Higher-order interactions in fitness landscapes are sparse

Abstract

Biological fitness arises from interactions between molecules, genes, and organisms. To discover the causative mechanisms of this complexity, we must differentiate the significant interactions from a large number of possibilities. Epistasis is the standard way to identify interactions in fitness landscapes. However, this intuitive approach breaks down in higher dimensions for example because the sign of epistasis takes on an arbitrary meaning, and the false discovery rate becomes high. These limitations make it difficult to evaluate the role of epistasis in higher dimensions. Here we develop epistatic filtrations, a dimensionally-normalized approach to define fitness landscape topography for higher dimensional spaces. We apply the method to higherdimensional datasets from genetics and the gut microbiome. This reveals a sparse higher-order structure that often arises from lower-order. Despite sparsity, these higher-order effects carry significant effects on biological fitness and are consequential for ecology and evolution.

Virus-immune dynamics determined by prey-predator interaction network and epistasis in viral fitness landscape

“…Note that because the sum of coefficients and weighted sum of sequences must vanish, along with a circuit being minimal, the number of binary sequences in a "(ones') complement" circuit must be four. In general, the number of circuits rapidly grows with n (there are 20 circuits for n = 3, 1348 circuits for n = 4 [16]) and can be interpreted geometrically in terms of shapes formed by vertices of the n-cube [5].…”

“…Indeed consider the nested equilibrium E n . We arrange the (persistent) nested virus components, together with the invading strain, in the vector v = ( y, 0) T , where the y is from (16) and the last component is the invading strain, y i , which is zero when at equilibrium E n . In the critical case, where the linear form A i corresponding to circuit C i is zero, there is a line of equilibria given by v * = v − αa where a is the (circuit) coefficients of A i and 0 ≤ α ≤ C with C = min {a k y k : a k > 0, k = 0, 1, .…”

“…2(b)). By (16), the strain which is replaced depends upon the sign of s 1 − 1 + Q2 R2 . In the case of n = 3, there are 2 3 − 3 − 1 = 4 circuits corresponding to a non-nested invading strain.…”

“…Thus, Theorem 2 implies the nested equilibrium is stable if and only if all of the quantities (i)-(iv) are positive. Furthermore, in each case that a single inequality fails, the following bifurcation occurs where the missing strain replaces a nested strain y j where j is determined by (i) arg min j=0,2 ( y j ), (ii) arg min j=0,3 ( y j ), (iii) arg min j=1,3 ( y j ), (iv) arg min j=0,3 ( y j ), where y j are defined in terms of viral and immune response fitness quantities in (16). For example, if a bifurcation from nested equilibrium E 3 occurs through inequality (iii) switching sign, then y 6 (101) replaces either y 1 or y 3 , depending on whether y 1 < y 3 , i.e.…”

Geometry of fitness landscapes: Peaks, shapes and universal positive epistasis

“…The triangulation is called the shape of the fitness landscape. Recent work on triangulations and fitness graphs includes [4,6,3,7,13,12,15,19,21].…”