A Parallel Implementation of the Modus Ponens Inference Rule in a DNA Strand-Displacement System

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1 A Parallel Implementation of the Modus Ponens Inference Rule in a DNA Strand-Displacement System Jack K. Horner P.O. Box 266 Los Alamos NM USA PDPTA 2013 Abstract Computation implemented in DNA reactions promises to advance high-performance computing (HPC) for at least three reasons. It (1) is inherently Amdahl-scalable by reactor-volume, (2) has a power/operationsper-second(ops) ratio that is potentially orders of magnitude smaller than that of silicon circuits, and (3) can provide a natural access-interface to DNA-based high-density information storage. In order to serve as general-purpose computing regime, DNA computing will have to support Boolean operations. Here, I describe an implementation of the modus ponens inference rule (commonly used in Boolean logic) in a DNA strand-displacement (DSD) system. Keywords: DNA computing, DNA strand displacement, modus ponens 1.0 Introduction Computing implemented in DNA reactions promises to advance high-performance computing (HPC) for at least three reasons. It: Is inherently Amdahl-scalable ([6]; [7], p. 39) by reactor-volume Has a power/operations-persecond(ops) ratio that is potentially orders of magnitude smaller than that of silicon circuits ([7], pp ; [8]) Provides a natural access-interface to DNA-based high-density information storage ([2]). DNA strand displacement A variety of information-processing circuits, including a catalytic gate ([3], an implementation of which has been tested in vitro), have recently been implemented in DNA strand displacement (DSD) reactions. In a DSD reaction, portions of a strand of DNA in one reactant displace portions of a strand in another DNA reactant. A DSD simulation system is described in [4] and [5]. Some examples of DNA molecules represented in the language defined in [5] follow. ( A "stroke" ( ) denotes the juxtaposition of multiple molecules.)

2 Figure 1. Some examples of DNA molecules expressed in the language of [5]. In Figure 1, the notation 1:2 represents a lower strand of DNA, where the 3 end of the strand is on the left, denoted by an arrowhead. The strand is divided into domains, which correspond to short DNA sequences. In the leftmost element of Figure 1, the domains are represented by numbers 1 and 2, where each number represents a distinct domain. In Figure 1, the red domain 1 represents a toehold domain, while the black domain 2 represents an ordinary specificity domain. Toehold domains are very short sequences, generally between 4 and 10 nucleotides in length, that enable one DNA strand to bind to another. Since toehold domains are short, two strands bound to one another will quickly unbind in the absence of further interaction along neighboring domains. In Figure 1, the notation <1 2> represents an upper strand of DNA, where the 3 end of the strand is assumed to be on the right. The strand consists of two domains that are complementary to domains 1 and 2 of the leftmost element of Figure 1, where two domains are complementary if their respective sequences are Watson-Crick complementary. Two complementary strands, 1:2 and <1 2>, can hybridize along their complementary domains to form a double-stranded molecule, denoted by [1 2], as shown in the third element from the left in Figure 1. A molecule can also consist of multiple upper strands bound to a single lower strand. For example, [1 2]:[3 4] denotes a DNA molecule that consists of upper strands <1 2>and <3 4> bound to a single lower strand 1:2:3:4. There can also be gaps between bound upper strands as in the molecule [1 2]:3:[4 5], where domain 3 of the lower strand is unoccupied (see rightmost element of Figure 1). Bound upper strands can also overhang to the left or right, as shown in Figure 2. Figure 2. Three examples of overhanging strands, expressed in the notation of [5]. In Figure 2, the molecule <1>[2 3]<4> consists of an upper strand< > bound to a lower strand 2:3. The region[2 3] of the molecule is double-stranded, while <1>and <4> represent single-stranded regions overhanging to the left and right. The molecule [1]<2>:[3] consists of an upper strand <1 2> bound to a molecule1:[3], where the single-stranded region <2> is overhanging the double-stranded region [3]. Multiple overhanging strands can be bound simultaneously along different regions, as in the case of the molecule<1>[2 3]<4>:<5>[6 7]<8>, which represents two upper strands, < > and < >, bound along regions [2 3] and [6 7], respectively. Notice

3 that the colon is used to separate the two bound upper strands. A given strand can also be displaced by another strand as a result of binding, as shown in Figure 3. Figure 3. An example of a strand displacement reaction in the notation of [5]. Although toehold domains are short enough to unbind rapidly in the absence of additional specificity domains, they are still long enough to greatly accelerate the initiation of strand displacement when additional specificity domains are present. In Figure 3, when the strand <1 2> becomes bound it initiates the displacement of its neighboring strand by a process of branch migration. Although this process involves a random walk of multiple elementary steps, these are relatively fast at experimental concentrations and can be omitted ([3]). This means that the unbinding reaction on toehold domain 1 in Figure 3 can be effectively ignored and the two consecutive reactions can be approximated by a single displacement reaction. In general, the DNA molecules are assumed to have no additional secondary structure. This can be achieved by careful selection of appropriate DNA sequences ([3],[4]). In addition, DNA sequences of distinct domains are assumed to be sufficiently different that they do not interfere with each other. For further detail, see [4] and [5]. Modus ponens Modus ponens, sometimes called the "rule of detachment"([9], p. 47), is an inference rule widely used in Boolean logic ([1]). It allows us to infer a proposition ψ from a conjunction of propositions of the form φ -> ψ and φ, where φ and ψ range over propositions and -> denotes Boolean implication ([1]). Implementing modus ponens in [4] requires mapping the elements of the rule (i.e., φ, φ -> ψ, and ψ) to elements of DNA circuits that are implementable in [4]. 2.0 Method Figure 4 shows a mapping of "φ" and " φ -> ψ" to the DSD strand-definition language of [5]. Figure 4. Mapping of φ and φ -> ψ to input species of [4]. The top species (the "substrate", in catalyst-gate nomenclature) represents the proposition φ -> ψ. The bottom species (the "fuel", in catalyst-gate nomenclature) represents the proposition φ.

