mossel@stat.berkeley.edu, Phylognetic trees: What to look for and where? Lessons from Statistical Physics Elchanan Mossel, U.C. Berkeley and Microsoft Research mossel@stat.berkeley.edu, www.stat.berkeley.edu/~mossel/ 11/22/2018

Statistical physics Statistical physics is a sub-field of mathematical physics studying complex systems with simple microscopic interactions. The Ising model on a graph G=(V,E) is a probability measure (“Gibbs distribution”) on the space of configurations σ : V  {-1,1} such that P[σ] is given by: exp(Σ(v, w) ε E σ(v)σ(w)/T)/Z = exp( Σ(v, w) ε E σ(v)σ(w))/Z Or, Weight() ~ exp( # { u ~ v : (u) = (v) } ) Traditionally studied on cubes in Zd. The Ising model on 200 x 200 grid 11/22/2018

Statistical physics - intuition The Ising model on the nxn grid is given by: exp(Σ(v, w) ε E σ(v)σ(w)/T)/Z = exp( Σ(v, w) ε E σ(v)σ(w))/Z We expect that: T small,  large ) strong correlations: Corr(boundary,0) >  > 0 for all n. T large,  small ) weak correlations: Corr(boundary,0) ! 0 as n ! 1. 2n boundary Onsager (1944) proved it where Critical  = c = ln(1+21/2)/2 For most other graphs, we know very little The Ising model on 200 x 200 grid  = c 11/22/2018

Statistical physics on trees The Ising model on a tree T=(V,E) is given by: exp( Σ(v, w) ε E (v,w) (v) (w))/Z It is equivalent to the following model: Let r be a root (chosen arbitrarily). Let (r) = § 1 with probability ½ and for Each edge (u,v) directed away from the root, let: (v) = (u) with probability (u,v). (v) is independent § 1 otherwise. (u,v) = ( e(u,v)-e-(u,v) )/ (e(u,v)+e-(u,v)) + + + + - + + - + - + - + + 11/22/2018

Ising Model on binary Trees low interm. high bias bias no bias bias no bias “typical” boundary “typical” boundary “Extermality” 8 e, 2(e) > 1 8 e, 2 2(e) · 1 Unique Gibbs measure 8 e, 2(e) · 1 “Non-Extermality” 8 e, 2(e)2 > 1 11/22/2018

Statistical physics on trees: History Uniqueness studied by Bethe (1930’s). Extremality phase more recently Spitzer 75, Higuchi 77, Bleher-Ruiz-Zagrebnov 95, Evans-Kenyon-Peres-Schulman 2000, Ioffe 99, M 98, Haggstrom-M 2000, Kenyon-M-Peres 2001, Martinelli-Sinclair Weitz- 2003, Martin-2003 Many problems are still open. Extremality has rich connections with Noisy computation/communication [von-Neumann53, Evans-Shculmann00,…] Mixing of Markov chains [Berger-Kenyon-Mossel-Peres01,Martinelli-Sinclair-Weitz05] Spinglasses and Random Sat problems [Parisi,Mezard,Montanari; Mezard-Montanari06] 11/22/2018

Phylogeny “Phylogeny is the true evolutionary relationships between groups of living things” Noah Shem Japheth Ham Cush Kannan Mizraim 11/22/2018

History of Phylogeny Intuitively: : “animal kingdom” or “plant kingdom.” More scientifically: morphology, fossils, etc. Darwin … But: Is a human more like a great ape or like a chimpanzee? No brain, Can’t move Stupid Walks Stupid Swims Stupid Flies Too smart Barely moves 11/22/2018

Molecular Phylogeny Molecular Phylogeny: Based on DNA, RNA or protein sequences of organisms. Mutation mechanisms: Substitutions Transpositions Insertions, Deletions, etc. Will only consider substitutions and assume sequences are aligned. Noah acctga Shem Japheth Ham acctaa acctga acctga Put Cush Kannan Mizraim acctga 11/22/2018 acctga agctga acctga

Simplifying assumptions: models Assumption: Letters of sequences (“characters”) evolve independently and identically. CFN model: The first stochastic model invented by Cavender, Farris and Neyman (70s): Let (r) = § 1 with probability ½ and for Each edge (u,v) directed away from the root, let: (v) = (u) with probability (u,v). (v) is independent § 1 otherwise. This is exactly the Ising model on the evolutionary tree! Dictionary: {A,C} = + (Pyrimidine group) {G,T} = - (Purine group). Some results can be generalized to other models. 11/22/2018

