DNA-Based Self-Propagating Algorithm for Solving Bounded-Fan-In ...

DNA-Based Self-Propagating Algorithm for Solving Bounded-Fan-In Boolean Circuits Mitsunori Ogihara

Animesh Ray

Department of Computer Science University of Rochester Rochester, NY 14627 email: [email protected]

Department of Biology University of Rochester Rochester, NY 14627
email: [email protected]

ABSTRACT This paper proposes a method for simulating Boolean circuits based on primer extension and DNA cleavage. The advantage of the current method is the requirement of little human intervention during the course of simulation. The paper also explores the potential of RecA-assisted DNA-DNA hybridization in DNA-based computation. The method could in principle allow simulation of many levels of large Boolean circuits in a single test tube.

1 Introduction DNA computing is an emerging field bridging the gap between computer science and biochemistry. Following seminal work by Adleman (Adleman 1994), the potential of DNA as an alternative device for massively parallel computation has been studied (see (Ogihara et al. 1997) for a survey). Among various topics in this field, exploring methods for simulating abstract parallel computation models seems important. There are two major abstract parallel computation models, the parallel random access machine model (the PRAM model) and the Boolean circuit model. The former captures computation carried out by a network of computers that communicate with others through shared-memory. The latter captures computation by a network of logic gates that pass information from the input level to the output level. Although these computational models look very different from each other, their computational capabilities are quite “similar” in the sense that one model can be simulated by the other model with a reasonably small overhead in resources (Fortune and Wyllie 1978). Regarding DNA-based simulations of abstract parallel computation models, Reif (Reif 1995) introduced the PAM model,

 Supported in part by the National Science Foundation CAREER Award CCR-9701911 and the NSF Grant CCR-9725021. 

Supported in part by the National Science Foundation Grant MCB9630402.

which is capable of simulating PRAM with a small overhead. Ogihara and Ray (Ogihara and Ray 1997) then presented a model that is capable of simulating semi-unbounded Boolean circuits with a small overhead. This latter model is simpler and more direct than the PAM model because a Boolean gate of circuits is simulated in such a way that there is a unique DNA strand corresponding to a gate and that the gate outputs one if and only if the strand is present in the test tube at some point during the computation. This model has been further simplified in a subsequent paper (Ogihara and Ray 1998), which proposes seemingly the smallest meaningful DNA computation model consisting of just five simple operations, i.e., synthesize, merge, anneal, separate-bylength, and detect, and characterizes its computational power in terms of Boolean circuit complexity classes. Furthermore, these fundamental operations are combined so that all uncertainty involved during the biochemical operations is handled to make the computation robust. There are a number of issues to be addressed when designing DNA-based methods for simulating Boolean circuits. The first one is the choice of a computational basis. The standard basis consists of the logical-AND, the logical-OR, and the negation. An alternate basis is the logical-NAND (the negation of logical-AND); it is a standard exercise to use only logical-NANDs to simulate the three functions in the standard basis. It is a reasonable assumption that DNADNA hybridization is inevitably used for simulating logical units. However, DNA-DNA hybridization is not exhaustive but rather reaches equilibrium, thereby leaving some DNA strands intact. Therefore, there may be some “noise” for each operation step, which could cumulatively fail the computation. Also, the equilibrium nature of hybridization suggests that simulation of any gate that converts negative information to positive, and vice versa, is a very difficult problem to implement—one will always have to worry about errors caused by strands that are not converted. An interesting paper (Amos et al. 1997) described a simple scheme for simulating the NAND-gate. Based on this scheme, the authors proposed a method for simulating Boolean circuits. However, the problem of coping with errors due to DNA strands that remain intact because of chemical equilibrium is yet to

