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Tropical Geometry for Biology

Tropical Geometry for Biology. Lior Pachter and Bernd Sturmfels Department of Mathematics U.C. Berkeley. Tropical arithmetic Annotation is sequence labeling Annotation is important for biology Annotation is tropical arithmetic Tropical geometry Tree basics

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Tropical Geometry for Biology

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  1. Tropical Geometry for Biology Lior Pachter and Bernd Sturmfels Department of Mathematics U.C. Berkeley

  2. Tropical arithmetic • Annotation is sequence labeling • Annotation is important for biology • Annotation is tropical arithmetic • Tropical geometry • Tree basics • Tree reconstruction is important for biology • Tree space is the tropical Grassmanian • Back to the data

  3. INPUT: ..t..r…o..p..i..c..a..a..l...g..e..e..t..r..y.. OUTPUT: ..t..r…o..p..i..c..a..a..l...g..e..e..t..r..y.. ome Annotation is the labeling of the input sequence, in this case with 3 colors: What is annotation?

  4. T A G T G A T A C G A T G G C G T T A T G A T G A A A T G A T G T T T A G A G C G A C G G A A C C T A C T T Leucine Biology example: gene annotation Input: TAATATGTCCACGGGTATTGAGCATTGTACACGGGGTATTGAGCATGTAATGAA Output:

  5. Finding a good annotation with tropical arithmetic Example: assign “scores”, say x,y,z to each color regardless of letter x y z Best annotation for TAAT is obtained by evaluating

  6. Tropical arithmetic • Annotation is sequence labeling • Annotation is important for biology • Annotation is tropical arithmetic • Tropical geometry • Tree basics • Tree reconstruction is important for biology • Tree space is the tropical Grassmanian • Back to the data

  7. What is a phylogenetic X-tree? In Darwin’s example X = {A,B,C,D,1}

  8. Tree basics 1 3 1 2 2 4 3 4 2 0.1 1 0.2 1 2 0.4 0.2 0.3 3 4 3 4 In general, the number of trees is the Schröder number (2n-5)!! = (2n-5)*(2n-7)*… 3*1

  9. Data

  10. Metrics and trees [ dij ] Distance between species i and j

  11. A primate tree from genome sequences

  12. Tree space is the tropical Grassmanian

  13. 1 2 4 5 Example: X={1,2,3,4,5} 3

  14. Back to the data

  15. Tree Markov models Generalized hidden Markov Phylogeny Evol. HMM Generalized Multi HMM Multi HMM Generalized HMM Final message: Tropical mathematics is important for comparative genomics. Phylogeny Graphical Models Alignment Annotation

  16. For more on mathematics and tropical geometry (and combinatorics and algebra and statistics…): L. Pachter and B. Sturmfels, Tropical Geometry of Statistical Models, PNAS 101, 2004 L. Pachter and B. Sturmfels, Parametric Inference for Biological Sequence Analysis, PNAS 101, 2004 D. Speyer and B. Sturmfels, The Tropical Grassmanian, Advances in Geometry 4, 2004. L. Pachter and B. Sturmfels, Mathematics of Phylogenomics, arxiv math.ST/0409132, 2004. and coming soon: Book (to be published by Cambridge University Press) Algebraic Statistics for Computational Biology edited by Pachter and Sturmfels

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