DNA computing Solving Optimization problems on a DNA computer Ka-Lok Ng Dept. of Bioinformatics Taichung Healthcare and Management University.

DNA computing Solving Optimization problems on a DNA computer Ka-Lok Ng Dept. of Bioinformatics Taichung Healthcare and Management University

Content 1.Why consider DNA computing ? 2.Basic molecular biology & Basic DNA operations 3.Molecular Computation - Solved problems 3.1 Hamiltonian Path Problem 3.2 Boolean formula 3.3 Integer knapsack problem 4.Limitations and Errors 5.Prospective

Why consider DNA computing ? Essences of computation 1.Massive Parallelism A test tube can contains 10 22 DNA strands, each reaction take place independently. 2.Number of operations/sec. Silicon-based computer is much better, ~ 10 operations/sec DNA computing needs human interception. 3.Extreme large associative memory Memory density ~ 1 bit/(nm) 3 >> video tape ~ 1 bit/10 12 (nm) 3 Human synapses ~ 10 14, each store a few bits. Associative memory – match a sub-sequence Store000000…. 110100…. 000111…. Input seq.*1*1……  retrieve & read the 2 nd strand

Basic Molecular Biology 5’ 3’ A G T C C ……………………. T C A G G …………………… 3’ 5’ The DNA Double Helix Purines(Double ring structure)Adenine, AGuanine, G Pyrimidines (Single ring structure) Thymine, TCytosine, C (A,T) Watson-Crick complement (C,G) pairwise attraction Hydrogen boning

Basic DNA Operations 1.DNA synthesiser – make arbitrary DNA strands, time ~ hours Notation : ATGC = 5’-ATGC-3’ (ATGC) C = 3’-TACG-5’ 2.Hybridization (annealing), by hydrogen boning, time ~ 30 sec. it is a 2 nd order kinetic reaction 3.Denature – by heating till the longest strands unstable, dsDNA  ssDNA 4.Ligation x y x C y C x y x C y C Ligase enzyme

Basic DNA Operations 5.Polymerase Extensions Polymerase enzyme attached to the 3’ end of the promer seq. & construct the x C of the longer seq. 3‘ 5‘ 3‘ Primer 3‘ 5‘ Polymerase enzyme x xCxC

Basic DNA Operations 6. Cut – Type II Restriction enzyme (endonucleases), cut ssDNA or dsDNA strand at a specific sub-seq., usually a 4 to 8 nucleotides seq. NomenclatureSpecific-siteExpected freq. EcoRI5’-GAATTC-3’4 6 = 4096 HaeIII5’-GGCC-3’4 4 = 256 PstI5’-CTGCAG-3’4 6 = 4096 HaeIIIPstIEcoRI GGCCCTGCAGGAATTC CCGGGACGTCCTTAAG Blunt3’-protruding5’-protruding

Basic DNA Operations 7.Merge – combine two or more test tubes of DNA solution 8.Separation by length – Gel Electrophoresis Agarose Gel Electrophoresis (AGE) Long seq. – 300 ~ 50,000 bp, t ~ 5 hours Polyacrylamide Gel Electrophoresis (PAGE) Short seq. – 1 ~ 1000 bp, t ~ 1 hour - gel buffer + gel buffer - + Shortest DNA strands

High Level Manipulations 1.Polymerase Chain Reaction (PCR) – Amplification developed by Kary Mullis (Noble Prize in medicine 1994) Prepare Primers x y z x C y C z C 3‘ 5‘ 3‘ xCxC zCzC zCzC 5‘ 3‘ xCxC 5‘3‘ Template

High Level Manipulations 1.Polymerase Chain Reaction (PCR) Polymerase  dsDNA  melting  polymerase …… Repeat the above two processes  amplification x y z xCxC 3‘ 5‘ polymerase z zCzC 5‘ 3‘ polymerase

High Level Manipulations 2.Separation by sub-sequence – by magnetic bead (affinity purification) 3.Append 3.1 Polymerase 3.2 Ligation s sCsC magnet A primer with attached bead anneal a short seq. s sCsC xy x C y C 3‘ 5‘ 3‘ x y

High Level Manipulations 4.Mark – for separation or operate selectively 4.1 appending a tag seq. 4.2 methylation or (de)phosphorylation 4.3 forming a dsDNA through hybridization or the action of polymerase 5.Unmark – removes the mark on the strand append a tag seq. ssDNA 3‘3‘ 5‘ Methylation or (de)phosphorylation of the 5’ end. Carry out by specific enzymes, it can Stop some restriction enzymes cutting to the Site.

