Genome Sequencing Algorithms

Transcription

1 Genome Sequencing Algorithms Phillip Compaeu and Pavel Pevzner Bioinformatics Algorithms: an Active Learning Approach Leonhard Euler ( ) William Hamilton ( ) Nicolaas Govert de Bruijn ( )

2 The Genome Sequencing Problem Determining the order of nucleotides in a genome Human genome contains about 3 billion nucleotides Ameoba dubia and Paris japonica contain 200 times more! Applications in Medicine, Agriculture, Biotechnology,...

3 The Genome Sequencing Problem There is no technology to read the genome from one end to another. Short snippets, called reads ( nucleotides), can be identified. No info about a location of a read is known. Assembling individual reads into the entire genome is akin to solving a giant overlapping puzzle. The newspaper explosion analogy

4 History of Genome Sequencing 1977: Walter Gilbert and Frederick Sanger developed independent DNA sequencing methods. 1990: Human Genome Project, Francis Collins. 1997: Celera Genomics, Craig Venter. 2000: Human genome is sequenced.

5 Next Generation Sequencing Illumina sequences human genomes for $10,000 Complete Genomics sequences 100s of genomes per month Beijing Genome Institute has 100s of sequencing machines. Is the world's biggest sequencing center.

6 Next Generation Sequencing Identification of mutations in personal genomes for health diagnosis Genome 10K project

7 Genome Assembly The Computational Problem CTA CTA CTA CTA CTA CTA CTA CTA

8 Genome Assembly The Computational Problem Sequencing Machine generates reads A String Reconstruction Problem

9 The Genome Sequencing Problem Reconstruct a genome from reads Input: A collection of of strings, Reads Output: A string, Genome, reconstructed from all the Reads

10 k-mer Composition Composition 3 (TATT) = A C C G G A G T Lexicographical ordering of k-mers A C G A G C G T

11 The String Reconstruction Problem Reconstruct a string from its k-mer composition. Input: A collection of of k-mers Output: A Genome, such that Composition k (Genome) k is is equal to to the collection of of k-mers

12 Naive String Reconstruction Approach A C G A G C G T A T No No 3-mer begins with TT!

13 Representing a Genome as a Path Composition 3 (TATT) = A C C G G A G T Nodes in a Graph Connect k-mer with k-mer 1 2 if if suffix(k-mer 1 ) ) = prefix(k-mer 2 ) ) A C C G G A The Genome T G

14 Path turns into a Graph Connect k-mer with k-mer 1 2 if if suffix(k-mer 1 ) ) = prefix(k-mer 2 ) ) A C C G G A T G

15 Path turns into a Graph Connect k-mer with k-mer 1 2 if if suffix(k-mer 1 ) ) = prefix(k-mer 2 ) )

16 Path turns into a Graph Nodes are ordered lexicographically. How does one find the genome string?

17 Genome Path in the Graph TATT A C G A G C G T The genome string is a Hamiltonian walk in the graph

18 Hamiltonian Path Problem Find a Hamiltonian path in in the graph Input: A graph. Output: A path visiting every node in in the graph exactly once. Hamiltonian Path: A path in a graph that traverses every node exactly once William R Hamilton ( )

19 A Different Path A C C G G A G T 3-mers as nodes C G A A C G G T 3-mers as edges

20 A Different Path C G A A C G G T 3-mers as edges and nodes as prefix and suffixes of the corresponding 3-mers C G A TA AA GT TT A C G G T

21 Glue Identical Nodes C G A TA AA GT TT A C G G T G A GT TT C G T C G TA AA A

22 Glue Identical Nodes G A GT TT C G T C G TA AA A G C G C G A TA AA A GT TT T

23 Glue Identical Nodes G A C G C G TA AA GT TT A T C C TA AA A G T G GT TT A G

24 De Bruijn Graph of the Genome C TA AA A G C T G GT TT A G

25 De Bruijn Graph of the Genome TA T T C TA AA A G C T G GT TT A G The genome string is an Eulerian walk in the De Bruijn graph

26 Eulerian Path Problem Find an Eulerian path in in a graph Input: A graph. Output: A path visiting every edge in in the graph exactly once. Eulerian Path: A path in a graph that traverses every edge exactly once. Leonhard Euler ( )

