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Selecting Genomes for Reconstruction of A ncestral Genomes. Louxin Zhang Department of Mathematics National University of Singapore. Boreoeutherian Ancestor. The Genome Selection for Reconstruction problem. - PowerPoint PPT Presentation

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  • Selecting Genomes for Reconstruction of Ancestral Genomes Louxin ZhangDepartment of MathematicsNational University of Singapore

  • Boreoeutherian Ancestor

  • The Genome Selection for Reconstruction problemInstance: 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)

  • OutlineIntroduction to reconstruction accuracy analysisMore genomes are not necessarily better for reconstructionGreedy 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 & Golstein96, Yang et al95)Bayesian methods (Yang et al95)

    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.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.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 WorksSimulation study (Martins99, Mooers04, Salisbury & Kim01, Zhang & Nei97, Yang et al95); Theoretical study (Mossel01, Lucena and Haussler05, Maddison95)

  • 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:BCA

  • 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

  • OutlineIntroduction to reconstruction accuracy analysisMore genomes are not necessarily better for reconstruction accuracyGreedy 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 lineageSuch a phylogeny is used when both fossil record and data at extant species are used for reconstruction (Finarelli and Flynn, 2006)

  • AYZp1p2Theorem 1: Aparsimony < p1 if < p2
  • AYZp1p2p1 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] + PA[{0, 1}|1] = p1 + (1-p1-p2) PZ[{1}|0] + (p2-p1) PZ[{0}|0] < p1 < p2
  • The reconstruction accuracy oncomb-shaped trees in the limit case

  • AYZp1p2Theorem 2: AML = p1 if < p20The marginal ML outputs 0 at A iff the state at Y is 0.

  • Another Example showing the Non-monotone Property of Reconstruction Accuracy

  • SimulationExperiment setupYule birth-death modelConservation probability along branches: 0.5~1Count 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 ML500 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.

  • OutlineIntroduction to reconstruction accuracy analysis

    More genomes are not necessarily better for reconstruction accuracy

    Greedy algorithms for genome selection problem

  • Genome selection for reconstructionthe problemInstance: 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 approachesThe 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 Algorithm1. Set S ;

    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 SS is the set of selected genomes

  • Backward Greedy Algorithm1. 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 gs;

    3. Output S

  • Validation test Trees with the same heightExperiment setupRandom 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 Steel89.)

  • Concluding remarks Formulate the genome selection for reconstruction problem Two greedy algorithms proposed for the problemValidation 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 ExampleBoreoeutherian ancestorFrom Encode project

    4 states at leaf nodes

    Expected accuracy at the root node

  • A Biological Example ResultsBackward 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

  • OutlineIntroduction 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 problemValidation 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:010{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 leavesThe 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
  • Reconstruction accuracy on complete phy