A Sticker Based Model for DNA Computation
A Sticker Bas e d Mo del for DNA Computation
Sam Roweis Er ik Winf ree
Richard Burgoyne Nickolas V Chelyap ov
Myron F Go o dman Paul W K Rothemund
y
Leonard M Adleman
Laboratory for Molecular Science Univers ityofSouther n Califor nia
and
Computation and Neural Systems Option Califor nia InstituteofTechnology
Departm entofBiome dical Engineer ing Univers ity of Souther n Califor nia
Department of Computer Science Univers ity of Souther n Califor nia
Department of Biological Science s Univers ity of Souther n Califor nia
May
Ab stract
Weintro duce a new mo del of molecular computation thatwe call the sticker model Likemany
previous prop o sals it make s us e of DNA strands as thephys ical sub strateinwhichinformation i s
repre s ented and of s eparation byhybr idization as a central mechani sm However unlike previous
estickers mo del has a random acce s s memory that require s no strandextens ion us e s mo dels th
no enzymes and at le ast in theory itsmater ials are reusable
The pap er de scr ib e s computation under thestickers mo del and di scus s e s p o s s ible me ans for
phys ically implementing each op eration We go on to propose a sp eci c machine architecture
for implementingthestickers mo del as a micropro ce s sor controlle d parallel rob otic workstation
Finally we di scus s s everal m etho ds for achieving acceptable overall error rate s for a computation
us ing bas ic op erations that are error prone
In the cours e of thi s development a number of previous general concer ns about molecular
computation Smith Hartmani s Letters to Science are addre s s e d First it i s cle ar thatgeneral
purp o s e algor ithms can be implemented by DNA bas e d computers potentially solving a wide
clas s of s e arch problems Second we nd that there are challengingproblems for which only
mo dest volume s of DNA should su ce Third wedemonstratethatthe formation and bre aking
of covalentbonds i s not intr ins ic to DNA bas e d computation Thi s means thatcostly andshort
lived mater ials such as enzyme s are not nece s sary nor are energetically co stly processes suchas
PCR Fourth weshowthatasingle essential biotechnology s equence sp eci c s eparation su ce s
h we illustratethatseparation errors for constructing a general purp o s e molecular computer Fift
can theoretically b e re duce d totolerable levels byinvoking a trade o b etween time space and
error rates atthe level of algor ithm de s ign wealsooutlineseveral sp eci c ways in whichthi s
can b e doneandpresentnumer ical calculations of the ir p erformance
De spite these encouragingtheoretical advance s we emphas ize that substantial engineer ing
challenge s remain at almo st all stage s andtha ttheultimatesucce s s or f ailure of DNA computing
will certainly dep endonwhether these challenge s can b e met in laboratory investigations
Repr int reque ststo roweis cns caltech edu TheMATLAB co de whichwas us e d togenerate all of the gure s in
Section of thi s pap er i s also available by reque st f rom roweis cns caltech edu Roweis is supporte d in part bythe
Center for Neuromorphic Systems Engineer ing as a part of theNational Science Foundation Engineer ingResearch
Center Program under grant EEC andbytheNatural Science s andEngineer ing Re s e arch Council of Canada
tal He alth NIMH TrainingGrant T MH Winf ree i s supp orte d in part byNational Institute for Men
also byGeneral Motors Technology Re s e archPartnership s program Adleman Chelyap ov andRothemundare
supporte d in part by grants f rom theNational Science Foundation CCR and Sloan Foundation
y
Towhom corre sp ondence should b e addre s s e d
Intro duction
Much of the recent intere st in molecular computation has been fuele d by the hope that it might
someday providetheme ans for constructinga massively parallel computational platform capable of
attacking problems whichhave b een re s i stanttosolution withconventional architecture s Mo del ar
chitecture s ha ve b een prop o s e d which sugge st that DNA bas e d computers may b e exible enough to
tackle a widerange of problems Adleman Adleman Amo s Lipton Boneh Be aver Rothemund
although fundamental issue s such as the volumetr ic scale of mater ials and delity of var ious labo
ratory pro ce dure s remain largely unanswere d
In thi s pap er we intro duce a new mo del of molecular comput ation that we call the sticker model
Like many previous proposals it makes us e of DNA strands as the phys ical sub strate in which
information is repre s ented and of s eparation by hybr idization as a central mechani sm However
unlike previous mo dels the stickers mo del has a ran dom acce s s memory that require s no strand
extens ion us e s no enzymes and at le ast in theory itsmater ials are reusable
The pap er b egins by intro ducing a new way of repre s enting information in DNA followed by an
ab stract de scr iption of the bas ic operations p ossible under thi s repre s entation Possible me ans for
phys ically implementing each o peration are di scus s e d We go on to propose a sp eci c machine
architecture for implementing the stickers mo del as a micropro ce s sor controlle d parallel rob otic
workstation employingonlytechnologie s which exi st today Finally we di scus s metho ds for achiev
ing acceptable error rate s f rom imp erfect s eparation units
The Stickers Mo del
Repre s entation of Information
Thestickers mo del employs two bas ic group s of s ingle strande d DNA molecule s in its repre s entation
of a bit str ing Cons ider a memory strand N bas e s in length subdivided into K non overlapping
regions e ach M bas e s long thus N MK Each region i s identi e d with exactly one bit p o s ition
computation We also de s ign K or equivalently one boole an var iable dur ing the cours e of the
di erent sticker strands or simply stickers Each sticker is M bas e s longand i s complementary to
oneandonlyoneofthe K memory regions Ifasticker i s anne ale d toitsmatching region on a given
memory strandthen the bit corre sp ondingthat particular region i s on for that strand If no sticker
is anneale d to a region then that region s bit i s o Figure illustrates thi s repre s en tation scheme
bit ... bit i bit i+1 bit i+2 bit ... (up to bit K) 5’
A TCG G T CATAG C A CT 3’ T T A G T A Memory G T M bases 0 0 0 A A G A Strands C G T G C G T A G C A G C C CTG G A G T A C T T A CC T Stickers 5’ G T A A TCG G T CATAG C A CT 3’ A
1 0 1
Figure Amemory strandandassociated stickers together calle d a memory complex repre s enta
bit str ing Thetopcomplex on the left has all three bits o thebottom complex has two anneale d
stickers andthus twobits on
Eachmemory strand along with its anne ale d stickers if any repre s entsone bit str ing Such partial
duplexe s are calle d memory complexes A large s et of bit str ings i s repre s ented by a large number
of identical memory strands e ach of whichhas stickers anne ale d only atthe require d bit p o s itions
We call such a collection of memory complexe s a tube Thi s di ers f rom previous repre s entations
of information us ing DNA in whichthe pre s ence or ab s ence of a particular sub s equence in a strand
corre sp onded t o aparticular bit b e ingonoro e g s ee Adleman Lipton In thi s new mo del
each p ossible bit str ing is repre s ented by a unique association of memory strands and stickers
where as previously e ach bit str ingwas repre s ented bya unique molecule
To give a feel for the numb ers involved a re asonable size problem for example bre aking DES as
di scus s e d in Adleman mightusememory strands of roughly bas e s N which repre s ent
binary var iable s K using bas e regions M
The information dens ityinthi s storage schemeis M bits bas e directly comparable tothedens ity
of previous scheme s Adleman Boneh Lipton Weremarkthat while information storage in DNA
has atheoretical maximumvalue of bits bas e exploitingsuchhighvalue s in a s eparation bas e d
molecular computer would require the ability to reliably separate strands us ing only single bas e
mi smatches Instead we choose to sacr i ce information dens ity in order to make the exp er imental
di cultie s le s s s evere
Op erations on Sets of Str ings
We now intro duce several p ossible operations on sets of bit str ings which together tur n out to be
quite exible for implementing general algor ithms The four pr inciple operations are combination of
twosetsofstrings intoonenew s et separation of one s et of str ings intotwonew s etsand setting or
th
clearing the k bit of every str ing in a s et Eachofthe s e logical s et operations has a corre sp onding
interpretation in terms of the DNA repre s entation intro duce d above Figure summar ize s these
require d DNA interactions
The mo st bas i c operation is to combine two sets of bit str ings into one Thi s pro duce s a
new set containing the multi s et union of all the str ings in the two input sets In DNA
thi s corre sp onds to pro ducing a new tube containing all the memory complexe s with their
ckers undi sturb e d f rom b oth inputtube s anne ale d sti
A set of str ings may be separated into two new sets one containing all the or iginal str ings
havingaparticular bit on andtheother all thos e withthe bit o Thi s corre sp onds toisolating
from the s et s tub e exactly tho s e complexe s witha sticker anne ale d tothe given bit s region
be is destroyed Theoriginal inputset tu
To set tur n on a particularbitinevery str ingofa set thesticker for that bit i s anneale d to
theappropr iate region on every complex in the s et s tube orleftinplace if alre ady anne ale d
Finally to clear tur n o a bit in every str ingofa set thesticker for that bit must b e removed
if pre s ent f rom every memory complex in the s et s tube
Computations in thi s mo del cons i st of a s equence of combination s eparation and bit s etting cle ar ing
ust ultimately pro duce operations Thi s s equence must b egin withsomeinitial s et of bit str ings andm
one p ossibly null s et of str ings deemed tobe the answers Wecallthetub e containingthe initial
