Using algebraic geometry for phylogenetic reconstruction...Title Using algebraic geometry for...

Using algebraic geometry for phylogeneticreconstruction

Marta Casanellas i Rius(joint work with Jesus Fernandez-Sanchez)

Departament de Matematica Aplicada IUniversitat Politecnica de Catalunya

IMA WorkshopApplications in Biology, Dynamics, and Statistics

March 8, 2007

M. Casanellas (UPC) Using algebraic geometry ... IMA, March 2007 1 / 36

Outline

1 Algebraic evolutionary models

2 Phylogenetic inference using algebraic geometry

3 The geometry of the Kimura variety

4 New results on simulated data

Outline

Phylogenetic reconstruction

Given n current species, e.g. HUMAN, GORILLA, CHIMP,given part of their genome and an alignment alignment

Goal: to reconstruct their ancestral relationships (phylogeny):

AlignmentMutations, deletions and insertions of nucleotides occur along thespeciation process.

Given two DNA sequences, an alignment is a correspondencebetween them that accounts for their differences. The optimalalignment is the one that minimizes the number of mutations,deletions and insertions.seq1 : ACGTAGCTAAGTTA... seq2 : ACCGAGACCCAGTA...

A possible alignment is:

seq1 A C − G − T A − G C T A A G T T Aseq2 A C C G A G A C− C C A − G T − A

AlignmentMutations, deletions and insertions of nucleotides occur along thespeciation process.

Given two DNA sequences, an alignment is a correspondencebetween them that accounts for their differences. The optimalalignment is the one that minimizes the number of mutations,deletions and insertions.seq1 : ACGTAGCTAAGTTA... seq2 : ACCGAGACCCAGTA...

A possible alignment is:

seq1 A C − G − T A − G C T A A G T T Aseq2 A C C G A G A C− C C A − G T − A

AACTTCGAGGCTTACCGCTG

AAGGTCGATGCTCACCGATG

AACGTCTATGCTCACCGATG

AACTTCGAGGCTTACCGCTG

AAGGTCGATGCTCACCGATG

AACGTCTATGCTCACCGATG

Algebraic evolutionary models

Assume that all sites of the alignment evolve equally andindependently.At each node of the tree we put a random variable taking values in{A,C,G,T }

ti = branch length(represents the number ofmutations per site along thatbranch)

Variables at the leaves are observed and variables at the interiornodes are hidden.At each branch we write a matrix (substitution matrix) with theprobabilities of a nucleotide at the parent node being substitutedby another at its child.

A C G T

P(A|A) P(C|A) P(G|A) P(T |A)P(A|C) P(C|C) P(G|C) P(T |C)P(A|G) P(C|G) P(G|G) P(T |G)P(A|T ) P(C|T ) P(G|T ) P(T |T )

Algebraic evolutionary models

Assume that all sites of the alignment evolve equally andindependently.At each node of the tree we put a random variable taking values in{A,C,G,T }

ti = branch length(represents the number ofmutations per site along thatbranch)

Variables at the leaves are observed and variables at the interiornodes are hidden.At each branch we write a matrix (substitution matrix) with theprobabilities of a nucleotide at the parent node being substitutedby another at its child.

A C G T

P(A|A) P(C|A) P(G|A) P(T |A)P(A|C) P(C|C) P(G|C) P(T |C)P(A|G) P(C|G) P(G|G) P(T |G)P(A|T ) P(C|T ) P(G|T ) P(T |T )

The entries of Si are unknown parameters.

Example (Group-based models. Kimura 3-parameters)Root node has uniform distribution: πA = ... = πT = 0.25

A C G T

Si =ACGT

(ai bi ci dibi ai di cici di ai bidi ci bi ai

), ai + bi + ci + di = 1

Kimura 2-parameters: bi = di .

Jukes-Cantor: bi = ci = di .

A C G T

Si =ACGT

), ai + bi + ci + di = 1

A C G T

Si =ACGT

), ai + bi + ci + di = 1

Hidden Markov processWe denote the joint distribution of the observed variables X1, X2, X3 aspx1x2x3 = Prob(X1 = x1, X2 = x2, X3 = x3).

px1x2x3 =∑

y4,yr∈{A,C,G,T}

πyr S1(x1, yr )S4(y4, yr )S2(x2, y4)S3(x3, y4)

px1x2x3 is a homogeneous polynomial on the parameters whosedegree is the number of edges.

