phylogeny of mixture models
DESCRIPTION
Phylogeny of Mixture Models. Daniel Štefankovič Department of Computer Science University of Rochester joint work with Eric Vigoda College of Computing Georgia Institute of Technology. Outline. Introduction (phylogeny, molecular phylogeny) Mathematical models (CFN, JC, K2, K3) - PowerPoint PPT PresentationTRANSCRIPT
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Phylogeny of
Mixture
Models
Daniel ŠtefankovičDepartment of
Computer ScienceUniversity of Rochester
joint work with
Eric VigodaCollege of ComputingGeorgia Institute of
Technology
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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Phylogeny
orangutan gorilla chimpanzee human
development of a group: the development over time of a species, genus, or group, as contrasted with the development of an individual (ontogeny)
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Phylogeny – how?development of a group: the development over time of a species, genus, or group, as contrasted with the development of an individual (ontogeny)
past – morphologic data (beak length, bones, etc.)
present – molecular data (DNA, protein sequences)
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Molecular phylogenyINPUT:aligned DNA sequences
OUTPUT:phylogenetic tree
Human:Chimpanzee: Gorilla:Orangutan:
ATCGGTAAGTACGTGCGAATTCGGTAAGTAAGTGGGATTTAGGTCAGTAAGTGCGTTTTGAGTCAGTAAGAGAGTT
orangutan gorilla chimpanzee human
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Universal phylogenyUniversal phylogeny
deduced from comparison of SSU deduced from comparison of SSU and LSU rRNA sequences (2508 and LSU rRNA sequences (2508 homologous sites) using Kimura’s 2-homologous sites) using Kimura’s 2-parameter distance and the NJ parameter distance and the NJ method. method.
The absence of root in this tree is The absence of root in this tree is expressed using a circular design.expressed using a circular design.
BacteriaBacteriaArchaeaArchaea
EucaryaEucarya
Source: Manolo Gouy, Introduction to Molecular Phylogeny
Example of a real phylogenetic tree
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Leaves = Taxa = {chimp, human, ...}Vertices = NodesEdges = BranchesTree = Tree
Dictionary
orangutan gorilla
chimpanzee human
Unrooted/Rooted trees
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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Cavender-Farris-Neyman (CFN) model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
0
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
0
1 with probability 0.32
0 with probability 0.68
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
0
1
1 with probability 0.32
0 with probability 0.68
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
0
1
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
0
1
1 with probability 0.15
0 with probability 0.85
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
0
0
0
0101
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
0
0
0
0101
1……….. 0……….. 1……….. 0………..
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
1
0
0
1001
11……….. 00……….. 10……….. 01………..
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
1
1
1
1110
110…….. 001…….. 101…….. 011……..
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CFN model
orangutan gorilla chimpanzee human
0.320.22
0.12 0.09
0.06
0.15
Weight of an edge = probability that 0 and 1 get flipped
Denote the distribution on leaves (T,w)T = tree topologyw = set of weights on edges
0000,0001,0010,0011,0100,0101,0110,0111,…
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Generalization to more states
orangutan gorilla chimpanzee human
Weight of an edge = probability that 0 and 1 get flipped
A
A
A
ACGT
transition matrix
0.9 0.05 0.03
0.02
0.05
0.87 0.07
0.01
0.03
0.07 0.89
0.01
0.02
0.01 0.01
0.96
AG
C
T
A G C T
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Models: Jukes-Cantor (JC)
0.9 0.05 0.03
0.02
0.05
0.87 0.07
0.01
0.03
0.07 0.89
0.01
0.02
0.01 0.01
0.96
AG
C
T
A G C T
there are 4 states
exp( t.R )
Rate matrix
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Models: Kimura’s 2 parameter (K2)
0.9 0.05 0.03
0.02
0.05
0.87 0.07
0.01
0.03
0.07 0.89
0.01
0.02
0.01 0.01
0.96
AG
C
T
A G C T
Rate matrix
exp( t.R )
purine/pyrimidine mutations less likely
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Models: Kimura’s 3 parameter (K3)
0.9 0.05 0.03
0.02
0.05
0.87 0.07
0.01
0.03
0.07 0.89
0.01
0.02
0.01 0.01
0.96
AG
C
T
A G C T
Rate matrix
exp( t.R )
take hydrogen bonds into account
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Reconstructing the tree?Let D be samples from (T,w).Can we reconstruct T (and w) ?
• parsimony• distance based methods• maximum likelihood methods (using MCMC)• invariants• ?
Main obstacle for all methods: too many leaf-labeled trees (2n-3)!!=(2n-3)(2n-5)…3.1
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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Maximum likelihood method
Likelihood of tree S is L(S) = maxw Pr(D | S,w)
Let D be samples from (T,w).