Molecular Computation - Solved problems Hamiltonian Path Problem 3.1Directed Hamiltonian Path Problem (DHPP) Adleman, Science, 266, 1024 (1994) Problem : Given 7 cities, is there a unique path every cities visit once ? 4 31 06 25 A possible solution : 0  1  2  3  4  5  6 Algorithm : 1.For each vertex V and edge E, create a 20-mer DNA strand V : x 0 y 0, x 1 y 1,……, x 6 y 6 and x 0 c,y 6 c E : y 0 c x 1 c, y 1 c x 2 c,…., y 5 c x 6 c, y 0 c x 3 c, y 0 c x 6 c,….. inout

Molecular Computation - Solved problems Hamiltonian Path Problem Algorithm (DHPP) 2.Hybridization – possibility of forming the following DNA strands Not x 0 begin, y 6 end ( 1  2  3  4  5  6 ) x 0 begin, not y 6 end ( 0  1  2  3  4  5 ) x 0 begin, y 6 end but not visit every cities once ( 0  3  2  3  4  5  6 )  consider to be the noise x 0 begin, y 6 end, and visit every cities once ( 0  1  2  3  4  5  6 )

Molecular Computation - Solved problems Hamiltonian Path Problem Algorithm (DHPP) 3.Separation Select those dsDNA start with x 0 and end with y 6 Amplify – use PCR to amplify the above type of DNA strands 4.Separate out all dsDNA that go through exactly 7 vertices (140-mer), by PAGE, for N<150, d = a – b ln N then amplify by PCR d N a

Molecular Computation - Solved problems Hamiltonian Path Problem Algorithm (DHPP) 5.Separate out all DNA strands go through all 7 cities – by affinity purification Melt the dsDNA strands from above Extract by affinity purification …………. 6.Detect if there are any DNA strands remain, Yes  solution of the DHPP No  No solution of the DHPP y 0 c x 1 c with attached bead y 5 c x 6 c with attached bead

Molecular Computation - Solved problems Hamiltonian Path Problem StepTime Create DNA strands Hybridization~30 sec PCR~ 2 hrs. Gel Electrophoresis~ 5 hrs. (AGE), ~ 1.2 hrs. (PAGE) Affinity Purification7 times ~ 1 hrs. Detect~ sec. Total~ 7 days !!

Molecular Computation - Solved problems Boolean formula Boolean Formula, B In particular, consider the Conjunctive Normal Form (CNF) B = C 1  C 2  C 3 …  C m where C k = x 1  x 2  x 3 ’… C is called the clause, x is called the literal,  is the logical AND,  is the logical OR and x’ is the negation of x  B = (x 1  x 2 )  ( x 1  x 2  x 3 ’ )  …  C m Satisfiability Problem ( B  True ) Determine a set of of the logical variables (x 1, x 2, x 3 …) such that B  T

Molecular Computation - Solved problems Boolean formula Algorithm (Boolean Formula) Example : B = (x 1  x 2 )  ( x 1 ’  x 2 ’ ) x 1 x 2 x 1  x 2 (x 1  x 2 )  (x 1 ’  x 2 ’) x 1 ’ x 2 ’ x 1 ’  x 2 ’ 0 0 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0

Molecular Computation - Solved problems Boolean formula 3.2Boolean Formula Lipton, Science, 268, 542 (1995) Encode an n bit binary number by a graph, G n x 1 x 2 x n a 1 a 2 a 3 …………a n a n+1 x 1 ’x 2 ’x n ’ Notation : x=1  True, x’=0  False, vertex a i, Edges E aixi, E aixi’, E xiai+1,E xi’ai+1 a 1 x 1 a 2 x 2 ’a 3 encode binary number 10 In general, graph G n represent {0,1} n X X

Molecular Computation - Solved problems Boolean formula Algorithm (Boolean Formula) 1.Create DNA strands to encode vertices and edges x 1 x 2 x n a 1 a 2 a 3 …………… a n a n+1 x 1 ’ x 2 ’ x n ’ Vertex (3n+1 strands)Edge (4n strands) a 1 E a1x1. a n+1 E a1x1’ x 1 E x1a2 x 1 ’ E x1’a2 p a1 q a1 5‘3‘ p an+1 q an+1 5‘3‘ q a1 c p x1 c 5‘3‘ q a1 c p x1’ c 5‘3‘ q x1 c p a2 c 5‘3‘ q x1’ c p a2 c 5‘3‘ p x1 q x1 5‘3‘ p x1’ q x1’ 5‘3‘

