Klaus-Peter Zauner and Michael Conrad- Parallel Computing with DNA: Toward the Anti-Universal...

8/3/2019 Klaus-Peter Zauner and Michael Conrad- Parallel Computing with DNA: Toward the Anti-Universal Machine

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P a r a l l e l C o m p u t i n g w i t h D N A : T o w a r d t h eA n t i - U n i v e r s a l M a c h i n e

K l a u s - P e t e r Z a u n e r a n d M i ch a e l C o n r a dW ayne S ta te U niversi ty , Dept . o f Co m pu ter Sc ience ,D e t ro i t M I 48202, U S A , e -ma ih b i oc o mp u t i ng~ c s . w a yne . e du ,w w w : h t t p : / / w w w . c s . w a y n e . e d u / b i o l a b /

A b s t r a c t . A D N A - b a s e d b i o m o l ec u la r s tr in g p ro c e ss in g sc h em e d em o n -s t r a t e d by A d l e ma n ha s a t t r a c t e d w i de a t t e n t ion . W hi l e i t is no t know nto w hat d egree the scheme can sca le up , i t never the les s in t roduc es a newand in te res t ing co ncep t which seems so fa r to have been over looked. T hekey poin t i s tha t the Ad leman scheme involves bui ld ing spec if ic hard wa refor a single pro blem ins tance . Th i s op ens a des ign degree of f reedom th a ti s not l imi ted to b iomolecula r a rchi t ec tures .

1 I n t r o d u c t i o nD N A c o m p u t i n g , f i rs t d e v e l o p e d b y L . A d l e m a n [ 1] , r a is e s n o v e l q u e s t i o n s a b o u tt h e r e l a t io n s h i p b e t w e e n t h e s t r u c t u r e a n d f u n c t i o n o f c o m p u t a t i o n a l s y st e m s .T h e s c h e m e h a s p r i m a r i l y b e e n d is c us se d f r o m t h e p o i n t o f v ie w o f i ts p r o -g r a m m a b i l i t y [1 3]. T h e i m p l i c a t i o n is t h a t i t s h a r e s t h e m a i n p r o p e r t y o f a c o n -v e n t i o n a l d i g i ta l c o m p u t e r , n a m e l y th e p r o p e r t y o f b e i n g g e n er M p u r p o s e a n di n t h e l im i t b e i n g c o m p u t a t i o n u n i v e rs a l.

T h e u n i v e r s a l i t y p r o p e r t y h a s i n f a c t b e e n d e m o n s t r a t e d f o r t h e s c h e m e a sa w h o l e [ 1 1 ] . H o w e v e r , c a u t i o n i s n e c e s s a r y , s i n c e t h i s i s a c l a s s p r o p e r t y , n o ta p r o p e r t y o f t h e a c t u a l s y s t e m s t h a t c a n b e c o n s t r u c t e d . T h i s s h a r p d e v i a t io nf r o m t h e u s uM c o n c e p t o f u n i v e r s a li ty i s d u e t o t h e f a c t t h a t t h e s t r u c t u r e o f a n yg i v e n r e a l i z a t i o n i s s p e ci fi c f o r a p a r t i c u l a r p r o b l e m i n s t a n c e . N e v e r t h e l e s s su c he x t r e m e s p e c ia l p u r p o s iv e n e s s c a n i n p r in c i p le a f fo r d c o m p u t a t i o n a l a d v a n t a g e sf o r c e r t a i n p r o b l e m d o m a i n s . W e w il l i l l u s t ra t e t h e r e a s o n s f o r t h i s b y d e v e l o p i n ga cl os e a n al y si s o f t h e m a n n e r i n w h i ch t h e A d l e m a n s y s t e m w o r ks . T h o u g h t h es c h e m e d o e s n o t h a v e p r a c t i c a l v a l u e a t t h e p r e s e n t t i m e [ 10 ], i t s e r v es a s a g o o ds p r i n g b o a r d f o r id e n t i fy i n g f e a t u re s t h a t s h o u l d a p p l y t o o t h e r p o s si b le m a c h i n e sa n d t o b i o l o g i c a l o r g a n i s m s .