The evolutionary model defines a polynomial map

ϕ : Rd −→ R4n

θ = (θ1, . . . , θd) 7→ (pAA...A, pAA...C, pAA...G, . . . , pTT...T)

ϕ : 4d−1 −→ 44n−1

ϕ : Cd −→ C4n

Algebraic variety V = imϕ, closure in Zariski topology.

Given an alignment, we can estimate the joint probability px1,x2,x3

as the relative frequency of column x1x2x3 in the alignment.

In the theoretical model, this would be a point on the variety V .

Goal: use the ideal of V , I(V ) ⊂ R[pAA...A, pAA...C, . . . , pTT...T], toinfer the topology of the phylogenetic tree.

ϕ : Rd −→ R4n

ϕ : 4d−1 −→ 44n−1

ϕ : Cd −→ C4n

ϕ : Rd −→ R4n

ϕ : 4d−1 −→ 44n−1

ϕ : Cd −→ C4n

Computing the ideal(Eriksson–Ranestad–Sturmfels–Sullivant ’05)

Some generators of I(V ) depend only on the model chosen (noton the topology). E.g. for Jukes-Cantor they are:

∑px1x2x3 = 1

pAAA = pCCC = pGGG = pTTT 4 terms

pAAC = pAAG = pAAT = · · · = pTTG 12 terms

pACA = pAGA = pATA = · · · = pTGT 12 terms

pCAA = pGAA = pTAA = · · · = pGTT 12 terms

pACG = pACT = pAGT = · · · = pCGT 24 termsFor this model, the unique polynomial that detects the phylogenyof the three species has degree 3.

Definition (Cavender–Felsenstein ’87)The generators of I(V ) are called phylogenetic invariants.

∑px1x2x3 = 1

Problem: computation of invariants

Computational algebra software fail to compute the ideal for ≥ 4species! Kimura 3-parameter, 4 species, 8002 generators like:

[Small trees webpage]

Problem: computation of invariants

Computational algebra software fail to compute the ideal for ≥ 4species! Kimura 3-parameter, 4 species, 8002 generators like:

[Small trees webpage]

Group-based models. Discrete Fourier transform

Group-based models (Kimura and Jukes-Cantor): G = Z2 × Z2,A=(0,0),C=(0,1),G=(1,0),T=(1,1) . Substitution matricesare functions on the group:

Si(gp, gc) = f i(gp − gc), gc , gp ∈ G.

pg1,...,gm = p(g1, . . . , gm) =14

∑ ∏e∈edges

f e(gp(e) − gc(e))

sum over all possible values at interior nodes.

Discrete Fourier transform f : G −→ C

f (χ) =∑g∈G

χ(g)f (g), χ ∈ Hom(G, C∗) ∼= G

Convolution:(f1 ∗ f2)(g) =∑h∈G

f1(h)f2(g − h) =⇒ f1 ∗ f2 = f1 · f2

Group based models.

Theorem (Evans–Speed)For a group-based model (i.e. Kimura 3, Kimura 2 or Jukes-Cantor) ona tree T, the discrete Fourier transform of the joint distributionp(g1, . . . , gn) has the following form

q(χ1, . . . , χm) =∏

e∈edges

f e( ∏

l∈leaves below e

Using a linear change of coordinates, one gets a monomialparameterization of the variety associated to the model

ϕ : Cd −→ C4n

θ = (θ1, . . . , θd) 7→ (qAA...A, qAA...C, qAA...G, . . . , qTT...T)

so that it is a toric variety.

The ideal is generated by binomials.

Fourier transform for Kimura 3-parameter model

Parameters Fourier parameters

ae be ce de

be ae de ce

ce de ae be

de ce be ae

A 0 0 00 Pe

C 0 00 0 Pe

G 00 0 0 Pe

A=ae+be+ce+de, PeC=ae−be+ce−de...

simplex: 43 ∆3

ae+be+ce+de=1 PeA=1

coordinates: px1...xn qg1...gn , g1+···+gn=0

simplex: 44n−1 ∆4n−1−1∑px1...xn = 1 qA...A = 1

Linear invariants: qg1...gn = 0 if g1 + · · ·+ gn 6= 0 in Z2 × Z2.

