzero epistatic terms is still large). This implies that $\mathrm{\ell }\gg 1$, $v_i\ll 1$ and $\overline{v}\mathrm{\ell }\gg 1$; for simplicity, we also assume strong-selection-weak-mutation (SSWM) selection dynamics and $s\ll 1,Ns\gg 1$, where $s$ are fitness effects and $N$ is the population size. Under these conditions, a mutation sweeps and fixes in a population before another one arises. The probability of fixation of a beneficial mutation, $p_{\text{fix}}$, is then proportional to its fitness effect (Haldane, 1927).

It is convenient to rescale fitnesses based on the total variance in fitness across all possible genotypes by defining $z=V^{-1/2}(y-\overline{y}),\sigma =V^{-1/2}s,\nu =V^{-1/2}\eta$. Note that $\nu$ is normally distributed with zero mean and unit variance. Here, $z$ has an intuitive interpretation as the ‘adaptedness’ of the organism. When the organism is neutrally adapted ($|z|\ll 1$), positive and negative epistatic contributions to the fitness are balanced and diminishing-returns is negligible. Diminishing-returns is relevant when the organism is well-adapted ($z\gg 1$). Below, we give the intuition behind our analysis, which is presented in full detail in the SI.

In the neutrally adapted regime, the linear negative feedback in Equation (5) is negligible and the DFE is determined by the distribution of $\simeq v^{1/2}\nu$. Loci with large $v$ can lead to a DFE with a long tail. If $\overline{v}$ is the typical VF of a locus, the fitness increases as $z\sim n_s{\overline{v}}^{1/2}$, where n_s is the number of substitutions. Since $\overline{v}$ is a measure of overall epistasis, this implies that epistasis speeds adaptation in the neutrally adapted regime by allowing access to more influential beneficial mutations.

Fitness increases until the effect of the negative feedback cannot be neglected. From Equation (5), this happens when $\overline{v}z\sim {\overline{v}}^{1/2}\nu$ (i.e., when $z^2\sim {\overline{v}}^{-1}$). Intuitively, fitness begins to plateau when its accumulated benefit from substitutions is comparable to the scale of the total genetic variance ($n_s\overline{v}\sim 1$) and further improvements are due to rare positive fluctuations. In this well-adapted regime, diminishing-returns and increasing-costs epistasis strongly constrain the availability of beneficial mutations, whose effects can be quantified in this model: for a mutation to have a fitness effect $\sigma$, we require from Equation (5) that $\nu \simeq \sigma /2v^{1/2}+v^{1/2}z$, which has probability $\sim e^{-{\nu }^2/2}$. Beneficial effects of large $\sigma$ arise when $\nu$ has a large positive deviation. The most likely $v$ that leads to a particular $\sigma$ is when $\nu$ is smallest (i.e., at $v^{*}\simeq \sigma /2z$), in which case $\nu \simeq \sqrt{2\sigma z}$, yielding a tail probability $\sim e^{-\sigma z}$. Remarkably, the beneficial DFE in the well-adapted regime is quite generally an exponential distribution independent of the precise form of the DVF (unless it is singular). In particular, we show in the SI that for the DFE, $\rho (\sigma |z)$,

which depends solely on the adaptedness of the organism. The exponential form arises because of the Gaussianity of $\nu$, but the argument can be easily extended to $\nu$ with non-Gaussian tails.

An exponential beneficial DFE has been previously proposed by Orr, 2003 but arises here due to a qualitatively different argument. Orr’s result instead follows from extreme value theory: suppose the fitness effects of $\mathrm{\ell }$ loci ($\mathrm{\ell }\gg 1$) are sampled from a DFE $\rho (\sigma )$ and $F(\sigma )\equiv {\int }_{-\mathrm{\infty }}^{\sigma }\rho ({\sigma }^{\prime })𝑑{\sigma }^{\prime }$. Then, the probability that a beneficial mutation has at least a certain effect size $\sigma$ is $P({\sigma }_b\ge \sigma )=\frac{1-F(\sigma )}{1-F(0)}\approx \frac{\ln F(\sigma )}{\ln F(0)}$, where the latter approximation holds when beneficial mutations are rare (i.e., $1-F(0)$ is small). A well-known result from extreme value theory (Gumbel, 2004; Majumdar et al., 2020) implies that for a large family of distributions $\rho (\sigma )$ and for $\mathrm{\ell }\gg 1$, we have $-\mathrm{\ell }\ln F(\sigma )\propto e^{-k\sigma }$ (for some constant $k$) and therefore $P({\sigma }_b\ge \sigma )=e^{-k\sigma }$. This argument is consistent with our results, but does not yield the dependence of $k$ on adaptedness and the rate of beneficial mutations without additional information about $\rho (\sigma )$.