2 U n i v e r s a l i t y v e r s u s Speci f ic i tyI n t h e a b s e n ce o f s p a c e a n d t i m e b o u n d s g e n e ra l p u r p o s e m a c h i n e s w o u l d b eu n i v e r s a l [ 7]. T h e u n i v e r s a l T u r i n g m a c h i n e is t h e s i m p l e s t a n d b e s t k n o w ne x a m p l e . S u c h a m a c h i n e ( o r f o r m a l i sm ) c a n e m u l a t e a n y o t h e r m a c h i n e. A c t u a lm a c h i n e s a r e , o f c o u r s e , n e v e r u n i v e r s a l i n th i s a r b i t r a r y s e ns e, s in c e t h e y a r e

In Parallel Problem Solving from Nature: PPSN IV, Berlin. H.-M. Voigt and W. Ebeling andI. Rechenberg and H.-P. Schwefel (Eds.), LCNS Vol. 1141, pp. 696--705, Springer-Verlag,Berlin 1996.


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a l w a y s s u b j e c t t o m e m o r y a n d t i m e l i m i t a t i o n s . W e c a l l s u c h f i n i t e m a c h i n e sq u a s i -u n i v e r sa l a n d a d o p t t h e c o n v e n t i o n t h a t t h e s t a t e i n c lu d e s t h e m e m o r y .T h e p r o g r a m c a n t h e n b e r e g a rd e d a s e n c o d e d i n th e s t a te .

U n i v e r s a l i t y i s n o t a l w a y s a n a d v a n t a g e . I n g e n e r a l h i g h e r p e r f o r m a n c e c a nb e a c h i e v e d w i t h s p e c i a l i z e d h a r d w a r e . A t y p i c a l e x a m p l e i s a n a n a l o g c o m -p u t e r u s e d t o s o l v e a d i ff e r e n ti a l e q u a t i o n . T h e e q u a t i o n is f ix e d in t h e p h y s i c a ls t r u c t u r e , b u t t h e p a r a m e t e r v a l u e s a n d i n i t i a l c o n d i t i o n s c a n b e v a r i e d . T h eh a r d w a r e i n th i s c a s e is c le a r l y p r o b l e m s p e c if ic , s in c e t h e p h y s i c a l s t r u c t u r e h a st o b e c h a n g e d t o d e a l w i t h d i f f er e n t d i f f e re n t i a l e q u a t i o n s .

T h e d i ff e r en c e b e t w e e n q u a s i - u n i v e r s a l a n d p r o b l e m s p e c i fi c m a c h i n e s is t h a tt h e p r o b l e m is r e p r e s e n t e d i n th e s t a t e i n t h e f o r m e r ca s e a n d i n t h e p h y s i c a ls t r u c t u r e in t h e l a t t e r. T h e p h y s i c al s t r u c t u r e m i g h t b e v i ew e d a s a m a t e r i a ls t a t e , b u t t h e t w o c a s es d if f er w i d e l y i n t h e a m o u n t o f e n e r g y r e q u i r e d t o c h a n g et h e s t a t e . T h u s w h e n a d i g i ta l c o m p u t e r i s r e p r o g r a m m e d w e o r d i n a r i ly v ie w i t a st h e s a m e m a c h i n e , w h e r e a s a n a n a l o g c o m p u t e r ( o r a s p e c i a l ly c u t d i g i t a l c i rc u i t )w o u l d m o r e n a t u r a l l y b e v i e w e d a s a d i f f e r e n t m a c h i n e . A s n o t e d a b o v e , b o t hm a c h i n e t y p e s c a n p r o c e s s d i f fe r e n t i n s t a n c e s o f a p r o b l e m , a n d i n b o t h c a s e st h i s i s a c h i e v e d b y s t a r t i n g f r o m d i f f e re n t i n i t ia l s t a te s . F o r p r e s e n t p u r p o s e sw e w i ll t h e r e f o r e d i s t in g u i s h t h e s t r u c t u r e o f a s y s t e m f r o m t h e s t a t e o f t h i ss t r u c t u r e , u s i n g t h e l a t t e r t o r e f e r t o c h a n g e s i n a g i v e n m a c h i n e s t r u c t u r e .