s et of bit str ings the mother tube for a computation Thus tocomplete our theoretical de scr iption of T A G C C A G TAT CTG G A Combine T A G C C A G TAT CTG G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT A G TAT A G TAT
A TCG G T CATAG C A CT A TCG G T CATAG C A CT T A G C C
A TCG G T CATAG C A CT T A G C C A G TAT
G A TCG G T CATA C A CT A TCG G T CATAG C A CT A G TAT
A TCG G T CATAG C A CT
T A G C C A G TAT CTG G A Separate on Bit 1 T A G C C A G TAT CTG G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT A G TAT CTG G A T A G C C A TCG G T CATAG C A CT T CATAG A T A G C C A TCG G C CT
A TCG G T CATAG C A CT A G TAT A G TAT CTG G A A TCG G T CATAG C A CT A TCG G T CATAG C A CT
A G TAT
A TCG G T CATAG C A CT
T A G C C A G TAT CTG G A Set Bit 3 T A G C C A G TAT CTG G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT CTG G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT CTG T A G C C T A G C C G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT A G TAT A G TAT CTG G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT
T A G C C A G TAT CTG G A Clear Bit 1 A G TAT CTG G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT A G TAT A G TAT
A TCG G T CATAG C A CT A TCG G T CATAG C A CT T A G C C
A TCG G T CATAG C A CT A TCG G T CATAG C A CT T A G C C CTG G A CTG G A
A TCG G T CATAG C A CT A TCG G T CATAG C A CT
Figure DNA manipulations require d for thefouroperations of thestickers mo del
howtocompute withthestickers mo del wemust describe howtocreateamother tub e of memory
complexe s and also how to re ad out at le ast one bit str ing f rom a p ossibly empty nal tube of
answers or recognize that the tube contains no strands We cons ider cre ation of themother tube
rst
It will su ce for our purp o s e s to cons ider cre ating a mother tube which corre sp onds to the
K L library set of str ings A K L library set contains str ings of length K generated by
takingthe s et of all p ossible bit str ings of length L followed by K L zero s There are thus
L
length K str ings in theset
Our paradigm of computation will generally b e tocasthard problems as large combinat or ial s e arches
L
over inputs of length L We s e archforthe few rare answer str ings by pro ce s s ingall p ossible
inputsinparalleland eliminatingthos e thatfailthe s e archcriter ia It i s imp ortantthatthememory
strandwedesign may havemorethan L bit regions The rstL bits repre s entthe enco dingofthe
input and are the random portion of the initial library The remainin g K L bits are us e d for
intermediate storage and answer enco ding and are initially o on all complexe s All bits can be
wr itten to and re ad f rom later in the computation as nee ded In thi s way cre ating a mother tube
g whichisa K L library set corre sp onds togeneratingallpossible inputs of length L and zeroin
theworkspace length K L
Lastly weindicatehowtoobtain a solution attheendofthecomputation
Toreadastring f rom the nal answer s et onememory complex must b e i solate d f rom the
answer tube and its anneale d stickers if any determined Alter nately it must be rep orted
that the answer tub e contains no strands
Example Problem
To illustrate the power of the operations de ned above we work through the solution of the NP
Complete Minimal Set Cover problem Garey within the stickers mo del Informally as sume we
are given acollection of B bags each containing some ob jects Th e ob jects come in A typ e s The
problem i s to ndthe smalle st subs et of the bags which between them contain at le ast one ob ject
of every typ e Formally the problem i s as follows Given a col lection C fC C g of subsets
B
S
C f Ag The of f Ag what is the smal lest subset I of f Bg such that
i
i I
solution of the problem in our mo del i s straightforward Wecreatememory complexe s repre s enting
B
all possible choice s of bags We mark all thos e which include bag i as containing every typ e
app e ar ing in the subs et C Then we s eparate out thos e complexe s which have b een marked as
i
containing all A t yp e s and re ad out the one s which us e s the fewest bags Formally the sticker
algor ithm for minimal s et cover is
Design a memory strand with K B A bit regions Bits B repre s entwhich bags are
Initialize a K B library set in a tube called T
chosen bits B B A which
ob ject typ e s are pre s ent
for i to B
Separate T into T and T based on bit i Mark the nal A positions of each
on
of f
complex to record which ob ject
for j to jC j
i
typ e s it contains
Set bit N C j in T
i on
Combine T and T into T
on
of f
For example the library set is theset f g
Technically the NP Completevers ion of thi s problemisthebinary deci s ion vers ion in whichwe ask if there exi sts
acollection of a particular size thatcovers the s et not for thecollection of thesmalle st s ize
for i B to B A Get rid of ones which do not have
Separate T into T and T based on bit i all A typ e s
bad
Discard T
bad
for i to B Count how many bags
for j i down to
were us e d Attheend
Separate T into T and T based on bit i
j j
j
of the outer lo op tube
Combine T and T into T
j j
j
T contains all com
i
Read T
plexe s which us e d ex
else if it was empty then Read T
actly i bags
else if it was empty then Read T
th
where above jC j is the number of items in subs et C and C j is the j item in subs et C Note
i i i i
t hattheabove algor ithm takes O AB step s andtheinputis O AB bits
We pointoutthat as wewillenvi s ion a rob otic system p erformingthe exp er imentsautomatically
we allow arbitrary s equential algor ithms for controlling the molecular operations However these
operations must b e p erformed blind the only in terf ace to molecular paralleli sm i s via initialize
combine separate set clear and read Thus the electronic algor ithms are re sp ons ible for exp er i
mentde s ign i e compiling higher level problem sp eci cations into conci s e s equence s of molecular
operations butthey cannot get any fee dbackfromthe DNA dur ingthe cours e of the exp er iment
As a nal comment we note that the stickers mo del is capable of simulating in parallel indep en
dent universal machines one per memory complex under the usual theoretical as sumption of an
unbounded number of sticker regions It should be noted that the stickers mo del is universal in
the s ens e di scus s e d even in theab s ence of the clear operation although more compact algor ithms
le us ing clear are p ossib
Phys ical Implementation of the Mo del
Eachlogical op eration in our mo del has a corre sp ondinginterpretation whichwegaveasweintro
duce d the op erations in terms of what must happen to the DNA memory strands and as so ciated
stickers when thatoperation i s carr ie d out In whatfollows we examinev ar ious phys ical pro ce dure s
which are candidates for implementing these requirements for all the operations described above
We sp e ak in terms of tubes instead of sets recall that a tube cons i sts of the collection of memory
complexe s that repre s entsa set of bit str ings
Often there are several p ossible implementations of a given operation each has its own as sumed
strengths andwe aknesses on whichwespeculate However which implementations if any tur n out
tobeviable will ultimately have tobedecided bylab oratory exp er iments
Combination
Combination of two tube s can be p erformed by rehydrating the t ube contents if not alre ady in
solution and then combining the uids together by p our ing or pumping for example to form a
Thi s can b e s een as the cons equence of two observations First a memory complexinthestickers mo del can
simulate a fee dforward circuit in the spir it of Boneh Us ingthe clear operation aclocke d fee dback circuit can
also be simulated Second allowingthe circuit to growwitheachclock cycle we can s imulatea universal machine
The electronic algor ithm i s re sp ons ible for de s igningthenew gates to tintothe circuit e achnew gate will require
d For concretene s s a fee dforward circuit C can b e anew bit andhence a new sticker region in thememory stran
t
automatically de s igne d which computes the instantaneous de scr iption of a TM attimestep t f rom thede scr iption at
L
t Thus thestickers mo del can s imulate in parallel the execution of a TM on all length L inputs
new tube It should be noted that even thi s s eemingly straightforward operation is plague d by
constraints if DNA i s not handle d gently theshe ar force s f rom p our ingandmixing it will f ragment
it into kilobas e s ections Kor nb erg
Also of concer n for thi s operation and indee d for all others is the amount of DNA which remains
stucktothewalls of tube s pump s pip et tetip s etc andthus i s lo st f rom thecomputation Even
if thi s lo st DNA is a minute f raction of the total which would be unimp ortant to molecular
biologi sts it is problematic for computation becaus e we are working with relatively few copie s of
eachrelevantmolecule
Separation
egoalofthe s eparation operation i s tophys ically i solatetho s e complexesinatube that Theultimat
have a sticker anneale d to some position f rom thos e that do not without di sturbingany anneale d
stickers Themechani sm of DNA hybr idization will b e central toanyproposal In general s epara
tion byhybr idization is is performed bybringingthesolution containingthe or iginal s et of memory
complexe s into contact with many identical single stranded probes In our cas e each bit p osition
has a particular typ e of prob e with a unique nucleotide s equence that is us e d when s eparation
on that bit is performed The prob e s equence is de s igned such that prob e s hybr idize only to the
region of the memory strand correspondingto their bit and nowhere els e Dur ing s eparation the
or iginal complexe s withthe key bit o will b e capture d on the prob e s while all thos e withthe bit
on will remain unboundinsolution b ecaus e the region i s covere d by a sticker Next the unbound
on complexe s are phys ically i solate d for example by conjugatingthe prob e s tomagnetic b e ads
