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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?

\vskip 10pt
\centerline{AGCAGTAG
\h
GAGCGAT}
\u
\centerline{
AGTCGTAG
\h
GATCACAG}
  

There are two Hamiltonian paths
with the largest weight sum:

GAGCGAT, GATCACAG, AGCAGTAG, AGTCGTAG giving an original sequence of \hfil \break
GAGCGATCACAGCAGTAGTCGTAG 

GAGCGAT, GATCACAG, 
AGTCGTAG 
AGCAGTAG, 
giving an original sequence of \hfil \break
GAGCGATCACAGTCGTAGCAGTAG, 


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?



\vskip 10pt

\centerline{ACTGGATT}

\centerline{ATTATG
\h
TGTT}

\u
\centerline{TTAACT
\h
TTCTACT}


There are two Hamiltonian paths
with the largest weight sum:

TTAACT, ACTGGATT, ATTATG, TGTT, TTCTACT 
giving an original sequence of \hfil \break
TTAACTGGATTATGTTCTACT 

TTCTACT, ACTGGATT, ATTATG, TGTT, TTAACT 
giving an original sequence of \hfil \break
TTCTACTGGATTATGTTAACT 


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



4.)  Is the solution always unique?  No. 

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

i.) TSP is a graph problem, while fragment assembly is a digraph problem.

ii.) In both cases, we are interested in the sum of the weights, but in TSP we usually want the 
minimum sum (i.e. minimized costs or distance --unless you want the maximum distance for more 
frequent flyer miles), while in fragment assembly
 We  want to maximize the weights of the arcs (i.e. maximize the overlaps between consecutive 
segments) in our simplified and altered problem.


Note in real life you may not want to maximize the weights, but instead you may use knowledge about 
the length of the original sequence.

ii.)  In both cases we want to visit each vertex exactly once, but in TSP, we also want to return to 
the starting point (i.e. we are looking for a closed path) while in fragment assembly we do not 
return to the starting point.

Side-note:  Both problems can be modified.  If the DNA is circular, we are looking for a Hamiltonian
cycle, and if the salesperson does not want to return home, we are looking for a Hamiltonian path.


\eject



\end