Canonical approximation of fitness landscapes

Higher-order epistasis and phenotypic prediction

“…In fact, such a family of priors is already well established in the literature in the form of random field models ( 33 , 49 , 56 ), which are parameterized in terms of the amount of variance due to each order of genetic interaction. Such models take the form of multivariate Gaussian distributions and can be constructed by drawing certain epistatic coefficients from zero mean normal distributions with appropriately chosen variances ( SI Appendix ).…”

“…1 G . In particular, the shape contributed by each of the variance components is given by a set of orthogonal polynomials known as the Krawtchouk polynomials ( 33 , 48 , 49 , 57 ), which we show visually in Fig. 1 E and F .…”

Symmetry, gauge freedoms, and the interpretability of sequence-function Relationships

“…A third symmetry group is H H am , which describes combinations of position permutations and position-specific character permutations. H H am is the largest symmetry group that preserves Hamming distances (33), and includes H PSCP , H PP , and H G C P as subgroups. Theorem 1 does hold for H H am , due the fact that H PSCP is a subgroup (see SI Sec.…”

Higher-order epistasis and phenotypic prediction

“…The crucial observation is that these pure k -th order components each have a very specific geometry, and as a result the distance correlation function for any such k -th order component must take a very specific shape. Technically, these shapes are given by a set of orthogonal polynomials known as the Krawtchouk polynomials [33, 56–58], but for our purposes it suffices to look at these functions visually, Figures 2A and 2B.…”

“…In particular, it is classically known that the three pictures in Figure 1A-C actually contain identical information, in the sense that for any given genotype-phenotype map, having any one of the panels in the top row of Figure 1 allows us to compute the other two (ref. [33, 53, 56], SI Appendix ). Moreover, recent results on the distance correlation function of mutational effects (previously denoted by γ , [12, 55]) have shown that Figure 1D can be derived from any of Figure 1A-C.…”