or a xingthe prob e s to solid supp ort and then washing Lastly the o memory complexe s are
recovere d f rom the prob e s thatboundthem byelution say byheatingandwashing The re sultis
twonew tube s one containingthememory complexe s for e achoftheoutputsetsoftheoperation
Notice that if heating is us e d to achieve the nal step of elution thi s must be done without also
removing all of the stickers from the memory strands Thi s nece ssitates that the prob e s have a
lower binding a nityforthe ir corre sp onding regions than do thestickers Thi s mightbeachieved
othe ir regions on thememory strands bymakingthe prob e s equence s not exactly complementary t
or merely shorter to cre ate a di erential between the temp erature of prob e strand and sticker
strand di s so ciation An alter nativeisto us e p erfectly complementary s equence s for b oththe prob e s
andstickers butto makethestickers outof an alter nate backb onemater ial such as PNA or DNG
Egholm Demp cy whichwould exhibit stronger and more sp eci c bindingto theDNA memory
strand than DNA prob e s PNA and DNG o er the additional advantage that decre as ing salt
concentration caus e s PNA DNA andDNG DNA tobind more strongly while the opposite i s true
for DNA DNA binding Thus the nal elution step might be achieved by washing in a zero salt
solution rat her than byheating There are other p o s s ibilitie s for cre ating di erential a nitybetween
thestickers andprobes
Setting and Cle ar ing
To set a bit in every str ingofasetthemostobvious choice i s direct anne aling An exce s s amount
of thesticker corre sp ondingtothe bit i s added tothetub e containingthe s et s memory complexe s
PNA clamp s Egholm havebeenshown to form PNA DNA tr iplexe s with remarkable a nityand sp eci city
The s e clamp s could also b e us e d as stickers
For example cro s slinkingtechnique s mightbeusedtocovalently b ondthestickers tothememory strands so that
they could not comeo dur ing elution although thi s confounds the clear operation and do e s not keep withthe reusable
spir it of themodel
Onesticker should anneal toevery complex that do e s not alre ady have one always in the p osition
oppositethe region corre sp ondingtothe bit b e ing s et Sub s equently theexcess unus e d stickers are
remove d p erhap s by ltration or by separating out all thememory complexe s Thi s latter proposal
could b e achieved byhavingauniversal region on every memory strand sayatthevery b eginning
or end that is never covere d by a sticker and de s igning a prob e for that region as described in
the s eparation operation above Sucha universal region i s a generally us eful ide a for recover ingall
memory complexe s f rom a given solution whichmay contain other sp ecie s
To clear a bit in every str ing of a set require s removing the stickers for only that bit f rom every
complex in a tube Simple heating will obviously not work since al l stickers from al l bit regions
will come o One p ossibility is to designate certain bit regions as weak regions These regions
have weak stickers which disso ciate more e as ily f rom thememory strandthan regular stickers By
heating to some interm ediate temp erature al l the weak stickers can be made to disso ciate at once
keeping all of the regular stickers in place
In order to implement the clear operation in full generality it may be p ossible to us e the phe
nomenon of PNA strandinvas ion bytriple helix formation Niels en It has b een shown thatunder
appropr iatecondit ions two single stranded oligo s of all pyr imidine PNA will invade an exi sting
complementary DNA DNA duplex toforma PNA DNA tr iple helix di splacingthe pyr imidine
DNA strand Thi s pro ce s s i s mo st e cient with PNA clamp s Egholm which contain boththe
Watson Cr ick and Ho ogsteen PNA strands in a single molecule We sugge st that if for example
nucleotide DNA st ickers are us e d then a bas e PNA clamp could be de s igned which forms
a tr iple helix with the central nucleotides of the DNA sticker By mixing PNA clamp s sp eci c
to a particular bit with a tube of memory complexe s and heating the PNA clamp s should form
ker destabilizingandthus prying it o atatemp erature lower tr iple helice s withthetargeted stic
than the di s so ciation temp erature for the una ected stickers The sp eci cityand reliabilityof thi s
operation are not yet known exp er imentally indee d the mechani sm of tr iplex formation Demidov
may be incompatible with the requirement that non targeted stickers remain in place In terms
of phys ical implementation pro sp ects clear seems to be the mo st problematic of our operations
Recall however that it can b e eliminated without s igni cantly sacr i cingthecomputational p ower
of themodel
Initialization and Final Output
Tomake a combinator ial library cont aining roughly onecopyofevery p ossible bit str ing of length L
L
followed by K L zero s it i s rst nece s sary tosynthe s ize roughly identical copie s of a properly
de s igned memory strand with K L regions Stickers must then be added randomly to these
he strands in p ositions L One pro ce dure that achieves thi s is outlined b elow Note that t
metho d require s only a s ingle step
The strands are split into two equal volumes To one volume is added an exce s s of stickers for
all bits L thi s re sults in all bits L be ing set on all strands The unus e d stickers are then
removed for example by ltration or by separating on a universal region of the memory strand
The two volumes are then recombined and heated caus ing all stickers to di s so ciate Finally the
mixture i s co ole d again caus ingthestickers to randomly anneal tothememory strands Since e ach
ds the re sultingmemory complexe s have any bit position has only one sticker for every two stran
given bit s et with probabilityonehalf very ne arly indep endently Under thi s mo del theoddsthat
L
L
any particular bit str ingisnot pre s entinthe nal library i s whichforthe L of intere st
i s almo st exactly e In other words e ach str ingiscreated at le ast once with probabilityroughly
L
Thi s p ercentage can obviously be incre as e d by synthe s izing more than strands initially
Notice thatthi s pro ce dure i s relatively robust to errors in stoichiometry For example if theoriginal
strands are split intovolumes whos e ratio i s not but then for say L a randomly chos en
str ing i s cre ated with probability still not vani shingly small
To obtain an output str ingit is nece s sary to be able to detect the pre s ence or ab s ence of memory
complexe s in a solution If any are pre s ent we also nee d tobeable toisolateatleastonememory
complex andthen identify whichstickers if any are anneale d toit
Detection of complexe s might be accompli shed by uore scent lab eling of each memory strand
Single molecule detection can then be p erformed by running the solution through a ne capillary
tube Such detection has alre ady b een achieved exp er imentally see for example Castro Thi s
technique may also b e e ective for i solatingasingle complex if thetimebetween detection eventsis
large enough In addition tothe capillary tube method mentioned above other prop o sals e g bas e d
on PCR for complex detection are p o s s ible
The nal step of identifying anne ale d stickers may be p ossible by direct imaging since we know
the order of bit regions w e could imagine just looking and re ading o the answer str ing p erhap s
us ing electron micro scopy Alter nately once a complex is isolated its stickers may be eluted and
p oure d over a detection hybr idization gr id Meade to determine which ones were pre s ent While
the s e p o s s ibilitie s are intr iguing more practical approache s bas e d on PCR are more likely towork
in the near term Adleman However we show below that detection alone is su cient to obtain
an outputstring Theapproachisto us e binary tree deco ding
Begin with the solution containing all putative answer complexe s of which there may be none
Detect complexe s in it If th ere are none then no answer has been found If there are some then
s eparatethembasedonthe rst bit of theanswer str ing Detect complexe s in e achofthe re sulting
solutions and retain the one which is not empty If neither is empty then there is more than one
answer andeither can b e retained Repeatthi s s eparation anddetection for all thebitsofthe answer
str ing
Memory Strand and Sticker De s ign
At several points in the above di scus s ion it was nece s sary to design the s equence of the memory
strandor stickers tohave certain propertie s In thi s s ection wesummar ize tho s e requirementsand
explore p o s s ibilitie s for achievingthem
amental requirement of s equence design is toachievesticker sp eci city It i s cr itical Themostfund
that the stickers only anneal to the memory strands when opposite their as s igned region and not
in any other position Thus the memory strand s equence must be de s igned so that any region s
complementary sticker i s only complem entary tothatone region andhas muchreduce d a nityatall
other alignments alongthe strand As a rst approximation tothi s we will require a certain minimum
numb er of bas e mi smatches at all other alignments Notice thatthi s i s a much stronger requirement
than s imply requir ingeachsticker tomismatch all bit regions butitsown It must mi smatchevery
other M longwindow p ossibly spanningtwo bit regions on the strand Mathematically wewish
tode s ign a s equence of length N suchthatthere exi st K non overlappingsub s equence s of length M
each call them regions withthefollowingproperty For e ach region itscomplementhas atleast
L
P
L
L
k L
k
The expre s s ion for the probabilityofa random bit str ingbeing cre ated is r r where
L
k
r is theratio of thevolumes into whichwesplit initially
The answer str ing whichwe are intere ste d in re adingoutmay b e a substringoftheentire str ingencoded bythe
memory strand in which cas e s eparation only nee ds to b e doneforthose bits
D mi smatche s with every other sub s equence of length M in theentire s equence The quantity D