For the Kimura 3-parameter model, we are interested inV := im(ϕ) where

ϕ :∏

e∈E(T )

C4 −→ C4n−1

(PeA,Pe

C ,PeG,Pe

7→ (qx1...xn){x1,...,xn|x1+···+xn=0}

But ∆4n−1−1 ⊂ {qA...A = 1}.

DefinitionThe Kimura variety of the phylogenetic tree T is W := V ∩ {qA...A = 1}.

Recursive construction of invariants

Computational algebra software: even in Fourier parameterization,fail for ≥ 5 leaves.

Theorem (Sturmfels – Sullivant ’05)For any group-based model, they give an explicit algorithm forobtaining the generators of the ideal of phylogenetic invariants of ann-leaved tree from those of an unrooted tree of 3 leaves. Thegenerators are binomials of degree ≤ 4.

Outline

Phylogenetic inference using algebraic geometry

1990-1995: Biologists claim that phylogenetic invariants are notuseful for phylogenetic reconstruction.Lake only used linear invariants!

In particular, a work of Huelsenbeck’95 reveals the inefficiency ofLake’s method of invariants.

(Eriksson ’05) for General Markov Model.

(C–Garcia–Sullivant ’05) for Kimura 3-parameter.

1990-1995: Biologists claim that phylogenetic invariants are notuseful for phylogenetic reconstruction.Lake only used linear invariants!

In particular, a work of Huelsenbeck’95 reveals the inefficiency ofLake’s method of invariants.

(Eriksson ’05) for General Markov Model.

(C–Garcia–Sullivant ’05) for Kimura 3-parameter.

Model: Unrooted tree of 4 leaves, Kimura 3-parameters.Naive algorithm: Take || · ||1 of the evaluation of all invariants onthe point given by the relative frequencies of the columns of thealignment. Obtain a score for each possible topology and choosethe one with smaller score.

Huelsenbeck ’95: assesses the performance of different phylogeneticreconstruction methods on the following tree space.

Results on simulated data (Huelsenbeck)

Huelsenbeck’s results:

Length 100 Length 500 Length 1000

Lake invariants

Huelsenbeck, Syst. Biol., 1995M. Casanellas (UPC) Using algebraic geometry ... IMA, March 2007 21 / 36

C–Fernandez-Sanchez studies on C–Garcia–Sullivantmethod

Why using phylogenetic invariants

1 We do not need to estimate parameters of the model.2 The algebraic model allows species within the tree to evolve at

different mutation ratesIn the non-algebraic version, one has: Si = exp(Q · ti) where Q is afixed matrix that represents the instantaneous mutation rate of allthe species in the tree.In the algebraic model, one is allowing different rates at eachbranch of the tree.

For instance using algebraic geometry one should be able toreconstruct the biologically correct tree:

dog human chimp rat mouse chicken

(Al-Aidroos, Snir ’05) chapter 21 in ASCB book.

dog human chimp rat mouse chicken

dog human chimp rat mouse chickendog human chimp rat mouse chicken

Simulations on non-homogeneous trees (different ratematrices)

Outline

The global geometry of the Kimura variety

Recall that V := im(ϕ) where

ϕ :∏

e∈E(T )

C4 −→ C4n−1

(PeA,Pe

C ,PeG,Pe

7→ (qx1...xn){x1,...,xn|x1+···+xn=0}

But ∆4n−1−1 ⊂ {qA...A = 1}.

DefinitionThe Kimura variety of the phylogenetic tree T is W := V ∩ {qA...A = 1}.

W = ϕ(∏

e∈E(T )(C4 ∩ {PeA = 1})).

Local complete intersection

V ⊂ AN algebraic variety, µ minimum number of generators ofI(V ), then

µ ≥ codim(V ) = N − dim(V )

.V is called a complete intersection if µ = codim(V ).