27 Hamiltonian Path vs. Eulerian Path C TA AA A G C T G GT TT A

28 Hamiltonian Path vs. Eulerian Path Euler has presented an efficient solution to the Eulerian path problem. No fast algorithm exists to solve the Hamiltonian Path problem. The Hamiltonian Path Problem is NP-Complete. C C TA AA A T GT TT G G A G

29 The Objective TA TT A A C C G G A A G G C C G G T T

30 To Do... TA TT A A C C G G A A G G C C G G T T A C C G G A G T TA AA GT TT

31 Constructing De Bruijn Graph The composition of the genome is known C G A TA AA GT TT A C C G G A G T AA GT A C G G T

32 Constructing De Bruijn Graph C G A TA AA GT TT AA GT A C G G T

33 Constructing De Bruijn Graph C G A GT TT TA AA A C G G T GT

34 Constructing De Bruijn Graph TA AA C G A GT TT A GT C G G T

35 Constructing De Bruijn Graph C G A TA AA GT TT A C G G T

36 Glue Identical Nodes C G A TA AA GT TT A C G G T G A GT TT C G T C G TA AA A

37 Glue Identical Nodes G A GT TT C G T C G TA AA A G C G C G A TA AA A GT TT T

38 Glue Identical Nodes G A C G C G TA AA GT TT A T C C TA AA A G T G GT TT A G

39 De Bruijn Graph of the Genome Composition C C TA AA A G T G GT TT A G De De Bruijn Graph(Genome Composition) == == De De Bruijn Graph(Genome)

40 Constructing the De Bruijn Graph De Bruijn graph of a collection of k-mers: Represent every k-mer as an edge between its prefix and suffix Glue all nodes with identical labels

41 Euler Cycle Problem Find an Eulerian cycle in in a graph Input: A graph. Output: A cycle visiting every edge in in the graph exactly once.

42 The Konigsberg Bridges

43 Eulerian Graph A graph is Eulerian if it contains an Eulerian cycle Every balanced and strongly connected graph is Eulerian

44 Algorithm to Find the Eulerian Cycle

45 Algorithm to Find the Eulerian Cycle

46 Algorithm to Find the Eulerian Cycle EulerianCycle(Graph) form a cycle Cycle by randomly walking in in Graph while there are unexplored edges in in Graph select a node newstart in in Cycle with unexplored edges form Cycle' by traversing Cycle (starting at at newstart) and then randomly walking Cycle Cycle' return Cycle

47 From Reads to De Bruijn Graph to Genome TATT C C TA AA A G T G GT TT A A C G A G C G T

48 Multiple Eulerian Paths TAT T C C TA AA A G T G GT TT A G

49 Multiple Eulerian Paths TATT C C TA AA A G T G GT TT A G C C TA TA AA A G A G T G GT TT

50 DNA Sequencing with Read-pairs Read-pair is a pair of reads separated by a fixed distance d. Genome: TATT. A- is a 3,1 read pair. A- represents the sequence A in the original debruijn graph

51 DNA Sequencing with Read-pairs Composition 3 (TATT) = A C C G G A G T PairedComposition 3,1 (TATT) = A C C G G C A G G G T A

52 Paired Composition A C C G G C A G G G T A Lexicographical order: A C G C A G G A G T C G

53 String Reconstruction from Read-pairs String reconstruction from read-pairs Input: A collection of of paired k-mers Output: A string Text such that PairedComposition(Text) is is equal to to the collection of of paired k-mers

54 Paired De Bruijn Graphs A C C G G C A G G G T A AA A C G C A GT TT GT G G A G T TA AA C G

55 Paired De Bruijn Graphs Combine nodes with identical labels A G G A G AA GT TT TA AA GT C C A G G T C

56 Paired De Bruijn Graphs C AA A G C A GT TT GT G G A G T TA C G

57 Paired De Bruijn Graphs G C A C A G G G T TA AA GT TT C G A Paired DeBruijn graphs obtained from the paired composition and the genome are identical.

58 Paired De Bruijn Graphs G C A C A G G G T TA AA GT TT C G A Unique genome string: TATT