Simplifying assumptions: trees Assumption 1: Evolution is on a tree. Assumption 2: Trees are binary -- All internal degrees are 3. Given a set of species (labeled vertices) X, an X-tree is a tree which has X as the set of leaves. Two X-trees T1 and T2 are identical if there’s a graph isomorphism between T1 and T2 that is the identity map on X. Most results to trees all of whose internal degrees are at least 3. u u Me’ v Me’ Me’’ Me’’ w w d a c b d a b c c a b d 11/22/2018

The Phylogenetic Challenge: Time Contemporary Genetic sequences Evolutionary model Genetic sequences ?? How to reconstruct Phylogenetic tree from genetic data at contemporary species?? 11/22/2018

Phylogeny Tree is unknown. Given sequences at the leaves of the tree. Want to reconstruct the tree (un-rooted). How “hard” is it as a function of n = “size of tree” = # leaves. k = length of sequences. 11/22/2018

Phylogeny 11/22/2018

n and k Length of sequence! Interested to know k = #characters needed to reconstruct the tree with n = #leaves. Erdos-Steel-Szekeley-Warnow96: If  < (e) < 1 -  for all e. Tree can be recovered from Sequences of length k = nc. In polynomial time. Question: How about shorter sequences? Previously, best lower bound on sequence length is k = (log n). However, in practice: Sometimes hard to find long sequences. Short sequences often suffice. 11/22/2018

Lesson 1: Phylogenetic lower bound for forgetful trees Th[M2004; Trans AMS]: If 2 2(e) < 1 for all e then we show A lower bound on sequence length of k = nc, where c > 0 is a function of  =maxe (e) and c ! 1 as  ! 0. Th [M2003; JCB] Similar theorem for general mutation models if mutation rates are high. Proofs are easy. 11/22/2018

Poly. lower bound for Phylogeny “Proof by coupling”: X=T ? ? L * k Known Known q-L * k If for all k characters we can couple bottom q-L levels, then X is independent of the data. By forgetfulness of tree, if k < nc, X is independent of data with high probability. Similar idea can be used to test trees (M+Riesenfeld) 11/22/2018

Lesson 2: Recent history is easy In the proof of lower bound, the “deep convergences” were hard to reconstruct. Theorem [M04]: If  < (e) < 1 -  for all e, then “most of the tree” can be reconstructed from sequences of length k = O(log n). “most of tree” := a forest F such that the true tree is obtained from F by adding o(n) edges. Result were refined + experiments in [Daskalakis-Hill-Jaffe-Miahescu-Mossel-Rao] Proof is *not* easy – based on Distorted Metrics. 11/22/2018

Lesson 3: Species that remember their past can reconstruct their history. Thm [Daskalakis-Mossel-Roch; To appear STOC06]: If 2 2(e) > 1 for all e then The tree can be recovered with high probability from sequences of length k = O( log n ). Solves M. Steel’s “Favourite conjecture” Builds on: [M2004; Trans AMS] Hard proof: Mixes probability, algorithms, statistical physics. 11/22/2018

Proof Sketch: Logarithmic reconstruction Two parts of the proofs: I. Statistical / algorithmic. II. Probability / statistical physics. By Forest result we may recover a forest containing 90% of the edges of the tree from O(log n) samples. Doesn’t use the 2 2 > 1 11/22/2018

Logarithmic Reconstruction II. Here we use the condition that 2  2 > 1 in order to estimate the characters at the inner nodes of the forest. “Like” I. 11/22/2018

Ising Model on binary Trees low interm. high bias bias no bias k = (nc) bias no bias Most tree from k = O(log n) k = O( log n ) “typical” boundary “typical” boundary “Extermality” 8 e, 2(e) > 1 8 e, 2 2(e) · 1 Unique Gibbs measure 8 e, 2(e) · 1 “Non-Extermality” 8 e, 2(e)2 > 1 11/22/2018

Many more challenges to come … We know very little … We don’t understand methods used in practice: Maximum Likelihood (NP hard on arbitrary data; [Chor-Tuller05; Roch05]) Markov Chain Monte Carlo (Can be exponentially slow on mixtures; M-Vigda05). In what sense Parsimony = Maximum – Likelihood? (2 Conjectures by Steel) Other mutation models: rates across sites, gene order etc. etc. + all the problems on Gibbs measures on trees 11/22/2018