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. What is annotation? 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.. Annotation is the labeling of the input sequence, in this case with 3 colors: ome

4. TAATATGTCCACGGTTGTACACGGCAG GTATTG AG GTATTG AG ATGTAACTG AA Input: TAATATGTCCACGGGTATTGAGCATTGTACACGGGGTATTGAGCATGTAATGAA Biology example: gene annotation Output: Leucine

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

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 13 24 12 34 12 43 In general, the number of trees is the Schröder number (2n-5)!! = (2n-5)*(2n-7)*… 3*1 1 2 3 4 0.1 0.2 0.4 0.2 0.3

9. Data

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

11. A primate tree from genome sequences

12. Tree space is the tropical Grassmanian

13. Example: X={1,2,3,4,5} 3 1 2 45

14. Back to the data

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

17. 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