Molecular Computation - Solved problems Boolean formula Algorithm (Boolean Formula) 2.Hybridization A path V-E-V-E-V……..V  denote an n bit binary number Example : a path a 1 x 1 a 2 x 2 ’a 3 denote a 2 bit binary number, 10 p a1 q a1 5‘ p x1 q x1 p x2’ q x2’ p a2 q a2 p a3 q a3 q a1 c p x1 c q x1 c p a2 c q a2 c p x2’ c q x2’ c p a3 c

Molecular Computation - Solved problems Boolean formula Algorithm (Boolean Formula) 3.Extraction Define E(t,i,a) to represent extracting test tube t, where the ith position has Boolean value, a = 0 or 1. OR – are done by using multiple tubes AND – are done by repeated extraction

Molecular Computation - Solved problems Boolean formula Algorithm (Boolean Formula) 3.Extraction Test tubeOperationValue present t0t0 Create DNA strands00, 01, 10, 11 t1t1 E(t 0,1,1)10, 11 t1’t1’Remainder of t 1 00, 01 t2t2 E(t 1 ’,2,1)01 t3t3 Merge, t 1  t 2 10, 11, 01  T t 4 (need to remove 11)E(t 3,1,0)01 t4’t4’Remainder of t 4 10, 11 t5t5 E (t 4 ’,2,0)10 t6t6 Merge, t 4  t 5 01, 10

Molecular Computation - Solved problems Boolean formula Algorithm (Boolean CNF Formula) 1.Create DNA strands to encode all n bit binary number 2.Hybridization 3.Extraction Let t k be the test tube satisfies C 1  C 2  C 3 …  C k and let C k+1 = x a  x a+1  … x m (where x is either 0 or 1), for simplification consider C k+1 = x a  x a+1 E(t k,a,1)  x a =1  T 1a T 1a R  E(T 1a R,a+1,1)  x a+1 =1  T a+1 T 1a  T a+1 satisfies C k+1 E(t k,a,0)  x a =0  T 0a  E(T 0a,a+1,1)  x a+1 =1  T a+1 T 0a R T 0a R  T a+1 satisfies C k+1

Molecular Computation - Solved problems Integer knapsack problem Integer knapsack problem Given a set of integers a i and integer A, does there exist a subset S  {1,…n}, s.t.  i  S ≦ A. 1.To solve this problem, make use of the synthesis, annealing and merging operations. 2.Prepare a starter, S, one strand end is blunt and blocked by 5’- biotinylation and the other end is sticky. 3.Use DNA double strands to encode integers a 1 ….a n. With length proportional to the magnitude and both ends are sticky. B S a1a1 xCxC x xCxC

Molecular Computation - Solved problems Integer knapsack problem 4.Generation of all possible combinations, 2 n, by concatenation of the DNA strands. 5.The final DNA solution consisting of 2 n different DNA double strands; the final answer is to check if the solution containing strands with length equal to A by agarose gel electrophoresis.

Molecular Computation - Solved problems Integer knapsack problem Limitation : This brute-force algorithm has an exponential time- complexity, O(2 n ). The concept of encoding all possible solutions by DNA strands is suffered from the exponential growth in the size of the solution space, for instance, a 70 cities of the DHPP will fit in a milliliter of solution (10 20 DNA strands). Hence, people consider to develop a parallel computation model. Dynamic programming approach 1.Parallelism : parallel algorithm, because of the principle of optimality applied, hence, a DNA computer might be useful for solving large instances of problems. 2.For the integer knapsack problem : the worst-case time complexity is O(minimum(2 n, nA)) [Ref. 1].