is theminimumnumber of mismatches nee de d for a sticker M bas e s long not toanneal
It i s also imp ortantto eliminate s econdary structure in thememory strandits elf Wemust prevent
thememory strand f rom anne alingtoitself and cre atinga hairpin structure as thi s make s regions
inacce s s ible for proper us e in the system Ful lling thi s requirement can be lo o s ely mo dele d by
the combinator ial problem of de s igning a N long sequence such that the complement of every
sub s equence of length M has at le ast D mi smatches with every other sub s equence of length M
he minimumnumber of mi smatches to prevent the memory strand f rom s elf The quantity D is t
anne aling
Finally wemust de s ign s eparation prob e s suchthatthey stick sp eci cally totheappropr iate region
and they have su ciently lower a nity there than the stickers Thi s ensure s that there exi sts a
w ash temp erature and salinity for which the prob e s will di s so ciate while the stickers will remain
in place Again as a rst approximation we require that the prob e s have at le ast D mi smatches
within the ir region andatleastD D mi smatches everywhere els e
These cr iter ia may seem daunting However t here are some ways to make thi s task potentially
easier Notice that in general we may le ave portions the memory strand unus e d that is we may
not identify tho s e p ortions withany regions so thattheproduct of K and M do e s not always equal
N but certainly still KM N In other words we le a ve gap s between the bit regions on the
memory strand In order toavoid the secondary structure problem it has b een sugge sted thatthe
memory strand be comp o s e d of only pyr imidines or pur ines and the stickers of only pur ine s or
pyr imidine s Mir The applie d mathematics literature on comma free co des and on de Bruijn
s equence s when D contains detaile d di scus s ions of manyofthe imp ortant i s sue s s ee Neveln
and Fre dr icks en for re asonable intro ductions Also Smith Baum have di scus s e d s equence de s ign
in thecontext of DNA computation
Finally D would b e re duce d if higher a nity PNA or DNG stickers were us e d furthermore D
would p o s s ibly b e re duce d to zero Other var iable s other than or in addition totemp erature could
t concentration and chemical solvent in order to achieve the relative be manipulated such as sal
a nitie s require d for e achop eration It i s worth sp eculatingaboutthe p o s s ibilityofusingnaturally
o ccurr ing s equence s e g plasmids for the memory strands becaus e of the obvious ease of their
mas s pro duction However it remains tobeseenifnatural s equence s can b e foundwhichmeet the
aboverestrictions
We emphas ize that the cr iter ia outlined above are for illustration only a more sophi sticated ap
proach would have to take into cons ideration the s equence dep endent thermo dynamic parameters
for oligonucleotidehybr idization There are s everal datasetsavailable for calculating H and S
for DNA DNA hybr idization Santalucia Bre slauer Petruska and s imilar data could b e obtained
for PNA and or DNG interactions Allowance s would also have to be made for potential bubble
mi smatches at incorrect sticker hybr idization s ites andsecondary structure due totriple helix for
intro duce additional constraints mation must be prevented The clear operation if us e d would
Although such sophi sticated de s ign approaches could sugge st potentially us eful memory strand
sticker and prob e s equence s correct operation will have tobetested experimentally
Our conclus ion i s that although design of thememory strandandthestickers may b e di cult the
de s ign space is large and once a strand with K regions is found it can be us e d and reus e d in
the stickers mo del for any problem requir ingK or fewer bits of memory Since the stickers mo del
K
us e s only a s ingle typ e of memory strand in contrast tothe di erentmolecule s require d in the
repre s entation of Boneh the de s ign pro ce s s i s s impli e d andthe functionalityofthe strand can
be tested exp er imentally once andforall
Experimental Fe as ibility
The stickers mo del as pre s ented above pre s ents challenging requirements for strand de s ign and
exp er imental implementation Several ob jections might b e rai s e d tothee ectthatitisunre asonable
to exp ect that these requirements can be met We attempt to br ie y addre s s some of the s e i s sue s
here
Obje ction No matter whatmetho ds are prop o s e d DNA bas e d technique s will su er f rom strands
be ingmisprocessed What error rates would b e require d in order tostill accompli sh us eful compu
tation
Response For many search problems including DES and NP complete problems probabili stic
algor ithms have practical value Answers sugge sted by the molecular computer so long as there
aren t too many can be ver i e d electroni cally To ensure that a complex carrying the solution to
the problem has a chance of ending up in the answers tube after a step computation
s eparation error probabilitie s of le s s than are require d To eliminate f als e p o s itive di strac
tors it may be nece s sary to re ne the answers tube by rep e ating the step s of the computation
Adleman Karp Thi s andother relate d error handlingstrategie s are di scus s e d in Section
Objection Pur ityand yield of for pur i cation of DNA are cons idere d excellentinmolecular
biology The conditions imp o s e d for s eparation of memory complexe s are much more challenging
s ince long strands may be us e d stickers must not be kno cked o and both sup er nate and eluant
are require d Yet DNA computation require s muchlower error rates bothforpurityand yield
Response Isolation of particular target DNA in complicated cDNA librar ie s is a routine task
in molecular biology fold enr ichment of target DNA with recovery has been rep orted
us ing for example tr iplex a nitycapture Ito The us e of PNA prob e s also shows some promi s e
pur i cation wit h yield us ing PNA mers has been rep orte d Orum However current
technique s do not meet our requirements for the separation operator We do not believe that thi s
is due toa fundamental limit So long as yield i s extremely high i e memory complexe s don t get
lo st our calculations s ee Section sugge st that a p o or s eparation can b e improve d dramatically
pp ortunitytode s ign our own s equence s that byautomate d pro ce ssing Furthermore wehave theo
can be e ectively s eparated for example by ensur ing that the memory strand has no secondary
structure We recognize thatattaining high step yield may b e a ma jor challenge however
Objection Even without trying to process them at all stickers will be f alling o their memory
strands atsomerate k Onceasticker di s so ciates it may then hybr idize toandthus corrupt some
d
other complex Dur ingoperations such as s eparate when memory complexe s must b e melted from
prob e s k surely incre as e s By the time the computation i s complete the contents of the memory
d
complexe s may b e completely scramble d
Response Suppose we would like to ensure th at fewer than of stickers fall o dur ingthe
cours e of a hour computation Thi s would require a k of le s s than sec A gener ic
d
DNA mer can b e e stimated to have the require d k at Cin M Na Wetmur PNA and
d
DNG stickers would b e exp ected tohaveaneven lower di s so ciation rate e sp ecially atlow salt High
wash temp erature s may b e avoided byusing DNA prob e s and PNA or DNG stickers andwashing
in lowsalt Additionally we must b e careful not to encourage other circumstance s suchasrough
phys ical handling which mightinduce sticker di s so ciation
Objection If DNA is sub jected to high temp erature s for a s igni cant portion of a hour
tion depur ination or strand bre akage byhydrolys i s computation it may b e damage d by de amina
thus render ing it non functional Such ob jections are di scus s e d br ie y in for example Smith
Response Under phys iological conditions of salinity pH andtemp erature thedepur ination half
life of a bas e is hours and the hydrolysis half life of a depur inated bas e is hours
Friedberg Thus after hours approximately of bas e s will be damage d and of
mer strands will remain unbroken Thi s last gure is very dep endent on the length of the
strands only of mers would survive While not go o d thi s indicates that for short
strands errors due to damage can be comp ensated for by a mild incre as e in the volume of DNA
us e d in a computation Additionally improved rates may be p ossible by carefully adjustingsolvent
salinity pH andcomposition and again minimizing rough phys ical handling
In summary although there are many serious engineer ing challenge s we do not see any as be ing
cle arly insurmountable
A Stickers Machine Proposal
Thi s section de scr ib e s the details of one p ossible machine that implements computation us ing the
ickers mo del The machineisa sort of parallel rob otic workstation for molecular computation st
in whichvar ious rob oticand uid owapparati are controlle d by a central programmable electronic
computer It contains of a rack of many test tube s asmall amountofrob otics some uid pump s
andheaters co olers andsomeconventional micro electronics For e achoftheoperations in t hemodel
wehavemade a sp eci c choice of phys ical pro ce dure s toimplementit Thus themachine repre s ents
one particular re alization of many possible var iations on the ideas di scus s e d above The proposal
is meant to provoke thought about the engineer ing i s sue s involved in eventually constructing a
molecular computer and not as a s er ious or viable construction plan
Theworkstation store s all DNA which repre s ents information dur ingthe computation in so calle d
data tubes Each data tube is a clo s e d cylinder with a nipple connector in either end that allows
uid to owinorout Ne ar oneendonthe ins ideisa permanent membranewhich pas s e s solv ent
butnotstickers or memory strands Thi s membrane gives a polar itytothedatatube the connector
