Page 41 - Tequio 11

On some polytopes in phylogenetics/Hoessly/27-40 39

15
It is interesting to note that each NNI-move on / corresponds to an edge of / (Haws et al., 2011).
Furthermore, partial results on facet inequalities exist, i.e., e.g. some facets from cherries (Forcey et al., 2016)
were characterised.

4. Conclusion and Outlook

We introduced notions from phylogenetics and mathematics that mostly relate to the distance-based approach
to phylogenetic reconstruction. The three polytopes are associated to either distances or the number of
species. While we focussed on tree-like metrics where tight span and fundamental polytope are well-
understood, for more general classes of metrics we still have limited knowledge about their structure. The
situation for BME polytopes is similar, where only small examples have complete characterisations. It will be
interesting to see in what ways structural properties of the introduced objects relate to each other and to other
notions from phylogenetics and mathematics in the future.

References

Bandelt, H.-J., & Dress, A. W. M. (1992). A canonical decomposition theory for metrics on a finite set. Adv. Math.,
92(1), 47–105.
Bollobas, B. (1998). Modern graph theory (corrected ed.). Heidelberg: Springer.
Delucchi, E., & Hoessly, L. (2020). Fundamental polytopes of metric trees via parallel connections of matroids.
European Journal of Combinatorics, 87, 103098.
Dress, A. (1984). Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups:
A note on combinatorial properties of metric spaces. Advances in Mathematics, 53, 321-402.
Dress, A., Huber, K. T., Koolen, J., Moulton, V., & Spillner, A. (2011). Basic phylogenetic combinatorics. Cambridge
University Press.
Eickmeyer, K., Huggins, P., Pachter, L., & Yoshida, R. (2008). On the optimality of the neighbor-joining algorithm.
Algorithms for Molecular Biology, 3(1), 5.
Forcey, S., Keefe, L., & Sands, W. (2016). Facets of the balanced minimal evolution polytope. Journal of
Mathematical Biology, 73(2), 447–468.
Gascuel, O., & Steel, M. (2006). Neighbor-Joining Revealed. Molecular Biology and Evolution, 23(11), 1997-2000.
Retrieved from https://doi.org/10.1093/molbev/msl072
Gordon, J., & Petrov, F. (2017). Combinatorics of the Lipschitz polytope. Arnold Math. J., 3(2), 205–218.
Haws, D. C., Hodge, T. L., & Yoshida, R. (2011). Optimality of the neighbor joining algorithm and faces of the
balanced minimum evolution polytope. Bulletin of Mathematical Biology, 73(11), 2627–2648.
Huson, D. H., Rupp, R., & Scornavacca, C. (2010). Phylogenetic networks: Concepts, algorithms and applications.
Cambridge University Press.
Isbell, J. R. (1964). Six theorems about injective metric spaces. Commentarii Mathematici Helvetici, 39(1), 65–
76.
Lefort, V., Desper, R., & Gascuel, O. (2015). FastME 2.0: A Comprehensive, Accurate, and Fast Distance-Based
Phylogeny Inference Program. Molecular Biology and Evolution, 32(10), 2798-2800.

15 A nearest-neighbour interchange (NNI) move on a phylogenetic tree rearranges the tree. Such moves are e.g. used in algorithms in order to optimise
A nearest-neighbour interchange (NNI) move on a phylogenetic tree rearranges the tree. Such moves are e.g. used in algorithms
15
over the set of trees.
in order to optimise over the set of trees.

Tequio, enero-abril 2021, vol. 4, no. 11