1. CSCI2950-C Genomes, Networks, and Cancer
Computability of Models for Sequence Assembly

2. Outline Some Terminology Other Algorithms
Assembly of Double-Stranded DNA with Bidirected Flow
Chinese Postman Problem ~ Eulerization Problem
Bidirected De Brujin Graph
Discussion

3. String Terminology Let v and w be two string over the alphabet 
v.w : concatenation of v and w
|v| : length of v
v[i] : ith character of v
v[i,j] : substring of v, beginning at the ith character, ending at the jth character
v : v concatenated with itself k times
 i,j s.t. v = w[i,j] : v is a substring of w k

4. String Terminology A string of length k is called a k-mer
The set all k-mers that are substring of v is called k-spectrum of v
A pair of reverse complement k-mers is called a k-molecule

5. Graph Terminology

6. Graph Terminology

7. The String Graph Framework & The De Brujin Graph Framework NP - HARD

8. Assembly of Double-Stranded DNA with Bidirected Flow

9. Recall that... Given a weighted bidirected graph G : Chinese Walk
~ cyclical walk that traverses each edge at least once
Chinese Postman Problem (CPP)
~ finding a minimum weight Chinese Walk of G or
reporting the non-existence of such a walk
Eulerization Problem (EP)
~ finding a minimum weight Eulerization Extension of G
or reporting the non-existence of such an extension

10. Theorem - 1 Given a bidirected graph G,
G contains an Eulerian tour if and only if it is connected and balanced

11. Theorem - 2 Given a weighted bidirected graph G,
There exists a Chinese walk of weight i if and only if there exists an Eulerian extension of weight i 1 2

12. 1 2 Proof (  ) W : a Chinese walk in G
Construct a new graph W2 , induced by W, in a way that the multiplicity of each edge is the number of time it is traversed by W

13. 1 2 Proof (  ) G W : a Chinese walk in G
Construct a new graph W2 , induced by W, in a way that the multiplicity of each edge is the number of time it is traversed by W G

14. 1 2 Proof (  ) W2 G W : a Chinese walk in G
Construct a new graph W2 , induced by W, in a way that the multiplicity of each edge is the number of time it is traversed by W W2 G

15. W visits every edge of G at least once  W2 is an extension of G
Proof (  ) 1 2 G W2
W visits every edge of G at least once  W2 is an extension of G +
W visits every edge of W2 exactly once  W is an Eulerian circuit of W2
W2 is an Eulerian extension of G

16. Proof (  ) 2 1
G2 : an Eulerian extension of G G2 G

17. 2 1 Proof (  ) G2 G W2 : AaBbCcBbCfDeCgAdDeCgA
W2 : an Eulerian circuit in G2 with weight w G2 G
W2 : AaBbCcBbCfDeCgAdDeCgA

18. 2 1 Proof (  ) G2 G W2 : AaBbCcBbCfDeCgAdDeCgA
Construct W from W2 by replacing every edge e’ G by an edge e G such that e’ is a duplicate of e.
W : AaBbCcBbCfDeCgAdDeCgA

19. W is a Chinese Walk with weight i
Proof (  ) 2 1 G
W : AaBbCcBbCfDeCgAdDeCgA
W is a cyclical walk in G which traverses every edge at least once and its weight is the same as the weight of W2 , i.
W is a Chinese Walk with weight i

20. Given a weighted bidirected graph G,
Theorem - 1 G contains an Eulerian tour if and only if it is connected and balanced Theorem - 2 There exists a Chinese walk of weight i there exists an Eulerian extension of weight i

21. A Polynomial Time Algorithm for CPP
~ based on the Theorem 1 & 2
Given a weighted bidirected graph G,
- If G is not connected, any extension will be not connected  No Chinese Walk exists

22. A Polynomial Time Algorithm for CPP
If G is connected, formulate EP as a min-cost bidirected flow problem as follows:
(G’ is the desired extension of G)
Constants
we : weight of edge e
Variables
fe: additional copies of edge e required to extend from G to G’

23. A Polynomial Time Algorithm for CPP
Constraints
~ using Theorem – 1
for each vertex x
for each edge e

24. A Polynomial Time Algorithm for CPP
Integer Programming Model:

25. A Polynomial Time Algorithm for CPP
Soundness of the Algorithm
G is connected  G’ is connected +
Constraint – 1  G’ is balanced 
G’ is Eulerian

26. A Polynomial Time Algorithm for CPP
Is G’ a min-weight Eulerian-extension?

27. A Polynomial Time Algorithm for CPP
Is G’ a min-weight Eulerian-extension?
Yes!
Objective Function minimizes total weight of inserted edges

28. A Polynomial Time Algorithm for CPP
Pseudo-code
IF G is not connected
RETURN “no Chinese walk exists”
ELSE
Solve it as a Min-Cost Flow Problem
IF there is no feasible solution,
RETURN “no Chinese walk exists”
...