on theend clo s e st tothemembraneisthe cle an s ide while theopposite connector i s dirty No
DNA i s ever pre s entonthe cle an s ideorinthe cle an connector When a datatub e i s not in us e it
is held cle an s idedown with all of the DNA in thetube re stingonthemembrane
ld e ither s etsofmemory complexe s or supplie s of unbound Thedatatube s whichmay b e empty ho
stickers Sp eci cally each set of bit str ings has as so ciated with it a data tube which holds the
memory strands and anneale d stickers repre s entingthos e strings Also e ach bit has as so ciated with
it a datatub e which contains a supply of stickers correspondingtothatbit
Whenever a new s et of complexe s i s cre a te d e g f rom a s eparation operation it i s place d in a new
data tube Whenever a s et of complexesisdestroyed e g f rom a combination op eration thedata
tube thatusedtocontain it i s di scarded or perhap s vigorously washed andster ilize d for reus e
In addition to data tube s there also exi st operator tubes of s imilar exter nal construction but with
or tube is merely an emptytub e withnipple connectors di erentinter nal contents A blank operat
on e achend A sticker operator tube is identical except for a p ermanent lter on its ins ide which
pas s e s stickers but not memory strands A s eparation operator tube contains many identical
copie s of one bit s oligo prob e There is a di erent s eparation operator tube for each bit It is
de s igned so thatthe prob e s cannot e scap e f rom thetube butunboundmemory complexe s can For
example the prob e s might be f astened to solid supp ort by biotinylatingthem and us inga biotin
bindingmatr ix or to large b e ads with lters that pas s memory strands but not b e ads For all of the
operator tub e s b othends are cons idere d dirty Figure illustrates thedataandoperator tube s Dirty Side Connectors
DNA (Memory Oligo Strands Probes Filter or (Passes Stickers) Stickers but not Membrane Memory Strands) (Passes only solvent) Clean Side Operator
Data Tube Tubes: Separation Sticker Blank
Figure Data andOperator Tube s in thestickers machine
At any time dur ing the operation of the machine some tube s are in us e and other are not All
tube s that are not in us e are store d on a large rack or carous el Any single operation takes place
as follows under controlofthe electroniccomputer twodatatub e s are s elected andremoved from
the rack by a rob ot One operator tube is also selected and removed The dirty sides of the data
tube s are connected totheoperator tube onedatatube ateachendoftheop erator The cle an s ides
of the data tube s are joined by a pump Solution i s cycle d through all three tube s The direction
of ow may be towards the rst data tube or vice versa or both intermingle d The temp erature
salinity direction andduration of the owiscontrolle d bythe electroniccomputer Once the ow
ube s stops oneormoreofthetub e s i s di sconnected andreplace d on therack or di scarded New t
then come in f rom therackuntil there are once again twodatatube s andoneop erator tube andthe
next operation b egins Notice that in general cle an connectors never touch dirtyones and only cle an
connectors contact thepumping system Thi s s etup for a gener icoperation i s shown in Figure
Data Data Tube Operator Tube Tube #1 #2
Pump & Heater/Cooler
Figure Setup for a gener icoperation in thestickers machine
We will now review howeach of our conceptual operations can b e p erformed as outline d gener ically
above Thede scr iptions b elow are summar ize d graphically in Figure
Tocombinetwosetsofcomplexe s s imply s elect thetwodatatube s andablank operator tube
Cycle cold solution towards say the rstdatatube Thi s catches all thememory complexe s
in the rst datatube The s econddatatube andtheblank op erat or are di scarded
To s eparateasetofcomplexe s bas e d on thevalue of somebit selectthedatatub e containing
the complexe s to be s eparated and also an empty data tube Select the s eparation operator
tube for the bit in que stion Cycle cold solution in b oth directions for some time thi s allows
theprobestobindtho s e complexe s thathavethe bit in que stion o Next cycle cold solution
to wards the empty data tube forcing all the unbound memory complexe s into it Detach
thi s or iginally empty tube and retur n it to the rack it holds the complexe s withthe bit in
que stion on Replace it with another empty data tube Cycle hot solution or p erhap s low
salinitysolu tion towards thi s new data tube Thi s rele as e s thememory complexe s b oundto
the prob e s and force s them intothenew datatube Detachthi s tube and retur n it tothe rack
also it contains complexe s withthe bit o Di scard the or iginal data tube now empty and
retur n theoperator tube tothe rack
To s et a bit add a sticker to a s et of complexe s s elect thedat atub e containingthe complexe s
and also the data tube containing the sticker supply for the sticker to be added Us ing the
sticker op erator tube cycle cold solution in both directions for some time Thi s washes the
stickers over thememory complexe s allowingthem to anneal Now cycle cold solution towards
the sticker data tube Thi s retur ns the unus e d stickers and le aves all the memory complexe s
caughtonthe lter in theoperator tube Di sconnect thesticker datatube and retur n it tothe
rack Replace it withanemptydatatube Cycle cold so lution towards thememory complex
data tube Thi s exp els the memory complexe s f rom the op erator tube and retur ns them to
their datatube Retur n the memory complex data tube tothe rack and di scard the operator
tube andemptydatatube
Additional paralleli sm can be added in many place s For example setting or cle ar ing bits might
be applie d tomanydatatube s at once bystacking all of them after theoperator tube Also many
copie s of therobotics might b e included to allowseveral operations to b e p erformed simultaneously
thi s would also require multiple copie s of for example the s eparator operator andsticker operator
tube s
vede scr ib e d it thestickers machine require s relatively rudimentary rob otics and electronics As weha
Simple uid pump s andheaters co olers are also nece s sary It can b e stocked witha gener icsupply
of empty data tube s blank operator tube s sticker operator tube s and salt solutions of var ious
concentrations It contains datatube s containingboththe or iginal s etsofmemory strands andthe
sticker supplie s for e ach bit It also nee ds tobeloaded withtheseparation operator tube s for each
bit An imp ortantfeatureisthatthese tub e s are reusable f rom problem to problem so longasthe
numberofbits require d do e s not excee d thenumb er of regions on thedesigned memory strand For
a problem of re asonable size on the order of afewthousand tube s might b e require d for example
ube be inga few m in s ize andoperator tube s DES as de scr ib e d in Adleman Witheachdatat
perhap s a hundre d times thi s s ize it i s not inconce ivable thatsucha machinemight tonade sktop
or lab b ench Thi s example directly addre s s e s the concer n that any us eful or hard computation
will require an enormous volume of DNA bydemonstrating botha sp eci c problem and a sp eci c
machineprop o sal for whichthi s s eems f ar f rom true Separation Combination
Data Source Source Tube Tube Tube #1
Blank Probes Probes Summary Cold of all Cold Warm Wash Wash Wash Data Operations Output Output Tube Tube Tube #2 (bit ON) (bit OFF)
Set Bit Clear Bit
Data Data Data Data Tube Tube Tube Tube
Filter Filter Filter Filter (Passes (Passes (Passes (Passes Stickers Stickers Stickers Stickers but not but not but not but not Memory Memory Memory Memory Strands) Strands) Strands) Strands) Cold Cold Cold Cold Wash Wash Wash Wash Sticker Blank Anti-Sticker Blank Tube Tube Tube Tube (PNA strand
invasion)
Figure Graphical synop s i s of all operations in thestickers machine
Re ducing Error Rates A Re nery Mo del
In thi s s ection weintro duce a s econd p ossible implementation of thestickers mo del In contrast to
the stickers machine di scus s e d above the stickers re nery addre s s e s the i s sue of howto p erform
reliable computation us inga very unreliable s eparation operator The re nery mo del also illustrates
the pr inciple of pip elining wherebyalargevolumeofmemory comp lexe s can b e pro ce s s e d bysmall
capacity op erators with minimal slowdown These advantage s come at the co st of a time space
trade o whichwe nd re asonable
An Error Framework
There are three fundamental typ e s of errors thatmightbemadebyanymolecular computer which
attempts to sort a huge library of initial candidate solution comp lexe s into thos e which enco de a
solution to a problem and thos e which do not It may give some false positives namely some of
the complexe s that it clas s i e s as solving the problem actually may not It may also have false
negatives which occur when complexe s that are clas s i e d as not solving the problem actually do
solve it Finally the machine may incur some strand losses some of the complexe s which were
pre s entinthe inputmay not appear in theoutputat all they may s imply get lo st somewhere ins ide
themachine What are the error requirementsto do us eful computation It is clear thatwewant
lowfalsepositiveandfalsenega tiverates and few strandlosses buthowlowdothey nee d tobe
Our mo del of a molecular computer i s a machinethattake s as inputa tub e enco ding a large number
of p otential solutions tosome problem andproduce s as outputtwo tub e s one lab ele d Yes andthe
other No In the Yes tube are all tho s e complexe s whichthemachine has decided encodesolutions
eproblem inthe No tub e are all tho s e complexe s whichithas decide d do not enco desolutions toth
Call a good complex one which actual ly does enco deasolution andabad complex onewhich actual ly
does not Becaus e themachine i s not p erfect there may b e some good complexe s in the No output
some bad complexe s in the Yes output as well as some losse s
Nowwe are in a p osition tostate our requirements for error rat es We want two things to b e true
