The concept of genetic epistasis defines an interaction between two genetic loci as the degree of non-additivity in their phenotypes. A fitness landscape describes the phenotypes over many genetic loci, and the shape of this landscape can be used to predict evolutionary trajectories. Epistasis in a fitness landscape makes prediction of evolutionary trajectories more complex because the interactions between loci can produce local fitness peaks or troughs, which changes the likelihood of different paths. While various mathematical frameworks have been proposed to calculate the shapes of fitness landscapes, Beerenwinkel, Pachter and Sturmfels (2007) suggested studying regular subdivisions of convex polytopes. In this sense, each locus provides one dimension, so that the genotypes form a cube with the number of dimensions equal to the number of genetic loci considered. The fitness landscape is a height function on the coordinates of the cube. Here, we propose cluster partitions and cluster filtrations of fitness landscapes as a new mathematical tool, which provides a concise combinatorial way of processing metric information from epistatic interactions. Furthermore, we extend the calculation of genetic interactions to consider interactions between microbial taxa in the gut microbiome of Drosophila fruit flies. We demonstrate similarities with and differences to the previous approach. As one outcome we locate interesting epistatic information on the fitness landscape where the previous approach is less conclusive.

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.

We introduce a new algorithm for enumerating chambers of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to physics and computer science.

A longstanding goal of biology is to identify the key genes and species that critically impact evolution, ecology, and health. Yet biological interactions between genes, species, and different environmental contexts change the individual effects due to non-additive interactions, known as epistasis. In the fitness landscape concept, each gene/organism/environment is modeled as a separate biological dimension, yielding a high dimensional landscape, with epistasis adding local peaks and valleys to the landscape. Massive efforts have defined dense epistasis networks on a genome-wide scale, but these have mostly been limited to pairwise, or two-dimensional, interactions. Here we develop a new mathematical formalism that allows us to quantify interactions at high dimensionality in genetics and the microbiome. We then generate and also reanalyze combinatorically complete datasets (two genetic, two microbiome). In higher dimensions, we find that key genes (e.g. pykF) and species (e.g. Lactobacillus plantarum) distort the fitness landscape, changing the interactions for many other genes/species. These distortions can fracture a landscape with one optimal fitness peak into a landscape with many local optima, regulating evolutionary or ecological diversification, which may explain how a probiotic bacterium can stabilize the gut microbiome.

The dual graph Γ(h) of a regular triangulation Σ(h) carries a natural metric structure. The minimum spanning trees of Γ(h) recently proved to be conclusive for detecting significant data signal in the context of population genetics. In this paper we prove that the parameter space of such minimum spanning trees is organized as a polyhedral fan, called the MST-fan of Σ(h), which subdivides the secondary cone of Σ(h) into parameter cones. We partially describe its local face structure and examine the connection to tropical geometry in virtue of matroids and Bergman fans.