T h e o p p o s i t e e x t r e m e o f u n i v e r s a li t y w o u l d b e a m a c h i n e t h a t e n c o d e s a si n-g le p r o b l e m i n s t a n c e i n it s s tr u c t u r e . W e w i ll c a ll s u c h a m a c h i n e a n instancemachine. T h e d i s t i n g u i s h i n g f e a t u r e o f a n i n s t a n c e m a c h i n e is t h a t i t is n o t p o s -s i b le t o p r o c e s s d i f fe r e n t i n s ta n c e s o f a p r o b l e m b y s t a r t i n g f r o m d i f f e re n t i n i ti a ls t a t e s . M o r e g e n e r a l ly , a n y c o m p u t i n g d e v i c e w i t h t h e f o l lo w i n g t w o p r o p e r t i e sq u a l i f i e s a s a n i n s t a n c e m a c h i n e :

1 . T h e p h y s i c a l s t r u c t u r e o f t h e m a c h i n e i s s p e ci fi c f o r a si n g le p r o b l e m i n s t a n c e( b u t w e e x c l u d e t h e t r i v i a l c a se o f a s y s t e m t h a t c a n o n l y b e u s e d t o a n s w e rq u e s t io n s a b o u t i ts o w n b e h a v i o r ) .

2 . T h e t i m e e v o l u t i o n o f t h e m a c h i n e le a d s to a s t a t e o r s t r u c t u r e t h a t c a n b ei n t e r p r e t e d a s a s o l u ti o n t o t h e p r o b l e m ( i. e. , r e p r e s e n t a t io n o f t h e p r o b l e mi n t h e s y s t e m s t r u c t u r e s h o u l d le a d t o t h e d e v e l o p m e n t o f a s o l u ti o n ) .

3 K e y f e a t u r e s o f D N A c o m p u t i n gT h e A d l e m a n s c h e m e m a y b e th e f ir st w h ic h e m p l o y s a c t u a l in s t a n c e m a c h i n e s.I n f a c t a ll re a l i z a t io n s o f i t a r e i n s t a n c e m a c h i n e s .

E a c h r e a l i z a t i o n i s a p r o b a b i l is t i c p a r a ll e l m a c h i n e w i t h a h i g h d e g r e e o f f in e -g r a i n e d p a r a l le l i sm . T h e n e w f e a t u r e i s t h a t t h e s e l f- a s s e m b l y p r o p e r t i e s o f D N Aa r e u s e d t o a c h i e v e t h i s h i g h d e g re e o f p a r a l l e l is m . P r o b l e m s a r e r e p r e s e n t e d w i t hD N A s e qu e n ce s . T h e s o l u ti o n o f t h e p r o b l e m p r o c e e d s t h r o u g h t h e f r e e e n e r g ym i n i m i z a t i o n a s s o c ia t e d w i t h t h e s e lf - as s em b l y ( o r a n n e a l in g ) o f t h e s e D N Am o l e c u l e s. T h a t t h e p h y s i c a l s t r u c t u r e o f t h e m a c h i n e i s s p e ci fi c f o r a s in g l ep r o b l e m is d u e t o t h e f a c t t h a t s p e c ia l D N A s e q u e n c e s a r e n e c e s s a r y fo r e a c hp r o b l e m i n s t a n c e .


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The actual computation proceeds in three phases (Fig. 1). First, in a synthesisphase, a structural representation of the input (i.e., the problem instance) iscreated. In the second phase the structures formed during the synthesis stageundergo a state to s tate development driven by free energy minimization. In thethird stage the result of the computation is extracted from an analysis of thefinal structure.

Let us concretize this picture by tracing an example through the three stages.Given a directed graph and a choice of a start and an end vertex in the graph,

Input

Synthesis _ I- I Free energy drivencomputation _ 1r I AnalysisOutput

Fig. 1. The three phases of the DNA based computation.

the question can be asked: does there exist a path from the start vertex to theend vertex that visits every vertex in the graph exactly once? Such a path, calleda (directed) Hamil tonian path, is frequently of interest in optimization problems.For the graph in Fig. 2 and the choice of A and D as the start and end verticesa Hamilton ian path exists and is indicated in the figure by solid arrow heads.