Under SSWM assumptions, from Equation (12), the typical effect size of a fixed mutation is ${\sigma }_{\text{fix}}\sim z^{-1}$, which typically has a VF,

The above relation makes precise the effects of increasing-costs epistasis on adaptation. As adaptation proceeds, the delicate balance of high fitness configurations constrains fixed beneficial mutations to have moderate VFs. A mutation of small VF is likely to confer small benefit and is lost to genetic drift, while one with a large VF is more likely to disrupt an established high fitness configuration.

This intuition is not captured in regular fitness landscape models, which assume statistically equivalent loci, that is, $v_i=\overline{v}$ for all $i$ and $P(v)=\delta (v-\overline{v})$ is singular. From Equation (9), we see that this leads to a Gaussian DFE whose mean decreases linearly with increasing fitness, in contrast to the exponential DFE in our theory. The key difference is the lack of loci with intermediate effect, which drive adaptation in the well-adapted regime. As a consequence, the rate of beneficial mutations declines exponentially ($U_b\sim e^{-\overline{v}{z}^2/2}$) and the fitness thus sharply plateaus at $z\sim {\overline{v}}^{-1/2}$. In contrast, our theory predicts a much slower depletion of beneficial mutations, $U_b\sim z^{-2}$ (SI). The rate of adaptation is $dz/dt\sim U_b{p}_{\text{fix}}{\sigma }_{\text{fix}}\sim z^{-4}$ (since $p_{\text{fix}}\sim {\sigma }_{\text{fix}}$), which leads to a slow but steady power-law gain in fitness, $z\sim t^{1/5}$. The rate of fixation of beneficial mutations is $d{n}_s/dt\sim U_b{p}_{\text{fix}}\sim z^{-3}\sim t^{-3/5}$, which gives $n_s\sim t^{2/5}$.

We verify our analytical results using numerics. As before, we generated a genotype-phenotype map using the CN model with an exponential DVF, $P(v)={\overline{v}}^{-1}{e}^{-v/\overline{v}}$ and $\mathrm{\ell }=400$ loci. The DFE can be calculated exactly by plugging in this $P(v)$ in Equation (9):

We simulated the evolution of randomly generated genotypes from $z=0$ to $z=2.5$ and $z=5$, and the DFE across all loci was measured (we chose $\overline{y}=0,V=1$ so that $y=z,s=\sigma$). The theoretical prediction for the DFE, Equation (14), closely aligns with the numerical results (Figure 4c).

Due to computational constraints, it is difficult to simulate evolution deep into the well-adapted regime. To compute the shape of adaptive trajectories and their variability, we instead simulated SSWM dynamics using the DFE directly from Equation (14), beginning from a neutrally adapted fitness ($z=0$). Typical trajectories (Figure 4d) show rapid adaptation to the well-adapted regime beyond which the fitness grows slowly as $t^{1/5}$, as predicted from theory. The predictions for the number of fixed beneficial mutation are also recapitulated (Figure 4e).

