Selecting Genomes for Reconstruction of Ancestral Genomes

Louxin ZhangDepartment of Mathematics

National University of Singapore

Boreoeutherian Ancestor

The Genome Selection for Reconstruction problem

Instance: Given a phylogeny P of a set of genomes, an integer k and a reconstruction method T

(say parsimony).

Solution: k genomes in the phylogeny that gives the highest accuracy of reconstructing the ancestral genome at the root of the phylogeny, using method T.

Two reasons

It is often impossible to sequence all descendent genomes below an ancestor;

More taxa do not necessarily give a higher accuracy for the reconstruction of ancestral character states in general (examples will be given below)

Outline

Introduction to reconstruction accuracy analysis

More genomes are not necessarily better for reconstruction

Greedy algorithms for genome selection

A joint work with G. Li, J. Ma and M. Steel

1. Reconstruction and its Accuracy

There are different methods for reconstructing the ancestral character states Parsimony Maximum likelihood methods (Koshi &

Golstein’96, Yang et al’95) Bayesian methods (Yang et al’95)

In this work, we study the problem with the Fitch parsimony and maximum likelihood in the Jukes-Cantor evolutionary model.

Jukes-Cantor Model

Characters evolve by a symmetric, reversible Markov process.

Probability of a substitution change of any sort is the same on a branch.

For simplicity, we assume there are two states 0 and 1.

Reconstruction Accuracy

In the symmetric Jukes-Cantor model,

the reconstruction accuracy of a method is independent of the prior distribution of the states at the root.

•D denotes a state configuration at leaves: it has one state for each leaf.•There are state configurations since thereare 2 possible states at each leaf.

n2

D

K KDIDA ),,0()0|Pr(

I(0, D, K) is 1 if the method K reconstructs state 0 from D and 0 otherwise.

Pr(D|0) is the probability that 0 at the root evolves into D.

•D denotes a state configuration at leaf nodes: it has one state for each leaf.•There are state configurations since thereare 2 possible states at each leaf.

n2

D

K KDIDA ),,0()0|Pr(

The reconstruction accuracy is the sum of generating Prob. of state configurations which allow the true state 0 to be recovered by the method K.

Previous Analysis Works

Simulation study (Martins’99, Mooers’04, Salisbury & Kim’01,

Zhang & Nei’97, Yang et al’95); Theoretical study (Mossel’01, Lucena and Haussler’05,

Maddison’95)

Fitch Method

Given a state configuration of the leaves, the Fitch method reconstructs a subset of states at each internal node (from leaves to the root ) recursively:

0 1 0

{0, 1}

{0}

B C

A

The unambiguous reconstruction accuracy:

PAccuracy= P[{1}|1]=P[{0}|0]

and the reconstruction accuracy

P[{1}|1]= the probability that Fitch method outputs true state at the root.

P[{0}|1], P[{1}|1], and P[{0, 1}|1] can be calculated bya dynamic approach (Maddison, 1995)

Calculating the Reconstruction Accuracy of Fitch Method

Outline

Introduction to reconstruction accuracy analysis

More genomes are not necessarily better for reconstruction accuracy

Greedy algorithms for genome selection

2. Reconstruction accuracy is not a monotone functionof the size of taxon sampling

umbalanced tree

• There is a large clade with a long stem• A short single sister lineage

Such a phylogeny is used when both fossil record and data at extant species are used for reconstruction (Finarelli and Flynn, 2006)

A

Y Z

p1 p2

Theorem 1: Aparsimony < p1 if ½< p2<= p1

p1 is the conservation probability on AYp2 is the conservation probability on AZ

0 {0}

{0}

0

{0}

{0, 1}

PA[{0}|0]

= PrAY[00] x (PrAZ[00] PZ[{0} or {0, 1}| 0] + PrAZ[01] PZ[{0} or {0, 1}| 1] = p1 (p2 (1- PZ[{1}| 0] ) + (1-p2) (1-PZ[{1}|1])

= p1 ( 1- p2 PZ[{1}|0] – (1-p2) PZ[{1}|1] )

Proof.

A

Y Z

p1 p2

p1 is the conservation probability on AYp2 is the conservation probability on AZ

0{1}

{0, 1}

1

{0, 1}

{0}

PA[{0, 1}|0]= [p1p2+(1-p1)(1-p2)] x PZ[{1}|0] + [ p1(1-p2)+p2(1-p1)] X PZ[{0}|0]

Aparsimony = PA[{0}|0] + ½ PA[{0, 1}|1] = p1 + ½ (1-p1-p2) PZ[{1}|0] + ½(p2-p1) PZ[{0}|0] < p1

½ < p2 <= p1

The reconstruction accuracy oncomb-shaped trees in the limit case

A

Y Z

p1 p2

Theorem 2: AML = p1 if ½< p2<= p1

p1 is the conservation probability on AYp2 is the conservation probability on AZ

0

DZ

DZ : a state configuration below Z.

PrA( 0DZ |s): the probability that s at A evolves into state configuration 0DZ, s=0,1.

Marginal ML method:

PrA(0DZ|0) = p1 x [ p2PrZ(DZ|0) + (1-p2)PrZ(DZ| 1)]

0

0, 1?