Fig . 2. Hamiltonian path problem.

The first phase of the computation is the encoding of the graph in DNA basesequences. To this end a short oligonucleotide of fixed size with an arbitrary,but unique base sequence is formally assigned to each vertex in the graph. Theoligonucleotides are illustrated in Fig. 3 as dashed boxes. The first half of thesequence is indicated by the lower case letter of the vertex and the second halfby the primed lower case letter.


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(~ - - 6 2 4F i g . 3 . E n c o d i n g o f t h e v e r t ic e s t h r o u g h b a s e s e q u e n ce s .

F r o m t h i s f o r m a l a ss i g n m e n t a m o l e c u l a r r e p r e s e n t a t i o n o f t h e g r a p h is d e -r ived a s fo l low s . For e ach edge in t he g raph a nuc l eo t ide sequ ence i s syn the s i zedw i t h t h e p r o p e r t y t h a t i t s fi rs t h a l f is id e n t ic a l w i t h t h e s e c o n d h a l f o f t h e s e-q u e n c e f o r m a l l y a s s ig n e d t o t h e v e r t e x f ro m w h i c h i t o r ig i n a te s . T h e s e c o n d h a l fo f t he seq uence t ha t r ep re sen t s t he edge i s i den t i c a l t o t he f i r st ha l f o f t he se-quence a s s igned to t he ve r t ex t ha t t he edge en t e r s . (See edge 2 i n F igs . 3 and 4a n d n o t e t h a t t h e s t a r t A a n d t h e e n d D a r e e x c e p t i o n s . )T h e n f o r e a c h i n t e r m e d i a t e v e r t e x , s u c h a s B a n d C i n t h e e x a m p l e g r a p h ,t h e o l ig o n u c l e o t id e c o r r e s p o n d i n g t o t h e c o m p l e m e n t o f t h e s e q u e n c e f o r m a l l ya s s igned to t he ve r t ex i s syn the s i zed . T he se t o f m olecu le s w hich rep re s en t s t heg r a p h d i s p l a y e d i n F i g . 2 is s h o w n i n F i g . 4 , w i t h c o m p l e m e n t s i n d i c a t e d b yb a r s o v e r t h e l o w e r c a s e l e tt e rs . T h e s t r u c t u r e o f t h e e i g h t m o l e c u l e s c o m p l e t e l yspec i f i e s t he g raph .

1 : 1 a I a ' ] b ] 4 : ] b ' I d I2:1 b ' I c I 5 :] c' ] b ]3 : l c ' l d I d ' I 6 : l d ' [ a IB: C:

d ' l

F i g . 4 . M o l e c u l ar r e p r e se n t a t io n o f t h e d i r e c t e d g r a p h s h o w n i n F ig . 2 a n d 3 .

A t t h i s p o i n t p h a s e 2 o f t h e c o m p u t a t i o n b e g i n s . T h e m o l e c u l es t h a t r e p r e -s e n t t h e g r a p h i n t e r a c t w i t h e a c h o t h e r t o f o r m s e l f - a s s e m b l e d s u p e r m o l e c u l a rc o m p l e x e s t h r o u g h t h e a s s o c i at i o n o f c o m p l e m e n t a r y b a s e s e q u e n c es . E a c h o f t h ec o m p l e x e s is a s t r u c t u r a l r e p r e s e n t a t i o n o f a p a t h i n t h e g r a p h . T h r e e e x a m p l e sf r o m t h e s e t o f p o s s i b le c o m p l e x e s ar e s h o w n i n F i g . 5 . I f a H a m i l t o n i a n p a t he x i s ts i n t h e g r a p h e n c o d e d i n t h e r e a c t a n t m o l e c u l e s t h e n w i t h h i g h p r o b a b i l i t ya s e lf - a ss e m b l e d c o m p l e x w il l b e f o r m e d t h a t r e p r e s e n ts t h i s p a t h .