be solved. In order to overcome this problem, we (Ogihara and Ray 1997, Ogihara and Ray 1998) used the deMorgan’s law to push all the negation gates to the input level1 where simulation is carried out by DNA synthesis without having to use DNA-DNA hybridization. This allows the specification of inputs  and  as two different DNA strands with no sequence relationship whatsoever. The second issue is feasibility of the methods. This issue is important because it is meaningless to discuss methods based on unreliable techniques. In all the three papers mentioned earlier, simulations are carried out by well-examined biochemical operations. The third issue is the speed of the simulation. The bottleneck of DNA-based computation is, no doubt, humanintervention. Whenever DNA strands are poured into or separated from test tubes, there is a need for a large amount of work by human (or by robots that do the same job with a higher accuracy), which could slow down simulation processes. The focus of the current work is the third issue. We investigate the possibility of automated simulation of multiple levels of Boolean gates. In the simulation methods proposed in previous papers (Amos et al. 1997, Ogihara and Ray 1997, Ogihara and Ray 1998), Boolean gates are stratified and the simulation is carried out level-wise from the input level to the output. These simulation designs mandated human-intervention, which incurs a large amount of overhead to computation time. A natural solution to the problem would be to design a “single-pot” method for circuit simulation in which the whole computation is conducted in a single test tube without having to separate intermediate products. Here we propose such a method. As in the previous methods (Ogihara and Ray 1997, Ogihara and Ray 1998), each gate of a simulated Boolean circuit is assigned a unique DNA single strand, and the presence of the strand at any time of the reaction is regarded as the gate outputting a  . Our novel approach is that the the strands corresponding to the outputs of gates are synthesized by primer extension and cleavage of phosphodiester bonds with restriction enzymes. The strands from one level of the circuit become templates and trigger primer extension for the next level, and the extended patterns are again cut out by restriction enzymes. In order to conduct a large scale circuit simulation with DNA, a crucial consideration is how rapidly sequencespecific DNA-DNA hybridization occurs when billions of DNA strands are present. This paper suggests the use of RecA protein (Kowalczykowski and Eggleston 1994) for assisting DNA-DNA hybridization. With the use of RecA protein, the reaction can be accelerated to the speed of one level per second if there are at most  

strands present in the test tube. This analysis leads to a plausible conclusion that the power of DNA parallel computation matches that of conventional com1 With





the deMorgan’s law is converted to and to . We repeatedly apply these conversion rules to move the negations to the input level. Such a conversion only doubles the size.

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2 Preliminaries First we introduce some notions and notation. A “pattern” is an ordered list of letters from &('*) +-,/.-,/0-,21-3 without directionality. For a pattern 45'64

#787 7 49:,/4; 4> , is its base-length ? , the reverse of 4 , denoted by 4@ , is 49 78787 4 , and the complement of 4 , denoted by 4 , is the letter-wise complement of 4 . A pattern with directionality is a single strand. Directionality is given by specifying which ends of the patterns are the ACB -end and the DEB -end; that is, ACB -4 -DEB is the single DNA strand with pattern 4 such that 4 is at the ACB -end and 4 9 at the DFB -end. On the other hand, in DEB -4 - ACB , the two ends are exchanged. For two patterns G and H , G H denotes the pattern constructed by concatenating 7 H after G . For patterns GI,/H$,KJ , we use

A B- G DFB - L H H

J

-D B - AEB

L

to denote the partial double strand produced by binding A B -G$H -D B and D B -HJ -A B (or, A B -JF@ H @ -D B ) by DNA-DNA hybridization between ACB -H - DEB and DEB -H - ACB . Primer extension works on such a partially double strand and completes the double strand toward the DFB -end. We will use

ACB - G DEB - N H

H

M J

-DEB - ACB

to denote the double strand produced by primer extension.

3 The circuit simulation 3.1 A formal description of the method

O be a bounded-fan-in circuit of depth P and QR' Q :7 787 -Q S an input to O , such that we want to compute OUTVQW . Assume that O is a full binary tree, P is even, levels of +XY gates and those of ZE[ -gates alternate in O , and the top level of O consists of a single ZC[ -gate. In general, an arbitrary Let

bounded-fan-in circuit with multiple outputs can be converted to a collection of such full-binary-tree circuits of equal depth. We work from the output level toward the input level and if there is a gate \ whose output is connected to more than on gate at the next level, we create multiple copies of \ so that for each copy of \ there is a unique gate at the next level to which the copy is connected. This conversion increases the size of the circuit by a multiplicative factor of at most ]F^ . Our simulation method can be concurrently run on all these trees. Let _`'a_