4 Figure 5 shows the DSD script for the catalytic gate described in [3] (without the "reporter" molecules of [3]), implemented in the language described in [5], using the input descriptors shown in Figure 4. The reaction product of interest is the sequence <1 2>, which represents ψ. (* DIRECTIVES AND DOMAIN-DEFINITION SEGMENT *) directive duration points 1000 (* run sim for 7000 sec, save 1000 points *) (* directive leak 1.0E-9, default leakage rate /nm/s *) (* directive tau , default merged rate /s *) (* directive migrate , default nucleotide migration rate /s *) (* directive lengths 6 20, default toehold and normal domain lengths *) (* directive tolerance 1.0E-6, default ODE tolerance for deterministic simulator *) (* directive time s, default time units *) (* directive concentration nm, default concentration units *) directive plot <2 3^ 4>; <1>[2]:<6>[3^ 4]:5^*; <1 2> (* plot a subset of strands *) directive scale (* multiply concentrations, divide binding and leak rates *) new 3@ 4.2E-4, 4.0E-2 (* initialize *) new 5@ 6.5E-4, 4.0E-3 (* initialize *) (* PROGRAM SEGMENT *) ( 13 * <2 3^ 4> 10 * <4 5^> 10 * <1>[2]:<6>[3^ 4]:5^* ) Figure 5. DSD code used in this study (adapted from [4]). The script shown in Figure 5 was executed as a "deterministic" system under [4] on a Dell Inspiron 545 with an Intel Core2 Quad CPU Q8200 clocked at 2.33 GHz, with 8.00 GB RAM, under Windows Vista Home Premium/SP Results Figure 6 shows the reaction graph produced by the method described in Section 2.0.

5 Figure 6. The reaction graph produced by the method described in Section 2.0. Rectangles with darker borders represent inputs; all other rectangles represent intermediates. Arrows represent reaction direction. The output of interest is the strand <1 2>, which represents ψ. ψ is "detached" by the system from the main reactant of the system, <1>[2]:<6>[3^ 4]:5^* (modus ponens is sometimes called the "rule of detachment"). Strand <4 5> is the catalyst of the system. ( See Figure 7 for concrete realizations of the sequences in the diagram.) Figure 7 shows a concrete DNA segment-description realization, generated by [4], of the species in Figure 6. Figure 7. A concrete DNA segment-description realization of the DNA species in Figure 6.

6 Figure 8 shows part of the output of DSD simulation of modus ponens under the conditions described in Section 2.0. Figure 8. Output of the simulation described in Section 2.0. The green line represents the proposition φ -> ψ (the "substrate", in catalytic-gate nomenclature). The red line represents φ (the "fuel", in catalytic-system nomenclature). The blue line represents the production of ψ (= " <1 2>") from modus ponens ("(φ and φ -> ψ) implies ψ "). The system is Amdahl-scalable by reactor-volume. In actual practice, a "reporter" species (typically, a dye) would be used to visualize the production of ψ. Compilation of the system described in Section 2.0 took ~3 seconds. The subsequent simulation utilized ~25% of the CPU and ~0.5 GB memory on the platform described in Section 2.0, as measured on the system monitor, and executed in ~0.1 second. 4.0 Conclusions and discussion The method described in Section 2.0 and the results described in Section 3.0 motivate several observations.: 1. Modus ponens can be implemented in a DSD system. 2. The implementation is Amdahlscalable by reactor-volume.

7 3. The implementation shown here could be easily abstracted to a DSD module ([5]) which would hide the representation internals of the implementation. 4. In principle, any catalytic system that, in the presence of a molecule A, "detached" a molecule B from a complex molecule C that contained both A and B, could be used to model modus ponens. 5. Other, perhaps simpler, implementations of modus ponens may be possible in [4]. The implementation described here, however, is known to work because the catalytic gate schema on which it trades has been implemented in vitro ([3]). 5.0 Acknowledgements This work benefited from discussions with Tony Pawlicki. For any problems that remain, I am solely responsible [5] Lankin MR, Petersen R, and Phillips A. Visual DSD user manual v0.14 beta [6] Amdahl G. Validity of the single processor approach to achieving large-scale computing capabilities. AFIPS Conference Proceedings 30 (1967), [7] Hennessy JL and Patterson D. Computer Architecture: A Quantitative Approach. Fourth Edition. Morgan Kaufmann [8] Amos M. Theoretical and Experimental DNA Computation. Springer [9] Tarski A. Introduction to Logic and to the Methodology of the Deductive Sciences (Second, revised edition, 1946). Dover republication, References [1] Church A. An Introduction to Mathematical Logic. Volume I. Princeton [2] Goldman N, Bertone P, Chen S, Dessimoz C, LeProust EM, Sipos B, and Birney E. Towards practical, high-capacity, low-maintenance information storage in synthesized DNA. Nature 494 (7 February 2013), [3] Zhang DY, Turberfield, AJ, Yurke B, and Winfree E. Engineering entropy-driven reactions and networks catalyzed by DNA. Science 318 (2007) [4] Phillips A. The DNA strand displacement simulator, DSD v Software is available for Windows (and other operating systems) at