P h a s e 3 in v o l v es e x t r a c t i n g t h e s o l u t i o n . T h e s e l f- a s s e m b l e d ( o r h y b r i d i z e d )c o m p l e x e s a r e h el d to g e t h e r b y h y d r o g e n b o n d s . S e q u e n c e s t h a t a r e li n e d u pn e x t t o e a c h o t h e r i n t h e s e c o m p l e x e s a r e f ir s t e n z y m a t i c a l l y li n k e d t o f o r m


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I a l a ' l b U b ' l d l ~ ' 1

I a I a ' l b I I b ' l ~ 1 1 r b Ir - - - - t i

I a l a ' l b l l b ' l c I I c ' l d I d ' l

Fig. 5. Examples of complexes produced in the self-assembly phase.

continuous strands. The strands are then analyzed to extract the molecules thatencode the Hamiltonian path, if such molecules exist. (More than one moleculemay be possible, first because the problem may have more than one solution andsecond because a given solution is typically encoded by more than one strand.)

Molecules corresponding to a Hamiltonian path must satisfy three conditions.The first is that the molecule must start with the sequence encoding the startvertex and end with the sequence encoding the end vertex. If in a polymerasechain reaction (PCR) the primers are chosen according to the sequences that en-code for the sta rt and the end vertex, only DNA segments tha t encode for a pathwith the correct start and end vertices will undergo exponential amplification(Fig. 6). Such specific amplification is the key feature of the PCR technique.

I a [ a ' I b I b ' l c I = ' l d l d ' l

I a l a ' l b I b ' l c I r d l d ' li t i " l

I : % a', > _Fi g. 6. Selective amplification by polymerase chain reaction (PCR).

The second necessary condition is that the molecules corresponding to prob-lem solutions have exactly the length of the sequences assigned to the verticesmultiplied by the number of vertices in the graph. Gel electrophoresis provides amethod of ensuring this by separating DNA strands migrat ing through a polymergel according to their length (Fig. 7).

The molecules that are known to encode the correct start and end verticesand to have the correct length are checked for whether they encode all inter-mediate vertices. Hybridizat ion probes of the sequences complementary to the


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a I a' [ b I b' ] c I c' I b I c' d l d' l" ' I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 . . . .

I a I a ' I b I b ' l c I c ' l d [ d 'l[ a I a ' I b I b ' I d I d ' I|

F i g . 7 . S e p a r a t i o n t h r o u g h g e l e l e c tr o p h o r e s i s .

o n e s f o r m a l l y a s s ig n e d t o t h e v e r t ic e s a r e u s e d t o e n s u r e t h i s t h i r d n e c e s s a r yc o n d i t i o n . E a c h v e r t e x is c h e c k e d s e p a r a t e l y . T h e p r o b e i s a t t a c h e d t o a s o l ids u p p o r t a n d D N A m o l e c u le s t h a t e n c o d e t h e v e r t e x i n q u e s t i o n w il l b i n d t o t h i sp r o b e ; o t h e r D N A m o l e c u le s c a n b e w a s h e d o u t . T h e p r o c es s is il l u s t r a t e d i nF i g . 8 f o r v e r t e x C .

r . a ] a ' ] b ] b ' ] c ] c ' ] d ] d ' - ]

F i g . 8 . I d e n t i f i c a t i o n w i t h h y b r i d i z a t i o n p r o b e s .

F i n a ll y , t h e m o l ec u l es th a t r e m a i n c a n b e s e q u e n c e d t o y i e l d t h e H a m i l t o n i a np a t h .

4 G e n e r a l I n s t a n c e M a c h i n e C h a r a c t e r i s t i c sA t f i rs t si g h t i t m i g h t s e e m t h a t i n s t a n c e m a c h i n e s , s u c h a s u s e d i n t h e A d l e m a ns c h e m e , w o u l d a l w a y s b e l e s s d e s i r a b l e t h a n q u a s i - u n i v e r s a l m a c h i n e s o r t h a nm a c h i n e s t h a t a r e d e s i g n e d f o r a s p e ci fi c c la s s o f p r o b l e m s . T h i s is n o t n e c e s s a r i l yt h e c a se . U n d e r s o m e c ir c u m s t a n c e s i n s t a n c e m a c h i n e s m a y b e a b l e t o s o lv el a r g e r p r o b l e m s o r b e a b l e t o d e a l w i t h a n i m p o r t a n t c a s e i n a s h o r t e r a m o u n to f t i m e .T h e D N A e x a m p l e c o n s id e r e d in t h e p r e v i o u s s e ct io n p r o v i d e s a u s e fu l p a r a d i g m .W e c a n u s e th i s e x a m p l e t o e li c it t h e m a i n c h a r a c t e r i s t i c s o f i n s t a n c e m a c h i n e s ,a n d t o c o n s i d e r h o w t h e s e b e a r o n i ss u es o f s iz e a n d s p e e d .