#787 7 _b9 be a binary string of length ?c,/(d ?edfP . Denote by \Eg the gate corresponding to the path _ from the output gate of O , where the branch to the left input is represented by a bit  and the branch to the right input is represented by a bit  . For each _ of even length, assign to \Eg patterns h-g and ig , each of length j , and define klgm'nA B -h-giog -D B , where j is much smaller than

pqErsltuv8w

TxOyW . These patterns are chosen so that the following conditions are satisfied: 1. Every h

g

starts with a pattern AEB -0+ - DEB .

2. These patterns are all distinct. 3. If _ and _ B differ only at the last bit, then A B - iog -D B and ACB - i gz -DFB are complementary antiparallel to each other; that is, i g @ '{i gz . 4. Other than the above, no patterns |},~ from these h ’s and i ’s are complementary antiparallel to each other. Also, for each _ of even length, define

€

€ g‚'ƒAEB - iog @ h-g @ h-g -FD B , and g 'ƒACB - iog @ h-g @ $h g K -ED B .

We call these strands preset templates. Note that the strand € B  g  is A h-g @ h-giog -D B , and complementary antiparallel to € @ that of g is ACB -h-g

/ h-giog -DFB . Also note that these com

plements are cut by the restriction enzyme „$…E†ˆ‡ 2 to yield kIg . These preset templates will be synthesized using methylated + (a methyl group attached to a nitrogen atom in the adenine) so that they are resistant to the cut by „$…E†ˆ‡ . Such a methylation does not affect Watson-Crick base-pairing rule. Our simulation proceeds so that the following condition is met: (*) For every _ of even length, the ZE[ -gate \ g outputs  if and only if at some point in the reaction, k € g is created by primer€ extension either on a template g or on a template g and by subsequent cuts by „…F†ˆ‡ .



For that matter, we first create a test tube in which (*) is met for all the input gates of O . Now we pour into this test tube copies of all the preset templates, which are methylated, together with unmethylated DNA molecules and the restriction enzyme „$…E†ˆ‡ , and let the primer extension and cleavage occur. We show that for every _ , (*) will be satisfied at some point of the reaction. In particular, once (*) is satisfied for the empty string (namely, for the output gate), the condition will hold until the end of the whole simulation. Let _ be an even-length string specifying an ZE[ -gate. Assume that (*) is satisfied for all _‰B under _ . We show that (*) holds for _ , too. First suppose that \Eg outputs  . Since \Eg computes ZE[ of two +XY -gates, without loss of generality, we may assume that the left input to \ g is  , and thus, \ gK and \ g both output  . So, by (*) k gK and k g are both created by primer extension. Since the second part of 2 Several restriction endonucleases cleave the unmethylated strand in a hemimethylated target, e.g., EcoRI (Jen-Jacobson et al. 1996), HaeIII, BsuR, and Sau3AI (Sambrook et al. 1989). However, these are not thermostable. Perhaps the most thermostable one is TaqI, which cleaves unmethylated - duplexes, but not a fully -methylated target. The hemimethylated target is cleaved but the strand preference is yet to be determined. It also cleaves single stranded unmethylated targets slowly, which may suffice for our purpose.

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

these strands are complementary antiparallel to each other, by DNA-DNA hybridization

ACB - $h gK“iogK L D B- L iogK”h$g @

-DEB -A B

is produced. Next the primer extension

ACB - $h gK“iogK DEB - N i gK

-DEB M h g  @ -ACB occurring on this strand produces AEB - h gK i g/ h g @ -DFB and

its complement€ DEB - h g/ i gK h g @ - ACB . The former will then

hybridize with g '6DEB -h g @ h g i g - AEB to form

ACB - h$gK“iogK h-g @ L L -DEB @ DEB - L L h g h g i g - ACB The primer extension of this will produce

