# Graph Algorithms in addition to Fragment Assembly Outline Introduction to Graph Theory Eule

## Graph Algorithms in addition to Fragment Assembly Outline Introduction to Graph Theory Eule

Fitzsimmons, Ian, Midday Host has reference to this Academic Journal, PHwiki organized this Journal Graph Algorithms in addition to Fragment Assembly Outline Introduction to Graph Theory Eulerian & Hamiltonian Cycle Problems Benzer Experiment in addition to Interval Graphs DNA Sequencing The Shortest Superstring & Traveling Salesman Problems Sequencing by Hybridization in addition to de Bruijn graphs Fragment Assembly in addition to Repeats in DNA Fragment Assembly Algorithms The Bridge Obsession Problem Bridges of Königsberg Find a tour crossing every bridge just once Leonhard Euler, 1735

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Eulerian Cycle Problem Find a cycle that visits every edge exactly once Linear time More complicated Königsberg Hamiltonian Cycle Problem Find a cycle that visits every vertex exactly once NP  complete Game invented by Sir William Hamilton in 1857 Mapping Problems to Graphs Arthur Cayley studied chemical structures of hydrocarbons in the mid-1800s He used trees (acyclic connected graphs) to enumerate structural isomers

Beginning of Graph Theory in Biology Benzers work Developed deletion mapping Proved linearity of genomes Demonstrated internal structure of the genome Viruses Attack Bacteria Normally bacteriophage T4 (a virus) kills bacteria However if T4 is mutated (e.g., an important gene is deleted) it gets disabled in addition to loses an ability to kill bacteria Suppose the bacteria is infected with two different mutants each of which is disabled  would the bacteria still survive Amazingly, a pair of disable viruses can kill a bacteria even if each of them is disabled. How can it be explained Benzers Experiment Idea: infect bacteria with pairs of mutant T4 bacteriophage (virus) Each T4 mutant has an unknown interval deleted from its genome If the two intervals overlap: T4 pair is missing part of its genome in addition to is disabled  bacteria survive If the two intervals do not overlap: T4 pair has its entire genome in addition to is enabled  bacteria die

Complementation between pairs of mutant T4 bacteriophages Benzers Experiment in addition to Graphs Construct an interval graph: each T4 mutant is a vertex, place an edge between mutant pairs where bacteria survived (i.e., the deleted intervals in the pair of mutants overlap) Interval graph structure reveals whether the T4 DNA is linear or branched Interval Graph: Linear Genome

Interval Graph: Branched Genome Interval Graph: Comparison Linear genome Branched genome DNA Sequencing: History Sanger method (1977): labeled ddNTPs terminate DNA copying at r in addition to om points. Both methods generate labeled fragments of varying lengths that are further electrophoresed. Gilbert method (1977): chemical method to cleave DNA at specific points (G, G+A, T+C, C).

Sanger Method: Generating Read Start at primer (restriction site) Grow DNA chain Include ddNTPs Stops reaction at all possible points Separate products by length, using gel electrophoresis DNA Sequencing (Shotgun) Shear DNA into millions of small fragments Read 500  700 nucleotides at a time from the small fragments (by e.g. Sanger method) Fragment Assembly Computational Challenge: Assemble individual short fragments (reads) into a single genomic sequence (superstring) Until late 1990s the shotgun fragment assembly of human genome was viewed as an intractable problem

Shortest Superstring Problem Problem: Given a set of strings, find a shortest string that contains all of them Input: Strings s1, s2, ., sn Output: A string s that contains all strings s1, s2, ., sn as substrings, such that the length of s is minimized Complexity: NP  hard Note: this as long as mulation does not take into account sequencing errors Shortest Superstring Problem: Example Reducing SSP to TSP Define overlap( si, sj ) as the length of the longest (proper) prefix of sj that matches a suffix of si. aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa What is overlap ( si, sj ) as long as these strings

Reducing SSP to TSP Define overlap( si, sj ) as the length of the longest (proper) prefix of sj that matches a suffix of si. aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa overlap=12 Reducing SSP to TSP Define overlap( si, sj ) as the length of the longest (proper) prefix of sj that matches a suffix of si. aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa Construct a complete graph with n vertices representing the n strings s1, s2, ., sn. Insert edges of length overlap ( si, sj ) between vertices si in addition to sj. Find the longest path which visits every vertex exactly once. This is the max Traveling Salesman Problem (TSP), which is also NP  hard. Reducing SSP to TSP (contd)

SSP to TSP: An Example S = { ATC, CCA, CAG, TCC, AGT } SSP AGT CCA ATC ATCCAGT TCC CAG ATCCAGT TSP ATC CCA TCC AGT CAG 2 2 2 2 1 1 1 0 1 1 Approximation Algorithms as long as SSP Sequencing by Hybridization (SBH): History 1988: SBH suggested as an an alternative sequencing method. Nobody believed it will ever work 1991: Light directed polymer synthesis developed by Steve Fodor in addition to colleagues. 1994: Affymetrix develops first 64-kb DNA microarray First microarray prototype (1989) First commercial DNA microarray prototype w/16,000 features (1994) 500,000 features per chip (2002)

How SBH Works Attach all possible DNA probes (oligos) of length l to a flat surface, each probe at a distinct in addition to known location. This set of probes is called the DNA array. Apply a solution containing fluorescently labeled DNA fragment (single str in addition to ) to the array. The DNA fragment hybridizes with those probes that are complementary to substrings of length l of the fragment. How SBH Works (contd) Using a spectroscopic detector, determine which probes hybridize to the DNA fragment to obtain the lmer composition of the target DNA fragment. Apply the combinatorial algorithm (below) to reconstruct the sequence of the target DNA fragment from the lmer composition. Hybridization on DNA Array

Conclusions Graph theory is a vital tool as long as solving biological problems Wide range of applications, including sequencing, motif finding, protein networks, in addition to many more References Simons, Robert W. Advanced Molecular Genetics Course, UCLA (2002). http://www.mimg.ucla.edu/bobs/C159/Presentations/Benzer.pdf Batzoglou, S. Computational Genomics Course, Stan as long as d University (2004). http://www.stan as long as d.edu/class/cs262/h in addition to outs.html

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