1. Maximum clique

2. 1Introduction 2Theoretical background Biochemistry/molecular biology 3Theoretical background computer science 4History of the field 5Splicing systems 6P systems 7Hairpins 8Detection techniques 9Micro technology introduction 10Microchips and fluidics 11Self assembly 12Regulatory networks 13Molecular motors 14DNA nanowires 15Protein computers 16DNA computing - summery 17Presentation of essay and discussion Course outline

3. NP complete continued

4.  Some problems are undecidable: no computer can solve them.  e.g. Turing’s “Halting Problem”  Other problems are decidable, but intractable: as they grow large, we are unable to solve them in reasonable time  What constitutes “reasonable time”? tractibility

5.  P =set of problems that can be solved in polynomial time  NP =set of problems for which a solution can be verified in polynomial time  P  NP  The big question: Does P = NP? P and NP summary

6.  The NP-Complete problems are an interesting class of problems whose status is unknown  No polynomial-time algorithm has been discovered for an NP-Complete problem  No suprapolynomial lower bound has been proved for any NP-Complete problem, either  Intuitively and informally, what does it mean for a problem to be NP-Complete? NP-complete problems

7.  A problem P can be reduced to another problem Q if any instance of P can be rephrased to an instance of Q, the solution to which provides a solution to the instance of P. This rephrasing is called a transformation  Intuitively: If P reduces in polynomial time to Q, P is “no harder to solve” than Q reduction

8.  Though nobody has proven that P != NP, if you prove a problem NP-Complete, most people accept that it is probably intractable  Therefore it can be important to prove that a problem is NP-Complete  Don’t need to come up with an efficient algorithm  Can instead work on approximation algorithms Why prove NP-completenss

9.  What is a clique of a graph G?  Answer: a subset of vertices fully connected to each other, i.e. a complete subgraph of G  The clique problem: how large is the maximum- size clique in a graph?  Can we turn this into a decision problem?  Answer: Yes, we call this the k-clique problem  Is the k-clique problem within NP? clique

10.  What should the reduction do?  Answer: Transform a 3-CNF formula to a graph, for which a k-clique will exist (for some k) iff the 3-CNF formula is satisfiable clique

11. The reduction:  Let B = C 1  C 2  …  C k be a 3-CNF formula with k clauses, each of which has 3 distinct literals  For each clause put a triple of vertices in the graph, one for each literal  Put an edge between two vertices if they are in different triples and their literals are consistent, meaning not each other’s negation  Run an example: B = (x   y   z)  (  x  y  z )  (x  y  z ) clique

12. Prove the reduction works:  If B has a satisfying assignment, then each clause has at least one literal (vertex) that evaluates to 1  Picking one such “true” literal from each clause gives a set V’ of k vertices. V’ is a clique (Why?)  If G has a clique V’ of size k, it must contain one vertex in each clique (Why?)  We can assign 1 to each literal corresponding with a vertex in V’, without fear of contradiction clique

13.  A clique of a graph G=(V,E) is a subgraph C that is fully-connected (every pair in C has an edge).  CLIQUE: Given a graph G and an integer K, is there a clique in G of size at least K?  CLIQUE is in NP: non-deterministically choose a subset C of size K and check that every pair in C has an edge in G. This graph has a clique of size 5 Clique problem, summary

14. Maximum clique with DNA

15.  Clique a set of vertices defined as a set of vertices in which every vertex is connected to every other vertex by an edge  Maximal clique problem Given a network containing N vertices and M edges, how many vertices are in the largest clique? Finding the size of the largest clique has been proven to be an NP- complete problem Introdcution

16. complete data pool Step 1 Make the complete data pool For a graph with N vertices, each possible clique is represented by an N-digit binary number 1: a vertex in the clique 0: a vertex out of the clique i.e. i.e. clique (4,1,0)  binary number 010011 Step 2 Find pairs of vertices in the graph that are not connected by an edge (0,2) (0,5) (1,5) (1,3) The complementary graph Algorithm

17. Step 3 Eliminate from the complete data pool all numbers containing connections in the complementary graph  xxx1x1 or 1xxxx1 or 1xxx1x or xx1x1x Step 4 Sort the remaining data pool to find the data containing the largest number of 1’s the largest number of 1’s size  the clique with the largest number of 1’s tells us the size of the maximal clique Algorithm

18. two DNA sections bit’s value bit’s value (Vi) V 0 ~V 5 0 bp when V i =1 10 bp when V i =0 position value position value (Pi) P 0 ~P 6 20 bp Longest = 6  10 + 7  20 = 200bp (000000) Shortest = 6  0 + 7  20 = 140bp(111111) dsDNA Construction of DNA molecules

19. sequence construction - randomly generated  to avoid mispairing, avoid accidental homologies longer than 4bp restriction sequences  embedded restriction sequences within each V i =1  POA (parallel overlap assembly) Construction of DNA molecules

20. POA (parallel overlap assembly) with 12 oligonucleotides P i V i P i+1 for even i for odd i P 0 V 0 P 1 P 2 V 2 P 3 P 4 V 4 P 5 PCR with P 0 and as primers (lane2 in fig3) POA

21. Construction of DNA molecules

23.  Break DNA : internal sequence V i =1  PCR with P 0 and as primers  broken sting were not amplified  Division of the data pool into two test tube  t 0 : Alf IIcut V o =1  t 1 : Spe Icut V 2 =1  combine t 0 and t 1 into test tube t, which did not contain xxx1x1 Digestion of restriction enzymes

