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Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture

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A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on n leaves, sample sequences of length Ω(log n) are needed. It was conjectured by Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than \({p^{\ast} = (\sqrt{2}-1)/2^{3/2}}\), then the tree can be recovered from sequences of length O(log n). The value p* is given by the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel’s conjecture was proven by the second author in the special case where the tree is “balanced.” The second author also proved that if all edges have mutation probability larger than p* then the length needed is n Ω(1). Here we show that Steel’s conjecture holds true for general trees by giving a reconstruction algorithm that recovers the tree from O(log n)-length sequences when the mutation probabilities are discretized and less than p*. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.


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S.R. thanks Martin Nowak and the Program for Evolutionary Dynamics at Harvard University where part of this work was done. E.M. and S.R. thank Mike Steel for his enthusiastic encouragement for studying the connections between the reconstruction problem and phylogenetics. We thank Satish Rao and Allan Sly for interesting discussions. We also thank the reviewers for their helpful comments. C.D. and S.R. performed this work at UC Berkeley and Microsoft Research.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Correspondence to Sébastien Roch.

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Partially supported by CIPRES (NSF ITR grant # NSF EF 03-31494). Supported by a Miller fellowship in Statistics and Computer Science, by a Sloan fellowship in Mathematics and by NSF grants DMS-0504245, DMS-0528488 DMS-0548249 (CAREER), ONR grant N0014-07-1-05-06 and ISF grant 1300/08.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Daskalakis, C., Mossel, E. & Roch, S. Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture. Probab. Theory Relat. Fields 149, 149–189 (2011). https://doi.org/10.1007/s00440-009-0246-2

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  • Phylogenetics
  • CFN model
  • Ising model
  • Phase transitions
  • Reconstruction problem
  • Jukes–Cantor

Mathematics Subject Classification (2000)

  • Primary 60K35
  • 92D15
  • Secondary 60J85
  • 82B26