DNA Computing

1. DNA Computing

2. Complementary parts (in nature) can “self-assemble”. A universal principle? • Elements of complementary nature abound in nature. • Elements of complementary nature spontaneously “stick” together. • This “complementary-attraction-principle” seems to pervade many aspects of life (both molecular and higher levels).

3. Cells: “atoms” that make up living things

4. DNA: “strings” that encode the traits of living organisms

5. DNA bases and their “complements”: Adenine (A) ----- Thymine (T) Guanine (G) ----- Cytosine (C) If DNA molecules (in a single strand) meet their complements (in another single strand), then the two strands will anneal (stick/coil together to form a double-helix). Complementary-attraction in DNA

6. DNA computing: Basic operations • Synthesize • Prepare large numbers of copies of any short single DNA strand. • Anneal • Create a double strand from complementary single strands. • Extract • “Pull out” those DNA sequences containing a given pattern of length l (from a test tube). • Detect • Determine whether or not there are any DNA strands at all in a test tube. • Amplify • Replicate all (or selectively, some) of the DNA strands in a test tube.

7. Hamiltonian Path Problem (HPP)

8. Dunedin Wellington Christ Church Dune_din Ton_Dune Church_Dune Auckland Church_Welling Welling_ton Christ_Church land_Dune Land_Christ Church_Auck Auck_land Adleman’s experiment

9. Dune_din Ton_Dune Church_Dune Church_Welling Welling_ton Christ_Church land_Dune Land_Christ Church_Auck Auck_land Adleman’s experiment

10. city flight Dune_din Ton_Dune Church_Dune Church_Welling Welling_ton Christ_Church Auckland ChristChurch Wellington Dunedin landChrist ChurchWelling tonDune land_Dune Land_Christ Church_Auck 8 x 3 = 24 Auck_land Adleman’s experiment

11. Both the “types” can duplicatesimultaneously. (Auck)land Dune(din) land land Dune Dune amplified land Dune { land,Dune} primers land land y { land,Dune} not amplified land y not amplified x Dune { land,Dune} x Dune Dune Adleman’s experiment: Filtering process I Getting rid of DNA strands that don’t start with Auckland, end with Dunedin (using PCR amplification)

12. DNA Shorter DNA strands move faster. Adleman’s experiment: Filtering process II Getting rid of DNA strands that don’t have length = 24 (using gel electrophoresis)

13. DNA Adleman’s experiment: Filtering process III Getting rid of DNA strands that don’t haveChrist_Church & Welling_ton(using probe molecules)

14. What is a natural algorithm? (prose version) Natural Algorithm: a “free”means to an end Traditionally, when computists solve problems, they try to achieve the desired end by painstakingly developing a suitable means---an “algorithm”. On the other hand, when natural computists solve problems, they try to discover a natural (computing) system, one that is bound to produce the desired end (or something “close” to such an end) and whose capacity to produce such an end is innate. (That is, the system’s ability to reach the desired end is not something the computist deliberately assigns to it, but something which the system has been endowed with.) The means by which natural systems realize an end is something that comes “for free”; the computist need not bother to know the exact means by which the system would achieve the desired end, but simply be aware of the fact that such an end will somehow be achieved.

15. What is a natural algorithm? (poem version) "What, my dear Sir, is a Natural Algorithm?"So asked Boswell."Bah, that is but a simple idea", said Dr. Johnson.An algorithm is nothing but a means,Not as hard as it seems;One which humans so meticulously design---And all that, my friend,Is for the computer---to achieve an 'end'. A natural algorithm is also a means,But one that you get “for free”:All you need, my dear Boswell, is to seekFor when you seek, you shall findThat piece of nature's machinery which does what you wantBe it sorting, be it searching or solving SAT!It's right there, neat and clean---The end you seek;Just take a peek. "But, Sir, by what means does nature reach its end?"Why bother, my dear Boswell,When nature does it well.The means is but free, andFor us (and for nature), it's the end that matters.What matters for starters,Though, is by one means or the other Will it reach its end!

16. 1 1 1 1 0 0 0 0 y=1 x=1 1 1 1 0 0 0 a1 a2 a3 1 1 1 y=0 x=0 0 0 0 Lipton’s SAT (x V y)^ (~x V ~y) Possible Paths: Paths satisfying Clause-1: Paths satisfying Clause-2:

17. 1 1 1 1 0 0 0 0 Possible Paths: 1 1 0 0 Paths of the form 1 1 get filtered off. 0 0 Paths satisfying Clause-1: Lipton’s SAT: Filtering process I Getting rid of DNA strands (paths) that do not satisfy Clause-1

18. 1 1 1 1 Paths satisfying Clause-1: 1 1 0 0 Paths of the form 1 1 get filtered off. 0 0 Paths satisfying BOTH Clause-1 and Clause-2 : Lipton’s SAT: Filtering process II Getting rid of DNA strands (paths) that do not satisfy Clause-2

19. 1 1 1 1 1 1 0 0 0 0 T0 : 1 1 1 1 Extract “x=1” 0 0 0 0 T1 : T1’: 1 1 1 Extract “y=1” 0 0 0 T2: + T3 : Lipton’s SAT: Filtering process I Paths satisfying Clause 1: (x = 1) OR (y = 1)

20. 1 1 1 1 Paths satisfying Clause-1: (x = 1) OR (y = 1) T3 : 1 1 1 Extract “x=0” 0 0 0 T4‘: T4: The working set for filtering process II 1 1 1 1 Extract “y=0” 0 0 0 0 T5: Is anything left (in T6)? + T6 : Lipton’s SAT: Filtering process II Paths satisfying BOTH clauses 1 & 2: (x = 1) OR (y = 1) AND (x = 0) OR (y = 0)

21. Universality of DNA computing What does a shuffle mean? Take any two strings x and y;we can form strings by just “cutting and pasting” pieces (substrings) from them in such a way that the resulting strings will preserve the order of letters in x and y. Call such a resulting string a shuffle of x and y. e.g. Take x = 0011 and y = 0011; 00001111 is a shuffle of x and y. But, 01100011 is not a shuffle of x and y. Twin-Shuffle language Pick x,an arbitrary string over the alphabet {0,1} and y,its underscored-version. Form the shuffles of ALL such x and y. The resulting (infinite) set of strings is the Twin-Shuffle language.

22. Universality of DNA computing DNA-strings = Twin-Shuffle language Every DNA double strand can be represented by a unique, valid shuffle, i.e. a string in TS. Also, for every string in TS, one can construct a (unique) double stranded DNA that mirrors such a string. In other words, the double-stranded DNA strings and the strings in TS can be put in one-to-one correspondence. (DNA) Universality Theorem For every computably enumerable language L, we can design a finite state machine (with outputs) that can generate exactly those strings in L when inputted with strings from TS.

23. x1 x2 x3 x4 x1 x2 x3 x4 Universality of DNA computing DNA-strings  Twin-Shuffle language Every DNA double strand can be represented by a unique, valid shuffle, i.e. a string in TS. Also, for every string in TS, one can construct a (unique) double stranded DNA that mirrors such a string. x1 x1x2 x2x3 x3 x4 x4 DNA strand shuffle x1 x2 x3 x4 x1 x2 x3 x4 x1 x1x2 x3x2 x3 x4 x4 shuffle DNA strand

24. References: • E. Schrödinger, What is Life: The Physical Aspect of the Living Cell (1944), Cambridge University press. • The Living Cell, Readings from Scientific American, W. H. Freeman and Company, 1965. Thank you!