Example: the Kimura variety W ⊂ C4n−1−1 has codimension4n−1 − 6n + 8, n = number of leaves.For n = 4, the minimum number of generators of the Kimuravariety is 8002 whereas its codimension is 48!On a neighborhood of a smooth point, any variety is a localcomplete intersection (i.e. it can be defined by codim(V )equations).Goal:

1 Find the singular points.2 Provide a set of local generators at non-singular points.

Case n = 3

The parameterization ϕ is:

ϕ : C9 −→ C15

((1,P1C ,P1

G,P1T ),(1,P2

C ,P2G,P2

T ),(1,P3C ,P3

G,P3T )) 7→ (qxyz=P1

x P2y P3

z ){x+y+z=0}

Let H = (Z2 × Z2, ∗) and let (ε, δ) ∈ H act on C9 sending(P1, P2, P3) to ((1,εP1

C ,δP1G,εδP1

T ),(1,εP2C ,δP2

G,εδP2T ),(1,εP3

C ,δP3G,εδP3

E.g. (ε, δ) = (1,−1), in probability parameters the group actionpermutes A ↔ C and G ↔ T .

Proposition

The Kimura variety W = im(ϕ) on T3 is the affine GIT quotient C9//H.

Arbitrary n

T unrooted n-leaved tree(2n − 3 edges, n − 2 interior nodes)

Extending the action of H to the n − 2 interior nodes of T , we have

TheoremThe Kimura variety W is isomorphic to the affine GIT quotient

(C3)2n−3//Hn−2

CorollaryW = im(ϕ), no closure needed.

|ϕ−1(q)| ≤ 4n−2 and there is just one preimage with biologicalmeaning, ∀q ∈ W.

We are able to determine the singular points. In particular,biologically meaningful points q ∈ W+ := ϕ(

∏∆3

+) are notsingular.

Arbitrary n

(C3)2n−3//Hn−2

∏∆3

+) are notsingular.

Arbitrary n

(C3)2n−3//Hn−2

∏∆3

+) are notsingular.

Arbitrary n

(C3)2n−3//Hn−2

∏∆3

+) are notsingular.

In Fourier parameters:

∆3+ =

In probability parameters this is transformed into:

A C G T

S =ACGT

(a b c db a d cc d a bd c b a

) a + b + c + d = 1

a + b − c − d > 0

a− b + c − d > 0

a− b − c + d > 0

In Fourier parameters:

∆3+ =

In probability parameters this is transformed into:

A C G T

S =ACGT

(a b c db a d cc d a bd c b a

) a + b + c + d = 1

a + b > 1/2

a + c > 1/2

a + d > 1/2

Case n = 3

(Sturmfels–Sullivant ’05): I(V ) is minimally generated by 16cubics and 18 quartics.

LemmaThe following six quartics

qAAAqATT qTCGqTGC−qACCqAGGqTAT qTTA, qCCAqCTGqTAT qTGC−qCACqCGT qTCGqTTA,

qAGGqATT qCACqCCA−qAAAqACCqCGT qCTG, qACCqATT qGAGqGGA−qAAAqAGGqGCT qGTC ,

qCACqCTGqGCT qGGA−qCCAqCGT qGAGqGTC , qGGAqGTCqTAT qTCG−qGAGqGCT qTGCqTTA

generate a local complete intersection that defines W at each pointq ∈ W+.

It does not depend on the point q. Skip proof

Proof.1 J = (f1 . . . , f6) defines a variety X containing W and both have the

same dimension.2 The points q ∈ W+ are smooth points of X and W .3 Both varieties coincide locally.

Arbitrary n

quartics:

quadrics: Q, non-redundant 2× 2 minors from flattening

TheoremJ3 ∪ Jn−1 ∪Q generate a local complete intersection that defines W atthe biologically meaningful points q ∈ W+.

Arbitrary n

quartics:

Arbitrary n

quartics: J3

Arbitrary n

quartics: J3

Arbitrary n

quartics: J3 ∪Jn−1

Arbitrary n

Outline

New results on simulated data

4 leaves

Instead of all generators, 48 polynomials that generate locally

4 leaves

(Eriksson–Yao’07) Machine learning approach, 52 invariants thatperform better on this parameter space.

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