A B -h$gKˆig/ h$g @ $h giog -D B , i.e., A B -klgK h-g @ kIg -D B , as well as its complement. Now the pattern straddling h-g @ klg is recognized by the restriction enzyme, and it is cut on the upper side only. (The lower side, namely the template side, has methylated DNA molecules at the cleavage site, so it is not cut.) Therefore, k g is produced. So, (*) is satisfied for _ . On the other hand, suppose that k g is produced. Since h g does not appear on any preset templates, the only posk € g is by primer extension on one of sible way to produce € the templates g and g . The only primers that can trigger primer extension on these templates are AEB - h$g @ -DFB and ACB -h$g K @ -DEB , respectively. These patterns do appear in the preset templates, but not as the DFB -ends; these templates are followed by some other patterns. Since all the preset templates are methylated, the primers cannot be produced from the preset templates, so they have to be synthesized by primer extension. Now the only template that allows synthesis of ACB -h g @ -DEB and ACB -h g / @ -DEB are ACB -k g -DFB and ACB -k g / -DFB , respectively, and the only primers that can attached to this are AEB - k gK -DEB and ACB - k g  - DEB , respectively. Therefore, either

ACB -k gK -DFB and AEB -k g -DEB are both present, or AEB -k g  -DEB and ACB -k g K -DFB are both present. By our assumption, this implies that \Eg outputs  . Thus, (*) holds for _ , too.

3.2 Schematic explanation of self-propagating circuit simulation The basic idea of this method is to let DNA strands interact among others to generate an output. Consider the following circuit of depth three in Figure 1. The bottom level of the circuit can be either all AND gates or all input gates. We assign unique single DNA strands ,K‰Bx,/• ,K•–B—,KO!,OB—,/P , and P B of equal length to the gates at the bottom. Here the strands are so designed that the second halves of  and ‰B (respectively, those of • and •–B , of O and OB , and of P and P B ) are equal and the second half of the  ’s (respectively, O ’s) is complementary antiparallel to that of the • ’s (respectively, P ’s). Assume that in our test tube, say ˜ , there are all, some, or none

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Figure 1 (a) the depth-3 circuit, (b) simulation of the AND gates, (c) synthesis of ä and å , (d) cutting ä and å , (e) synthesis of æ of these eight patterns so that each pattern is present in ˜ if and only if the gate corresponding to it outputs  . Suppose we let the strands in ˜ anneal and let primer extension occur. Then the gate \ outputs  if and only if at least one of  ’s

and at least one of • ’s are present, so the sequence with the pattern complementary antiparallel to the • (or •–B ) is synthesized. We s call the output | (or |B ). A similar situation holds for \ with O in place of  . We call the outputs ~ and ~–B . We will synthesize from either | or |B a new strand ä . This is done by two steps. We prepare a methylated template, which is the complementary antiparallel of ä followed by the complementary antiparallel of | as well as another with |B in place of | . Here the pattern ä is chosen so that its AEB end is recognized by a restriction enzyme that cuts only the methylated strand of a hemi-methylated duplex target. Now if | or |B has been synthesized, then annealing and primer extension creates ä , and this part is not cut due to the nature of the restriction enzyme. Suppose we have now put into the test tube a new methylated template with a blocked D B -end, which is the complementary antiparallel of ä . Then the synthesized ä will hybridize with this methylated template and the restriction enzyme will cut the upper half to unleash ä . A similar construction is done for å . Now the OR of the presence of ä and å are examined by a new template with the complementary antiparallel of a pattern æ at the AEB -end. The whole process in the above can be done in a single test tube. If we mix all the templates in ˜ and set the temperature so that DNA strands repeatedly renature and denature at equilibrium (a suggestion of J. Wetmur and L. Adleman), then in the case when \ outputs  , the strand corresponding to \ will

Figure 2

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A small example

be synthesized with high probability, but in the case when \ outputs  , the probability is  . It is expected that physical implementation of a large Boolean circuit will require substantial manipulation of reaction conditions. For this purpose, the output of any gate within the circuit can be monitored individually and optimized empirically.

