Algebraic Invariants of Phylogenetic Trees

Transcription

1 Harvey Mudd College November 21st, 2006

2 The Problem DNA Sequence Data Given a set of aligned DNA sequences... Human: A G A C G T T A C G T A... Chimp: A G A G C A A C T T T G... Monkey: A A A C G A T A C G C A...

3 The Problem We wish to determine the tree that corresponds to their shared history.

4 Some Definitions Human: A G A C G T T A C G T A... Chimp: A G A G C A A C T T T G... Monkey: A A A C G A T A C G C A... Definition A pattern σ is the sequence of characters we get when we look at a single site (column) of our sequence data.

5 Some Definitions Human: A G A C G T T A C G T A... Chimp: A G A G C A A C T T T G... Monkey: A A A C G A T A C G C A... Definition A pattern σ is the sequence of characters we get when we look at a single site (column) of our sequence data.

6 Some Definitions Human: A G A C G T T A C G T A... Chimp: A G A G C A A C T T T G... Monkey: A A A C G A T A C G C A... Definition A pattern frequency p σ is the percentage of time that σ appears in our set of sequence data.

7 Some Definitions Human: A G A C G T T A C G T A... Chimp: A G A G C A A C T T T G... Monkey: A A A C G A T A C G C A... Definition A pattern probability p σ for a given tree and model of sequence evolution is the percentage of time that we would expect to see a given pattern σ.

8 Invariants Definition An invariant f for a phylogenetic tree and a model of sequence evolution is a polynomial in indeterminates x σ such that f ((p σ )) = 0.

9 Invariants Definition An invariant f for a phylogenetic tree and a model of sequence evolution is a polynomial in indeterminates x σ such that f ((p σ )) = 0. Example One example of an invariant for all trees and all models of sequence evolution is the trivial invariant. ( xσ ) 1

10 Why are invariants useful? Suppose we had all of the invariants for a phylogenetic tree and model of sequence evolution.

11 Why are invariants useful? Suppose we had all of the invariants for a phylogenetic tree and model of sequence evolution. ˆ From the sequence data, we could obtain the pattern frequencies p σ. These serve as estimators for the pattern probabilities p σ.

12 Why are invariants useful? Suppose we had all of the invariants for a phylogenetic tree and model of sequence evolution. ˆ From the sequence data, we could obtain the pattern frequencies p σ. These serve as estimators for the pattern probabilities p σ. ˆ We could evaluate all of the invariants by plugging in the pattern frequencies p σ.

13 Why are invariants useful? Suppose we had all of the invariants for a phylogenetic tree and model of sequence evolution. ˆ From the sequence data, we could obtain the pattern frequencies p σ. These serve as estimators for the pattern probabilities p σ. ˆ We could evaluate all of the invariants by plugging in the pattern frequencies p σ. ˆ If the sequence data comes from the tree (and model), then all of the evaluated invariants should be approximately zero.

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15 ˆ The sequences we will be looking at are 0/1 sequences.

16 ˆ The sequences we will be looking at are 0/1 sequences. ˆ We will assume that the characters are uniformly distributed at the root.

17 ˆ The sequences we will be looking at are 0/1 sequences. ˆ We will assume that the characters are uniformly distributed at the root. ˆ Along each edge, the probability of the character changing is p; the probability of staying the same is q = 1 p.

18 (cont d) We can explicitly calculate the pattern probabilities for this tree and model of sequence evolution.

19 (cont d) We can explicitly calculate the pattern probabilities for this tree and model of sequence evolution. p 00 = 1 2 ( q 3 + p 3) ( pq 2 + qp 2)

20 (cont d) We can explicitly calculate the pattern probabilities for this tree and model of sequence evolution. p 00 = 1 2 ( q 3 + p 3) ( pq 2 + qp 2) p 11 = 1 2 ( q 3 + p 3) ( pq 2 + qp 2)

21 (cont d) We can explicitly calculate the pattern probabilities for this tree and model of sequence evolution. p 00 = 1 2 ( q 3 + p 3) ( pq 2 + qp 2) p 11 = 1 2 ( q 3 + p 3) ( pq 2 + qp 2) p 01 = pq 2 + qp 2

22 (cont d) We can explicitly calculate the pattern probabilities for this tree and model of sequence evolution. p 00 = 1 2 ( q 3 + p 3) ( pq 2 + qp 2) p 11 = 1 2 ( q 3 + p 3) ( pq 2 + qp 2) p 01 = pq 2 + qp 2 p 10 = pq 2 + qp 2

23 (cont d) We can think about this as a parameterization of a curve in a four-dimensional space.

