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

2. 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

3. 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

4. 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

5. 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

6. 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

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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. Phylogeny
11/22/2018

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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