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DNA Computing
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).
Complementary parts (in
nature) can “self-assemble”.
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)
Adleman’s experiment
Dunedin
Christ Church
Auckland
Dune_din
Church_Welling
Welling_ton
land _Dune
Wellington
Auck_land
Christ_Church
Adleman’s experiment
Dune_din
Church_Welling
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 ChurchWellington
TGACCCGA
Christ ChurchDunedin
TGACGGCT
Christ ChurchAuckland
TGACTGAA
Wellington-Dunedin
CCGAGGCT
Christ_Church
land _Dune
Welling_ton
CITY
Auck_land
Adleman’s experiment
city
Dune_din
Church_Welling
Christ_Church
land _Dune
Welling_ton
Auck_land
Auckland ChristChurch Wellington Dunedin
landChrist ChurchWelling tonDune
8 x 3 = 24
flight
Adleman’s experiment: Filtering process I
Getting rid of DNA strands
that don’t start with Auckland, end with Dunedin (using PCR amplification)
Both the “types” can duplicate
simultaneously.
(Auck)land  Dune(din)
land
land
Dune
land
Dune
Dune
amplified
{ land, Dune}
primers
land
land
y
{ land, Dune}
land
not amplified
y
x
Dune
x
Dune
Dune
{ land, Dune}
not amplified
Adleman’s experiment: Filtering process II
Getting rid of DNA strands
that don’t have length = 24 (using gel electrophoresis)
DNA
Shorter DNA strands move faster.
Adleman’s experiment: Filtering process III
Getting rid of DNA strands
that don’t have Christ_Church & Welling_ton (using probe molecules)
DNA
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.
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 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!
Lipton’s SAT
y=1
x=1
(x V y) ^ (~x V ~y)
a1
a2
a3
y=0
x=0
1
1
1
Possible Paths:
Paths satisfying Clause-1:
0
1
1
0
1
0
0
1
1
Paths satisfying Clause-2:
0
0
0
0
1
0
0
1
Lipton’s SAT: Filtering process I
Getting rid of DNA strands (paths) that do not satisfy Clause-1
1
1
1
1
Possible Paths:
0
0
0
0
Paths of the form
0
0
get filtered off.
1
1
1
1
Paths satisfying Clause-1:
0
0
Lipton’s SAT: Filtering process II
Getting rid of DNA strands (paths) that do not satisfy Clause-2
1
1
1
1
Paths satisfying Clause-1:
0
0
Paths of the form
1
1
get filtered off.
1
1
Paths satisfying BOTH
Clause-1 and Clause-2 :
0
0
Lipton’s SAT: Filtering process I
1
1
1
T0 :
0
0
1
0
0
Extract “x=1”
1
1
1
1
T 1’ :
T1 :
0
0
0
0
Extract “y=1”
1
T2 :
0
+
1
1
1
1
Paths satisfying Clause 1:
T3 :
0
0
(x = 1) OR (y = 1)
Lipton’s SAT: Filtering process II
1
1
Paths satisfying Clause-1:
1
1
(x = 1) OR (y = 1)
T3 :
0
0
Extract “x=0”
1
1
1
1
T 4‘ :
T 4:
The working set for
filtering process II
0
0
Extract “y=0”
1
T 5:
Is anything left (in T6)?
0
+
1
1
Paths satisfying BOTH clauses 1
& 2:
T6 :
0
0
(x = 1) OR (y = 1) AND
(x = 0) OR (y = 0)
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 TwinShuffle language.
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.
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 x2 x3 x4
x 1 x2 x3 x4
DNA strand
x1 x1
x 2 x3
x2 x3 x4 x4
shuffle
x1 x1
x2 x2
shuffle
x1 x2 x3 x4
x1 x2 x3 x4
DNA strand
x3 x3 x4 x4
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.
Thank you!