E���cient Turing-Universal Computation with DNA Polymers Lulu Qian1, David Soloveichik4, and Erik Winfree1,2,3 1 Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA luluqian@caltech.edu 2 Computer Science 3 Computation & Neural Systems, California Institute of Technology, Pasadena, CA 91125, USA winfree@caltech.edu 4 Computer Science & Engineering, University of Washington, Seattle, WA 98195, USA dsolov@u.washington.edu Abstract. Bennett���s proposed chemical Turing machine is one of the most important thought experiments in the study of the thermodynam- ics of computation. Yet the sophistication of molecular engineering re- quired to physically construct Bennett���s hypothetical polymer substrate and enzymes has deterred experimental implementations. Here we pro- pose a chemical implementation of stack machines ��� a Turing-universal model of computation similar to Turing machines ��� using DNA strand displacement cascades as the underlying chemical primitive. More specif- ically, the mechanism described herein is the addition and removal of monomers from the end of a DNA polymer, controlled by strand displace- ment logic. We capture the motivating feature of Bennett���s scheme: that physical reversibility corresponds to logically reversible computation, and arbitrarily little energy per computation step is required. Further, as a method of embedding logic control into chemical and biological systems, polymer-based chemical computation is significantly more e���cient than geometry-free chemical reaction networks. 1 Introduction With the birth of molecular biology 70 years ago came the realization that the processes within biological cells are carried out by molecular machines, and that the most central processes involved the manipulation of information-bearing polymers. Roughly 30 years ago, Charles Bennett took that vision one step further by recognizing that arbitrarily complex information processing could be carried out, in principle, by molecular machines of no greater complexity than those already observed in nature [5,6]. Based on the intrinsic reversibil- ity of chemical reactions, Bennett used this insight to give a thermodynamic argument that there is no fundamental energetic cost to computation ��� only Y. Sakakibara and Y. Mi (Eds.): DNA 16, LNCS 6518, pp. 123���140, 2011. c Springer-Verlag Berlin Heidelberg 2011

124 L. Qian, D. Soloveichik, and E. Winfree a cost to erase data. This conclusion derives from four principles: (1) as Lan- dauer observed [15], making a logically irreversible decision entails an energetic expenditure of kT ln 2, and thus there is an unavoidable cost to irreversible log- ical operations (2) being logically reversible is not enough to ensure low-energy computation, since it is possible to implement reversible logic using irreversible mechanisms (3) a physically reversible system with an essentially linear state space can be biased ever-so-slightly forward, in which case progress is made despite involving a Brownian random walk, with the mean speed being linear in the (arbitrarily near zero) energy expended per step and (4) any logically irreversible computation can be recast with a minimal number of extra compu- tational steps [5,7] as a logically reversible computation that requires irreversible operations only when preparing input and output during repeated use. It���s in- triguing to ask whether Landauer���s and Bennett���s principles have any bearing on the remarkable e���ciency of living things, but cellular processes typically use several times more energy than needed for logical irreversibility. On the other hand, modern electrical computers expend many orders of magnitude more en- ergy than required by logical irreversibility, presenting the challenge of building computers that have the e���ciency Bennett argued is possible. Direct implementation of Bennett���s hypothetical chemical Turing machine has been hampered by our inability, as yet, to engineer molecular machinery to spec. Len Adleman���s laboratory demonstration of a DNA computing paradigm for solving NP-complete problems [1] ignited renewed interest in the molecu- lar implementation of Turing machines. Early theoretical proposals made use of existing enzymes but required a series of laboratory manipulations to step the molecular Turing machines through their operational cycle [20,2,23], while later theoretical proposals suggested how autonomous molecular Turing ma- chines could be built but made use of hypothetical enzymes or DNA nanostruc- tures [14,4,11,28,12]. Experimental demonstrations of autonomous biomolecular computers implemented weaker models of computation such as digital circuits or finite state machines [22,3]. Two-dimensional molecular self-assembly is Turing universal [26], implementable with DNA tiles [21], and can be physically and logically reversible [27], but it has the distinct disadvantage of storing the entire history of its computation within a supramolecular complex ��� it���s bulky. Recent work has pointed to an alternative to geometrical organization (in polymers or crystals) as the basis for Turing-universal molecular computation: abstract chemical reaction networks (CRNs) with a finite number of species in a well-mixed solution are structurally simple enough (essentially geometry-free) that in principle arbitrary networks can be implemented with DNA [25], yet they are (probabilistically) Turing universal [24]. This Turing universal computation using geometry-free chemical reaction networks is theoretically accurate and reasonably fast (only a polynomial slowdown), but requires molecular counts (and therefore volumes) that grow exponentially with the amount of memory used [16,24]. In contrast, reaction networks using heterogeneous polymers ��� the simplest kind of geometrical organization, as in Bennett���s vision ��� can store all information as strings within a single polymer, therefore requiring volume that

E���cient Turing-Universal Computation with DNA Polymers 125 grows only linearly with the memory usage [6,14]. Further, geometry-free mod- els are not energy e���cient, requiring much more than Landauer���s energy limit because the computation must be driven irreversibly forward to avoid error. Here, we combine the advances in geometry-free CRN implementation [25] with a simple DNA polymer reaction primitive to obtain a plausible DNA implemen- tation of time- and space- and energy-e���cient Turing-universal computation. Our construction requires a small fixed number of polymers, thereby having the same e���cient linear memory/volume tradeoff as Bennett���s hypothetical scheme it also has a time complexity nearly as good (only a quadratic slowdown). As in Bennett���s scheme, the time complexity scales linearly with energy use (for small energies). Both constructions can perform irreversible computation using the minimum achievable amount of energy per step, kT ln m, where m is the mean number of immediate predecessors to the logical states of the Turing machine simulation. (This energy bound is 0 for reversible Turing machines). Our constructions will consist of two parts. First, a geometry-free chemical re- action network, and secondly, reactions involved in polymer modification. While the polymer modification reactions will perform the essential job of information storage and retrieval, the geometry-free reaction network will perform the logic op- erations. We describe the necessary elements for the implementation of relevant geometry-free chemical reaction networks in section 2. In the following section 3 we describe the polymer reactions. Based on these two DNA implementation schemes, section 4 shows how they can be used to e���ciently simulate stack machines. Fi- nally, in section 5 we show how Bennett���s logically reversible Turing machines can be implemented with physically reversible DNA reactions. We conclude by evalu- ating our contributions and pointing out room for further improvement. 2 Irreversible and Reversible Chemical Reaction Networks In this section we discuss the components necessary for the implementation of the geometry-free chemical reaction network part of our constructions. Recent work has proposed a DNA implementation of arbitrary (geometry-free) chemical reaction networks [25], with which we assume the reader is familiar. (Even so, the construction given here is self-contained.) However, thermodynamic re- versibility was not considered. Indeed, reversible reactions would simply corre- spond to two separate forward and reverse reactions, with both reactions having to be independently driven irreversibly by chemical potential energy provided by DNA fuels. This is wasteful for the consumption of both energy and fuel reagents. The construction we develop in this section is entirely physically reversible in the sense that firing a sequence of forward reactions and then the reverse sequence of the corresponding reverse reactions brings the chemical system into the same exact physical state as it was in the beginning, including recovery of fuel reagents and any energy used. The major challenge in adapting the scheme of ref. [25] to implement reversible chemical reactions in a physically reversible manner, is that the exact DNA strand representing a particular signal species is a function of not just the signal, but also