Lattice polytopes of phylogenetic trees
Kaie Kubjas
DFG Research Training Group
M ETHODS FOR D ISCRETE
S TRUCTURES
Department of Mathematics, Freie Universität Berlin, Germany
kubjasmath.fu-berlin.de
www.math.tu-berlin.de/MDS
Supervisor: Christian Haase
Phylogenetic models
Jukes Cantor binary model
Let T be a rooted tree with each vertex a random variable with 2
possible states (binary states {0, 1}).
Theorem
T2
(Buzy«ska and Wi±niewski [BW07℄). Let T1 and
be 3-valent trees with the same number of leaves. Then
ehrPZ2 ,T1 (t) = ehrPZ2 ,T2 (t).
π
b
a
c
1
d
2
Example.
ehrPZ2 ,snowake (t) = ehrPZ2 ,3-aterpillar (t)
3
1
(t + 1)(t + 2)(t + 3)
22680
1942t4 + 5616t3 + 9511t2 +
=
INPUT:
root distribution π = (π0 , π1 ) and edge transition matrices representing the probabilities of transition between
the states
Ma =
Mc =
OUTPUT:
` a00
a
a01
a
´
10 11
` c00
c01 ´
c10 c11
Mb =
Md =
` b00
b01
d10 d11
=
π0 a0i b00 c0j d0k + π0 a01 b01 c1j d1k
+
π1 a1i b10 c0j d0k + π1 a11 b11 c1j d1k
• Eight polynomials Φijk give the map
snowflake
• Fixing a parameter space P ⊆ C18 specifies the model.
Combinatorial proof (together with Haase and Paffenholz).
• The polytope of the 3-valent 4-leaf tree T0 is the join of two squares.
To each pair of non-isomorphic labelings of the leaves of T0 we
associate triangulations of the associated polytopes together with
an isomorphism of the triangulations.
• Object of interest is the Zariski closure of Φ(P ).
T1
Jukes Cantor binary model has uniform root
distribution ( 12 , 21 ) and transition matries of the form
Example.
Ma =
Mc =
a00
1−a00 ´
a00
00
` 1−a
c00 1−c00 ´
1−c00
c00
Mb =
`
Md =
`
b00
1−b00 b00
d00 1−d00
1−d00 d00
3-caterpillar
• Given four pointed trees Ti , where i = 1, . . . , 4, we can produce a
tree T by grafting the tree Ti along the i-th leaf of the labeled T0 .
There is a natural projection from PZ2 ,T to PZ2 ,T0 . The preimage
of a triangulation of PZ2 ,T0 gives a subdivision of PZ2 ,T .
Φ : C18 → C8 .
`
8988t + 3780)
´
10 b11
` db00
d01 ´
probability of observing the letter i at the leaf 1, the letter
j at the leaf 2, and the letter k at the leaf 3
Φijk
(31t6 + 372t5 +
General setting (see [ERSS05]): rooted tree T , k states (usually 2
for the binary states or 4 for the nucleotides), root distribution π and
edge transition matrices Me .
T3
T1
T1
e
e
T3
T4
T4
T2
T2 T4
trees differing by an elementary mutation along e
T2
1−b00 ´
´
T3
e
• Consider polytopes of two trees that differ by an elementary
mutation. Their subdivisions are isomorphic via a piecewise
unimodular transformation.
• Buczyńska and Wiśniewski showed that any two 3-valent trees with
the same number of leaves differ by a sequence of elementary
mutations. Composing corresponding maps gives a piecewise
unimodular transformation between the associated polytopes.
Group-based models
• In case of group-based models transition matrices are invariant
under a group action of G.
• Varieties Φ(P ) are toric and give lattice polytopes PG,T in the
lattices LG,T .
• Ehrhart polynomial is an important invariant of group-based models:
ehrPG,T (k) = |kPG,T ∩ LG,T | for k ∈ Z≥0 .
Transition matries of the Jukes Cantor binary
model are invariant under the Z2 ation.
Example.
The polytope PZ2 ,T of the Jukes Cantor binary
model assoiated to a tree T is dened by
Kimura 3-parameter model
• Transition matrices of the Kimura 3-parameter model are invariant
under the Z2 × Z2 action.
• Can we generalize the theorem of Buczyńska and Wiśniewski to the
Kimura 3-parameter model?
Theorem
ehrPZ2 ×Z2 ,snowake (t) 6= ehrPZ2 ×Z2 ,3-aterpillar (t).
Example.
onv(a ∈ Z : ae ∈ {0, 1} and
E
X
ae ∈ 2Z for every v ∈ I),
v∈e
(K. [K10℄).
Proof.
ehrPZ2 ×Z2 ,snowake (3) =
where E is the set of edges and I is the set of inner verties of
the tree T .
6=
69248000
69324800 = ehrPZ
2 ×Z2 ,3-aterpillar
References
[BW07] W. Buczyńska and J. Wiśniewski. On the geometry of binary symmetric models of phylogenetic trees. J. Eur. Math. Soc., 9:609–635, 2007.
[ERSS05] N. Eriksson, K. Ranestad, B. Sturmfels and S. Sullivant. Phylogenetic algebraic geometry. In Projective Varieties with Unexpected Properties. Berlin, 2005.
[K10] K. Kubjas. Hilbert polynomial of the Kimura 3-parameter model. http://arxiv.org/abs/1007.3164, 2010.
(3).