Molecular Computation - Solved problems Integer knapsack problem Given a set of integers w 1,w 2 ……w n and W, with the corresponding profit integers p 1,p 2 ……p n, is it exist that a sub-set S  {1,2,….n}, that satisfy  i  S w i ≦ W max and maximize  i  S p i. Dynamic programming Solution Let f i (x) be the optimal solution to the integer knapsack problem, f i (x) = max { f i-1 (x), p n + f n-1 (x-w n ) } where x is the capacity remaining, and f i (x) = 0 for x>0 and f i (x) = ﹣ ∞ for x  0. Notice that f i (x) is an ascending function, i.e. 0=x 1  x 2  …..  x n ， f i (x 1 )  f i (x 2 )  …..  f i (x k ) ； f i (x) = ﹣ ∞ ， x  x 1 ； f i (x) = f i (x k ) ， x  x k ； and f i (x) = f i (x j ) ， x j  x  x j+1 。 To solve this problem, we make use of the method suggested by Horowitz etc. [Ref. 2] to compute f i (x j ) for 1  j  k. Let the ordered set S 1 i = { （ P,W ） | （ P - p i+1, W - p i+1 ）  S i } to represent f i (x), where P = f i (x j ) ， W = x j and S 0 ={ （ 0,0 ） }.

Molecular Computation - Solved problems Integer knapsack problem S i+1 can be computed from S i by first computing S 1 i = { （ P,W ） | （ P - p i+1, W - w i+1 ）  S i } where S i+1 = S i ∪ S 1 i. If S i contains （ P j,W j ） and （ P k,W k ） with P j  P k and W j  W k then the pair （ P j,W j ） can be discarded from S i, and this condition is known as the dominance rules. For example, consider the case n=3, (w 1, w 2, w 3 )=(2,3,4), (p 1, p 2, p 3 )=(1,2,5) and W max =6. For this case, we have S 0 ={(0,0)}; S 1 0 ={(1,2)} S 1 ={(0,0),(1,2)}; S 1 1 ={(2,3),(3,5)} S 2 ={(0,0),(1,2), (2,3),(3,5)}; S 1 2 ={(5,4),(6,6),(7,7),(8,9)} S 3 ={(0,0),(1,2), (2,3),(3,5),(5,4),(6,6),(7,7),(8,9)} The pair (3,5) is discarded because of the dominance rules.

Molecular Computation - Solved problems Integer knapsack problem Implementation of dynamic programming Consider the case n = 3, (w 1, w 2, w 3 ) = (2,3,4), (p 1, p 2, p 3 ) = (1,2,5) and W max = 6. DNA OperationTest Tubes, T P and T W S 0 = {(0,0)} CopyS 0 = {(0,0)} Addition : (p,w) = (1,2)S 0 1 = {(1,2)} Merge S 1 = S 0  S 0 1 = {(0,0), (1,2)} CopyS 1 = {(0,0), (1,2)} Addition: (p,w) = (2,3)S 1 1 = {(2,3), (3,5)} Merge S 2 = S 1  S 1 1 = {(0,0), (1,2), (2,3), (3,5)} CopyS 2 = {(0,0), (1,2), (2,3), (3,5)} Addition: (p,w) = (5,4)S 2 1 = {(5,4), (6,6), (7,7), (8,9)} Merge S 3 = S 2  S 2 1 = {(0,0), (1,2), (2,3), (3,5), (5,4), (6,6), (7,7), (8,9)}

Molecular Computation - Solved problems Integer knapsack problem Implementation of dynamic programming Difficulties 1.Do not know how to communicate between DNA strands. This operation is required in order to match P k and W k. 2.Do not know how to compare numbers between DNA strands. This operation is required in order to test the dominance rules.

Limitations and Errors 1.DNA synthesis ~ 90% efficiency 2.Long strands of DNA decay quickly, 10000 is the maximum base length can be kept in vitro without significant breakage. 3.Extraction A good path were lost during extract Take a bad path as if a good one 4.Undesirable hybridization 5.Seq. s could anneal with a similar seq. s c

Prospective 1.There appears little theoretical difficulty in creating a functional DNA computer. 2.Depend on finding killer applications uniquely suitable for computation by DNA. 3.Improvements in reducing errors and operation costs.

DNA computing Solving Optimization problems on a DNA computer Ka-Lok Ng Dept. of Bioinformatics Taichung Healthcare and Management University.

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DNA computing Solving Optimization problems on a DNA computer Ka-Lok Ng Dept. of Bioinformatics Taichung Healthcare and Management University.

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Presentation on theme: "DNA computing Solving Optimization problems on a DNA computer Ka-Lok Ng Dept. of Bioinformatics Taichung Healthcare and Management University."— Presentation transcript:

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