Maximum clique

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

NP complete continued

 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

 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

 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

 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

 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

 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

 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

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

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

 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

Maximum clique with DNA

 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

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 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

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

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   20 = 200bp (000000) Shortest = 6   20 = 140bp(111111) dsDNA Construction of DNA molecules

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

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

Construction of DNA molecules

 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

 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

 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

correct answer Readout

 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

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

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

Clique in microreactors

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

Positive selection

Negative selection

Logical operations

logical NOT operations Logical operations

a  ba  b logical AND operations Logical operations

a  ba  b logical OR operations Logical operations

magnet Microreactor structure

magnet Microreactor structure

Selection principle

DNA input and transport principle

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

ABCDEF A B C D E F Maximal cliques – connectivity matrix

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

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

DNA in DNA out Optical control DNA computer design

DNA computer design – selection modules

DNA information flow

100  m Flow separation – laminar flow

100  m Flow separation – laminar flow

Micro fabrication

DNA computer design – 20 nodes

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

The DNA library

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

DNA hybridisation

100  m DNA hybridisation

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

3.5mm