24.  Elimination all strings connected by edges xxx1x1, 1xxxx1, 1xxx1x, xx1x1x  PCR amplification of remaining data DNA ( Fig 3),  Lane 5: digestion result  Lane 6: PCR result Digestion and PCR amplification

25.  Reading the size of the largest clique(s) shortest length : 160bp  four vertices  What is the maximal clique?  6 C 4 = 15, 15 different strings  read the answer by molecular cloning 1 insertion the DNA into M13 bacteriophage through site-directed mutagenesis 2 transfection of the mutagenized M13 phase DNA into E.coli 3 cloning 4 DNA extraction and sequencing Readout

26. correct answer 111100 Readout

27.  Production of ssDNA during PCR  cannot be cut by restriction enzymes  solution : digestion of the ssDNA with S1 nuclease before restriction digestion  Incomplete cutting by restriction enzymes  repetition of digestion-PCR process  increase the signal-to-noise discussion - major error

28. Strengths  high parallelism Weaknesses  limitation on the number of vertices that this algorithm can handle  maximum number of vertices with picomole operations = 27 (36 vertices with nanomole)  exponential increase in the size of the pool with the size of the problem  Further scale-up becomes impractical  New algorithms are needed Discussion - strengths and weaknesses

29. Rapid and accurate data access is needed  biotin-avidin purification  electrophoresis  DNA cloning  too slow/ too noising  biochip is needed to accelerate readout Discussion – future direction

30. Clique in microreactors

31. all possible solutions {000} {001} {010} {011} {100} {101} {110} {111} clauses (x=1)^(y=0)^(z=1) Selection principle

33. Positive selection

34. Negative selection

35. Logical operations

36. logical NOT operations Logical operations

37. a  ba  b logical AND operations Logical operations

38. a  ba  b logical OR operations Logical operations

39. magnet Microreactor structure

40. magnet Microreactor structure

41. Selection principle

42. DNA input and transport principle

43. 6 nodes, 2 initial answers 6 Max: S ABCDE =101001 Maximal cliques

44. ABCDEF A101001 B011100 C111011 D010110 E001110 F101001 Maximal cliques – connectivity matrix

45. SA=0SA=0 SE=0SE=0 SD=0SD=0 SC=1SC=1SC=0SC=0 SB=0SB=0 SA=0SA=0SA=1SA=1 SF=0SF=0 SF=1SF=1 Maximal cliques – flow diagram

46. 0xxxxx 00xxxx 0xx0xx 00x0xx 0xxx0x 00xx0x 0xx00x x0x00x 00x00x 0xxxxx 00xxxx 0xx0xx x0x0xx 00x0xx 0xxxxx x0xxxx 00xxxx 0xxxxx xxxxxx XXXXXX with x={0,1} SA=0SA=0 SA=0SA=0 SA=0SA=0 SA=0SA=0SE=0SE=0 SD=0SD=0 SC=1SC=1SC=0SC=0 SB=0SB=0 SA=0SA=0SA=0SA=0SA=1SA=1 SA=0SA=0SF=0SF=0SF=1SF=1 Maximal cliques – flow diagram

47. DNA in DNA out Optical control DNA computer design

50. DNA computer design – selection modules

51. DNA information flow

52. 100  m Flow separation – laminar flow

53. 100  m Flow separation – laminar flow

54. Micro fabrication

61. DNA computer design – 20 nodes

63.  word codes  optical programmability  usage of masks to programme  immobilisation of DNA to paramagnetic beads  hybridisation of DNA-strands DNA sequence handling

64. The DNA library

65. PBS1: 5'-GCCCTAAAGGATCCACGTAAGGTCCTATGC V0-1: 5'-AACCACCAACCAAACC V0-0: 5'-AAAACGCGGCAACAAG V1-1: 5'-TCAGTCAGGAGAAGTC V1-0: 5'-TCTTGGGTTTCCTGCA V2-1: 5'-TTTTCCCCCACACACA V2-0: 5'-TTGGACCATACGAGGA V3-1: 5'-CGTTCATCTCGATAGC V3-0: 5'-AGAGTCTCACACGACA V4-1: 5'-AAGGACGTACCATTGG V4-0: 5'-CTCTAGTCCCATCTAC V5-1: 5'-CAACGGTTTTATGGCG V5-0: 5'-GCGCAATTTGGTAACC V6-1: 5'-TAGCAGCTTCCTTACG V6-0: 5'-ACACTGTGCTGATCTC V7-1: 5'-CACATGTGTCAGCACT V7-0: 5'-TGTGTGTGCCTACTTG V8-1: 5'-GATGGGATAGAGAGAG V8-0: 5'-AATCCCACCAGTTGAC V9-1: 5'-ATGCAGGAGCGAATCA V9-0: 5'-GCTTGTTCAACCTGGT V10-1: 5'-CCCAGTATGAGATCAGV10-0: 5'-CTGTCCAAGTACGCTA V11-1: 5'-ATCGAGCTTCTCAGAGV11-0: 5'-TGTAGAGGCTAGCGAT PBS2: 5'-TGGTTTGGCGGCTTTAGAATTCTGTGACAC The DNA library

66. DNA hybridisation

67. 100  m DNA hybridisation

71. liquid handling DNA computer robotics detection system sorting module computer control DNA computer control

72. 3.5mm