We use a fitness landscape model with $\mathrm{\ell }$ loci to generate the genotype-fitness map. Each locus is assigned a sparsity µ from $P(\mu )$, which is an exponential distribution with mean $\overline{\mu }$. Each of $M$ independent pathways sample loci with each locus $i$ having probability ${\mu }_i$ of being selected to a pathway. We choose $\mathrm{\ell }=400,\overline{\mu }=0.02,M=500$ so that $\overline{\mu }\mathrm{\ell }=8$ ensures significant epistasis. All loci in a pathway interact with each other, where additive and higher-order coefficient terms of all orders were drawn independently from a standard normal distribution. The total fitness is the sum of contributions from the $M$ pathways. We normalize the coefficients so that the sum of squares of all coefficients is 1, that is, the total variance across genotypes is 1. The mean, $\overline{y}$, is close to zero from our sampling procedure. The above procedure is a simple and efficient way to generate epistatic terms to order $\sim$20, beyond which the computational requirements are limited by the exponentially increasing demand. Note that the effects described in the paper were also observed with only pairwise and cubic epistatic terms.

The VFs shown in Figure 2a can be calculated numerically from the definition. From the theory, given our choice of $P(\mu )$, these should follow an exponential distribution with mean $\overline{v}\approx \overline{\mu }/(1+\overline{\mu })$. There may be deviations since $M$ is finite whereas the calculations assume $M\to \mathrm{\infty }$. To generate Figure 2b, in order to get a range of background fitnesses, we first sample 128 random genotypes. These have fitnesses close to zero; in order to obtain a range of fitness values, we simulated the evolution of these 128 genotypes up to $y=1,2,3,4,5$ under SSWM assumptions to get $128\times 5=640$ genotypes at roughly five fitness values. The fitness effect of applying a mutation (i.e., flipping its sign) is measured for 30 randomly chosen loci (which are kept fixed) over each of the 640 genotypes. This is shown for 5 of the 30 and for the mean over the 30 loci in Figure 2b.

To generate Figure 2c, we sampled 64 random genotypes and 12 replicates of each. The evolution of these 768 genotypes was simulated for a total of 50 generations with a mutation rate of 1 per generation. The mean fitness gain over the 12 replicates is plotted for each of the 64 founders against their initial fitness.

To generate Figure 2d, the residuals are measured using the same procedure as for the experimental data analysis described below for the initial 128 genotypes at $y\approx 0$ and the 30 loci with the largest VF.

Double mutants were created by mutating all pairs of the 30 randomly chosen loci on the 640 evolved genotypes. Their mean fitness effect was computed and plotted along with the mean fitness effect for single mutants, shown in Figure 2e. The VF of the pair of loci for the double mutant was estimated as before and compared to the sum of the estimated VFs of the corresponding single mutants. This is shown in Figure 2f.

To generate the plots in Figure 4c, we simulated the evolution of 128 randomly sampled genotypes to $y=2.5$ and $y=5$. The fitness effect of 200 randomly sampled loci was measured and the distribution is plotted.

The data from Johnson et al., 2019 consists of the fitness after the addition of 91 insertion mutations on each of 145 background genotypes. The fitness of a particular mutation at locus $i$ can be modeled as

where $y_i,y$ are the mutant and background fitnesses, respectively, $c_i,b_i$ are constants for each locus, and the residual ${\text{Residual}}_i(g)$ depends on the background genotype $g$.

We estimate the VF $v_i=(1-{\widehat{\rho }}_i)/2$, where the Pearson correlation ${\widehat{\rho }}_i=\text{Corr}(y_i\oplus y,y\oplus y_i)$, where the symbol ⊕ denotes that the mutant and background fitness datasets are concatenated. ${\tilde{v}}_i$ is estimated as the negative one-half of the slope of the best linear fit of $s_i=y_i-y$ and $y$. The residuals for each of the 145 genotypes for each of the 91 mutations are simply