T a b l e 1 s u m m a r i z e s t h e m a i n c h a r a c te r i st i c s a n d c o m p a r e s t h e m t o t h o s eo f c o n v e n t i o n a l ( d ig i ta l ) c o m p u t e r s . R e c a l l t h a t s e l f -a s s e m b l y ( o r a n n e a l i n g ) i st h e k e y t o D N A c o m p u t i n g . T h i s m e a n s t h a t t h e s t a t e t r a n si t i o n s a r e p r i m a r i l y


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Table 1. Contrasts between universal and instance machine computingConventional Computer Instance Machine

State transi tion Constraint controlled Free energy dominatedDissipation Must be high enough to af-Cont ributes to integrity offord speed and reliability, low machine structure, allows forenough to sustain integrity ot higher speed and reliabilitymachine structure

Initial conditions Initial state is part of problemrepresentation, course of com-putation narrowly prescribedProgramming requires precisecontrol of system state, po-tential barriers limit numberof states

Potential size ofstate set

Minimum size ofstate set

Machine life cycle

State set must be largeenough to accommodate allclasses of tasks, typically onlya fraction of states used for agiven problem instanceUnlimited reuseability de-sired, reset mechanism mustbe built in

Initial structure (or familyof states) specifies problem,course of computation notsensitive to initial stateProgramming through con-trol of system structure al-lows for closely similar states,expanded state set possibleOnly states relevant to prob-lem instance need to besupported

Only limited reuseability nec-essary, in deterministic caseone run sufficient

driven by either energy minimizat ion or entropy maximization. In a digital com-puter, by contrast, energy is irrelevant to the course of the computation. Thedesigner takes great care to ensure that the different states of the machine are assimilar as possible from the energy point of view and that any differences thatdo occur are precluded from affecting the state transitions. This feature is keyto conventional programmability. The programmer therefore has the freedom toprescribe the course of the computation by setting constraints that restrict thedynamic degrees of freedom without considering energy/entropy aspects [12].Constraints play a role in all computational systems, including DNA instancemachines, but in the latter the energy differences between different states arethe main controlling factor. Consequently the dynamics are self-organizing.

Of course digital computers must be plugged into a source of energy andmust export heat to the environment. The energy serves to push the systemover the potential barriers that separate the different possible states. For thisreason dissipation is closely connected to both speed and reliability. It is possibleon paper to construct universal computing models that are practically reversible[2], but these systems are nearly as likely to run backwards as forwards. Also,to ensure that the system undergoes correct state transitions it is importantfor states to be separated by significant potential barriers. The constraints that


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encode the program followed by a digital machine limit the amount of dissipationthat is possible, however. Clearly the rate of heat export mus t be high enough sothat these constraints do not melt. This l imitation is much less severe for systemswith self-organizing dynamics, since the course of the computation as a whole,including structural changes, is driven by dissipation. The speed and reliabilityattainable is therefore potentially greater than for conventional machines.

The high parallelism of DNA computing actually has its origin in this speedand reliability property, since it depends on the high speed and reliability of DNAhybridization. The scheme also illustrates the distinction between programmingby structure preparation as opposed to state preparation. The outcome of a con-ventional computation is highly sensitive to the initial state, since this encodesthe program. The DNA computer, by contrast, encodes the problem it solvesin its initial structure, and therefore in a large family of states. The course ofthe computa tion is accordingly highly stable to perturbations, since it follows abasin of attraction. Unlike a conventional constraint controlled machine, it is notnecessary to support the existence of possible states that are never relevant tothe problem at hand. Furthermore, the number of possible states that a conven-tional machine can assume is limited by the requirement for significant potentialbarriers.