3.3 A small example for experimental simulation We illustrate with a small experimental circuit how the simulation is designed. Consider the circuit in Figure 2. The simulation of this circuit involves chaining the outputs from level 1 through level 3. This simple circuit does not require cleavage of primer-extended product by a restriction endonuclease, but every other step is necessary. The four input variables at the bottom are represented as:

Q

s Q

' '

Q$úû' Q

'

The gates \

A B -ôöõ÷ùøbø#ôöõ÷ùôoøöõyõ÷}õ÷ùøIôoøöõ÷ A B -õyõôoø#ôoø‰÷}õôoøöõyõ÷}õ÷ùøIôoøöõ÷ A B - ÷}õôoøbø#ôöõ÷ùøöõyõ÷ùôoøIôöõ÷}õ÷b÷ A B -ô‚ôoøöõøöõô‰ô‰ô‰÷}õ÷ùôoø#ôöõ÷}õ÷b÷ and \

-D -D -D -D

B, B, B , and Bxü

s

are

A B - ÷ùøöõøöõô‰÷b÷ùôöõôöõ÷}õøIô‰÷ùø‰÷b÷ùøbø#ô‰÷ùô‰÷ùøbøbøöõ -D B

and

A B - ÷ùøöõøöõô‰÷b÷ùôöõôöõ÷}õø#ô‰÷ùø‰÷b÷}õ÷ùøIô‚ôoø‰÷}õô‰÷ -D B ü Q$ý

, one of the input to the level 3 AND gate, is

A B - ÷ùô‰ôöõyõ÷b÷ùô‰ôöõyõø‰÷b÷ùøƒþ ÿ þ õyõôoø#ôöõô‰÷ùøöõøöõô‰÷b÷ùôöõ -D B ü Note that in this simulation model, the two inputs of an AND gate together define the gate. s In the first step, either Q anneals to Q-ú or Q anneals to

Q$ú , and the primer extension synthesizes the new sequence:

AEB - 78787 ÷ùôö s õ÷ùøIô‚ôoø‰÷}õ -DFB . Also, either Q anneals to Q or Q anneals to Q , and primer extension product is the new sequence: A B 787 7 õøbøbø‰÷ùô‰÷ùôoøbø -D B . If either or both new sequences are produced, then they anneal to s either \ or \ , respectively, and primer extension pro

duces: ACB - 787 7 ÷b÷ùø‰÷ùôoøöõ÷}õôöõô‰÷b÷ùôöõøöõø‰÷ -DEB . If this sequence is produced, then it anneals to Q-ý , and further primer extension produces: AEB - 78787 ôöõôoøIôöõ÷ùø‰÷b÷ùøöõyõô‚ô‰÷b÷}õyõô‰ô‰÷ -DEB . The synthesis of this sequence in the test tube is equal to an output of  , which may be detected by hybridizing the content to its complement that has been affinity tagged. The entire circuit can be simulated by mixing all the oligonucleotides (Q , ü8ü8ü ,2Q-ý , \ , ü ü8ü ,2\ ), a thermostable



polymerase for primer extension, dNTP mixture, appropriate salt and buffer, and by cycling the temperature several times. In this small-scale simulation, the length of complementary  pairing is 10 bases (˜9
DFA ÷ ). However, there is no special reason for choosing only 10 base overlap for primer annealing; in fact, longer overlap should decrease error and be useful for RecA assisted search (see below).

4 Limits to the Simulation and RecAAssisted Search What is the practical limit of our DNA simulation? Since  our circuits are full binary trees, the size of those sof depth is  

 . With ' ]F , the size is around   . If we hold the initial amount of each oligonucleotide at 100 femtomole s per base, for a circuit of depth ] , we will use at  ú mole) (    most  

 ´ ú , i.e.,  ü  mole of bases. The second order rate constant for DNA-DNA annealing is approximately   M´  s´  (Cantor and Schimmel 1980, page 1272). There s fore the initial rate of annealing of two oligonucleotides, each present ins     ú M concentration (per base) is µ T2 ´  ú W ,   . Thus, approximately   ú duplex per liter i.e., µ  Ms ´ will be formed per second, which is a very fast rate. If on the other hand, a circuit of depth ] (of size approximately µ