with high probability say each time we run the molecular computer there is at le ast one
good complex in the Yes tube and the ratio of good to bad complexes in the Yes tube i s re asonable
say Informally when we get the answer tube we will sh around in it pull out a random
complex if there are any andreadthesolution thatitencodes We will b e di sappointed if either
a we do not ndany complexe s in theanswer tube or b the complex we re ad do e s not actually
enco deasolution Our goal i s to b e di sappointe d withlow probability
Wewould liketobeable toanswer the que stion Howgooddoindividual operations haveto b e for
tmentto berare Unfortunately itisvery complicated to expre s s theabove requirements di sappoin
in terms of conditions on the delityoftheindividual operations suchas separate In f act even for
re asonably s imple error mo dels theanswers are extremely dep endentontheparticular architecture
of the molecular computer and on the problem be ingsolved Instead we will work wit h a mo del
For example one could imaginea s imple mo del of errors whichischaracter ize d byonlythree numb ers e ach
between and a f als e p o s itiverate R afalsenegativerate R andalossrate R Anygiven complex i s
fp fn loss
lo st with probability R If not lo st good complexe s go to the Yes tub e withprobability R and bad
loss fn
complexe s go toth e No tub e with probability R regardle s s of the sp eci c bit str ingthey enco de Under sucha
fp
mo del if our inputtub e contains G good complexe s and B bad complexe s typically G B then we require a f als e
negativerate R whichislessthan somefunction f G B a f als e p o s itiverate R whichislessthan f G B
fn fp
is the f raction of runs of the exp er imentthat will re sultin anda lossrate R whichislessthan f G B where
loss
which allows us tocharacter ize the fraction of complexes not yet correctly processed denoted simply
atsometime T after we b egin thecomputation Thi s quantity can be easily understo o d as follows
we tur n on our molecular computer at time and fee d it its input It works away placing some
complexe s in the Yes tube andsomeinthe No tube Attime T westopthemachineandcollect the
Yes and No tube s Atthi s p oint or iginal inputcomplexe s f all intothree categor ie s tho s e which
have b een correctly place d intoeither Yes or No thos e whichhave b een incorrectly place d into
en Yes or No and thos e whichwere e ither lo st or were still b e ing pro ce s s e d bythemachinewh
we tur ned it o The fraction of complexe s not yet correctly pro ce s s e d is the fraction of the
or iginal input complexe s which f all intoeither categor ie s or aboveattime T Wewould like
tobevery ne ar zero Belowwedevelop a mo del which allows us to compute for var ious machine
architecture s and also var ious time and space tradeo factors in terms of only the delity of the
atomicoperations which are us e d bythemachine indep endentofthe problem b e ingsolved
Computing
We will cons ider a very s imple mathematical mo del of amolecular computer as a s er ie s of exactly
S identical s eparation operations The s eparation operation is us e d becaus e it is a fundamental
operation in thestickers mo del b oththe set bit and clear bit operations can b e de scr ib e d in terms of
only s eparations s ee Section Thi s mo del as sumes thatthe algor ithm us e d to pro ce s s complexe s
has the e ect of passing each one though at mo st S separations an as sumption which is true for
all algor ithms that terminate within a known time It further as sumes that complexe s do not
interfere withone another nor do di erentbitpositions on a s ingle strand For themoment let us
also as sumethatthere are no strandlosses we will retur n tothi s crucial i s sue later
As sume that regardle s s of which bit is be ing us e d to s eparate and of the value s of any other
bits each s eparation op eration takes one unit of time to complete and has a probability p of
correctly pro ce s s ing each complex in its input Notice that we exp ect p to be near unity In
every s eparation weassumethateach complex ends up in oneortheother of theoutputtub e s no
strands are phys ically lo st Now any computat ion will take S units of time and when it is done
S
the fraction of complexe s not yet correctly pro ce s s e d will be a depre s s ingly high p
For example if p and S then The main point of thi s section is that
without changing p i e withoutimprovingthe bas ic biotechnology us e d to implementop erations
di sappointment However it tur ns outthateven when f f and f havebeendetermined theconvers ion f rom these
three numbers to a requirementonthe delityofindividual operations i s highly architecture dep endent compare for
example the s imple OR of all bitsinabitstringwiththesimple AND
Notethat good complexe s can b e incorrectly pro ce s s e d at somestep s yet still endupinthe Yes tub e s imilarly
b ad complexe s can end up in No after incorrect pro ce s s ing Westill countthe s e cas e s as incorrect
Recall that s ince answer re adout and stranddetection are not p ermitted dur ingthecourseofthe computation
the algor ithm which controls the pro ce s s ing cannot get any fee dbackand so cannot do any ifthen els e typ e branching
To s ee thatthemodel as sumption i s not as re str ictiveasitmay s eem cons ider architecture s that are of theformof
fee dforward layere d circuitswith S layers Eachlayer rece ive s somenumber of inputtube s from the previous layer and
pro duce s some possibly di erent number of outputtub e s which it passes tothenext layer No tube may go through
more than one s eparation p er layer In thi s way for anyindividual complex sucharchitecture s lo ok like a series of S
identical s eparation op erations although di erent complexe s may take di erentpaths through the circuit The rst
layer rece ive s as its inputthesingle tub e whichwas the inputtotheentire problem The nal layer S produce s as
itsoutputthe nal outputtub e s for the problem Any terminating algor ithm for doinga stickers computation can
b e converted into a fee dforward circuit of thi s kind
In practice op erations like separation havea muchhigher probability of correctly processingsome inputsthan
others For example if hybr idization i s us e d it i s muchharder for prob e s toerroneously capture complexe s than it
is for them to let through complexe s whichthey should capture All of th emathematics whichfollows can e as ily b e
done for theassymetr ic probability cas e although it i s somewhatmorecomplicated
and withoutreducing S i e withoutmovingto e as ier problems the fraction can b e made much
smaller us ingintelligent space andtime tradeo s
Imaginethatyou haveinhand enough hardware i e unitsthat p erform s eparations andtest tube s
to p erform a given computation A spacetradeo of factor H involve s obtaining H extra identical
copie s of thathardware whic hwemay us e in parallel A time slowdown of factor M involves taking
M S unitsoftime instead of merely S to p erform the computation Howcanthe s e f actors b e us e d
toreduce errors Given anyalgorithm A for p erforminga computation andfactors H and M we
would liketoinvestigate algor ithm transformations which giveusanew algor ithm A thatruns in
no more than M S timeand require s no more than H copie s of thehardware thathasasmaller
than the or iginal A
Rep e ating the Computation
A bas ic transformation repeating was proposed by Adleman in Adleman It makes us e of a
slowdown f actor of M by proposing A as follows
Repeat M times
Run A on input I producing tubes Y and N
Discard tube N and rename tube Y to tube I
Return tube I as the Yes tube and an empty tube as No
Thi s approach is of value when the or iginal algor ithm A was known to very reliably place good
complexe s into its Yes output i e low f als e negatives but to often also place bad complexe s into
Yes i e high f als e p ositives Note tha t if the or iginal algor ithm was known instead to have high
f als e negatives andlow f als e p o s itives then thefollowingvers ion of repeating can b e us e d
Make an empty tube Z
Repeat M times
Run A on input I producing tubes Y and N
Combine tube Y into tube Z destroying Y
Rename tube N to tube I
Return tube Z as the Yes tube and tube I as No
repeating re duce The p erformance of thi s transformation i s b ounded bythe By howmuchdoes
performance of an imaginary transformation calle d repeating with an oracle which makes us e of a
new oracle op eration The oracle takes as inputtwo tube s Y and N and pro duce s as outputthree
tube s Y N and X In Y are all t he good complexe s thatwere in theinputtube Y inN are all
the bad complexe s thatwere in theinputtube N andinX are all the bad complexe s f rom Y along
with all the good complexe s f rom N In other words the oracle xe s up Y and N by putting
any incorrectly pro ce s s e d complexe s into X Us ingthi s magical o p eration repeatingwithanoracle
transforms A intothefollowing A
Make an empty tube Z
Repeat M times
Run A on input I producing tubes Y and N
Run the oracle on Y and N producing Y N and X
Discard tube N and combine tube Y into tube Z destroying Y
Rename tube X to tube I
Return tube Z as the Yes tube and tube I as No
S S M
Thi s transformation improves f rom p to p Thevanilla repeating transformations can
approach but never excee d thi s improvement The re ason that plain repeating works well at all is
that for very di sparatefalsepositiveandnegativerates one can approximatethe action of the oracle
easily While the s e transformat ions do yield somereduction in they require enormous slowdowns
toimproveeven mo de st s ize d problems For larger problems the slowdowns the s e transformations
require are enormous Figure shows theslowdown f actors require d toachievevar ious p erformance
levels for thecaseinwhich p and S or S
Slowdown vs. Performance (Atomic = 90%, 100 layers) Slowdown vs. Performance (Atomic = 90%, 1000 layers) 0 0 10 10
−1 −1 10 10
−2 −2 10 Horizontal Scale in units of 10^5 ! 10 Horizontal Scale in units of 10^46 !!