24 (cont d) We can think about this as a parameterization of a curve in a four-dimensional space. w(p) = p 00 = 1 2 ( (1 p) 3 + p 3 + p(1 p) 2 + (1 p)p 2) x(p) = p 11 = 1 ( (1 p) 3 + p 3 + p(1 p) 2 + (1 p)p 2) 2 y(p) = p 01 = p(1 p) 2 + (1 p)p 2 z(p) = p 10 = p(1 p) 2 + (1 p)p 2

25 (cont d) We can think about this as a parameterization of a curve in a four-dimensional space. w(p) = p 00 = 1 2 ( (1 p) 3 + p 3 + p(1 p) 2 + (1 p)p 2) x(p) = p 11 = 1 2 ( (1 p) 3 + p 3 + p(1 p) 2 + (1 p)p 2) y(p) = p 01 = p(1 p) 2 + (1 p)p 2 z(p) = p 10 = p(1 p) 2 + (1 p)p 2 We should be able to describe this curve as the roots of three different polynomials in the variables w, x, y, and z. These polynomials are the invariants.

26 (cont d) This problem is known as the implicitization problem.

27 (cont d) This problem is known as the implicitization problem. The Groebner package in Maple is able to perform this calculation.

28 (cont d) This problem is known as the implicitization problem. The Groebner package in Maple is able to perform this calculation. 2x x x x 11 1 x 00 x 11

29 (cont d) This problem is known as the implicitization problem. The Groebner package in Maple is able to perform this calculation. 2x x x x 11 1 x 00 x 11 Notice that these invariants do not depend on the probability p. Thus, we do not need to know anything about the specific rates (probabilities) of change along each edge.

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31 In the previous example, we could think of the group Z 2 as acting on the characters 0 and 1. Along each edge, we randomly select the group element 1 Z 2 with probability p.

32 Kimura s 3 Parameter Model The group acting on the nucleotides is Z 2 Z 2.

33 Kimura s 3 Parameter Model The group acting on the nucleotides is Z 2 Z 2. ˆ The element (0, 0) leaves the nucledotides fixed.

34 Kimura s 3 Parameter Model The group acting on the nucleotides is Z 2 Z 2. ˆ The element (0, 0) leaves the nucledotides fixed. ˆ The element (1, 0) flips the nucleotides across the vertical axis.

35 Kimura s 3 Parameter Model The group acting on the nucleotides is Z 2 Z 2. ˆ The element (0, 0) leaves the nucledotides fixed. ˆ The element (1, 0) flips the nucleotides across the vertical axis. ˆ The element (0, 1) flips the nucleotides across the horizontal axis.

36 Kimura s 3 Parameter Model The group acting on the nucleotides is Z 2 Z 2. ˆ The element (0, 0) leaves the nucledotides fixed. ˆ The element (1, 0) flips the nucleotides across the vertical axis. ˆ The element (0, 1) flips the nucleotides across the horizontal axis. ˆ The element (1, 1) flips the nucleotides across both axes.

37 Fourier Transform With the Fourier Transform, we can use the group structure inherent in these models to help us determine the invariants.

38 Fourier Transform With the Fourier Transform, we can use the group structure inherent in these models to help us determine the invariants. If G is an abelian group f : G C, then the Fourier Transform ˆf of f is the function defined by ˆf (χ) = g G χ(g)f (g) where χ is a homomorphism from G to the complex numbers.

39 Fourier Transform With the Fourier Transform, we can use the group structure inherent in these models to help us determine the invariants. If G is an abelian group f : G C, then the Fourier Transform ˆf of f is the function defined by ˆf (χ) = g G χ(g)f (g) where χ is a homomorphism from G to the complex numbers. The Fourier Transform has the nice property of turning convolution into multiplication.

40 Linear Change of Coordinates The Fourier Transform provides a linear change of coordinates.

41 Linear Change of Coordinates The Fourier Transform provides a linear change of coordinates. Under this new coordinate system, each parameterization simply becomes a product (monomial) of the new Fourier parameters.

