Phylogenetic trees:

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Phylogenetic trees What to look for and where?
Lessons from Statistical Physics
Elchanan Mossel, U.C. Berkeley and Microsoft
Research mossel_at_stat.berkeley.edu,
www.stat.berkeley.edu/mossel/
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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 s V ? -1,1 such that
  Ps is given by
• exp(S(v, w) e E s(v)s(w)/T)/Z exp(? S(v, w) e
  E s(v)s(w))/Z
• Or, Weight(?) exp(? u v ?(u) ?(v) )
• Traditionally studied on cubes in Zd.

The Ising model on 200 x 200 grid
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Statistical physics - intuition

• The Ising model on the nxn grid is given by
• exp(S(v, w) e E s(v)s(w)/T)/Z exp(? S(v, w) e
  E s(v)s(w))/Z
• We expect that
• T small, ? large ) strong correlations
• Corr(?boundary,?0) gt ? gt 0 for all n.
• T large, ? small ) weak correlations
• Corr(?boundary,?0) ! 0 as n ! 1.

2n
0
boundary

• Onsager (1944) proved it where
• Critical ? ?c ln(121/2)/2
• For most other graphs, we know very little

The Ising model on 200 x 200 grid ? ?c
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Statistical physics on trees

• The Ising model on a tree T(V,E) is given by
• exp( S(v, w) e 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))

-



-



-
-

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Ising Model on binary Trees
low
interm.
high
bias
bias
no bias
no bias
bias
typical boundary
typical boundary
Unique Gibbs measure 8 e, 2?(e) 1
Extermality 8 e, 2?(e) gt 1 8 e, 2 ?2(e) 1
Non-Extermality 8 e, 2?(e)2 gt 1
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Statistical physics on trees History

• Uniqueness studied by Bethe (1930s).
• 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-Pere
  s01,Martinelli-Sinclair-Weitz05
• Spinglasses and Random Sat problems
  Parisi,Mezard,Montanari Mezard-Montanari06

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Phylogeny

• Phylogeny is the true evolutionary relationships
  between groups of living things

Noah
Shem
Ham
Japheth
Cush
Mizraim
Kannan
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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, Cant move
Stupid Walks
Stupid Swims
Stupid Flies
Too smart Barely moves
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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
Ham
Japheth
acctga
acctaa
acctga
Put
Cush
Mizraim
Kannan
acctga
acctga
agctga
acctga
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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.

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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 theres 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
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The Phylogenetic Challenge
Time
Contemporary Genetic sequences
Evolutionary model
Genetic sequences
??

• How to reconstruct Phylogenetic tree from genetic
  data at contemporary species??

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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.

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Phylogeny
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n and k
Length of sequence!

• Interested to know k characters needed to
  reconstruct the tree with n leaves.

• Erdos-Steel-Szekeley-Warnow96
• If ? lt ?(e) lt 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.

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Lesson 1 Phylogenetic lower bound for forgetful
trees

• ThM2004 Trans AMS
• If 2 ?2(e) lt 1 for all e then we show
• A lower bound on sequence length of k nc, where
• c gt 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.

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Poly. lower bound for Phylogeny

• Proof by coupling

XT
?
?
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 lt nc, X is
  independent of data with high probability.
• Similar idea can be used to test trees
  (MRiesenfeld)

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Lesson 2 Recent history is easy

• In the proof of lower bound, the deep
  convergences were hard to reconstruct.
• Theorem M04
• If ? lt ?(e) lt 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.

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Lesson 3 Species that remember their past can
reconstruct their history.

• Thm Daskalakis-Mossel-Roch To appear STOC06
• If 2 ?2(e) gt 1 for all e then
• The tree can be recovered with high probability
  from sequences of length
• k O( log n ).
• Solves M. Steels Favourite conjecture
• Builds on M2004 Trans AMS
• Hard proof Mixes probability, algorithms,
  statistical physics.

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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.
• Doesnt use the 2 ?2 gt 1

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Logarithmic Reconstruction

• II. Here we use the condition that 2 ? 2 gt 1 in
  order to estimate the characters at the inner
  nodes of the forest.

Like I.
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Ising Model on binary Trees
low
interm.
high
bias
bias
no bias
k ?(nc)
no bias
bias
Most tree from k O(log n)
k O( log n )
typical boundary
typical boundary
Unique Gibbs measure 8 e, 2?(e) 1
Extermality 8 e, 2?(e) gt 1 8 e, 2 ?2(e) 1
Non-Extermality 8 e, 2?(e)2 gt 1
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Many more challenges to come

• We know very little
• We dont 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