−3 −3 10 10
−4 −4 10 10
−5 −5 10 10
−6 −6 10 10 −7 −7 10 10 −8 10 −8 10 −9 10 −9 10 −10 Fraction Not Yet Correctly Processed (delta) 10 −10 1 2 3 4 5 6 7 8 9 Fraction Not Yet Correctly Processed (delta) 10 2 4 6 8 10 12 14
Slowdown Factor Required (in units of 10^5 !) Slowdown Factor Required (in units of 10^46 !!)
Figure Performance vs Slowdown for repetition with an oracle
A New Op eration Comp ound Separation
It is p ossible to make much better us e of space and time tradeo s than the above transformations
do Shortly wewilldevelop new transformations whichdothi s but rstwemust intro duce a new
ound separation operation whichthey employknown as comp
Thecentral ob s ervation i s thatthefollowing algor ithm analogous to countercurrent cascadestage s
in chemical engineer ing Wankat will exp onentially improve up on the accuracy of the Separation
step
Begin with a tube T whose contents we wish to separate based on bit k
Begin also with N extra tubes called T T and T T initially empty
N N
for t to Q
for j N to N s t t j mod
Separate T into T and T based on bit k
j on
of f
Combine T and T into T
on j j
Combine T and T into T
j j
of f
Notice that for odd t odd numb ere d tube s start o empty and for even t even numbere d tube s
start o empty
Thus e ach complex will p erform a bias e d random walk in t ube s T through T withab sorption
N N
at the boundar ie s Mo st memory complexe s which have bit k on will end up in T while mo st
N
memory complexe s whichhave bit k o will endupinT A graphical illustration of the pro ce s s
N
t s ee the is shown in Figure The statistics of such pro ce s s e s have been thoroughly worked ou Original Input
Final bit i ? bit i ? bit i ? bit i ? bit i ? Final "Off" "On" off on off onoff on off on off on Output Output 2N - 1 identical
units
Figure A Comp ound Separator
Gambler s Ruin problem Feller Let p be theprobabilitythat a s eparation step correctly moves
a complex into T or T At the end of the algor ithm we would like to know the probabilitie s
on
of f
that a complex with bit k on or o will e ither be in tube T T or still stuck in some other
N N
tube Let us rst cons ider the cas e Q i e each complex continue s to be pro ce s s e d until it it
ab sorb e d at either T or T Then a complex has probability p of be ing correctly processed
N N
where
p
p
N
p
For example if p wechoose N andthen p It i s cr itical tothi s argument
that no memory complexe s are lo st in thewoodwork However it i s not crucial that Q be The
exp ected time t for a complex to arr iveineither T or T is
N N
compound
N
r N
ht i
compound
N
p r
p
In the example ht i In f act in thi s example Q ensure s that where r
compound
p
fewer than of the complexe s are not correctly pro ce s s e d Figure shows the p erformance
of compound separation as a function of number of step s Q forvar ious chain lengths N
Wehaveshown that byapplyingthe compound separation algor ithm above we can achieve excellent
error ra tes even when thefundamental s eparation op eration i s not reliable Thi s comes atthecost
of a small linear slowdown and a few extra tub e s in general wenee d to p erform Q N s eparations
instead of one
Notice that thi s algor ithm can be e as ily parallelize d if N atomic s eparator units are available
instead of just onethen the slowdown f actor can b e re duce d to Q by p erforming all the s eparations
simultaneously i e do all iterations of the inner for j lo op in parallel We will call thi s
parallelize d algor ithm paral lel compound separation
Although the bas ic mathematics are not new to our knowle dge the rst application of thi s idea
Sup er Extract op eration is very s imilar to molecular computation app e are d in Karp Their
although not identical to the comp ound s eparation wehave proposed above We refer thereader
to the excellent di scus s ion and detaile d analys i s including some intere sting bounds contained
there in Compound Separator Performance (Atomic = 90%) 0 10 N=1
−2 N=2 10 N=3
−4 N=4 10 N=5 N=6 −6 10 N=7 N=8 −8 10 N=9 N=10 −10 Fraction Not Yet Correctly Processed (delta) 10 0 10 20 30 40 50 60 70 80 90 100
Steps Performed
Figure Comp ound Separation Performance The bars on the left show the mean time one
standard deviation for complexe s tobeab sorb e d ataboundary
Better Transformations The Re nery Idea
What if we were to replace every separate op eration in our or iginal algor ithm A witha compound
separation Thi s would incur a slowdown factor of M Q N but would give an enormous
re duction in since the delityof each s eparation has improved exponentially Thi s i s exactly the
idea behindthe serial re nery transformation whichexploits a slowdown f actor of M It proposes
A tobe
Run A on input I replacing each separation operation with a compound separation operation
where Q N M of chain size N and duration Q
Notice that if a space tradeo f actor of H is also available then we can employthe one layer re nery
transformation which makes us e of the available paralleli sm H and slowdown M by sp ecifying A
tobe
Run A on input I replacing each separation operation with a parallel compound separation
operation of chain size N and duration Q where Q N M H
Th e one layer re nery is so named becaus e if A or iginally processed one layer in parallel before
moving on to the next layer withsu cient paralleli sm A may now process each layer in parallel
for Q step s andthen moveontothenext layer
For the momentwedefer the i s sue of howtodecideontheoptimal f actor ization of M or M H into
Q N although weretur n toitshortly Theobvious choice i s tochoose N H and Q M First
let us ndouthowmuchimprovementin thi s transformation buys us The exact expre s s ion for
i s complicated but e as ily computable Theplots in Figure showthe p erformance ofthe one
S
For the a cionado p N Q where p is the probabilityofgettingab sorb e d atthe correct b oundary in
Q step s or le s s in a bias e d random walk bias probability p withab sorption atboundar ie s N and N In tur n
i h
P P
i
N Q
i N iN i
v v v
p p co s sin sin where the expre s s ion in square brackets p N Q
N N N
v i
is the probabilityofab sorption in exactly i step s All of themathematics can b e extended tothe cas e when the random
walk bias i s di erentineach direction s ee Feller
layer re nery transformation as a function of slowdown f actor M for var ious comp ound s eparator
chain lengths N and for S and S The plotsassume that we have chos en Q M
and N H
One Layer Performance (Atomic = 90%, 100 steps) One Layer Performance (Atomic = 90%, 1000 steps) 0 0 10 10 N=4 −1 N=3 −1 10 10 N=4 N=5 −2 −2 10 10 N=5 N=6 −3 −3 10 10 N=6 N=7 −4 −4 10 10 N=7 N=8 −5 −5 10 10 N=8 N=9 −6 −6 10 10 N=9 N=10 −7 −7 10 10 N=10 −8 −8 Fraction Not Yet Correctly Processed (delta) 10 Fraction Not Yet Correctly Processed (delta) 10 0 10 20 30 40 50 60 0 10 20 30 40 50 60
Slowdown Factor Slowdown Factor
Figure One Layer Re nery p erformance for S and S Plots as sume that we have
chos en Q M and N H s ee text
A Fully Parallel Re nery Architecture
In the remainder of thi s section we show how by exploiting the ideas above a new machine ar
ecture calle d the stickers re nery which achieves the same low error rates as the one layer chit
re nery andgreater sp ee d up bycontinuously pro ce s s ingallstep s in the computation atthecost
of cours e of additional space The re nery architecture may have other advantage s as well which
we will commentonbelow
ime for a complex to get through a single comp ound s eparation As shown in Figure the mean t
chain i s cons iderably le s s than the Q require d to obtain maximal p erformance typically by a f actor
of about Mo st of thetimedur ing a computation i s sp entwaiting for a few stragglingcomplexe s to
comeout of a s eparator chain We can avoid thi s wasted timeby pro cee dingto pro ce s s complexe s
as so on as they are absorbe d in T or T The p aral lel re nery transformation cre ates A by
N N
replacingeach separation operation in A byaparal lel compound separation of chain length N and
then iteratively pro ce s s ingthe entire computation in parallel for T iterations
Sp eci cally suppose A has S W s eparations S fee dforward layers atmostW per layer anduses
j j j J
on i in i
of f i
and T bas e d on bit k Then the into T s eparates T tube s T T where s eparation i
i
paral lel re nery transformation given as A whichisde ne d as
j
j j
and T for N n N and T Begin with S W N tubes T
on n n
of f n
Initially T contains the mother tube complexes
for t to T
for j to S W do all j in parallel
for n N to N do all n in parallel
j
j j
Separate T into T and T based on bit k
j
n on n
of f n
for j to S W do all j in parallel
n N to N do all n in parallel for
j j
j
Combine T and T into T
n
on n
of f n
j
j
of f j
Combine T into T
of f N
j
j
on j
Combine T into T
on N
Compare d to the or iginal A the fully parallel re nery require s a space tradeo factor of H