Since an instance machine by definition deals with only one instance of aproblem it is in principle unnecessary to reset it to its initial structure and runit again. Some rerunning would be useful for randomized computations and fortesting the machine. The potential advantage is that it is unnecessary to restrictthe design to materials with high reversibility (i.e., reuseability) and unnecessaryto support reset mechanisms. The latte r could be costly with free energy drivendevices. In the case of the Adleman system the effort and energy required to usethe same DNA bases for a different computation would be much greater thanthat required to start with a new batch of material. The machine is not only aninstance machine, but a throw away instance machine.

5 I s t h e r e a N i c h e f o r A n t i - U n i v e r s a l i t y ?For an instance machine to be worthwhile the problem instance would have tobe very important . The number of instances could be combinatorially explosive,but in practice it is often a particular instance that is of interest. Economicdecision makers, for example, often are presented with particular instances oflarge graph problems. Another example would be the need to rapidly recognize aparticular pa tte rn in a complex background. In some domains it may be sufficientto have solutions for a small number of arbitrary instances. This is the casewhen it is required to judge the quality of a particular heuristic, approximate,or suboptimal solution procedure. The availability of a small number of optimalsolutions provides a useful benchmark. Developing test sets against which toassess genetic algorithms would be an example [14].

DNA computing is far from being competitive with conventional machines inany of these domains, and may never become so. The Hamilton ian path problem


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originally used to illustrate its operation is NP complete. Thus it is almost cer-tainly the case (though not yet proved) that the number of resources required tosolve the problem increases exponentially with problem size. The Adleman sys-tem cannot overcome this combinatorial explosion, jus t as conventional systemscannot. The advantage that it could conceivably have would be its enormousparallelism that converts the exponential time burden that conventional ma-chines face into an exponential materials burden in terms of the amount of DNArequired. Currently, however, it is not possible to exploit this tradeoff due tonumerous practical limitat ions connected with the biochemical techniques avail-able.

But the instance machine approach illustrated by DNA computing carriesover to other technologies, where aspects of it may find more immediate applica-tion. This would be the case for conventional electronic and optical computers,where limits are set by the effect of state changes on machine structure. Forexample, the lifetime of a transistor is limited by electromigration (i.e., the ef-fect of switching operations on the distribution of atoms). This puts a limit notonly on transistor size, but on the materials and geometries that can be used.These limitations would be irrelevant to an instance machine since the numberof switching operations could be extremely small. Similarly, many materials withhighly desirable optical computing properties have been discovered but cannotbe used for conventional purposes because of low reversibility. The course of thecomputation in these designs would not be driven by free energy minimization,but admitting changes in the machine structure opens up a new design degreeof freedom.

Device proposals that utilize protein self-assembly for pattern recognition goa step further [5, 6]. Input signals are coded as molecular shapes, which then self-assemble to form higher level structures whose shape features represent differentclasses of input patterns. Enzymes specific for these shape features control theoutput of the device. The pa ttern recognition problem is thus converted to a freeenergy minimization process. Unlike Adleman DNA computing, which is alsobased on self-assembly, the protein self-assembly device is not programmable ina conventional sense, since a fixed set of formal (physics independent) rules is notavailable for ascertaining the effect of protein modifications. This is actually anadvantage from the s tandpoint of potential computing power, since the numberof interactions that can contribute to the computation is much less restricted.Such systems must be bred to perform desired functions through an evolutionaryprocedure [4, 3].

Biological evolutionary systems also make use of structure creating and de-stroying processes to maintain the potentiality of solving new problems withouthaving to pay the price of maintaining the capacity to solve all problems. Thisis possible because their problem solving capabilities are represented in theirstructure as determined by strong chemical bonds [15, 8], as it is in DNA com-puting or in the protein self-assembly design. The structure-based computingprinciple illustrated by the instance machine concept is arguably better suitedto the analysis of natural biological systems than is the state-based computingconcept utilized in conventional comDutin~ models.


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Klaus-Peter Zauner and Michael Conrad- Parallel Computing with DNA: Toward the Anti-Universal...

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