where the overline represents an average over the 145 genotypes, which is used as an estimate of the constant term and $c_i=2{\tilde{v}}_i-1$. In Figure 3b, we plot the distribution of estimated v_i and ${\tilde{v}}_i$. In Figure 3c, we compute the AoIL for each locus using Equation (7), which we show in the SI to be $|\text{Cov}(s_i,y_i+y)|/(|\text{Cov}(s_i,y_i+y)|+\text{Var}(s_i))$. In Figure 3d, we compute the additivity using Equation (8). The additive effect is $f_i=\overline{(y_i-y)}/2$, and $\text{Var}(s_i)/4$ gives the sum of squares of the epistatic terms (SI). In Figure 3e, we compute the variance of the residuals across the 145 genotypes for each locus and plot it against the locus’ estimated ${\tilde{v}}_i(1-{\tilde{v}}_i)$. In Figure 3f, we plot the distribution of residuals over all genotypes and loci. The prediction is that in the WE limit the distribution of residuals is determined by $2{\tilde{v}}_i(1-{\tilde{v}}_i)\eta$, where $\eta$ is a Gaussian random variable. We multiply $\sqrt{{\tilde{v}}_i(1-{\tilde{v}}_i)}$ for each locus with 10,000 i.i.d. standard normal random variables, pool the resulting numbers for all loci, and plot the predicted distribution in Figure 3f. The distributions are variance-matched. While Figure 3e shows that the variance of the residuals aligns with the theoretical prediction of being proportional to slope, Figure 3f shows that the data is also consistent with the predicted Gaussianity of the background-genotype-dependent contribution.

In the main text, we propose that the fitness effect after a mutation at locus $i$ can be written as

Here, ${\eta }_i$ is a genotype and locus-dependent contribution, which is distributed as a mean-zero Gaussian with variance equal to the total genetic variance, $\overline{y}$ is a constant, and v_i is the variance fraction, which is defined further below. Equation (17) corresponds to the symmetric case where the fitness effects of the mutations $x_i=-1\to +1$ and $x_i=+1\to -1$ are simultaneously regressed against their respective background fitness. The directed case when the mutation is specified to change from $x_i=-1\to +1$ (or $x_i=+1\to -1$) will be considered in the directed mutation section below.

We show that Equation (17) arises under certain conditions in a generic model of a complex trait with $\mathrm{\ell }\gg 1$ biallelic loci. Essentially, we would like to compute the distribution of the new fitness, y_i, across genotypes after a mutation at locus $i$ given the current fitness $y$ and the set of all parameters $\mathrm{\Theta }$ of the model (i.e., $P(y_i|y,\mathrm{\Theta })$, with $s_i=y_i-y$). Using the chain rule of probability, we can write

where the sum is over all possible genotypes. While $P(y_i|g,\mathrm{\Theta })$ is determined by the genotype-fitness map, $P(g|y,\mathrm{\Theta })$ is the crucial factor that gives weight only to the genotypes that yield the current fitness $y$. If the fitness is much larger than the mean fitness over all possible genotypes, Equation (18) implicitly ensures that weight is given to only those genotypic configurations that lead to such an unusually large fitness. We will analyze the case of a ‘microcanonical ensemble’ where every genotypic configuration that leads to a particular fitness is equally likely (with no linkage), that is, the prior $P(g)$ across genotypes is uniform (see the section 'Maximum entropy interpretation' for a discussion).

It is difficult to directly evaluate the sum in Equation (18). In the following sections, we give a simple derivation but elaborated to highlight the key assumption that leads to Equation (17). In short, the negative correlation between $s_i=y_i-y$ and $y$ implied by Equation (17) is a trivial consequence of y_i and $y$ having the same distribution w.r.t. ${\displaystyle P(g)}$. ${\displaystyle {\eta }_i}$ is in general arbitrary and determined by the genotype-fitness map. However, ${\eta }_i$ is normally distributed if we make certain assumptions about the structure of epistasis in the genotype-fitness map. The emergence of the negative linear relationship for a directed mutation $x_i=-1\to +1$ is subtler.

The fitness $y(g)$ for a genotype of length $\mathrm{\ell }$, $g=\{x_1,x_2,\mathrm{\dots },x_{\mathrm{\ell }}\}$, where $x_i=\pm 1$, can be generally written as Poelwijk et al., 2016; Neher and Shraiman, 2011

The symmetric choice of $x_i=\pm 1$ is chosen for mathematical convenience. In this form, the total variance contribution from the qth-order epistatic terms is ${\sum }_{i_1>i_2>\mathrm{\dots }{i}_q}{f}_{i_1{i}_2\mathrm{\dots }{i}_q}^2$, and the total genetic variance, defined as

is the sum of variance contributions from orders $q=1$ to $\mathrm{\ell }$, that is, the sum of squares of all the $f$’s. The sum over the $2^{\mathrm{\ell }}$ genotypes is an expectation value assuming all genotypes are equally likely; we will denote this expectation using an overline hereafter. We use this expectation value as a proxy for empirical averages over the ‘background’ genotypes in a population. With a sufficient number of background genotypes, the empirical average should converge to this expectation value.

