epistasis

Paixão T & Barton N 2016 The effect of gene interactions on the long-term response to selection. PNAS 113:4422-4427.

• a population under selection can reach trait values far beyond the initial trait variance it displays
• the epistasis among variants in the existing population may not predict longer-term evolution

• u(p₀) = (1 − exp(−4N_esp₀)) / (1 − exp(−4N_es)) = p₀ + 2p₀(1 − p₀)N_es + O((N_es)²)
• R^∞ = ∑_iα_i(u(p₀) − p₀)
• R^∞ ≈ 2∑_iα_ip₀(1 − p₀)N_es_i
• R^∞ ≈ 2∑_iα_i²p₀(1 − p₀)βN_e = 2βN_eV_A = 2N_eR⁰
• V_A⁰ = 2∑_iα_i²p₀(1 − p₀) is the initial additive genetic variance
• R⁰ = βV_A⁰ is the response in the first generation
• this calculation is valid in the limit of weak selection at every locus
• the probability of fixation is approximated by the first-order perturbation to neutrality
• it is also only strictly valid when the trait is additive, excluding any form of interaction between genes
• it seems very hard to generalize to allow for epistasis
• then, the calculation of the fixation probability of an allele depends on the trajectories of the alleles at all other loci it interacts with
• R^∞ = ∑_{t = 0}^∞βV_A⁰(1 − 1 / 2N_e)^t = β2N_eV_A⁰

• the long-term response for haploids
• <R^∞> = β∑_{t = 0}^∞V_A^t = β∑_{t = 0}^∞(1 − F_t)∑_{k = 1}^∞kF_t^{k − 1}V_A(k)⁰ = βN_e∑_{k = 1}^∞V_A(k)⁰ = βN_eV_G⁰ ... [1]
• the total response to directional selection is proportional to the initial total genotypic variance, V_G⁰ = ∑_{k = 1}^∞V_A(k)⁰
• —a simple extension of Robertson's (17) result
• the change in mean can thus be very much greater than the change in the first generation, if N_e is large (Fig. 1)
• for a given initial total genetic variance, V_G⁰, the initial response is slower with epistasis, because it is proportional only to the additive component, V_A⁰
• epistasis generates additional additive variance
• the total selection response is the same

• for a given initial genotypic variance, epistasis slows the initial response, but dues not affect its final value
• for a given initial additive variance, epistasis increases the final response by V_G⁰ / V_A⁰, compared with an additive genetic architecture

• Robertson's (17) result ... suggests an alternative view
• as alleles fluctuate in frequency, they change the average effects of alleles at other loci, thereby changing their probability of fixation
• alleles that are initially deleterious but become beneficial as the response unfolds should contribute to this increase in long-term response
• this view predicts that an increase in long-term response should be correlated with an increase in the number of alleles that are beneficial at the end of the response

• the long-term response for diploids
• <R^∞> = β∑_{t = 0}^∞V_A = β∑_{t = 0}^∞(1 − F_t)∑_{k = 1}^∞k(2F_t)^{k − 1}V_A(k)⁰ = 2N_eβ∑_{k = 1}^∞2^{k − 1}V_A(k)⁰ > 2N_eβV_G⁰
• epistasis can increase the total response disproportionately
• it is unlikely that higher-order components will be large enough for this effect to be substantial

• Wright (1) argued that epistasis would cause populations to become trapped at local "adaptive peaks" and proposed that a "shifting balance" between selection and random drift could allow them to explore alternative peaks, so as to move toward the global optimum
• it remains unclear whether adaptation is significantly slowed by trapping on local peaks
• Mayr (23) criticized the supposed neglect of epistasis by "bean-bag genetics," provoking a robust defense by Haldane (24)
• the failure of large genome-wide association studies to assign much heritable variance to specific loci (the so-called "missing heritability") has been attributed to epistasis
• this explanation is unnecessary
• the practical success of the additive model in quantitative genetics appears hard to reconcile with the strong molecular interaction between genes