) is simulated, in order to keep the DNA concentration to  ü  M we must use  femtomole of each oligonucleotide. Because of the second order kinetics of DNA annealing, this will produce approximately  duplex per second per liter. In order to overcome the kinetic barrier, we propose to examine the use of RecA protein (Kowalczykowski and Eggleston 1994) from Escherichia coli to speed up the homology search process. RecA protein subunits bind to singlestranded DNA and the complex searches for sequence homology by a second order reaction kinetics but catalyzes strand transfer by a first order reaction with a rate constant of A   ú s ´ (Gonda and Radding 1986, Bazemore et al. 1997). The rate of primer extension is approximately 300 nucleotides per second. With these considerations, we estimate that the best rate for a circuit of depth at most 40, which is of size at most µ

, will be approximately 500 seconds. There is one other requirement: the target for the RecA

search must be approximately 150 bases long. Since 150 base long oligonucleotide is difficult to synthesize, these can be made by PCR amplification of selected naturally occurring sequences, followed by strand separation. PCR amplification can be in the presence of modified DNA base precursors for synthesis of the template strands.

5 Biochemical Implementation Each gate oligomer is synthesized as before (Ogihara and Ray 1997) except that the N6 of adenine residues are methylated and the D B -ends are blocked (using dideoxyribonucleotides at the DFB -terminus). The DNA computer will consist of a single tube containing equimolar amounts of each gate oligonucleotide, nanomolar amounts of the four deoxyribonucleotide triphosphates (unmethylated), appropriate salts and empirically determined amounts of a thermostable DNA polymerase, and the „$…E†ˆ‡ restriction endonuclease. RecA assisted searching becomes important only for circuit sizes exceeding   / (corresponding to a depth of 36). In those cases, thermostable RecA protein (Yu et al. 1995) will also be present in the mixture. Note that ATP is not essential for searching and strand transfer of short (15-30 bases) oligomeric sequences by RecA (Hsieh et al. 1992). The computation will be initiated by adding a mixture of all the input variable nucleotides corresponding to the value of 1 to the computation tube, and raising the temperature to optimum for primer extension. For  15-25 nucleotide overlaps, approximately 65 C should be optimal. At this temperature annealing (assisted by RecA when present), primer extension and restriction of the newly synthesized molecules occur. The temperature can be slowly varied around the average ˜ 9 , at an empirically determined rate, such that dissociation of the primer extended product and reannealing to the new template can be facilitated. If one of the dNTPs is fluorescent-labeled, then the output can be detected by pouring the reaction content over a ‘DNA-chip’ with the complements of the outputs immobilized on the surface (Chee et al. 1996). Annealing of the fluorescent labeled molecules will be read out by a confocal microscope. The outputs of one circuit computation may be used to determine the input of another round of computation in a fresh tube. In this way circuits of depth exceeding 40 can be processed in series. Whether computation of such massive parallel circuits will be feasible in practice depends on the practical problems in biochemical implementation and on the cost of making DNA strands.

6 Conclusion We proposed a DNA-based method for evaluating multiple levels of a Boolean circuit in a single test tube. The methods previously proposed required manual intervention between every level of the circuit. By contrast, the present method computes the entire circuit automatically from the input to the output. Although this “self-propagation” method appears theoretically possible, its practical feasibility remains to be

tested.

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Reif, J. (1995). Parallel molecular computation. In: Proceedings of 7th ACM Symposium on Parallel Algorithms and Architecture. ACM Press. pp. 213–223. Sambrook, J., E. F. Fritsch and T. Maniatis (1989). Molecular Cloning: a Laboratory Manual. 2nd ed.. Cold Spring Harbor Press. NY. Yu, X., E. Angov, R. Carmerini-Otero and E. Egelman (1995). Structural polymorphism of the RecA protein from the thermophilic bacterium Thermus aquaticus. Biophysical J. 69(6), 2728–2738.