The representation in Equation (19) is a Fourier representation of the fitness function on the $\mathrm{\ell }$-dimensional hypercube and makes calculations much simpler. For instance, to get the terms from each order, we have

It is useful to define the variance contribution due to a particular locus $i$ as (symmetry w.r.t. interchanging indices is used for each term throughout)

We also define the variance fraction of locus $i$,

which plays a key role in the model.

We would like to relate the fitness before and after locus $i$ is flipped (i.e., $x_i\to -x_i$), denoted by $y$ and y_i, respectively. We have from Equation (19),

where $y_{\overline{i}}$ and ${\xi }_i$ respectively contain all the terms not containing and containing locus $i$, that is,

We have $y_{\overline{i}}=(y+y_i)/2-\overline{y}$ and ${\xi }_i=(y-y_i)/2$. Since $y_{\overline{i}}$ and ${\xi }_i$ contain all the genotype dependence, we can write

From the properties of the Fourier representation in Equation (21), it is easy to see that the means are $\overline{y_{\overline{i}}}=0,\overline{{\xi }_i}=0$, the variances are ${\displaystyle \overline{y_{\overline{i}}^2}=V-V_i}$, ${\displaystyle \overline{{\xi }_i^2}=V_i}$, and the covariance is $\overline{y_{\overline{i}}{\xi }_i}=0$.

The calculations so far have been exact. We now make the key assumption that $y_{\overline{i}}$ and ${\xi }_i$ are both normal-distributed across genotypes. This assumption is similar to that of Fisher’s infinitesimal model, where the distribution of trait values across strains for a complex trait is argued to be normal-distributed since the trait value is due to infinitesimal independent contributions from many loci. While $y_{\overline{i}}$ is easily seen to be normal-distributed for large $\mathrm{\ell }$, an argument can be made for ${\xi }_i$ only if locus $i$ has a large number of independent, nonzero epistatic terms and the additive term f_i is smaller in magnitude than the epistatic terms; specifically, we require that $f_i^2\lesssim {\sum }_{j\ne i}{f}_{ij}^2+{\sum }_{k<j\ne i}{f}_{ijk}^2+\mathrm{\dots }$. If instead $f_i^2\gg {\sum }_{j\ne i}{f}_{ij}^2+{\sum }_{k<j\ne i}{f}_{ijk}^2+\mathrm{\dots }$, then ${\xi }_i$ is bimodal, where the two modes correspond to ${\xi }_i\approx \pm f_i$ at $x_i=\pm 1$. For loci with pairwise and third-order epistasis, the number of pairwise and third-order epistatic terms scale $\propto \mathrm{\ell }$ and $\propto {\mathrm{\ell }}^2$, respectively, which justifies the normality assumption for large $\mathrm{\ell }$ even if individual epistatic terms are smaller in magnitude than the additive terms.

Under these assumptions and since $y_{\overline{i}}$ and ${\xi }_i$ are linearly independent, we have

where $\phi$ is the standard normal pdf. Therefore, y_i is normal-distributed across genotypes and from the above equation can be written as

where the VF $v_i=V_i/V$ was defined previously and ${\displaystyle \overline{{\eta }_i}=0,\overline{{\eta }_i^2}=V}$. This leads to the form in Equation (17),

The above derivation was presented to clarify the basic assumptions. Simply computing the covariance between $y$ and y_i in Equations (24) and (25), we get ${\displaystyle \overline{(y-\overline{y})(y_i-\overline{y})}=\overline{y_{\overline{i}}^2}-\overline{{\xi }_i^2}=V-2V_i}$. The correlation is then $1-2v_i$. Equation (32) follows if additionally $y_i,y$ are jointly Gaussian, which is true if locus $i$ has many independent, nonzero epistatic terms.