PrA(0DZ|1) = (1-p1) x [ (1-p2)PrZ(DZ|0) + p2PrZ(DZ| 1)]

PrA(0DZ|0)-PrA(0DZ|1)=(p1+p2-1)PrZ(DZ|0) + (p1-p2)PrZ(DZ|1) >0

The marginal ML outputs 0 at A iff the state at Y is 0.

Another Example showing the Non-monotone Property of Reconstruction Accuracy

Simulation

Experiment setup Yule birth-death model Conservation probability along branches:

0.5~1 Count the number of random trees in

which the ambiguous accuracy of using a single (longest or shortest) path is better than that from the full phylogeny

Simulation results:Counting the bad trees

+: using the shortest path

Comparison of Parsimony, Joint ML and Marginal ML

500 random trees with 12 leaves generated: • Yule birth-death model• branch length is uniform from 0 to 1

• MML outperforms JML, MP. • In 80% of instances, MML is strictly better than JML• In 99% of instances, JML is strictly better than MP.

Outline Introduction to reconstruction

accuracy analysis

More genomes are not necessarily better for reconstruction accuracy

Greedy algorithms for genome selection problem

Genome selection for reconstructionthe problem

Instance: A phylogeny P over n genomes, integer k and a reconstruction method T

Question: Find k genomes that allows the ancestral genome of the root of P to be reconstructed with the maximum accuracy, using method T.

Our approaches

The genome selection problem is unlikely polynomial-time solvable

(no hardness proof yet) As a result, we propose two greedy

algorithms for the problem: Forward greedy algorithm & Backward greedy algorithm

Forward Greedy Algorithm

1. Set S ← φ;

2. For i = 1, 2, · · · , k do 2.1) for each genome g not in S, compute the accuracy A(g) of the reconstruction by applying method T to S ∪ {g}; 2.2) add g with the max accuracy A(g) to S ;

3. Output S

S is the set of selected genomes

Backward Greedy Algorithm

1. Let S contain all the given genomes;

2. For i = 1, 2, · · · , n − k do 2.1) for each genome g in S, compute the accuracy A’(g) of the reconstruction by applying T to S − {g};

2.2) remove g from S if A’(g) is the max over all g’s;

3. Output S

Validation test– Trees with the same height

Experiment setup Random trees with N (9, or 16)

leaves generated by program Evolver in PAML with the following parameters:

Birth rate=10; Death rate=5; Sampling fraction=1. Tree height = 0.1, 0.2, 0.5, 1, 2, or 5.

Performance of the selection method for reconstruction with Parsimony

Performance of the selection method for reconstruction with Marginal Maximum Likelihood

Performance of the selection method for reconstruction with Joint Maximum Likelihood

Marginal Maximum Likelihood

Parsimony Method

Concluding remarks Reconstruction accuracy is not monotone

increasing with the taxon sampling size in unbalanced trees for Parsimony method

--- Another kind of “inconsistency”

1. One implication of this observation is that Parsimony, ML method might not explore the full power of incorporating fossil record into current data.

Hence, modification might probably be needed.

2. Caution should be used in drawing conclusion on testing hypothesis with ancestral state reconstruction.

3. Is the reconstruction accuracy function monotone in ultrametric phylogeny? It seems true when the number of taxa is large.

Consider the complete binary tree when conservation prob on each branch is less than 7/8,

(The ambiguous reconstructionaccuracy)= (the accuracy of using just one taxa )=1/2

in the limit case. (Rormula exists, see Steel’89.)

Concluding remarks Formulate the genome selection for

reconstruction problem Two greedy algorithms proposed for the

problem Validation test shows that the

reconstruction accuracy of using the genomes selected by the greedy algorithms are comparable to the

the max reconstruction accuracy.

Thanks You!

A Biological Example

Boreoeutherian ancestor From Encode project

4 states at leaf nodes

Expected accuracy at the root node

A Biological Example – Results

Backward algo is always similar as the exhaustive search

With 8 leaf nodes, the accuracy from Backward algo is 93.6%, near to the accuracy 94.6% with full phylogeny

Outline Introduction to phylogeny reconstruction

accuracy analysis

More genomes are not necessarily better for reconstruction accuracy

Greedy algorithms for genome selection problem

Validation test

Conclusion

Conclusion Formulate the genome selection for

reconstruction problem Two greedy algorithms proposed for the

problem Validation test shows that the

reconstruction accuracy of using the genomes selected by the greedy algorithms are comparable to the

the max reconstruction accuracy.

Fitch Parsimony method

Given character states in the leave nodes the method reconstructs a subset of states at each internal nodes by the following rule:

0 1 0

{0, 1}

{0}

More Genomes Are Not Necessarily Better – An example with 4 leaves

The complete tree

More Genomes Are Not Necessarily Better – An example with 4 leaves

The unambiguous reconstruction accuracy of using one genome is

Ppath= p2+(1-p)2;

The unambiguous reconstruction accuracy of using all the 4 genomes is

Pwhole= Ppath – 3p2(1-p)2;

More genomes give more noise!

A Small with Six Leaves

(The ambiguous reconstruction accuracy)< (The unambiguous accuracy on the shortest path) When 0.5<p<0.65

Reconstruction accuracy on complete phylogeny in limit case

When conservation rate on each branch is less than 7/8,

(The ambiguous reconstructionaccuracy)= (the accuracy of using just one genome )=1/2