For |D|!1 then the maximum likelihood tree is T
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MCMC Algorithms for max-likelihoodCombinatorial steps:
Numerical steps (i.e., changing the weights)
NNI moves (Nearest Neighbor Interchange)
Move with probability min{1,L(Tnew
)/L(Told
)}
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MCMC Algorithms for max-likelihoodOnly combinatorial steps:
NNI moves (Nearest Neighbor Interchange)
Does this Markov Chain mix rapidly?
Not known!
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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Mixtures
one tree topologymultiple mixtures
Can we reconstruct the tree T?
The mutation rates differ for positions in DNA
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Reconstruction from mixtures - ML
maximum likelihood: fails to for CFN, JC, K2, K3
Theorem 1:
For all 0<C<1/2, all x sufficiently small:(i) maximum likelihood tree ≠ true
tree(ii) 5-leaf version: MCMC torpidly
mixingSimilarly for JC, K2, and K3 models
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Reconstruction from mixtures - MLRelated results: [Kolaczkowski,Thornton] Nature, 2004. Experimental results for JC model [Chang] Math. Biosci., 1996.
Different example for CFN model.
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Reconstruction from mixtures - ML
Proof:Difficulty: finding edge weights that maximize likelihood.
For x=0, trees are the same -- pure distribution, tree achievable on all topologies. So know max likelihood weights for every topology.
(observed)T log (T,w)
If observed comes from \mu(S,v) then it is optimal to take T=S and w=v (basic property of log-likelihood)
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Reconstruction from mixtures - ML
Proof:Difficulty: finding edge weights that maximize likelihood.
For x=0, trees are the same -- pure distribution, tree achievable on all topologies. So know max likelihood weights for every topology.
For x small, look at Taylor expansion bound max likelihood in terms of x=0 case and functions of Jacobian and Hessian.
=
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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Reconstruction – other algorithms?
GOAL: Determine tree topology Duality theorem: Every model has either:
A) ambiguous mixture distributions on 4 leaf trees
(reconstruction impossible)
B) linear tests (reconstruction easy)
The dimension of the space of possible linear tests: CFN = 2, JC = 2, K2 = 5, K3 = 9
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For all 0<a,b<1/2, there is c=c(a,b) where: above mixture distribution on tree T is identical to below mixture distribution on tree S.
Ambiguity in CFN model
Previously: non-constructive proof of nicer ambiguity in CFN model [Steel,Szekely,Hendy,1996]
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What about JC?
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What about JC?Reconstruction of the topology from mixture possible.
Linear test = linear function which is >0 for mixture from T
2
<0 for mixture from T3
There exists a linear test for JC model.
Follows immediately from Lake’1987 – linear invariants.
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Lake’s invariants ! Test
f=μ(AGCC) + μ(ACAC) + μ(AACT) +μ(ACGT) - μ(ACGC) - μ(AACC) - μ(ACAT) - μ(AGCT)
For μ=μ(T1,w), f=0
For μ=μ(T2,w), f<0
For μ=μ(T3,w), f>0
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Linear invariants v. Tests
Linear invariant = hyperplane containing mixtures from T1
Test = hyperplane strictly separating mixtures from T2 from mixtures from T3
pure distributions from T2
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Linear invariants v. Tests
Linear invariant = hyperplane containing mixtures from T1
Test = hyperplane strictly separating mixtures from T2 from mixtures from T3
pure distributions from T2
mixtures from T2
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Linear invariants v. Tests
Linear invariant = hyperplane containing mixtures from T1
Test = hyperplane strictly separating mixtures from T2 from mixtures from T3
mixtures from T2
mixtures from T3
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Linear invariants v. Tests
Linear invariant = hyperplane containing mixtures from T1
Test = hyperplane strictly separating mixtures from T2 from mixtures from T3
mixtures from T2
mixtures from T3
test
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Separating hyperplanes Duality theorem: Every model has either:
A) ambiguous mixture distributions on 4 leaf trees
(reconstruction impossible)
B) linear tests (reconstruction easy)
ambiguous mixture separating hyperplane
Separating hyperplane theorem:
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Duality theorem: Every model has either:
A) ambiguous mixture distributions on 4 leaf trees
(reconstruction impossible)
B) linear tests (reconstruction easy)
ambiguous mixture strictly separating hyperplane?
Separating hyperplane theorem ?:
Strictly separating hyperplanes ???
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ambiguous mixture strictly separating hyperplane?
Separating hyperplane theorem ?:
{ (x,y) | x>0 } [ { (0,y) | y>0 }
NO strictly separating hyperplane
{(0,0)}
Strictly separating not always possible
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When strictly separating possible?
