1. Sorting Cancer Karyotypes by Elementary Operations Michal Ozery-Flato and Ron Shamir School of Computer Science, Tel Aviv University

2. Outline Introduction Modeling the evolution of cancer karyotypes The karyotype sorting problem Combinatorial Analysis Results 2

3. 3 Introduction

4. 4 http://www.ncbi.nlm.nih.gov/sky/skyweb.cgi Normal female karyotype

5. 5 The "Philadelphia chromosome"

6. 6 http://www.ncbi.nlm.nih.gov/sky/skyweb.cgi Breast cancer karytype (MCF-7)

7. 7 Chromosomal Instability A phenotype of most cancer cells. –Losses or gains of chromosomes result from errors during mitosis –Chromosome rearrangements are associated with "double strand breaks" multi-polar mitoses

8. 8 Double Strand Breaks Constitute the most dangerous type of DNA damage –A successful repair ligates two matching broken ends –Mis-repair can result in rearrangements (e.g. translocations) or deletions M.C. EscherM.C. Escher, 1953 Double strand break

9. The Challenge Analyze the evolution of aberration events in cancer karyotypes 9

10. 10 The Mitelman Database of Chromosome Aberrations in Cancer Over 55,000 cancer karyotypes, culled from over 8000 scientific publications Can be parsed automatically (CyDAS parser www.cydas.org) www.cydas.org The largest current data resource on cancer genomes' organization

11. 11 Modeling the Evolution of Cancer Karyotypes

12. 12 The Normal Karyotype Band = basic unit observable in karyotype. A unique region in the genome, identified by integer Normal Chromosome = interval of bands –Two normal chromosomes are either disjoint or equivalent Normal karyotype = a collection of normal chromosomes –Usually contains two copies of each chromosome (with the possible exception of the sex chromosomes)

13. 13 The Cancer karyotype Fragment = a sub-interval (>1 bands) of a normal chromosome Chromosome = –One fragment, or a concatenation of several fragments –Orientation-less: [1,4]::[37,40]  [40,37]::[4,1] Cancer karyotype = a collection of chromosomes concatenation (breakpoint)

14. 14 Elementary Operations Breakage Fusion duplication deletion  These operations can generate all known chromosomal aberrations!

15. 15 The Karyotype Sorting Problem

16. 16 The Karyotype Sorting (KS) Problem Find a shortest sequence of elementary operations that transforms the normal karyotype into given cancer karyotype Find the elementary distance = #operations in such a solution to KS. ???

17. 17 The Karyotype Sorting (KS) Problem (inverse formulation) Find a shortest sequence of inverse elementary operations that transforms the given cancer karyotype into the normal karyotype ???

18. 18 Inverse Elementary Operations Breakage Fusion duplication deletion  c-deletion addition

19. 19 Assumptions ~95% of the karyotypes in the Mitelman Database have no recurrent breakpoints Assumptions: –The cancer karyotype contains no recurrent breakpoints –Every added chromosome contains no breakpoints [20,39]::[12,1] Breakpoint ID={39 0,12 0 }

20. 20 The Reduced Karyotype Sorting (RKS) Problem Assumptions  reduced problem: –No breakpoints in the cancer karyotype (i.e every chromosome is an interval) –No breakpoints created by fusions / additions –  All the normal chromosomes are identical 12345678910110 12345678910110 The normal karyotypeThe cancer karyotype breakage, fusion, c-deletion, addition identical chromosomes

21. 21 Combinatorial Analysis (RKS Problem)

22. 22 Extending the karyotypes 12345678910110 The normal karyotype 12345678910110 The cancer karyotype

23. 23 Parameter 1: f = #disjoint pairs of complementing interval ends Observation: –  f = -1 for fusion;  f = 1 for breakage –  f  {0,-1,-2} for c-deletion –  f  {0,1,2} for addition f = 5 12345678910110

24. 24 The histogram Parameter 2: w = #bricks Observations: –w is even –  w = 0 for breakage / fusion –  w  {0,  2} for addition / c-deletion 12345678910110 The cancer karyotype 12345678910110 The histogram A wall with 2 bricks A brick

25. 25 Simple Bricks A brick is simple if –no lower brick (in the same wall), and –no complementing interval ends Parameter 3: s = #simple bricks Observation: –  s  {0,-1} for breakage –  s =0 for constrained-deletion –|  s|  2 for addition Simple bricks 12345678910110

26. 26 The Weighted Bipartite Graph of Bricks Parameter 4: m = the minimum weight of a perfect matching weightv -,v + :simplev -,v + :non-simpleotherwise v - < v + 201 v + < v - 021 12345678910110 Positive bricks

27. Results

28. 28 Main Theorem The elementary-distance, d, satisfies: w/2+f+s+m-2N  d  3w/2+f+s+m-2N N = #intervals in the normal karyotype

29. 29 Results (2) Used the main theorem to devise a polynomial-time 3-Approximation algorithm Combined with a greedy heuristic on real data (95% of Mitelman DB) optimal solutions computed for 100% of karyotypes –99.99% cases : lower bound is achieved (hence solution is optimal) –30 cases: lower-bound+2 but actually optimal (manual verification)

30. 30 Summary A new framework for analyzing chromosomal aberrations in cancer A 3-approximation algorithm when there are no recurrent breakpoints –100% success on 57,252 karyotypes (with no recurrent breakpoints) from the Mitelman DB. Future work: handle recurrent breakpoints –Analyze the remaining 5% of the karyotypes in the Mitelman DB.

31. 31 Thank for your attention. Questions?