1. MAPS OF DNA AND INTERVAL GRAPHS by Akshita Gurram

2. OUTLINE Problem Statement Graph Construction Depicting the Graph Problem NP-hard problem Special Properties Graph Solution

3. Problem Statement Shear DNA into millions of small fragments. These fragments are known as “clones”.

4. Problem statement The sequence of the clones remains unknown. The relative order of the probes is identified in clones. The sequence of the probes is known in advance yxzw Probes Clones

5. Problem Statement The main goal of mapping of DNA is to reconstruct the relative positions of clones along the original DNA. This problem has much in common with interval graphs and consecutive 1s property of incidence matrices.

6. Graph Construction Assume that the clones overlap when the probes sequence matches with each other. By sequencing the clones accordingly with the probes,the original DNA is obtained back.

7. Graph Construction Consider the clones a,b,c,d,e and let the adjacency matrix tells which pairs of clones overlap in a DNA. a b c d e a 0 1 0 1 0 b 1 0 1 1 0 c 0 1 0 1 1 d 1 1 1 0 0 e 0 0 1 0 0

8. Graph construction T h is problem of finding whether pairs of clones overlap in a long DNA strand can be modeled by an interval graph. Let G be a graph with vertices corresponding to clones and edges corresponding to overlapping pairs of clones. b a d ce

9. Relation to a graph problem The problem in graphical terms can be defined as vertices represent clones and two vertices are adjacent if and only if the clones overlap. From the graph we construct a set of intervals on the real line corresponding to probes using which the clones were sequenced.

10. Graph problem Interval model for the given problem is

11. An NP-Hard Problem The proper sequence of the clones is hard to determine without the probes. The problem is solvable in polynomial time on interval graphs that is NP-hard.

12. Special Properties The properties that define interval graphs: The graph should not contain cycles formed by more than 3 vertices in a graph. It is chordal and its compliment G is a comparability graph. It contains no induced C4 and G is transitively orientable. It is chordal and contains no asteroidal triple.

13. Graph Solution From the interval graph we get the sequencing order of the clones with following intervals: a: (1,4) b: (2,8) c: (5,10) d: (3,6) e: (7,9)

14. Comments Interval graphs are extensively used in bioinformatics. It is difficult to determine the clones sequence without the help of probes order.

15. References Golumbic, M. C. (1980). Algorithmic graph theory and graphs. New York: Academic Press. Graph algorithms in bioinformatics. http://sydney.edu.au/engineering/it/~mcharles/ teaching/compx456/slides/L07_GraphsDNAse q.pdf

16. References Michael S.Waterman and Jerrold R.Griggs-Interval graphs and maps of DNA. http://link.springer.com/article/10.1007%2FBF0246 0022 http://www.powershow.com/view/80865- MTVlN/Physical_Mapping_of_DNA_powerpoint_ ppt_presentation

17. Thank You!!

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