Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker
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Chapter 8: Graph Algorithms July/23/2012 Name: Xuanyu Hu Professor: Elise de Doncker. Outline. Graphs Graphs and Genetics DNA Sequencing Shortest Superstring Problem. 1: Graphs. Diagrams with collections of points connected by lines are examples of graphs .

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Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

Chapter 8: Graph AlgorithmsJuly/23/2012Name: Xuanyu HuProfessor: Elise de Doncker


Outline
Outline

  • Graphs

  • Graphs and Genetics

  • DNA Sequencing

  • Shortest Superstring Problem


1: Graphs

  • Diagrams with collections of points connected by lines are examples of graphs.

  • The points are called vertices and lines are called edges.



How to use graph knights problem 1
How to Use Graph: Knights Problem 1 of

  • This upper picture shows two white and two black knights on a 3*3 chessboard.

  • Can they move, using the usual chess knight's moves, to occupy the positions shown in the below picture?






How to use graph knights problem 2
How to Use Graph: Knights Problem 2 to a neighboring point in the diagram, in either a clockwise or counterclockwise direction.

  • This picture represents anohter chessboard obtained from a 4*4 chessboard by removing the four corner squares.

  • Can a knight travel around this board, pass through each square exactly once, and return to the same square it started on?




Connected and disconnected
Connected and Disconnected the correct sequence of moves.

  • A graph is called connected if all pairs of vertices can be connected by a path, which is a continuous sequence of edges, where each successive edge begins where the previous one left off.

  • Graphs that are not connected are disconnected.


Cycles
Cycles the correct sequence of moves.

  • Paths that start and end at the same vertex are referred to as cycles.

  • For example, the paths(3-2-10-11-3), and paths(3-2-8-6-12-7-5-11-3) are cycles.


The bridge obsession problem
The Bridge Obsession Problem the correct sequence of moves.

Find a tour crossing every bridge just once

Leonhard Euler, 1735

Bridges of Königsberg


Eulerian cycle problem
Eulerian Cycle Problem the correct sequence of moves.

  • Find a cycle that visits every edgeexactly once.

  • Graph theory was born when Leonhard Euler solved the famous Königsberg Bridge problem.

More complicated Königsberg


Hamiltonian cycle problem

Can you travel from any one of the the correct sequence of moves.vertices in this graph, visit every other vertex exactly once, and end up at the original vertex?

Hamiltonian Cycle Problem

Game invented by Sir

William Hamilton in 1857


Trees
Trees the correct sequence of moves.

  • Arthur Cayley studied chemical structures of hydrocarbons in the mid-1800s

  • Structures of this type of hydrocarbon are examples of trees, which are simply connected graphs with no cycles.


  • Every the correct sequence of moves.tree has at least one vertex with degree 1, called leaf.

  • Every tree on n vertices has n-1 edges, regardless of the structure of the tree.


  • Every tree on n the correct sequence of moves.vertices has n-1 edges, regardless of the structure of the tree.

  • Every tree has a leaf, we can remove it and its attached edge. We keep this up until we are left with a graph with a single vertex and no edges.


2 graphs and genetics

Seymour Benzer, 1950s the correct sequence of moves.

2: Graphs and Genetics

Benzer’s work

  • Developed deletion mapping

  • “Proved” linearity of the gene

  • Demonstrated internal structure of the gene


Viruses attack bacteria
Viruses Attack Bacteria the correct sequence of moves.

  • Normally bacteriophage T4 kills bacteria

  • However if T4 is mutated (e.g., an important gene is deleted) it gets disable and looses 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
Benzer the correct sequence of moves.’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 and is disabled – bacteria survive

  • If the two intervals do not overlap: T4 pair has its entire genome and is enabled – bacteria die


Benzer s experiment and graphs
Benzer the correct sequence of moves.’s Experiment and 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 DNA is linear or branched DNA


Interval graph linear genes
Interval Graph: Linear Genes the correct sequence of moves.


Interval graph branched genes
Interval Graph: Branched Genes the correct sequence of moves.


Interval graph comparison
Interval Graph: Comparison the correct sequence of moves.

Linear genome

Branched genome


3 dna sequencing history
3: the correct sequence of moves.DNA Sequencing: History

  • Gilbert method (1977):

  • chemical method to cleave DNA at specific points (G, G+A, T+C, C).

Sanger method (1977): labeled ddNTPs terminate DNA copying at random points.

  • Both methods generate labeled fragments of varying lengths that are further electrophoresed.


Sanger method generating read

Start at primer (restriction site) the correct sequence of moves.

Grow DNA chain

Include ddNTPs

Stops reaction at all possible points

Separate products by length, using gel electrophoresis

Sanger Method: Generating Read


Dna sequencing
DNA Sequencing the correct sequence of moves.

  • Shear DNA into millions of small fragments

  • Read 500 – 700 nucleotides at a time from the small fragments (Sanger method)


Fragment assembly
Fragment Assembly the correct sequence of moves.

  • 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 intractable problem


4 shortest superstring problem
4: the correct sequence of moves.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

  • Note: this formulation does not take into account sequencing errors


Shortest superstring problem example
Shortest Superstring Problem: Example the correct sequence of moves.

  • Concatenating all eight strings results in a 24-letter superstring

  • the shortest superstring contains only 10 letters.


Conclusion and qustions
Conclusion and Qustions the correct sequence of moves.

  • Graphs

    graphs, vertex(vertices), edges, connected, disconnected, cycles, trees, degree, leaf

  • Graphs and Genetics

  • DNA Sequencing

  • Shortest Superstring Problem


References
References the correct sequence of moves.

1.http://bix.ucsd.edu/bioalgorithms/slides.php

2.http://en.wikipedia.org/wiki/Graph_theory

3.http://simple.wikipedia.org/wiki/Genetics

4.http://seqcore.brcf.med.umich.edu/doc/educ/dnapr/sequencing.html

5.http://www.wiley.com/college/pratt/0471393878/student/animations/dna_sequencing/index.html

6.http://math.mit.edu/~goemans/18434S06/superstring-lele.pdf


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