MAPS OF DNA AND INTERVAL GRAPHS

by Akshita Gurram

OUTLINE

• Problem Statement• Graph Construction• Depicting the Graph Problem• NP-hard problem• Special Properties• Graph Solution

Problem Statement

• Shear DNA into millions of small fragments.

• These fragments are known as “clones”.

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

y x z w

Probes

Clones

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.

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.

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

Graph construction• This 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

c e

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.

Graph problem

• Interval model for the given problem is

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.

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.

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)

Comments

• Interval graphs are extensively used

in bioinformatics.

• It is difficult to determine the clones

sequence without the help of probes

order.

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_Gra

phsDNAseq.pdf

References

• Michael S.Waterman and Jerrold R.Griggs-

Interval graphs and maps of DNA.

http://

link.springer.com/article/10.1007%2FBF02460022

• http://www.powershow.com/view/80865-MTVlN/

Physical_Mapping_of_DNA_powerpoint_ppt_present

ation

Thank You!!

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