N S s ince every s eparation i s expanded and a slowdown f actor not nece s sar ily integer of
T
M Thequestion i s what paralleli sm H andslowdown M are require d to obtain a de s ire d
S
performance Weanswer thi s que stion by calculating given N and T as b efore First wenote
thattheprobabilitythat a given complex i s correctly pro ce s s e d after T step s can b e decomp o s e d into
the probability p N T that itisineither the Yes tube or the No tub e after T step s i e not
done
stillinthemachinewhen westop andthe probability p N thatacomplex arr ivingina nal
cor r ect
N
p
tube has b een correctly pro ce s s e d Recall thatacomplex has probability p
p
of having been correctly s eparated every time it le aves a comp ound s eparator so p N
cor r ect
S
p The di str ibution of emergence times can be obtained by convolving the di str ibution for a
p p single comp ound s eparation thus numer ically calculating p N T Then
cor r ect
done done
The re sult of doing such a computation for p N and S and S are
shown in Figure b elow
Refinery Performance (Atomic = 90%, 100 layers) Refinery Performance (Atomic = 90%, 1000 layers) 0 0 10 10 N=4 −1 N=3 −1 10 10 N=4 N=5 −2 −2 10 10 N=5 N=6 −3 −3 10 10 N=6 N=7 −4 −4 10 10 N=7 N=8 −5 −5 10 10 N=8 N=9 −6 −6 10 10 N=9 N=10 −7 −7 10 10 N=10 −8 −8 Fraction Not Yet Correctly Processed (delta) 10 Fraction Not Yet Correctly Processed (delta) 10 0 5 10 15 0 5 10 15
Slowdown Factor Slowdown Factor
Figure Fully Parallel Re nery p erformance for S and S Thebarsonthe left show
themean time onestandard deviation for complexe s toemerge from the entire re nery
Thi s s econd probabilityisindep endentofwhen the complex emerge s
Advantage s of the Full Re nery
With the fully parallel re nery we can obtain the same target error p erformance and a roughly
fold smaller slowdown factor then the one layer re nery at the co st of S fold more space and
paralleli sm Thi s may not s eem likea bene cial trade o s ince S can b e p otentially large and is
small In f act it tur ns out t hat the fold sp ee dup can be achieved with an extra space tradeo
of much le s s than S times However the fully parallel re nery a ords a number of intere sting
p o s s ibilitie s For example suppose our fundamental s eparation units can handle limited volume
butwenee d to pro ce s s a fold larger volu me of DNA We can pip eline the computation by
inputtingsmall aliquotsofthemother tube ateachstep andwaitinguntil the last aliquot getsout
Nowmostofthemachineisbeingutilize d mo st of thetime inste ad of idly pumpingsolution around
If the non pip eline d parallel re nery would havetaken step s then after about step s the
han the non pip elined entire computation will be ni shed p erforming a fold larger s e arch t
vers ion while taking only twice the time In other words we are now exploiting for computation
all of theadditional paralleli sm andtimeemployed beyondthatusedbythenaive algor ithm while
gainingvastly improve d error rates for free
The parallel re nery mo del do e s not require re us e of any s eparation unit toserveatmultiple p oints
in the algor ithm and thus a general purp o s e rob otic workstation such as the stickers machine is
unnece s sary Weenvi s ion a sp ecial purp o s e re nery system b e ingassemble d f rom standard units
for each problem to be solved A s eparation unit cons i sts of a re s ervoir into whichcomplexe s are
rece ived an a nity column with DNA prob e s on solid support pump s and heaters for the wash
andelution andtwoexitchannels lab ele d on and o whichleadpermanently through piping
or tubing to the re s ervoirs of other separation units We refer to such a machine as embodying
e the the stickers re nery architecture It is our hope that a re nery architecture will alleviat
problem of lo st strands b ecaus e the phys ical permanence of all connections allows temp orar ily
stuckstrands toeventually b ecomeunstuckandstill completethe computation
Original Input Re-process Strands
Recycle Stickers
universal ?universal ? universal ? universal ? universal ?
noyes noyes no yes no yes no yes Stickers Reservoir
bit i ? bit i ? bit i ? bit i ? bit i ? Final "Bit Set"
off on off onoff on off on off on Output
Figure A Reliable set Op eration
If we cons ider where the complexe s are at some time t we see that the vast majority of them are near layer
t ht i leavingtherestofthemachineempty awaste Thi s ob s ervation le ads toanintermediate clas s of
compound
re nery algor ithms in whicha movingwindowof L S layers of the circuit are b e ingcontinually pro ce s s e d as in the
full parallel re nery algor ithm Since the di str ibution of complexe s i s f airly thin L can b e small thus requir ingless
space while achievingnearly identical p erformance
Notethatthe p erformance of theop erations set and clear can also b e improve d us ingthese ideas A
set operation can b e implemented bytwo comp ound s eparations the rst s eparating bas e d on the
universal tag and the second s eparating bas e d on the bit be ingset as diagrammed in Figure
The starting tube is seeded at the b eginning of the computation with an exce s s of stickers which
theuniversal s eparation recycle s Complexe s which f aile d to acquire thesticker are retur ned tothe
startingtube where they have another chance tohybr idize witha sticker A s imilar technique could
b e us e d for clear addinga step to pur ify stickers f rom PNA clamp s
Us ing re ner ie s
It is illustrative to cons ider us in g the re nery to solve a particular problem We will cons ider
bre aking DES for whichthe naive algor ithm A has S and W Let s suppose p
Us ingtheonelayer re nery algor ithm and N we incur a space f actor of and a slowdown of
no further slowdown help s thi s achieves We started with keys exactly
oneofwhic h is good We can b e sure except for in a million thatthegoodkey will endupin
the Yes tube but bad keys will b e incorrectly pro ce s s e d Will the incorrectly
processed complexe s also endupinthe Yes tub e as di stractors In thecaseofthe DES algor ithm
we argue thatthey won t endupinthe Yes tub e Adleman However we cannot makethesame
argument for gener ic algor ithms and so we cons ider the worst cas e scenar io in which all of the
incorrectly pro ce s s e d complexe s are di stractors In thi s cas e we nee d toachieve to get
the number of di stractors below With the one layer re nery thi s could either be re alize d by
incre as ing the space f actor to N and the slowdown to or by simply re running
the N vers ion mentioned above three times in a row giving a space factor of and a
slowdown of Thi s last approach is an intere sting example of what can b e further achieved
by composing thevar ious algor ithm transformations wediscussedabove
Conclus ions
ical molecular computer A number of previous In thi s pap er we have tr ie d to vi sualize a pract
concer ns Smith Hartmani s Letters to Science have been addre s s e d First it is now cle ar from
our own work and that of others that general purp o s e algor ithms can be implemented by DNA
bas e d computers potentially solvinga wide clas s of s e arch problems Second we nowunderstand
that there are challenging problems such as bre aking DES for which only mo dest volumes of
DNA e g grams should su ce Third we demonstrated that the formation and bre aking of
covalentbonds i s not intr ins icto DNA bas e d computation Thi s me ans thatcostly andshort lived
mater ials such as enzymes are not nece s sary nor are energetically co stly pro ce s s e s such as PCR
All the mater ials in the stickers mo del are potentially reusable from one computation to the next
Fourth we have shown that a single essential biotechnology s equence sp eci c s eparation su ce s
for constructingageneral purp o s e molecular computer Fifth wenowknowthatseparation errors
can theoretically be re duce d to tolerable levels by invoking a trade o between time space and
error rates at the level of algor ithm de s ign we have also illustrated several sp eci c ways in which
thi s can b e doneand pre s ente d encouragingnumer ical calculations of the ir p erformance
ve been overcome at a theoretical level sugge sts that re al appli That several ma jor roadblo cks ha
cations of molecular computation may be fe as ible in the future Nonethele s s we emphas ize that
Thus the exp ected numb er of di stractors will b e rst run s econdrun third run
substantial engineer ingchallenge s remain atalmostallstage s andthattheultimatesucce s s or f ail
ure of DNA computing will certainly dep endonwhether these challenge s can b e met in lab oratory
investigations
Acknowle dgments
Theauthors would liketoexpresstheir appreciation to Profe s sor John Baldeschwieler for hi s contr i
butions tothi s pap er through e arly di scus s ions of thi s work Sam Roweis and Er ik Winf ree are also
grateful tothe ir advi sor Profe s sor John Hop eld for hi s p erp etual wi sdom andlongterm advi ce
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