{ (x,y) | x>0 } [ { (0,y) | y>0 }
NO strictly separating hyperplane
{(0,0)}
(x,y2 – xz) x¸0, y>0
standard phylogeny models satisfy the assumption
Lemma: Sets which are convex hulls of images of open sets under a multi-linear polynomial map have a strictly separating hyperplane.
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
WLOG linearly independent
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
Have s1,…,sn such that s1 P1(x) + … + sn Pn(x) ¸ 0 for all x 2 O
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
Have s1,…,sn such that s1 P1(x) + … + sn Pn(x) ¸ 0 for all x 2 O
Goal: show s1 P1(x) + … + sn Pn(x) > 0 for all x 2 O
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
Have s1,…,sn such that s1 P1(x) + … + sn Pn(x) ¸ 0 for all x 2 O
Goal: show s1 P1(x) + … + sn Pn(x) > 0 for all x 2 O
Suppose: s1 P1(a) + … + sn Pn(a) = 0 for some a 2 O
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
linearly independents1 P1(x) + … + sn Pn(x) ¸ 0 for all x 2 Os1 P1(0) + … + sn Pn(0) = 0Let R(x)=s1 P1(x) + … + sn P(x) - non-zero polynomial
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
linearly independents1 P1(x) + … + sn Pn(x) ¸ 0 for all x 2 Os1 P1(0) + … + sn Pn(0) = 0Let R(x)=s1 P1(x) + … + sn P(x) - non-zero polynomial
R(0)=0 ) no constant monomial
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
linearly independents1 P1(x) + … + sn Pn(x) ¸ 0 for all x 2 Os1 P1(0) + … + sn Pn(0) = 0Let R(x)=s1 P1(x) + … + sn P(x) - non-zero polynomial
R(0,…,0,xi,0…,0) ¸ 0 ) no monomial xi
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ProofLemma: For sets which are convex hulls of images of open sets under a multi-linear polynomial map – strictly separating hyperplane.
P1(x1,…,xm),…,Pn(x1,…,xm), x=(x1,…,xm) 2 O
Proof:
linearly independents1 P1(x) + … + sn Pn(x) ¸ 0 for all x 2 Os1 P1(0) + … + sn Pn(0) = 0Let R(x)=s1 P1(x) + … + sn P(x) - non-zero polynomial
R(0,…,0,xi,0…,0) ¸ 0 ) no monomial xi
…. ) no monomials at all, a contradiction
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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Duality application: non-constructive proof of mixturesDuality theorem: Every
model has either:
A) ambiguous mixture distributions on 4 leaf trees
(reconstruction impossible)
B) linear tests (reconstruction easy)
For K3 model the space of possible tests has dimension 9 T = 1 T1 + … + 9 T9
Goal: show that there exists no test
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Duality application: non-constructive proof of mixtures
transition matrix P = exp(x.R)
rate matrix
entries in P = generalized polynomialspoly(,,,x) exp(lin(,,,x))
LEM: The set of roots of a non-zero generalized polynomial has measure 0.
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Non-constructive proof of mixtures
transition matrix P(x) = exp(x.R)
T1,…,T9 are generalized polynomials in ,,,x
Wronskian det Wx(T1,…,T9) is a generalized polynomial ,,,x
det Wx(T1,…,T9) 0
P(x)
P(2x)
P(3x)
P(4x) P(0)
Test should be 0 by continuity.
Wx(T1,…T9) [1,…9]=0) NO TEST !
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Non-constructive proof of mixtures
LEM: The set of roots of a non-zero generalized polynomial has measure 0.
The last obstacle: Wronskian W(T1,…,T9) is non-zero
Horrendous generalized polynomials, even fore.g., 1,=2,=4
plug-in complex numbers
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OutlineIntroduction (phylogeny, molecular phylogeny)
Mathematical models (CFN, JC, K2, K3)
Maximum likelihood (ML) methods
Our setting: mixtures of distributionsML, MCMC for ML fails for mixtures
Duality theorem: tests/ambiguous mixtures
Proofs (strictly separating hyperplanes, non-constructive ambiguous mixtures)
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M a semigroup of doubly stochasic matrices (with multiplication). Under what conditions on M can you reconstruct the tree topology?
Open questions
* x x x
x * x x
x x * x
x x x *
* x y y
x * y y
y y * x
y y x *
* x y z
x * z y
y z * x
z y x *
* x
x *
0<x<1/4
yes
0<y·x<1/4
yes
0<x<1/2no
0<z·y·x<1/2
no
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Idealized setting: For data generated from a pure distribution (i.e., a single tree, no mixture):
Are MCMC algorithms rapidly or torpidly mixing?
How many characters (samples) needed untilmaximum likelihood tree is true tree?
Open questions