29. A Polynomial Time Algorithm for CPP
ELSE
Construct the G’
Find an Eulerian circuit of G’
Find the corresponding Chinese walk

30. A Polynomial Time Algorithm for CPP
Running Time?
O(|E| log (|V|))

31. A Polynomial Time Algorithm for CPP
Integer Programming Model:

32. A Polynomial Time Algorithm for CPP

33. A Polynomial Time Algorithm for CPP
Optimal Solution:
fb = 1
fe = 1
fg = 1
all other variables are zero

34. A Polynomial Time Algorithm for CPP?
A Polynomial Time Algorithm for Sequence Alignment

35. A Polynomial Time Algorithm for Sequence Alignment
Input: k-molecule spectrum of the genome
ATT TTG TGC GCC CCA CAA AAC
TAA AAC ACG CGG GGT GTT TTG

36. A Polynomial Time Algorithm for Sequence Alignment
Arbitrarily label one k-mer as positive, one k-mer as negative
- ATT - TTG +TGC +GCC
TAA + AAC+ ACG - CGG-
+CCA +CAA +AAC
GGT- GTT TTG -

37. A Polynomial Time Algorithm for Sequence Alignment
Construct nodes from all possible (k-1) molecules
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
For every k-molecule in the spectrum, let z be one of its two k-mers
Let x and y be (k-1)-mers corresponding to z[1..k-1] and z[2..k] respectively

38. A Polynomial Time Algorithm for Sequence Alignment
Insert edges according to the following criteria:
- ATT TAA+
An edge is positive incident to x, if x is from the positive strand, and negative incident otherwise
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
An edge is negative incident to y, if y is from the positive strand, and positive incident otherwise

39. A Polynomial Time Algorithm for Sequence Alignment
Insert edges according to the following criteria:
- TTG AAC+
An edge is positive incident to x, if x is from the positive strand, and negative incident otherwise
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
An edge is negative incident to y, if y is from the positive strand, and positive incident otherwise

40. A Polynomial Time Algorithm for Sequence Alignment
Insert edges according to the following criteria:
+ TGC ACG-
An edge is positive incident to x, if x is from the positive strand, and negative incident otherwise
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
An edge is negative incident to y, if y is from the positive strand, and positive incident otherwise

41. A Polynomial Time Algorithm for Sequence Alignment
Insert edges according to the following criteria:
+ GCC CGG-
An edge is positive incident to x, if x is from the positive strand, and negative incident otherwise
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
An edge is negative incident to y, if y is from the positive strand, and positive incident otherwise

42. A Polynomial Time Algorithm for Sequence Alignment
Insert edges according to the following criteria:
+ CCA GGT-
An edge is positive incident to x, if x is from the positive strand, and negative incident otherwise
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
An edge is negative incident to y, if y is from the positive strand, and positive incident otherwise

43. A Polynomial Time Algorithm for Sequence Alignment
Insert edges according to the following criteria:
+ CAA GTT-
An edge is positive incident to x, if x is from the positive strand, and negative incident otherwise
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
An edge is negative incident to y, if y is from the positive strand, and positive incident otherwise

44. A Polynomial Time Algorithm for Sequence Alignment
Insert edges according to the following criteria:
+ AAC TTG-
An edge is positive incident to x, if x is from the positive strand, and negative incident otherwise
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
An edge is negative incident to y, if y is from the positive strand, and positive incident otherwise

45. A Polynomial Time Algorithm for Sequence Alignment
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-

46. A Polynomial Time Algorithm for Sequence Alignment
How to read the sequence?
If a positive incident edge is used to enter the node, read negative k-mer
If a negative incident edge is used to enter the node, read positive k-mer
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-

47. A Polynomial Time Algorithm for Sequence Alignment
-AT TA+
- TT AA+
+ TG AC-
+ GC CG-
+CC GG-
+CA GT-
+ AC TG-
ATTGCCAAC

48. Future Work ... NP – hardness ? Optimal solution ?
~ parsimony assumption

49. Any questions / comments?
Thanks...
Any questions / comments?