DNA Computing

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

Cells: “atoms” that make up living things

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

Complementary-attraction in DNA 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).

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.

Hamiltonian Path Problem (HPP)

Welling_ton Auck_land Ton_Dune Dune_din Land_Christ Church_Auck Christ_Church Church_Dune Church_Welling land _Dune Adleman’s experiment Auckland WellingtonChrist Church Dunedin

CITYDNA name Complement AucklandACTTGCAGTGAACGTC Christ ChurchTCGGACTGAGCCTGAC WellingtonGGCTATGTCCGATACA DunedinCCGAGCAAGGCTCGTT FLIGHTDNA flight number Auckland-Christ Church CGTCAGCC Auckland-DunedinCGTCGGCT Christ Church- Wellington TGACCCGA Christ Church- Dunedin TGACGGCT Christ Church- Auckland TGACTGAA Wellington-DunedinCCGAGGCT Welling_ton Auck_land Ton_Dune Dune_din Land_Christ Church_Auck Christ_Church Church_Dune Church_Welling land _Dune Adleman’s experiment

Welling_ton Auck_land Ton_Dune Dune_din Land_Christ Church_Auck Christ_Church Church_Dune Church_Welling land _Dune Adleman’s experiment cityflight Auckland ChristChurch Wellington Dunedin landChrist ChurchWelling tonDune 8 x 3 = 24

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

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

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

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. Natural Algorithm : a “free” means to an end What is a natural algorithm? (prose 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 seek For when you seek, you shall find That piece of nature's machinery which does what you want Be 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, and For 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! What is a natural algorithm? (poem version)

Lipton’s SAT (x V y) ^ (~x V ~y) Paths satisfying Clause-1: Paths satisfying Clause-2: Possible Paths: a1a1 a2a2 a3a3 x=1 y=1 x=0 y=0

Lipton’s SAT: Filtering process I Paths satisfying Clause-1: Possible Paths: 00 Paths of the form get filtered off. Getting rid of DNA strands (paths) that do not satisfy Clause-1

Lipton’s SAT: Filtering process II Paths satisfying Clause-1: Paths of the form get filtered off. 11 Paths satisfying BOTH Clause-1 and Clause-2 : Getting rid of DNA strands (paths) that do not satisfy Clause-2

Lipton’s SAT: Filtering process I T 0 : Extract “x=1” T 1 : T1’ :T1’ : Extract “y=1” T2 :T2 : T 3 : Paths satisfying Clause 1: (x = 1) OR (y = 1)

Lipton’s SAT: Filtering process II T 3 : Paths satisfying Clause-1: (x = 1) OR (y = 1) Extract “x=0” 0 1 T4:T4: T4‘ :T4‘ : Extract “y=0” T5:T5: T 6 : + Paths satisfying BOTH clauses 1 & 2: (x = 1) OR (y = 1) AND (x = 0) OR (y = 0) Is anything left (in T 6 )? The working set for filtering process II

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; is a shuffle of x and y. But, 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.

Universality of DNA computing (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. 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.

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. x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 x 1 x 2 x 3 x 4 DNA strandshuffle x 1 x 1 x 2 x 3 x 2 x 3 x 4 x 4 shuffle x 1 x 2 x 3 x 4 DNA strand

Thank you! References: 1.E. Schrödinger, What is Life: The Physical Aspect of the Living Cell (1944), Cambridge University press. 2.The Living Cell, Readings from Scientific American, W. H. Freeman and Company, 1965.