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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Assarf B, Gawrilow E, Herr K, Joswig M, Lorenz B, Paffenholz A, Rehn T (2017) Computing convex hulls and counting integer points with polymake. Math Program Comput 9(1):1–38
Beerenwinkel N, Pachter L, Sturmfels B (2007) Epistasis and shapes of fitness landscapes. Stat Sin 17(4):1317–1342
Beerenwinkel N, Pachter L, Sturmfels B, Elena SF, Lenski RE (2007) Analysis of epistatic interactions and fitness landscapes using a new geometric approach. BMC Evol Biol 7(1):60
Benjamini Y, Yekutieli D (2001) The control of the false discovery rate in multiple testing under dependency. Ann Stat 29(4):1165–1188
Brenner J, Cummings L (1972) The Hadamard maximum determinant problem. Am Math Monthly 79:626–630
Dauxois J, Hassenforder C (2004) Toutes les probabilités et les statistiques: cours et exercices corrigés. Ellipses
De Loera JA, Rambau J, Santos F (2010) Triangulations, vol 25 of algorithms and computation in mathematics. Springer, Berlin (Structures for algorithms and applications)
de Visser JAGM, Krug J (2014) Empirical fitness landscapes and the predictability of evolution. Nat Rev Genet 15(7):480–490
Edelsbrunner H, Harer JL (2010) Computational topology. American Mathematical Society, Providence (An introduction)
Everitt BS, Skrondal A (2010) The Cambridge dictionary of statistics, 4th edn. Cambridge University Press, Cambridge
Gawrilow E, Joswig M (2000) polymake: a Framework for Analyzing Convex Polytopes. In: Kalai G, Ziegler GM (eds) Polytopes — Combinatorics and Computation. DMV Seminar, vol 29. Birkhäuser, Basel
Gentle JE (2002) Elements of computational statistics. Statistics and computing. Springer, New York
Gould AL, Zhang V, Lamberti L, Jones EW, Obadia B, Gavryushkin A, Carlson JM, Beerenwinkel N, Ludington WB (2017) High-dimensional microbiome interactions shape host fitness. bioRxiv
Hallgrímsdóttir IB, Yuster DS (2008) A complete classification of epistatic two-locus models. BMC Genet 9(1):17
Herrmann S, Joswig M (2008) Splitting polytopes. Münster J Math 1:109–141
Herrmann S, Joswig M, Speyer D (2014) Dressians, tropical Grassmannians, and their rays. Forum Math 26(6):1853–1882
Kreznar JH, Keller MP, Traeger LL, Rabaglia ME, Schueler KL, Stapleton DS, Zhao W, Vivas EI, Yandell BS, Broman AT, Hagenbuch B, Attie AD, Rey FE (2017) Host genotype and gut microbiome modulate insulin secretion and diet-induced metabolic phenotypes. Cell Rep 18(7):1739–1750
Maclagan D, Sturmfels B (2015) Introduction to tropical geometry, volume 161 of graduate studies in mathematics. American Mathematical Society, Providence
Schervish MJ (1996) Theory of statistics. Springer Series in Statistics. Springer, New York
Seidel R (2018) Convex hull computations. In: Tóth CD, Goodmann JE, Rourke JO (eds) Handbook of discrete and computational geometry, chapter 26, 23rd edn. CRC Press, Boca Raton
Tsagris M, Beneki C, Hassani H (2014) On the folded normal distribution. Mathematics 2(1):12–28
Wright Sewall (1932) The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Proceedings of the sixth international congress of genetics, vol 1, pp 356–366
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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.
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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
- Fitness landscape
- Polyhedral subdivision
- Dual graphs
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