Cluster partitions and fitness landscapes of the Drosophila fly microbiome

Abstract

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 investigate properties of fitness landscapes, Beerenwinkel et al. (Stat Sin 17(4):1317–1342, 2007a) 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.

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Acknowledgements

The data was collected by Alison Gould and Vivian Zhang in Will Ludington’s Lab at UC Berkeley and is now published in Gould et al. (2017). We are indebted to Christian Haase and Günter Rote for a fruitful discussion concerning epistatic weights, leading to the definition (5).

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Correspondence to Lisa Lamberti.

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Research by M. Joswig is partially supported by Einstein Stiftung Berlin and Deutsche Forschungsgemeinschaft (EXC 2046: “MATH\(^+\)”, SFB-TRR 109: “Discretization in Geometry and Dynamics”, SFB-TRR 195: “Symbolic Tools in Mathematics and their Application”, and GRK 2434: “Facets of Complexity”). W. B. Ludington acknowledges support by the NIH Office of the Director’s Early Independence Award Grant DP5OD017851.

Appendix

Appendix

See Tables 6, 7, 8, 9 and Figs. 7, 8.

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Eble, H., Joswig, M., Lamberti, L. et al. Cluster partitions and fitness landscapes of the Drosophila fly microbiome. J. Math. Biol. 79, 861–899 (2019). https://doi.org/10.1007/s00285-019-01381-0

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Keywords

  • Fitness landscape
  • Epistasis
  • Polyhedral subdivision
  • Dual graphs
  • Filtration
  • Microbiome

Mathematics Subject Classification

  • Primary: 52B55 Computational aspects related to convexity
  • Secondary: 92B05 General biology and biomathematics
  • 92B15 General biostatistics
  • 57Q15 Triangulating manifolds
  • 68U05 Computer graphics; computational geometry