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_

Dun

e

Dune_din

Land

_Chr

ist

Chu

rch_

Auc

k

Christ_Church

Church_DuneChurch_Welling

land

_D

une

Adleman’s experiment

Auckland

Wellington Christ Church

Dunedin

CITY DNA name Complement

Auckland ACTTGCAG TGAACGTC

Christ Church TCGGACTG AGCCTGAC

Wellington GGCTATGT CCGATACA

Dunedin CCGAGCAA GGCTCGTT

FLIGHT DNA flight number

Auckland-Christ Church

CGTCAGCC

Auckland-Dunedin CGTCGGCT

Christ Church-Wellington

TGACCCGA

Christ Church-Dunedin

TGACGGCT

Christ Church-Auckland

TGACTGAA

Wellington-Dunedin CCGAGGCT

Welling_ton

Auck_land

Ton_

Dun

e

Dune_din

Land

_Chr

ist

Chu

rch_

Auc

kChrist_Church

Church_DuneChurch_Welling

land

_D

une

Adleman’s experiment

Welling_ton

Auck_land

Ton_

Dun

eDune_din

Land

_Chr

ist

Chu

rch_

Auc

k

Christ_Church

Church_Dune

Church_Welling

lan

d _

Du

ne

Adleman’s experiment city flight

Auckland ChristChurch Wellington Dunedin landChrist ChurchWelling tonDune

8 x 3 = 24

Adleman’s experiment: Filtering process I

Getting rid of DNA strands Getting rid of DNA strands

that don’t start with that don’t start with AucklandAuckland, end with , end with Dunedin Dunedin (using (using PCR amplificationPCR amplification))

landland DuneDune

{ { landland,, Dune}Dune}landland DuneDune

landland

DuneDune

(Auck)land Dune(din) Both the “types” can duplicate simultaneously.

primers

amplified

landland yy

{ { landland,, Dune}Dune}landland yy

landland

not amplified

xx DuneDune { { landland,, Dune}Dune}

xx DuneDuneDuneDune

not amplified

Adleman’s experiment: Filtering process IIGetting rid of DNA strands Getting rid of DNA strands

that don’t have length = 24 (using that don’t have length = 24 (using gel electrophoresisgel electrophoresis))

Shorter DNA strands move faster.

DNA

Adleman’s experiment: Filtering process IIIGetting rid of DNA strandsGetting rid of DNA strands

that don’t havethat 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 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!

What is a natural algorithm? (poem version)

Lipton’s SAT

(x V y) ^ (~x V ~y)

Paths satisfying Clause-1:1 1

0

11

0

Paths satisfying Clause-2:

0

11

00 0

0 0

1 1

0

1 1

0Possible Paths:

a1 a2 a3

x=1 y=1

x=0 y=0

Lipton’s SAT: Filtering process I

Paths satisfying Clause-1:

1 1

0

11

0

0

1

0 0

1 11

0

Possible Paths:

0 0

Paths of the form

get filtered off.

Getting rid of DNA strands (paths) that do not satisfy Clause-1Getting rid of DNA strands (paths) that do not satisfy Clause-1

Lipton’s SAT: Filtering process II

Paths satisfying Clause-1:

1 1

0

11

0

Paths of the form

get filtered off.

1 1

Paths satisfying BOTH Clause-1 and Clause-2 : 0

11

0

Getting rid of DNA strands (paths) that do not satisfy Clause-2Getting rid of DNA strands (paths) that do not satisfy Clause-2

Lipton’s SAT: Filtering process I

0 0

1 1

0

1 1

0

T0 :

Extract “x=1”

1 11

0

T1 : T1’ :0 0

0

1

Extract “y=1”

T2 :0

1

++

1 1 1

0 0

1

T3 :Paths satisfying Clause 1:

(x = 1) OR (y = 1)

Lipton’s SAT: Filtering process II

1 1 1

0 0

1

T3 :

Paths satisfying Clause-1:

(x = 1) OR (y = 1)

Extract “x=0”

0

1

T4:1 1 1

0

T4‘ :

Extract “y=0”

T5:1

0

0

1 1

0

T6 :

++

Paths satisfying BOTH clauses 1 & 2:

(x = 1) OR (y = 1) AND

(x = 0) OR (y = 0)

Is anything left (in T6)?

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; 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.

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.

x1 x1 x2 x2 x3 x3 x4 x4x1 x2 x3 x4

x1 x2 x3 x4

DNA strand shuffle

x1 x1 x2 x3 x2 x3 x4 x4

shuffle

x1 x2 x3 x4

x1 x2 x3 x4

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.