42 Linear Change of Coordinates Example Original Parameterization p 000 = π 0 α 0 β 0 γ 0 + π 1 α 1 β 1 γ 1 p 001 = π 0 α 0 β 0 γ 1 + π 1 α 1 β 1 γ 0 p 010 = π 0 α 0 β 1 γ 0 + π 1 α 1 β 0 γ 1 p 011 = π 0 α 0 β 1 γ 1 + π 1 α 1 β 0 γ 0 p 100 = π 0 α 1 β 0 γ 0 + π 1 α 0 β 1 γ 1 p 101 = π 0 α 1 β 0 γ 1 + π 1 α 0 β 1 γ 0 p 110 = π 0 α 1 β 1 γ 0 + π 1 α 0 β 0 γ 1 p 111 = π 0 α 1 β 1 γ 1 + π 1 α 0 β 0 γ 0

43 Linear Change of Coordinates Example Change of Coordinates Greek Latin: p q π 0 = 1 2 (r 0 + r 1 ) π 1 = 1 2 (r 0 r 1 ) α 0 = 1 2 (a 0 + a 1 ) α 1 = 1 2 (a 0 a 1 ) β 0 = 1 2 (b 0 + b 1 ) β 1 = 1 2 (b 0 b 1 ) γ 0 = 1 2 (c 0 + c 1 ) γ 1 = 1 2 (c 0 c 1 ) p ijk = 1 1 r=0 s=0 t=0 1 ( 1) ir+js+kt q rst

44 Linear Change of Coordinates Example New Parameterization q 000 = r 0 a 0 b 0 c 0 q 001 = r 1 a 0 b 0 c 1 q 010 = r 1 a 0 b 1 c 0 q 011 = r 0 a 0 b 1 c 1 q 100 = r 1 a 1 b 0 c 0 q 101 = r 0 a 1 b 0 c 1 q 110 = r 0 a 1 b 1 c 0 q 111 = r 1 a 1 b 1 c 1

45 How Does This Help?

46 How Does This Help? In this new coordinate system, the invariants are much easier to describe.

47 How Does This Help? In this new coordinate system, the invariants are much easier to describe. The set of all invariants is known as a toric ideal and is generated by binomials.

48 How Does This Help? In this new coordinate system, the invariants are much easier to describe. The set of all invariants is known as a toric ideal and is generated by binomials. {q 001 q 110 q 000 q 111, q 010 q 101 q 000 q 111, q 100 q 011 q 000 q 111 }

49 How Does This Help? In this new coordinate system, the invariants are much easier to describe. The set of all invariants is known as a toric ideal and is generated by binomials. {q 001 q 110 q 000 q 111, q 010 q 101 q 000 q 111, q 100 q 011 q 000 q 111 } Taking the inverse, each of these binomials turns into a quadratic with eight terms such as p 001 p p 001 p 100 p 000 p 011 p 000 p p 100 p 111 p 101 p p 010 p 111 p 001 p 110

50 My Research My research looks at what happens when we use a non-abelian group to model the sequence evolution.

51 My Research My research looks at what happens when we use a non-abelian group to model the sequence evolution. In particular, I am looking at using the symmetric group S 4 to find invariants for the symmetric model of sequence evolution.

52 How Is This Different? In this case, the Fourier Transform of f at a representation φ becomes ˆf (φ) = g G φ(g)f (g). Here, φ(g) is not a complex number, but instead a complex matrix.

53 Progress I proved that, even in this non-abelian case, the parameterization becomes a product of Fourier parameters, but the parameters can now be matrices.

54 Work To Do ˆ Because the matrices do not commute, we cannot use the same toric ideal techniques to find the invariants.

55 Work To Do ˆ Because the matrices do not commute, we cannot use the same toric ideal techniques to find the invariants. ˆ I am looking into a noncommutative analogue of the toric ideals.

56 Work To Do ˆ Because the matrices do not commute, we cannot use the same toric ideal techniques to find the invariants. ˆ I am looking into a noncommutative analogue of the toric ideals. ˆ I am currently looking into free associative algebras which are noncommutative analogues of polynomial rings.

57 Thank you! Questions?

58 One of the results looks like the following: Theorem ˆp φ = ˆf e (φ e ). e E(T )