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Fragment Assembly of DNA (Ch 4 of Setubal and Meidanis)

Goal:  Sequence DNA of length 30,000 to 100,000 base pairs

Problem:  Can only sequence DNA of length less than 700 base pairs.

Solution:  Break the long DNA sequence into 500 to 2000 fragments of length 200-700 base pairs,
sequence these shorter fragments, and use discrete mathematics to determine the sequence of the
original long piece of DNA.


We will illustrate this process with an example.  Note, however, the problem and techniques have
been greatly simplified and altered from actual practice.

Example 1:

The DNA sequence 

\centerline{AGACCGCGTATAGCATCAGCATGACGCGTAGCACTA}
\u
 is randomly broken into the 
following 5 pieces

AGACCGCGTATAG
\u
CGTATAGCATCAG
\u
TAGCATCAGCATGACGCGT
\u
GCGTAG
\u
TAGCACTA

In order to sequence a long piece of DNA, it must be cut into many smaller pieces.  These smaller
pieces are then sequenced.  The order of the smaller pieces within the longer original piece must
then be determined in order to determine the sequence of the original longer piece of DNA.

Suppose you did not know the original DNA sequence, but you did know the sequences of the 5 pieces.  
Could you reconstruct the original DNA sequence?
\eject


We can construct a directed graph as follows:

i.)  The vertices represent the DNA pieces


ii.)  If there is an overlap between the end of piece $A$ and the beginning of piece $B$, then an
arc is drawn from piece $A$ to piece $B$.

iii.) Each arc is given a weight corresponding to the length of the overlap.

\vskip 15pt

\centerline{AGACCGCGTATAG}

\vskip .9in

\centerline{CGTATAGCATCAG \hskip 1.5in TAGCATCAGCATGACGCGT}

\vskip .9in

\centerline{GCGTAG \hskip 2in TAGCACTA}

\vskip 10pt


To find the original DNA sequence, we look for a Hamiltonian path (a Hamiltonian path is a path that
visits each vertex exactly once).  We will assume that the Hamiltonian path with the largest sum of
arc weights corresponds to the correct sequence.  You can read more about how fragment assembly is
really done in Ch 4 of Setubal and Meidanis.

\eject

Fragment Assembly of DNA Homework

1.)  Draw the directed graph corresponding to the following fragments.  Find a Hamiltonian path
with the largest weight sum.  Use this path to determine the sequence of the original DNA sequence.
Is there more than one Hamiltonian path  
with the largest weight sum?

AGCAGTAG
\u
GAGCGAT
\u
AGTCGTAG
\u
GATCACAG



2.)  Draw the directed graph corresponding to the following fragments.  Find a Hamiltonian path
with the largest weight sum.  Use this path to determine the sequence of the original DNA sequence.
Is there more than one Hamiltonian path  
with the largest weight sum?

ACTGGATT
\u
ATTATG
\u
TGTT
\u
TTAACT
\u
TTCTACT

3.)  To find the original DNA sequence, what are you looking for (i.e., what kind of graph theory
problem is this?)?

4.)  Is the solution always unique?

5.)  How is this problem similar to and different from the traveling salesperson problem?



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