Previously, we considered the symmetric flip $x_i\to -x_i$ and averaged over all $\mathrm{\ell }$ loci including $i$. Here, we consider the case when the mutation is specified to change either from $x_i=-1\to +1$ or $x_i=+1\to -1$. In this case, we should average over all loci except $i$.

We consider $x_i=-1\to +1$ (the opposite case is similar). The fitness before the mutation is

and the fitness after the mutation is

so that

The tilde and hat are used here to distinguish the fitness from the y_i defined previously for the symmetric case corresponding to the flip $x_i\to -x_i$. We can easily compute averages over genotypes using the Fourier orthogonality relations. To compute the expectation value of products of two quantities, say s_i and $y_i^{-}$, we simply compute a dot product where the components of the vectors are the coefficients of terms in the expansion above. For example, $\overline{s_i{y}_i^{-}}$ is $(\overline{y}-f_i)(2f_i)+{\sum }_{j\ne i}(f_j-f_{ij})(2f_{ij})+{\sum }_{j>k\ne i}(f_{jk}-f_{ijk})(2f_{ijk})+\mathrm{\dots }$. We have for the means, variances, and covariances, 

The slope when s_i is regressed against $y_i^{-}$ is ${\displaystyle \overline{(s_i-{\overline{s}}_i)(y_i^{-}-\overline{y_i^{-}})}/\overline{(y_i^{-}-\overline{y_i^{-}}{)}^2}}$. We can define a ‘modified’ VF ${\tilde{v}}_i$ as half the negative slope, 

Writing the linear form based on this correlation, we get

where ${\eta }_i$ is again normally distributed (in the WE limit) with zero mean and variance ${\displaystyle \overline{(y_i^{-}-\overline{y_i^{-}}{)}^2}}$ and ${\displaystyle K_i^2=\frac{\overline{(s_i-{\overline{s}}_i{)}^2}}{\overline{(y_i^{-}-\overline{y_i^{-}}{)}^2}}-4{\tilde{v}}_i^2}$. Note that, unlike v_i, ${\tilde{v}}_i$ can be negative.

However, we argue that ${\tilde{v}}_i$ is typically positive in the WE limit, which leads to a negative linear trend. The second term in the numerator on the right-hand side of Equation (42) has the same number of terms as ${\displaystyle \overline{(s_i-{\overline{s}}_i{)}^2}}$, but these terms appear as products of Fourier coefficients that may have opposing signs. In particular, if ${\sum }_{j\ne i}{f}_{ij}^2>{\sum }_{j\ne i}{f}_j{f}_{ij}$, ${\displaystyle \sum _{j>k\ne i}{f}_{ijk}^2>\sum _{j\ne i}{f}_{jk}{f}_{ijk}}$, and so on, then ${\tilde{v}}_i$ is guaranteed to be positive. If we denote the typical magnitude of qth-order epistasis terms as e_q (e₁ corresponds to additive effects), each of this relationships has the form $e_{q+1}^2\mathrm{\ell }>e_q{e}_{q+1}\sqrt{\mathrm{\ell }}$ when $\mathrm{\ell }\gg 1$, that is, $e_q<e_{q+1}\sqrt{\mathrm{\ell }}$. If the number of loci is sufficiently large, then these relationships will hold even if the typical magnitude of individual higher-order epistasis terms is smaller than the lower-order terms. We therefore expect that the second term in the numerator on the right-hand side of Equation (42) is smaller than ${\displaystyle \frac{\overline{(s_i-{\overline{s}}_i{)}^2}}{2}}$ when $\mathrm{\ell }\gg 1$. A similar argument can be made for the cross terms $f_j{f}_{ij},f_{jk}{f}_{ijk},\mathrm{\dots }$ once the squares in the denominator of the right-hand side of Equation (42) are expanded.

When $\mathrm{\ell }\gg 1$, we can then write

Further, in the WE limit, $K_i^2\approx 4{\tilde{v}}_i-4{\tilde{v}}_i^2$ so that the variance of $K_i{\eta }_i$ is $\propto {\tilde{v}}_i(1-{\tilde{v}}_i)$.

To estimate the ratio of the magnitudes of the second and first terms in the numerator on the right-hand side of Equation (42) from data, we use the expression for the covariance,